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by Copyright Dmitriy Evgenievich Protsenko 2002 Electrosurgical Tissue Resection: A Numerical Study by Dmitriy Evgenievich Protsenko, MS Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin August 2002 Acknowledgements Of the many people who have played roles in my graduate education I am most thankful for the efforts of and contributions of Dr. John A. Pearce that he made toward my studies. Dr. Pearce made available to me his broad expertise on the electro-magnetic fields and their application in biomedical engineering. He also provided me with a set of world-class laboratory equipment necessary for successful pursuit of my education and research. He has given me the opportunity to formulate my research project described in this dissertation and assisted with it development and successful completion. I will always be grateful to Dr Pearce for most valuable experience that I have gained working under his supervision. In no lesser degree I am grateful to my co-supervisor Dr. Massoud Motamedi. During my first year as a research assistant in UTMB at Galveston Dr. Motamedi played a key role in my acquaintance with the field of biomedical engineering and provided me with valuable support and encouragement in my studies. Under his guidance I was able to acquire valuable experience not just in biomedical optics but also in various topics from writing a paper to organizing and conducting an animal study. I am very thankful for the efforts that each of the other of my dissertation committee members have put into my research and education. Dr. Sharon Thomsen taught me valuable lessons on tissue thermal damage analysis and evaluation. Dr. Jonathan Valvano provided valuable discussions on my research. I am especially thankful to Dr. Philp Schmidt for filling in vacant position in the committee on such a short notice. I owe a special note of thanks to Dr. A. J. Welch who taught me a grate deal on lasertissue interaction as well as on general principles of biomedical engineering. Dr. iv Welch also gave me full access to all facilities of the optics laboratory, which were instrumental in most of my research projects. Much of my education was obtained in the contacts in collaborations with other graduate students all of whom are doctors now. During my first year in Galveston Dr. Wei-Chang Lin was of immense help in setting up experiments as well as in providing friendly support and advice. Dr. Jorge Torres has taught me most of what I know about careful experimental planning and data collection. During my years in UT Austin I received valuable lessons in experimental technique from Drs. Eric Chen, Kin Fong Chan, Dan Hammer, Brian Sorg and Jennifer Barton. Drs. Joshua Pfefer and Mathieu Ducros provided assistance with numerical modeling. Joni Burks, Chris Humphries, Ann Armstrong have also been very kind and helpful during my graduate studies, and I owe many thanks to Margaret Fitch for the valuable assistance during my final weeks before graduation. Finally, I sincerely appreciate the support I have received from the following companies and agencies: C. R. Bard Inc., ValyLab Inc., Ethicon Inc., MURI project and Temple Foundation. v Electrosurgical Tissue Resection: A Numerical Study Publication No.____________________ Dmitriy Evgenievich Protsenko, Ph.D. The University of Texas at Austin, 2002 Supervisor: John A. Pearce The nature of the electrical, thermal mechanical and chemical phenomena associated with an electrosurgical resection of biological tissues is an important aspect of general surgery and other specialized medical treatments. A better understanding of the phenomena and the ability to model them are indispensable if advancements in the state of the art are to be achieved. This study particularly emphasizes two of the phenomena that have significant influence on the outcome of the electrosurgical procedure. These are the nature of the electric contact between tissue and electrosurgical scalpel and the mechanism of tissue water vaporization and vi subsequent mechanical damage to the tissue due to interstitial formation of the vapor micro bubbles and vacuoles. A numerical model of the interaction between tissue and electrosurgical scalpel was used to study the vaporization process at a number of power settings and for different scalpel geometries. An electric discharge striking between tissue and electrode was investigated and incorporated into an analytical model used for numerical simulation. For the water vaporization effect, surface evaporation at the tissue scalpel contact area and bulk vapor nucleation are introduced to facilitate the modeling of the change in tissue thermal and electric properties and tissue mechanical and thermal damage. A number of physical experiments were performed on beef muscle and saline and water samples to establish experimental values for the numerical model and observe electric circuit parameters, temperature variations and thermal damage cause by the electrosurgical current. These results are compared to those obtained from the simulations performed for the tissue-scalpel electric contact achieved by means of electric sparks, pure mechanical and mixed spark-mechanical contact. The simulation results for the contact through sparks alone are in least agreement and for the pure mechanical contact are in reasonably good agreement with those observed experimentally. It is reasonable to conclude that the sparks do not dominate the process of electrosurgical tissue resection though they contribute to formation of tissue thermal damage. vii Table of Contents List of Figures List of Tables Chapter 1: Introduction and Problem Statements 1.1 Introduction 1.2 Literature Survey 1.3 Problem Statement 1.4 Dissertation Outline Chapter 2: Background 2.1 Outline of ETR Model 2.2 Volume Power Generation 2.2.1 Electric Conductivity and Permittivity 2.2.2 Continuity Equation and Boundary Conditions 2.2.3 Power Generation Term 2.3 Tissue Heat Transfer 2.3.1 Heat Diffusion Equation 2.3.2 Boundary Conditions 2.4 Coagulation Damage 2.5 Phase Change Phenomena 2.5.1 Surface Water Vaporization 2.5.2 Interstitial Water Vapor Nucleation 2.5.2.1 Homogeneous Nucleation 2.5.2.2 Effect of Dissolved Gases 2.5.2.3 Heterogeneous Nucleation 2.5.2.4 Effect of Interfacial Tension Gradient 2.5.3 Vapor Bubble Growth 2.6 Electric Discharge in Gases 2.6.1 Electric Breakdown and Spark Discharge xiii xxii 1 1 4 12 13 17 17 19 19 21 22 23 23 25 27 29 31 36 38 41 42 43 44 45 46 viii 2.6.2 Arc Discharge Chapter 3: Thermal Ablation Model 3.1 Abstract 3.2 Anatomy of Skeletal Muscle 3.3 Histological Evaluation of Electrosurgical Thermal Damage 3.4 Ablation model 3.4.1 Phenomenological Formulation of Ablation Process 3.4.2 Functional Analysis of Ablation Process 3.4.2.1 Estimation of Nucleation Threshold and Nucleus Size 3.4.2.2 Vapor Bubble Growth and Additional Nucleation 3.4.3 Limitations of Ablation Model 3.5 Summary Chapter 4: Evaluation of Water Vaporization Rate from Muscle Tissue Surface 4.1 Abstract 4.2 Introduction 4.3 Methods 4.3.1 Experimental Methods 4.3.2 Numerical Methods 4.4 Results 4.4.1 Temperature Measurements 4.4.2 Water Vaporization Rates 4.4.3 Vaporization Coefficients 4.5 Discussion 4.5.1 Temperature in the Vapor Surface Layer 4.5.2 The Accuracy of Snelling s Formula 4.5.3 Vaporization Rates from Water and Muscle Tissue 4.5.4 Vaporization Coefficients for Water, Saline and 49 51 51 51 53 55 55 57 57 61 65 65 67 67 68 71 71 73 74 74 76 78 79 79 80 80 ix Muscle Tissue 4.6 Conclusion Chapter 5: Numerical Study of the Electric Sparks in the ETR Process 5.1 Abstract 5.2 Introduction 5.3 Methods 5.3.1 Tissue and Cutting Process Representation 5.3.2 Variable Step Grid 5.3.3 Governing Differential Equations in Variable Step Grid 5.3.3.1 Heat Source 5.3.3.2 Temperature Field 5.3.4 Water Vaporization 5.3.4.1 Surface Vaporization 5.3.4.2 Interstitial Vapor Nucleation 5.3.4.3 Tissue Material Properties as a Function of Water Content 5.3.5 Macroscopic Parameters of Electrosurgical Circuit 5.4 Results 5.4.1 Typical Simulation Results 5.4.1.1 Dynamics of the Simulated ETR Process 5.4.1.2 The Sparks and Computational Noise 5.4.2 Heat Source and Temperature Distribution 5.4.2.1 Two-Dimensional Distribution 5.4.2.2 Comparison with Theoretical Prediction 5.4.3 Electric Conductivity and Water Content 5.4.4 Total Current, Tissue Impedance and Cutting Speed 5.4.4.1 Effect of the Interface Power Density 5.4.4.2 Effect of Nucleation Temperature 5.4.5 Comparison with Data from Literature 82 83 84 84 84 88 88 92 93 94 98 99 100 101 103 104 105 107 108 113 116 116 119 124 124 124 128 131 x 5.5 Discussion 5.5.1 Effect of Tissue and Cutting Process Representation 5.5.2 Effect of Local Power Density 5.5.3 Effect of Contact Spot Type 5.5.3.1 The Heat Source and Temperature Distributions 5.5.4 Current, Voltage and Impedance 5.5.5 Effect of Interface Power Density 5.5.6 Effect of Nucleation Temperature 5.5.7 Computational Considerations 5.6 Conclusion Chapter 6: Numerical Study of thermal Damage in the ETR Process 6.1 Abstract 6.2 Introduction 6.3 Methods 6.3.1 Experimental Methods 6.3.1.1 Tissue Samples 6.3.1.2 Experimental Procedure 6.3.1.3 Data Statistical Analysis 6.3.2 Numerical Methods 6.3.2.1 Tissue and Cutting Process Representation 6.3.2.2 Rectangular and Elliptical Grids 6.3.2.3 Heat Source Calculation in Elliptical Grid 6.3.2.4 Interpolation of the Heat Source into Rectangular Grid 6.3.2.5 The Temperature Field 6.3.2.6 Estimation of Coagulation Damage 6.3.2.7 Interpolation of Electric Conductivity into Elliptical Grid 6.4 Results 132 132 133 135 135 137 138 140 141 141 143 143 144 146 146 146 146 149 150 151 152 154 155 157 158 159 159 xi 6.4.1 Experimental Results 6.4.1.1 Circuit Parameters and Cutting Speed 6.4.1.2 Lesion Pathology 6.4.1.3 Lesion Factorial Analysis 6.4.2 Numerical Simulation Results and Thermal Parameters 6.4.2.2 Circuit Parameters and Cutting Speed 6.4.2.3 Thermal Damage 6.5 Discussion 6.5.1 General Representation of ETR Process by the Numerical Model 6.5.2 Experiment and Model Comparison 6.5.2.1 Electrosurgical Circuit Parameters 6.5.2.2 Cutting Speed 6.5.2.3 Thermal Damage 6.5.2.3.1 Vaporization Damage 6.5.2.3.2 Coagulation Damage 6.6 Conclusion Chapter 7: Summary and Conclusion 7.1 Thermal Ablation and Water Vaporization Model 7.2 Implementation of Numerical Models 7.3 Geometry of Scalpel-Tissue Electric Contact 7.4 Parameters of Electrosurgical Circuit 7.5 Vaporization and Coagulation Thermal Damage 7.6 Future Directions 7.7 Conclusion Appendix Bibliography Vita 159 160 161 163 169 174 179 180 180 182 182 183 185 186 188 190 192 192 193 194 196 196 197 198 200 203 212 xii List of Figures Figure 1.1. Electric field distribution: a) uniform tissue - electrode contact, b) contact through electric discharge. Note that the distances rc and rs are counted from the centers of the scalpel and spark impact site respectively. Minimum values of rc and rs are equal to R0, and spark, r0 respectively. ..6 Figure 1.2. Power density as a function of distance from the tissue-scalpel interface for the semicylindrical and hemispherical source models. Power densities at the distance of 1 mm are indicated. 7 Figure 1.3. Time to reach water boiling temperature as a function of distance from the tissue-scalpel interface for the semicylindrical and hemispherical source mode. Time to reach water boiling temperature at the distance of 1 mm is indicated for bot models. 8 Figure 1.4. Sparking as a side effect of tissue evaporation. 9 Figure 2.1 Mass fluxes at liquid-vapor interface. mv is the vaporization mass flux, mc is the condensation mass flux, u0 is the net vapor velocity, corresponding, in this case, to net vaporization. .32 Figure 2.2 Breakdown voltage in air as a function of product of pressure and discharge gap withd [Raizer, 1991]. Vmin = 300 V, (pd)min = 0.83 Torr cm. ...47 Figure 3.1. A sarcomere. The thick filaments produce the dark A (Anisotropic) band. The thin filaments extend in each direction from the Z line. Where they do not overlap the thick filaments, they create the light I (Isotropic) band. The H zone is that portion of the A band where the thick and thin filaments do not overlap. ..52 Figure 3.2 Transmission polarizing Microscopy (TRM) cross-section of the electrosurgical lesions in beef muscle produced by the loop urological resectors: top - wire loop (Storz, Inc.), bottom - Max Blade (C. R. Bard, Inc.). Cutting was performed under deionized water with the output power of the ESU at 200 Watts and cutting speed 2 mm/sec.. .54 xiii Figure 3.3: Model of the ETR process. Upper and lower rows are side and top views of the cut. Scalpel is in electric contact with tissue (a). Vapor bubbles appear in the muscle fibers in front of the scalpel (b). When they have grown to the size of the cell (c) the cell bursts open and scalpel advances forward (d). At the same time smaller bubbles are growing further away from the scalpel (c); they are filled with water after the temperature dropped (d). ..56 Figure 3.4 Nucleation rate. Traces correspond to a) pure water, b) gas-contaminated water, c) low surface tension and d) combination of gas contamination and low surface tension respectively. A level of 1.6 1015 nuclei m-3 s-1 corresponds to appearance of one nucleus inside a 50 m cross-sectional volume during one ms time interval. ..59 Figure 3.5: Nucleus radius. Traces correspond to a) pure water, b) low surface tension, c) gas-contaminated water and d) combination of gas contamination and low surface tension. Levels of 50 and 1 m correspond to the radiuses of myofibril and muscle fiber respectively ..61 Figure 3.6 Deposited, Ec, vs vaporization, Ev, energies. Traces correspond to the power densities of a) 1010, b) 1011 and c) 1012 W/m3 respectively. ..64 Figure 4.1. Experimental setup for measurement of water vaporization rate from water and saline surfaces. For the measurement of vaporization rate from muscle tissue a tissue sample fixed to a Styrofoam lid was placed on the water surface in the beaker. ..72 Figure 4.2. Difference between surface and vapor temperatures as a function of the surface temperature. ..75 Figure 4.3. Vaporization rates from water, saline and muscle tissue surfaces. The points represent experimental data. The lines show vaporization rates for water and saline predicted by the Snelling s formula and for muscle tissue predicted by the kinetic theory based on results of this study. ...77 xiv Figure 4.4. Vaporization coefficients of water, saline and muscle tissue. The points represent values calculated from experimental data and kinetic theory prediction. The lines show least squares fit. .. ..79 Figure 5.1. Total current as a function of distance from the tissue-scalpel interface to the return electrode for the semicylindrical and hemispherical source models. Total current in the semicylindrical model is 200 mA and total power is the same for both models. .. 86 Figure 5.2 Tissue sample and contact spot representation. ..90 Figure 5.3 Flow chart of the algorithm used in numerical modeling of ETR. .91 Figure 5.4 Numerical grid. Grid step is constant in X direction and variable in Y and Z directions. ..93 Figure 5.5 Histories of output parameters for the simulation with the circilar electrode tissue contact spot. Cutting through first five 50 m muscle layers is reproduced. The indicated features are present, though might not be clearly visible, on all six reproduced histories. All indicated features, but the sparks , are common to all three contact spot types. The sparks are organic for the circular and elliptical types only. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx = Ry =100 m. .110 Figure 5.6 Histories of output parameters for the simulation with the elliptical electrode tissue contact spot. Cutting through first five 50 m muscle layers is reproduced. The indicated features are present, though might not be clearly visible, on all six reproduced histories. All indicated features, but the sparks , are common to all three contact spot types. The sparks are organic for the circular and elliptical types only. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx=250 m, Ry = 50 m. 111 xv Figure 5.7 Histories of output parameters for the simulation with the line electrode tissue contact spot. Cutting through first five 50 m muscle layers is reproduced. The indicated features are present, though might not be clearly visible, on all six reproduced histories. All indicated features, but the sparks , are common to all three contact spot types. The sparks are organic for the circular and elliptical types only. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx= m, Ry = 50 m. ... 112 Figure 5.8 Histories of output parameters for the simulation with the elliptical electrode tissue contact spot. Events occurring during cutting through fiber are reproduced. All indicated features are common to all three contact spot types. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx=250 m, Ry = 50 m. .... .114 Figure 5.9 Histories of output parameters for the simulation with the line electrode tissue contact spot. Events occurring during cutting through fiber are reproduced. All indicated features are common to all three contact spot types. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx= m, Ry = 50 m. ... .115 Figure 5.10 Distributions of heat source and temperature in XY (along the scalpel, across the cut) and XZ (along the scalpel, along the cut) planes recorded in simulation with circular contact spot. The temperature distributions are plotted for the square areas as indicated on the heat source plots. The distributions were recorded at the moment just before tissue layer was destroyed. Approximate locations of the grid elements with maximum heat source and temperature and corresponding maximum values are indicated. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx=250 m, Ry = 50 m. ... 117 xvi Figure 5.11 Distributions of heat source and temperature in XY (along the scalpel, across the cut) and XZ (along the scalpel, along the cut) planes recorded in simulation with elliptical contact spot. The temperature distributions are plotted for the square areas as indicated on the heat source plots. The distributions were recorded at the moment just before tissue layer was destroyed. Approximate locations of the grid elements with maximum heat source and temperature and corresponding maximum values are indicated. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx = Ry =100 m. . .118 Simulated and analytically predicted heat source as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest heat source made in direction along the scalpel a) and across the scalpel b) shown. .. .120 Simulated and analytically predicted temperature as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest temperature made in direction along the scalpel a) and across the scalpel b) shown. ...121 Simulated electric conductivity as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest temperature made in direction along the scalpel a) and across the scalpel b) shown. 122 Simulated water content as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest temperature made in direction along the scalpel a) and across the scalpel b) shown. ....123 Tissue impedance as a function of the interface power density. Simulations were performed for all range of power and contact spot sizes as indicated in Table 5.3. The nucleation temperature was 112.5 C in all cases. Data ranges corresponding to unrealistically large circular spot area or, in the case of the line spot, unrealistically high ESU power are indicated. ...125 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 xvii Figure 5.17 Total current as a function of the interface power density. Simulations were performed for all range of power and contact spot sizes as indicated in Table 5.3. The nucleation temperature was 112.5 C in all cases. Data ranges corresponding to unrealistically large circular spot area or, in the case of the line spot, unrealistically high ESU power are indicated. ..126 Cutting speed as a function of the interface power density. Simulations were performed for all range of power and contact spot sizes as indicated in Table 5.3. The nucleation temperature was 112.5 C in all cases. Data ranges corresponding to unrealistically large circular spot area or, in the case of the line spot, unrealistically high ESU power are indicated. .. .127 Tissue impedance as a function of nucleation temperature for the circular a) and line b) contact spots. Each data point corresponds to a given ESU power and contact spot sizes and represents an average value obtained during a steady-state cutting through 10 consecutive layers. The simulations were performed in the interval of the interface power density, Pint, from 5x107 to 5x109 W/m2. Arrow in the right portion of the plots shows general increase in interface power density. .129 Cutting speed as a function of nucleation temperature for the circular a) and line b) contact spots. Each data point corresponds to a given ESU power and contact spot sizes and represents an average value obtained during a steady-state cutting through 10 consecutive layers. The simulations were performed in the interval of the interface power density, Pint, from 5x107 to 5x109 W/m2. Arrow in the right portion of the plots shows general increase in interface power density. ..130 Schematic diagram of measurement connections in experiments. The voltage probe is connected to a capacitive voltage divider with the probe capacitance. Electrode voltage is calculated from the measured divider ratio (determined from an open circuit measurement of probe and actual voltage). 147 Figure 5.18 Figure 5.19 Figure 5.20 Figure 6.1 xviii Figure 6.2 Evaluation of electrosurgical lesions. Three types of thermal damage can be identified on each lesion: melted proteins, vapor bubbles and birefringence loss. The depth of thermal damage was measured from the edge of the cut at six sites, from A to F, approximately located as shown. The damage zones were determined from the edge of the cut; D1 -vapor bubble (water loss) zone, D2 birefringence loss zone. ..148 Elliptical and rectangular grids. Actual sizes of grid steps, dy0, dz0 and d 0, are much smaller in relation to the scalpel than shown. The cutting direction is Z. Scalpel radiuses Ry = Rz correspond to a polar grid. . 151 Arrangement for Bessel interpolation: a) from elliptical into rectangular and b) from rectangular into elliptical grids respectively. In the case of elliptical to rectangualr interpolation unit steps in radial, R, and angular, , direction are taken as shown. ... ..156 Typical coagulation and vaporization damage obtained in experiment. H&E stain. Damage types of melted proteins, vapor bubble and birefringence loss are visible in all cases. ....162 Typical vapor bubbles and melted protein thermal damage. Note a relatively sharp boundary between melted proteins and relatively cohesive myofibrils. . ..164 Results of factorial analysis of vaporization thermal damage obtained with cylindrical wire electrodes. Means Diamonds are shown for each data set. Top and bottom endpoints of a diamond indicate 95% confidence interval of the data set. Two horizontal marks inside indicate 95% overlapping level of the data sets. ... 165 Results of factorial analysis of coagulation thermal damage obtained with cylindrical wire electrodes. Means Diamonds are shown for each data set. Top and bottom endpoints of a diamond indicate 95% confidence interval of the data set. Two horizontal marks inside indicate 95% overlapping level of the data sets. ...166 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figures 6.7 Figures 6.8 xix Figure 6.9 Distributions of heat source, temperature, electric conductivity and water content in XY plain recorded in simulation with circular electrode. The records correspond to the time just before tissue layer was destroyed. The input parameters are ESU output power, PESU = 25 Watts, Nucleation threshold temperature, Tn = 107.5 C, Electrode diameter 0.45 mm. .. 170 Distributions of heat source, temperature, electric conductivity and water content in XY plain recorded in simulation with elliptical electrode. The records correspond to the time just before tissue layer was destroyed. The input parameters are ESU output power, PESU = 25 Watts, Nucleation threshold temperature, Tn = 107.5 C, Electrode diameters 0.45 x 0.9 mm. . .171 Simulated for circular electrode a) heat source and b) electric conductivity as a function of distance from the scalpel s center. Distributions along the cutting (6 o clock), across the cutting (3 o clock) and 4:30 are shown. .172 Simulated for elliptical electrode a) heat source and b) electric conductivity as a function of distance from the scalpel s center. Distributions along the cutting (6 o clock), across the cutting (3 o clock) and 4:30 are shown. .173 Cutting speed as a function of surface power density. Simulations were performed for all range of power and electrode sizes as indicated. The nucleation temperature was 107.5 C in all cases. Experimental data and corresponding linear regression fit are shown together with simulation results. .176 Vaporization thermal damage. Depth of the vaporization zone is determined at 50% out of 100% initial water content. ...177 Coagulation thermal damage. Depth of the birefringence loss zone is determined at > 1 threshold level. . ..178 a) Water saturation pressure as a function of temperature. Data points correspond to tabulated values, solid line represent a fifth order polynomial fit. b) Relative error of polynomial approximation ..201 Figure 6.10 Figure 6.11 Figure 6.12 Figure 6.13 Figure 6.14 Figure 6.15 Figure A.1. xx Figure A.2. a) Water phase change enthalpy as a function of temperature. Data points correspond to tabulated values, solid line represent a fifth order polynomial fit. b) Relative error of polynomial approximation .. ..201 a) Water phase change enthalpy as a function of temperature. Data points correspond to tabulated values, solid line represent a fifth order polynomial fit. b) Relative error of polynomial approximation ... .202 Figure A.3. xxi List of Tables Table 3.1 Nucleation parameters of water and muscle cell. . ..51 Table 3.2 Number of bubbles formed in 50 m cell crossectional volume during expansion of initial bubble. . .. .56 Table 4.1. Vaporization rates from water and tissues and corresponding surface temperature and room humidity levels. 73 Table 5.1. Temperature independent model parameters. Temperature dependent parameters are given in Appendix.. 80 Table 5.2 Simulation input parameters. ....95 Table 5.3. Simulation results. Average value standard deviation for the cutting speed, maximum and average temperatures, tissue impedance and total current. Circular contact spot radius: Rx =50 m, elliptical contact spot radiuses: Rx = 500 m, Ry = 50 m, line contact spot width: Ry=50 m, The nucleation temperature Tn=112.5 C. Typical experimental data taken from [Pearce, 1986]. ... ... .120 . Table 6.1. Electrosurgical circuit parameters and cutting speed. Values in table are expressed as mean one standard deviation. ...147 .. Table 6.2. Evaluation of width of the water loss zone for the wire electrodes .153 Table 6.3. Evaluation of width of the birefringence loss zone for the wire electrodes. . ..153 Table 6.4. Evaluation of width of the water loss zone for the blade electrodes. ...153 Table 6.5. Evaluation of width of the birefringence loss zone for the blade electrodes. . ..154 xxii Table 6.6 Comparisons of electrosurgical circuit parameters obtained in experiment with the predicted by the numerical model. Mean values one standard deviation are shown for the experimental data. Simulated values were obtained in steady-state cutting. The simulated values are the average values one standard deviation obtained during cutting trough ten consecutive layers of muscle fibers. Scalpel diameters: E1 experiment: 0.45 x 2.35 mm, simulation: 0.45 x 2.25 mm; C1: 0.45 mm, C2: 0.7 mm, C3: 1.1 mm. . .. ..160 Table A.1. Coefficients for 5th order polynomial fit of temperature dependant model parameters. The temperature is in K .. 200 xxiii Chapter 1: Introduction and Problem Statements In this chapter, a general introduction to the nature of this work and problem statements are presented. A literature survey of theories relevant to the analysis of electrosurgical tissue resection and tissue thermal ablation in general is presented. 1.1 Introduction There are many surgical procedures that utilize Electrosurgical Tissue Resection (ETR). The most numerous applications are in general, gynecologic, urologic and neurosurgical procedures [Kelly and Ward, 1932] and [Mitchel, et al., 1962] as well as in dental [Oringer, 1975] and dermatologic [Burdick, 1966] procedures. In general, ETR is used for simultaneous cutting and coagulating of the tissue at the surgical site. Electrosurgery consists of applying radio frequency (RF) electric current to a specific surgical site in order to destroy tissue by cutting or coagulating. The usual circuit arrangement used in tissue cutting procedures consists of an Electrosurgical Unit (ESU) connected with the scalpel electrode and return electrode or patient plate. The technique of ETR is to bring the sharp edge of the electrosurgical scalpel in contact with the tissue and apply RF current between the scalpel and return electrode. High electric current density at the tip of the scalpel causes rapid heating of the tissue in its immediate vicinity by the Joule heat. Cutting is accomplished by disrupting or ablating of the tissue adjacent to the scalpel electrode due to rapid temperature rise up and above water evaporation temperature [Honig, 1986]. Coagulating is accomplished by denaturation of proteins due to thermal damage [Pearce, 1986]. It should be noted that in this study the term ablation describes a general process of physical tissue removal caused by the temperature rise and subsequent vaporization of tissue water. In literature on electrosurgery the term cardiac ablation refers to the 1 specific surgical procedure that involves destruction of the electric conduction pathways on the myocardial surface [Kalbfleisch, et al., 1992]. Such destruction is usually achieved by the thermal coagulation of the conducting tissue by coagulation and may or may not be accompanied by tissue removal by vaporization [Shimko, et al., 2000]. In general, tissue damage caused by disruption and ablation is extremely shallow and limited only to a few or even one cell layer adjacent to the scalpel [Pearce, 1986]. Coagulation damage reaches deeper into the tissue and it is generally responsible for sealing off the smaller blood vessels and arterioles. To seal off the larger vessels and provide complete hemostasis an electrosurgical technique different from simple cutting is required. It is accomplished by bringing the side of the scalpel in contact with the bleeding site and applying somehow lower RF current to it. Much lower current density in this case causes blood coagulation without disruption of the adjacent tissue. Most of the action of ETR takes place at the interface between the electrosurgical scalpel and tissue because it is the region where the current density is the highest. The processes at the interface are extremely complex involving all of electrical, thermal, thermodynamic, mechanical and chemical action that are difficult to study by direct experiment. Due to this complexity, only limited theoretical understanding of the physics of the ETR exists. The complete understanding of physical events that occur at the tissue-scalpel interface and in the immediate vicinity of it and the ability to model these phenomena play an important role in advancing the state of the art in electrosurgical technique. Since the 1920 s [Bovie, 1928, repr. 1995], a large number of studies on clinical applications as well as on experimental and theoretical analysis of electrosurgery had been published [Hall, 1976], [Wicker, 1990]. The analysis of 2 electrical, thermal mechanical and other problems involving tissue disruption and coagulation represents a serious technical challenge to system designers as well as to physicians. The challenge is to solve the complete electrostatic problem to obtain the distribution of the electric field, current density, temperature, mechanical force and thermal damage accurately and efficiently using a computer facility. In addition, the importance of using computational program to analyze experiments related with ETR systems is that several parameters (current density, pressure, temperature) that govern the process cannot be measured directly in the course of experiment, and must be therefore be calculated from the computational programs from measurable quantities. An analytical model of the process consists of several sets of equations. For the electrostatic part, the governing equations are the Laplace equation with appropriate boundary conditions and Ohm s law. The heat transfer equation, mass, energy and momentum conservation laws and kinetic equations are the thermodynamic part of the governing equations. Hook s law and mechanical equilibrium conditions are responsible for the mechanical governing equations. In addition, the phenomenological Arrhenius formula describes coagulation damage. It is difficult to solve these sets of equations since these systems involve tissue electric, thermal and mechanical properties that are transient and nonuniform. Moreover, such a system involves tissue properties, which are dependent on temperature and water content. The nature and complex geometry of electric contact along the tissue-scalpel interface pose another fundamental difficulty involved in finding the numerical solution in problems with ETR. For simplicity, most simulations of electrosurgical treatment assume perfect mechanical and, as a consequence, perfect electric contact across all interface area. But, it has been suggested; that in the case of ETR the electric circuit at the interface is closed by a series of electric sparks striking at discrete location [Pearce, 1986]. The crossectional area of such sparks is reactively 3 small (few dozens of microns [Raizer, 1977]) and, respectively, the current density at the sparking site is extremely high. It is also possible that the contact is mechanical but not uniform so the actual current density at the interface region is still higher than the nominal current density determined as ratio of the output ESU current over the total area of the scalpel. In addition, the exact nature of the mechanism causing tissue destruction or ablation under the scalpel should be identified. Most analytical models involving ETR [Shimko, et al., 2000] assume that tissue destruction at the treatment spot occurs immediately as the tissue temperature reaches boiling point of 100 C at normal atmospheric pressure. However, numerous simulations and experimental studies of laser tissue ablation [Welch and van Gemert, 1995], point out that much more complex processes are governing tissue destruction. The laser-tissue and RF currenttissue interactions have different mechanisms of heat generation. Nevertheless, the energy density and deposition rate in the applications involving Ar+ ion and Nd:YAG continuous wave (CW) laser ablation [Welch and van Gemert, 1995] are remarkably close to those that might be expected in the case of electrosurgical ablation [Pearce, 1986]. That makes it possible to utilize analytical models of laser tissue vaporization and ablation as well as some experimental data for the ETR simulation. 1.2 Literature Survey Two models of RF tissue cutting were proposed respectively by Honig [Honig, 1975] and Pearce [Pearce, 1986, pp. 72-79]. Honig s model suggests that the entire front part of an RF scalpel is in uniform electrical contact with the tissue, Figure 1 a), and, therefore, may be described as a semicylindrical current source. The power density under such a scalpel is [Pearce, 1986]: 4 1 P(r ) = I Lr c , 2 (1.1) where: I is total cutting current, L is length of an electrode in contact with the tissue, is tissue electric conductivity and rc is distance from the center of the cylindrical scalpel. The second model, suggested by Pearce [Pearce, 1986], incorporates electric sparks randomly striking between the electrode and the tissue surface, Figure 1.1 b). Each spark strikes during some part of the RF cycle and occupies only a small portion of the scalpel-tissue interface area. The resulting power density under the site of a spark impact is approximated as that of a hemispherical source [Pearce, 1986]: 1 P(r ) = 4 I 2 , r s 2 (1.2) where: rs is distance from the center of the spark strike site. It should be noted that minimum values of rc and rs in Equations (1.1) and (1.2) correspond respectively to radiuses of the scalpel, R0, and spark, r0, as shown on Figure 1.1. For relatively high cutting speeds the effects of heat transfer, perfusion and metabolic heat can be neglected; the bioheat equation reduces to [Pearce 1986, p. 240]: dT P(r ) = , dt c p (1.3) where: is tissue density and cp is tissue specific heat. From that, the time to reach water boiling point can be expressed as: 5 t= c p T , P(r ) (1.4) where: T is the difference between initial temperature and boiling temperatures. It should be noted that, in the case of the hemispherical source, Equation 1.4 estimates time to heat tissue around only a single source. Therefore, in order to estimate the time required for rising temperature along the entire scalpel the result should be multiplied by a factor equal to the ratio of the scalpel length to the spark diameter. a) b) Figure 1.1. Electric field distribution: a) uniform tissue - electrode contact, b) contact through electric discharge. Note that the distances rc and rs are counted from the centers of the scalpel and spark impact site respectively. Minimum values of rc and rs are equal to R0, and spark, r0 respectively [Pearce 1986]. Figures 1.2 and 1.3 show the power density and the time to reach water boiling point (temperature of 100 C) as a function of distance from the scalpel-tissue interface for the semicylindrical and hemispherical source models. Calculations were 6 made assuming a typical total cutting current of 200 mA (r.m.s.), tissue electrical conductivity of 0.25 S/m, cylindrical scalpel length of 7 mm and diameter of 0.43 mm, and spark diameter of 0.17 mm [Pearce, 1986]. Figure 1.2 demonstrates that the power density for the hemispherical source model exceeds that of the semicylindrical model. However, Figure 1.3 shows that while at the scalpel-tissue interface the time to reach water boiling point is higher for the hemispherical source model by a factor of 100 - at the depth of 1 mm it is higher for the semicylindrical model by a factor of 4.5. Figure 1.2. Power density as a function of distance from the tissue-scalpel interface for the semicylindrical and hemispherical source models. Power densities at the distance of 1 mm are indicated. The semicylindrical and hemispherical source models predict clearly different speeds of heating for various distances from the tissue-scalpel interface. It appears that for the same total current the semicylindrical source model might predict relatively low cutting speed and deep tissue thermal damage while the point source model must give higher cutting speed and shallower damage. However, the 7 hemispherical source model suggests a smaller area of a contact between scalpel and tissue and, therefore, higher total impedance and, for the same total current, higher ESU output power than the semicylindrical model. Figure 1.3. Time to reach water boiling temperature as a function of distance from the tissue-scalpel interface for the semicylindrical and hemispherical source mode. Time to reach water boiling temperature at the distance of 1 mm is indicated for both models. Another model is described by Wurzer et al. [Wurzer et al., 1987]. This model suggests that, at least initially, the front portion of the scalpel is in more or less uniform contact with the tissue, Figure 1.4 a). RF current heats up the tissue and causes partial evaporation of the tissue water and creation of a vapor film in front of the scalpel, Figure 1.4 b). Then, in that vapor-filled gap between scalpel and tissue electric discharges (sparks) begin to strike, Figure 1.4 c). The model does not specify whether electric sparks are the primary mechanism of the steady state RF cutting process or a side effect. However, it is suggested that sparks contribute to the tissue 8 desiccation and are responsible for extensive thermal damage and tissue carbonization at the walls of the cut [Wurzer et al., 1996, Wurzer et al., 1997]. Electrode Electric Discharge Tissue a) b) c) Figure 1.4. Sparking as a side effect of tissue evaporation [Wurzer et al., 1987]. Shimko et al developed a numerical model for the RF perforation of cardiac tissue that accounts for the heat transfer effect, thermal convection at the tissue surface and temperature dependence of tissue electric conductivity [Shimko et al., 2000]. The model describes a catheter electrode which cylindrical tip brought in contact with cardiac tissue at the right angle. Though the geometry of the tissueelectrode interface is clearly different from that discussed by Honig and Pearce, nevertheless, the parameters of the model make it closely related to the general case of ETR. The authors do not consider thermodynamics and mechanics of water vaporization and make the simple assumption that when tissue in front of the perforating catheter reaches vaporization temperature of 100 C it is destroyed immediately. Simulation performed for a catheter tip of 0.15 0.22 mm in radius and RF voltages of 80 120 Volts. which correspond to 15 34 Watts of applied power. Results obtained show that the time to reach 100 C in front of the catheter varies from less than 50 ms for the highest power to more then 500 ms for the lowest. With 9 this model, the perforation speed also depends on the applied voltage and decreases with increase in voltage from 7 mm/s to 2.2 mm/s respectively. The model incorporates convective cooling of the tissue and electrode surface by the blood. It is shown that due to the presence of surface convective heat flux, the surface temperature is always lower than the temperature at a depth of several hundred microns, which can reach 110 160 C. There is substantial literature on the subject of tissue thermal ablation by laser radiation. The mechanism of the heat generation by laser radiation is based on photon absorption by tissue molecules and subsequent energy relaxation through vibrational states. In the case of electrosurgery, conductive current generates the heat (Joule heating). Though these mechanisms are different, in many cases the strength and geometry of the laser and electrosurgically generated heat sources are similar. In soft tissues, the penetration depth of the laser radiation and, therefore the thickness of the heat generating layer, varies with the laser wavelength from several microns for the CO2 laser to several millimeters for the Argon and Nd:YAG lasers [Welch and van Gemert, 1995]. Combining physical effects specific for particular laser of energy application it is useful to identify two ablation models: the moving front and distributed heat source models. In the case of the shallow penetration depth of the CO2 laser it is reasonable to neglect the thickness of the surface layer altogether and treat laser radiation as an incident heat flux [Zweig, et al., 1987]. This assumption allows modeling the ablation problem as a so-called moving boundary problem [Wrobel, 1995], which can be solved by applying mass, momentum and energy conservation laws together with the condition of thermodynamic equilibrium at the phase interface. Laser radiation impinges on the tissue surface and, being absorbed, effects a phase transformation within the surface layer. For pulses of 500 s duration and 100 mJ of energy focused 10 into a spot of 250 m diameter the subsurface power density is about 1012-1013 W/m3. This is the same order of magnitude of the power density as predicted by Pearce for the electrosurgical sparks [Pearce, 1986]. Such high power densities cause extremely rapid water vaporization, intense flux of vapor leaving the surface and, as a consequence, pressure rise in tissue reaching up to a tens of MPa, Lane, et al. [Lane, et al., 1987]. Walsh and Deutsch [Walsh and Deutsch, 1989] have shown that the heat of ablation, i.e. thermal energy required for initiation of tissue destruction, correlates with the tissue ultimate tensile strength (UTS). Cummings and Walsh [Cummings and Walsh, 1993] demonstrated that mechanical disruption of tissue occurs when the ablation pressure exceeds the UTS of irradiated tissue at sites of intrinsic weaknesses. LeCarpenter et al. [LeCarpentier, et al., 1993] have investigated the heating of a tissue phantom up to and above the vaporization temperature by a distributed heat source produced by continuous Nd:YAG laser radiation. It is demonstrated, both experimentally and numerically, that a distributed heat source and equilibrium vaporization at the tissue surface leads to the increase of subsurface temperature above the value corresponding to the equilibrium vaporization at the surface. The authors suggested that significant elevation of subsurface temperature is responsible for generation of interstitial vapor bubbles and subsequent tissue tearing and mechanical destruction. Sagi et al. [Sagi, et al., 1992] in a model of the CO2 laser ablation assume that equilibrium vaporization occurs not just at the surface but in any tissue region where the temperature reaches the vaporization threshold. Tissue temperature does not change during the vaporization process and it is suggested that further temperature rise is entirely due to the heating of dehydrated tissue. Gerstmann et al. [Gerstmann, et al., 1994] modified this model by introducing a layer of char at the irradiation site. The char layer represents a completely dehydrated tissue heated to a temperature 11 significantly higher than the vaporization level. It acts as a heat source for the tissue beneath, causing it to boil. The model assumes that water vapor leaves the upper level of the boiling tissue through perforations in the char layer. The power densities in the described distributed heat source models range from 106 to 108 W/m3, which is close to that predicted by Honig and Pearce for a semi-cylindrical source model [Honig, 1975, Pearce, 1986]. Though these models identify subsurface temperature rise they account only for surface equilibrium vaporization and are not concerned with non-equilibrium phase change events. 1.3 Problem Statement The drawbacks of the approaches described above lead to the conclusion that there is a need to develop a general and efficient computational model of the electrosurgical tissue resection for analyzing accurately and physically the transient phenomena in surgical procedures, which utilize the electrosurgery. This helps provide a clear picture of the tissue resection process, and explains and predicts the ablation and thermal damage. The objective of this dissertation is to develop a numerical computational model that describes the ETR process in terms of local variables. Since this objective is broad and, the physics involving the ETR are complex, a more feasible and attainable goal is defined. The goal is to formulate a numerical model of the ETR that will satisfy the following criteria: 1. The model should be capable of phenomenological modeling of ETR effects accurately and efficiently in terms of local variables since electric field, temperature and thermal damage distributions around the scalpel-tissue interface are more important than average values. The model can be useful for analyses 12 associated with surgical procedures such as the choice of optimal power setting and prediction and analysis of thermal damage. 2. The model can be described in terms of macroscopic electric parameters such as ESU output power, current and voltage and total impedance. 3. The model should be in a form suitable for incorporation of different electrode geometries at least in very general representation. 4. The results of the model should be easy to compare with experimental results. Temperature rise, thermal damage and macroscopic electric parameters will be compared. A study of the ETR model should lead to better understanding of the tissue ablation and thermal damage processes associated with electrosurgical procedures. 1.4 Dissertation Outline A general description of the research effort is provided here to introduce the reader the contents of this dissertation. Chapter 1 is a general introduction concerning the focus of this work and why this research is undertaken. A literature survey of the analysis of the ETR process and tissue thermal ablation, including the laser tissue ablation, is presented; and the statement of the objective of this research is presented. Chapter 2 provides background for the numerical model of ETR. It deals with most of the theoretical aspects of this work. Six of the phenomena that have a significant influence on the ETR process are dealt with in this chapter. These are electric current flow, temperature rise, water surface evaporation, interstitial vapor nucleation and growth of vapor bubbles and coagulation damage. In addition, the model of electric sparking in the gap between the tissue and scalpel is described. 13 Governing equations for the effects involved in the simulation are derived and explained briefly. A qualitative model of the ablation mechanism of electrosurgical resection of skeletal muscle tissue is introduced in Chapter 3. The anatomy of skeletal muscle is outlined briefly. Histological samples of the lesions produced by electrosurgical resectors are examined and a hypothesis of tissue ablation due to water vapor generation is proposed. Simplified numerical analysis of nucleation rate and expansion of a vapor bubble in muscle fiber is conducted and conclusions on the ablation mechanism caused by interstitial water phase change and manifested in disruption of cell membrane and extracellular matrix are made. Experiments to determine the vaporization coefficient for surface vaporization of muscle tissue are described in Chapter 4. The vaporization mass flux predicted by kinetic theory was compared with the experimentally documented fluxes for water and muscle and the values of the vaporization coefficients were determined. It was found that vaporization rate from the muscle surface is reduced in comparison with vaporization from open water surface. A numerical study of the role of electric sparks in the ETR process was conducted in Chapter 5. A finite-difference numerical model is developed in a threedimensional Cartesian coordinate system with variable grid step and corresponds to the hypotheses of semicylindrical and hemispherical heat sources. The finitedifference representation of the governing electrical, thermodynamic and mechanical equations is formulated and numerical implementation of the thermodynamic and mechanical events leading to tissue destruction is given. The models allow for the setting of total ESU power, voltage or current as an input macroscopic parameter. The electric sparks are modeled as a sequence of contact spots randomly appearing along the scalpel. The contact spot can be circular or elliptical, elongated along the scalpel. 14 A number of simulations are performed for different values of ESU power settings and different sizes and shapes of the contact spot. Macroscopic parameters of the electric circuit: total dissipated power, circuit impedance, voltage, current and phase shift between them are calculated. The simulation results are compared with typical values reported in literature. It is shown that the hypotheses of sparks playing leading role in the ETR is not feasible as it does not provide for the typically observed macroscopic parameters of the operational electric circuit. Based on the results of the previous chapter a two-dimensional model simulating the cutting of muscle tissue in-vitro by an infinite straight electrosurgical scalpel with perfect electric and thermal conductivity and elliptical cross-section is introduced in Chapter 6. The second model is developed in a two-dimensional elliptical coordinate system with variable steps and corresponds to the hypotheses of a semi-cylindrical heat source. The model accounts for the scalpel s geometry (elliptical or cylindrical cross-section) and allows more detailed study of vaporization and coagulation thermal damage. Simulations are performed for different values of ESU power settings and different geometry of the scalpel. Macroscopic parameters of the electric circuit: total dissipated power, circuit impedance, voltage, current and phase shift between them are calculated. Thermal damage in terms of local values for tissue water content, vapor vacuoles concentration and size and coagulation damage as well as the cutting speed are presented. Detailed study of the ETR process is performed with the twodimensional elliptic coordinate system model. The cross-section of the scalpel is varied from circular to elliptical (to simulate more realistically typical electrosurgical blades) and effect on the circuit parameters, thermal damage and cutting speed is studied. It is demonstrated that after the scalpel begins its advance through tissue, the scalpel-tissue contact area begins to shrink, even if initially the entire scalpel was in perfect electric contact with tissue. 15 A number of experiments were performed to verify the numerical models are presented. Beef skeletal muscle was used as a tissue model. Cuts several centimeters long and about one centimeter deep were performed under different power settings and with wire and blade electrodes of different diameters. Circuit current, voltage and phase shift between them as well as cutting speed were recorded. After the experiment histological slides of the tissue samples were prepared and extent of coagulation damage and density and size of vapor vacuoles were measured. Macroscopic circuit parameters and tissue thermal damage are compared with those predicted by the models. Finally, Chapter 7 presents an overview of the results and a conclusion to the entire study. The importance of the phase-change phenomena and tissue mechanics introduced here is discussed and specific improvements are suggested. The direction of the future work required for understanding mechanism of interstitial vapor nucleation and tissue destruction is indicated. 16 Chapter 2: Background The background information contained in this chapter provides an overview of most of the theoretical aspects of this work. First of all, basic physics of the interaction between RF current and tissue, tissue thermal behavior and thermal coagulation is outlined. Then detailed description of the surface and interstitial phasechange phenomena is followed. Finally, description of electric sparks and hypotheses on their formation during ETR is given. 2.1 Outline of ETR Model In order to develop a realistic and accurate numerical model of the electrosurgical tissue resection (ETR) several electromagnetic, thermodynamical, mechanical and chemical processes originating from the interaction between tissue and electromagnetic field and electrosurgical scalpel have to be considered in detail. The fundamental processes are: (1) flow of electric current and distribution of volumetric heat source in tissue, (2) temperature rise in tissue, (3) tissue thermal coagulation, (4) phase change of tissue water, (5) tissue mechanical reaction associated with the water vaporization. The challenge in developing a comprehensive ETR model arises not only from the variety and complexity of phenomena that have to be considered but also from the mutual connections and interactions between these phenomena. In overview, the complex process of ETR is hypothesized to consist of the following interdependent stages: 1. Electromagnetic Interaction: Strength and distribution of the electric current as well as onset of the sparking at the tissue scalpel interface depend on the local tissue electric conductivity and permittivity, which depend on temperature and 17 water content. Output electric characteristics of the ESU depend on circuit impedance derived from the local electric properties of the tissue. 2. Temperature Rise: Tissue is heated via generation of the Joule heat and by the heat conduction. These two processes are dependent on the local values of electric conductivity and thermal diffusivity, which are temperature and water content dependent. Along with the heat generation there is a heat sink due to the energy required for the pressure and density change and the phase change. 3. Thermal coagulation: Coagulation or protein denaturation occurs as a function of temperature rise. Coagulation tissue damage is one of the two clinical effects of the ETR procedures. 4. Phase Change of Tissue Water: As the tissue temperature increases, the saturation pressure of water increases as well and when it exceeds the ambient pressure vaporization begins. At the tissue surface the vaporization rate depends upon temperature and saturation pressure. Interstitial vaporization depends also on vapor nucleation rate and the rate of the growth of the vapor bubbles. Primarily the interfacial tension, concentration and solubility of the inert gases, all dependent on temperature, determine the nucleation rate, and the bubble growth rate is dependent on tissue elastic properties. 5. Tissue Mechanical Reaction: Subsurface vapor nucleation and subsequent rapid expansion of vapor bubbles generate mechanical stress on surrounding tissue structures and ultimately results in tissue rupture and allows the scalpel to advance. This is the second of the two clinical effects of ETR procedures. It is likely that thermal coagulation and decrease in water content affect the elastic properties of tissue. Also, when the electrosurgical scalpel advances it may 18 compress surrounding tissue layers and effectively change tissue water content and, therefore, tissue physical properties. The following sections of this chapter contain detailed description of all of the processes mentioned and, due to extreme complexity of the phenomena in many cases, explain necessary assumptions and simplifications. 2.2 Volume Power Generation This section of the chapter covers primary mechanisms of electrosurgery including conduction of electric current, volumetric power generation and matching of electric circuit. Although studies concerned primarily with electrosurgical tissue coagulation are cited, it is assumed that their results involving thermal dependence of tissue electric properties can be applied to the present case. 2.2.1 Electric Conductivity and Permittivity Liquid component of an average skeletal muscle tissue accounts for approximately 80% of its volume and 60% of its mass. The rest are various proteins, fats and minerals. About two thirds of the fluids is enclosed in the muscle cells and the remaining one third is divided between extra cellular liquid and blood plasma [Ganong, 1995]. Various ions dissolved in the tissue fluids are the prime factor determining electric conductivity of muscles and other soft tissues. Semipermeable to ions cell membranes, non-liquid components as well as organic molecules bounded in the tissue structures are responsible for the polarization or dielectric properties of muscle. 19 Generally, tissue conductivity, , and dielectric permittivity, , are defined as complex values [Pearce, 2000]: = j (2.1 a) and = j (2.1 b) separating them into non-dissipative, and , and dissipative, and , parts respectively. The non-dissipative parts originate from the reversible energy change due to oscillation of ions and dipolar momentum of molecules in the electric field. The elastic collisions of ions and absorption of energy by the molecules and irreversible polarization determine the dissipative parts [Roussy and Pearce, 1995]. At the typical electrosurgical frequencies the contribution of the non-dissipative part of the electric conductivity and the dissipative part of the permittivity are small and can be neglected [Pearce, 2000]. In this study only the dissipative part of the conductivity and non-dissipative part of the permittivity are considered. Electric conductivity of soft tissue is a function of temperature, T, and water content, W. Generally, it can be represented in the form as [Ryan, 1996]: (T ,W ) = 0W exp(rT (T T0 )) (2.2) where 0 is conductivity at initial temperature T0 and rT is the rate of conductivity increase with temperature. In this study it is assumed that tissue water content is decreases due to the mass loss in the surface vaporization process and due to the 20 growth of vapor volume in the interstitial nucleation process. Respectively, the water fraction is given as a function of mass and volume loss: W = 1 Wm Wv (2.3) where Wm and Wv are water fraction lost through surface vaporization and bubble expansion respectively. 2.2.2 Continuity Equation and Boundary Conditions The integral form of the continuity equation for electric current in an arbitrary macroscopic closed volume U, bounded by surface S is given by: I= S J ds = t U dU (2.4) where I is net electric current, J is vector of volumetric current density, ds is vector surface element and is charge density. Assuming that there is no net electric current and no electric charge generated inside a control volume and applying divergence Gauss theorem to the left side of the equation the differential form of Equation (2.4) is obtained: J = 0 (2.5) The current density is related to electric field, E, by the Ohm law: J = E (2.6) 21 and, since electric field is minus gradient of electric potential, V: E = V (2.7) the differential form of the continuity equation can be given as: ( V ) = 0 (2.8) where = (x,y,z) is tissue electric conductivity, a function of temperature and water content. The boundary conditions for the electric potential are as follows: the potential is given at the scalpel-tissue interface and null at the outer tissue boundaries. The last condition roughly corresponds to the null potential at the return electrode. The functional dependence of the electric potential over the scalpel-tissue interface for two developed models is discussed in the next chapter. 2.2.3 Power Generation Term The local power density, q, can be obtained from the Poynting power theorem [Roussy and Pearce, 1995]: P= E J (2.9) where E is the electric field vector, J is the conductive current density vector. Substitution of Equation (2.6) into the expression for the power density (2.9) and replacement of the electric conductivity and permittivity by their complex representation (2.1 a) and (2.1 b) yields: 22 q = ( + j 0 ) E 2 (2.10) where is electric field frequency and the non-dissipative part of the electric conductivity and the dissipative part of the permittivity are neglected. The heating of the tissue results from the dissipation of energy of the translational motion of the charge carriers. Therefore, only the dissipative or real part of the power density contributes to the heating, or: Re{q} = E 2 (2.11) 2.3 Tissue Heat Transfer Thermal energy generated in the tissue volume leads to the temperature rise and phase change of the tissue components. The energy is redistributed to the neighboring regions of tissue by the heat conduction mechanism and part of heat is inevitably lost through radiation, convection and vaporization mass flux from the surface. This study models tissue in-vitro and, therefore, contributions of the metabolic heat and blood perfusion into the heat balance are neglected. 2.3.1 Heat Diffusion Equation Heat conduction can be described by the Fourier diffusion equation, [Welch and van Gemert, 1995]: P P T dm = (k T ) h fg + q c p + T T t dt (2.12) 23 where T is temperature, is tissue density, cp is tissue heat capacity, P is saturation pressure, k is tissue thermal conductivity, dm/dt is volumetric rate of tissue water loss due to phase change, hfg is phase change enthalpy, q is the heat source and t is time. The left-hand side of the equation (2.12) represents change in the tissue enthalpy. It is easy to show that in the temperature range expected in this model (from the room temperature of 20 to about 150 C) the enthalpy change due to the variation of pressure and density with temperature consumes very little thermal energy and, therefore, can be neglected: c p T dm = (k T ) h fg + q t dt (2.13) Water is the major component of muscle tissue, and muscle thermal properties are not significantly different from those of water. However, vaporization and ablation processes reduce tissue water content and alter thermal properties. The prime subject of this study is a steady state process of tissue cutting during which the water inside the tissue matrix is dynamically replaced by vapor rather than completely removed. Such an assumption allows defining the tissue thermal conductivity, k, heat capacity, cp, and density, , as linear functions of the water content: k (m w ) (k w k v )m w + k v (2.14) c p ( m w ) (c w c v ) m w + c v p p p (2.15) (mw ) ( w v )m w + v (2.16) 24 where mw is the mass fraction of water and superscripts w and v indicate the values corresponding to the liquid and vapor states respectively. 2.3.2 Boundary Conditions To solve Equation (2.13) for the temperature distribution it is necessary to specify boundary conditions at the tissue surface. Two surface regions are considered: the surfaces that are relatively far away from the vaporization region and the immediate tissue-scalpel interface. Though the distant surfaces are open to convective heat flux, its contribution to the overall heat balance is insignificant and many contemporary numerical models replace it with a constant temperature condition. The boundary conditions at the scalpel-tissue interface are more complex. Here, at least three types of heat fluxes must be considered: 1. 2. conductive heat flux between tissue and metal scalpel, convective heat flux caused by the vapor and water flow in the tissue-scalpel gap 3. vaporization heat flux associated with the vaporization mass flux The vaporization heat flux removes thermal energy from the interface and the conductive and convective fluxes may act in either way, depending on the temperature gradient across the interface. It is useful to estimate the relative contribution of each component. Usually, boiling heat transfer accounts for the largest heat loss [Incropera and DeWitt, 1990]. In the temperature range expected in this model (up to 150 C), attainable vaporization heat flux is about 1.3 103 kW/m2 [Cole, 1979, p.20]. Convective heat 25 flux is one or two orders of magnitude lower [Incropera and DeWitt, 1990 p. 529] and, assuming steady-state cutting conditions, can be neglected. The effect of the conductive heat flux can be estimated using the following consideration. Thermal diffusivity of tissue, t=6 10-8 m2/s is more then sixty times lower than thermal diffusivity of steel, s=4 10-6 m2/s and, initially, while the temperature of the scalpel is low, the heat flow into it from the tissue is significant. Common surgical techniques of the ETR consist of several strokes delivered one after another, each taking about a second to complete [Pearce, 1986]. Initially, the scalpel is at a room temperature and after prolonged application it can become extremely hot. Using the lumped capacitance method [Incropera and DeWitt, 1990 pp. 225229] the temperature rise of the scalpel can be estimated as: t T (t ) = Ts T exp( ) (2.17) where Ts is temperature of the tissue surface, T is initial temperature difference between scalpel and the tissue surface and is a thermal relaxation constant, equal: = Vc p As k L (2.18) where V is the scalpel volume, As is the scalpel surface area, and L is the scalpel length. For the estimations a typical scalpel electrode can be approximated as a cylinder with length, L=2 cm and radius, r=1.5 mm The thermal properties of the scalpel can be taken as equal to those of stainless steel (k=15 W/mK, =8,000 kg/m3, cp=480 J/kgK). If initially scalpel and the interface are at temperatures of 20 C and 100 C, respectively, then during the first second of cutting, Equation 2.17 yields a 26 temperature rise of only 5 C. Similarly, if initially the scalpel was at a high temperature, bringing it in contact with cold tissue will not result in relatively rapid drop of its temperature. This result makes it possible to neglect the temperature changes of the scalpel and treat the interface as a constant temperature boundary considering two cases: the scalpel at a room temperature, and at the maximum obtainable in the simulation temperature. 2.4 Coagulation Damage Temperature rise leads to initiation and acceleration of the irreversible biochemical processes such as protein denaturation [Thomsen, 2000] and ultimate cell necrosis. Although thermal coagulation is a general name for a number of simultaneous chemical and morphological processes, the dynamics of damage accumulation can be accurately represented by a first order kinetic model. It is assumed that a bimolecular collision process activates the reactants into a complex in a higher energy state which may either relax back to inactivated condition or progress to a damaged state [Welch and van Gemert, 1995, pp. 561-570]. The rate of disappearance of undamaged material relates to its concentration C through an overall reaction velocity, k: dC = kC , dt (2.19) which in turn relates to the equilibrium constant for formation of the activated complex, K*: G * k = A K * = A exp R T g (2.20) 27 where G* is the Gibbs free energy of formation of the activated complex, Rg is the universal gas constant, T is absolute temperature and A is proportionality factor generally dependent on temperature and concentration of the reactants. In turn, the free energy of formation is given as: G * = H * + T S * = E a + R g T + T S * (2.21) where H* and S* are the activation enthalpy and entropy respectively and Ea is the activation internal energy. Solving Equation (2.19) it is convenient to introduce a parameter characterizing coagulation damage in the form of Arrhenius integral equal to the logarithm of the ratio of the initial concentration of the undamaged material, C(0), to the remaining undamaged material, C( ), at the moment of time, , [Welch and van Gemert, 1995, p. 566]: C (0) ( ) = ln = A C ( ) e Ea R g T (t ) dt (2.22) 0 Note that in this representation A and Ea can be treated as calibration constants, representing molecular collision factor and reaction activation energy respectively. The Arrhenius equation demonstrates that coagulation damage accumulates lineally with time and exponentially with the rise in temperature. The later factor makes the equation extremely sensitive to the choice of calibration constants, which determine coagulation threshold. It is said that coagulation had 28 occurred when is reached value of 1.0, which corresponds to a tissue damage concentration of 63.2%. 2.5 Phase Change Phenomena In the previous sections addressing the processes of heat and electric conduction in tissue it was mentioned that the thermodynamic and electric properties of tissue fluids play the leading role in shaping these processes. In fact, as long as the water phase change can be neglected, it is a reasonably good choice for a thermoelectric model of a tissue to approximate skeletal muscle as a volume of tissue fluids and ignore its protein, fat and mineral content. It is clear that the liquid constituents have the most formidable impact on tissue thermodynamic and electric behavior and it is absolutely necessary to consider the process of vaporization and tissue drying when modeling the electrosurgical procedures. Water vaporization influences the tissue heating process as an energy sink. Its effect becomes important around the saturation point (for water it is 100 C at standard atmospheric pressure). Above the saturation point water becomes metastable and nucleation and subsequent growth of vapor bubbles inside the tissue becomes possible. The present study suggests that a high local pressure gradient and tissue stress generated due to vapor bubbles is the prime mechanism of the electrosurgical tissue resection [Pearce, 1986]. Though the amount of work done in the general area of liquid to vapor phase change is extremely abundant it is surprising that to-date only a few studies has been applied to biological tissues. By far the most thoroughly researched topics on water phase change as applied to biomedical sciences are: 29 1. 2. 3. surface vaporization to the atmosphere at normal body temperatures, interstitial nucleation during decompression, cavitation due to laser or ultrasound irradiation. The first topic and its applications is reflected in work on the subject of precipitation and body temperature regulation [Ayling, 1986], fluid loss through burn injuries [Gwosdow, et al., 1993] and tears evaporation in ocular tissues [Tsubota, 1992], [Mathers, 1996,]. There is a growing number of studies of interstitial nucleation as applied to decompression sickness. Theoretical studies discussing nucleation on dissolved inert gases [Srinivasan, et al., 1999] as well as some experimental work [Liew and Burkard, 1994] are of particular interest. The cavitation phenomena are covered in studies relevant to the laser tissue ablation [Jansen, et al., 1994], [Frenz, et al., 1998] and ultrasound ablation [Hubert, et al., 1998] and ultrasound tissue imaging [Fujishiro, et al., 1998]. All three areas correspond to special cases of the phase change phenomena: first to relatively low temperature and pressure gradients across the liquid-vapor interface, second to isothermal conditions and third to very high temperature and pressure gradients. Alternatively, these special cases can be described as slow and uniform, and extremely rapid and highly localized energy deposition processes respectively. The vaporization processes in tissue in between those limiting cases was described using Snelling s formula derived from experimental measurements of vaporization rates from lake Nukus in Uzbekistan [Brutasaert, 1982]: hm = (C1 + C 2 u )( Psat Patm ) , '' (2.23) were hm is the vaporization rate in units of water depth per unit of time, Psat is the saturation vapor pressure at the temperature of the water surface, Patm is the vapor 30 pressure in the atmosphere, u is the free stream velocity of air 7.5 m above the surface and C1 and C2 are calibration constants equal to 7.31x10-11 m/s Pa and 1.2x1011 1/Pa, respectively [Torres, 1994]. The modified equation (2.23) was derived to describe vaporization from solar ponds containing salt water from the Dead Sea and the Mediterranean Sea [Salhotra, et al., 1985]. In the case of the absence of wind the equation is: hm = C1 ( S Psat Patm ) , '' (2.24) where S is the salinity factor that accounts for the presence of dissolved ions. This study considers the following phase-change thermodynamic phenomena: 1 2 3 equilibrium surface vaporization, interstitial vapor nucleation interstitial bubble growth An analytical description of these processes is given in the framework of the developed model. Their relative contribution to the gross tissue behavior during electrosurgical resection is examined and numerical implementation is discussed. 2.5.1 Surface Water Vaporization Figure 2.1 shows a liquid-vapor interface. The dashed line represents a control surface in the vapor phase just above the actual interface. To determine net vaporization flux through the control surface it is necessary to consider the interface at the molecular level. The classical kinetic theory of gases gives the Maxwell 31 Figure 2.1 Mass fluxes at liquid-vapor interface. mv is the vaporization mass flux, mc is the condensation mass flux, u0 is the net vapor velocity, corresponding, in this case, to net vaporization [Van Carey, 1992]. velocity distribution for the fraction of molecules with Cartesian velocities in a certain range [Van Carey, 1992, p.113]: dn m = n 2 k B T 1/ 2 m 2 2 2 exp 2k T u + v + w B ( ) dudvdw , (2.25) where n is the total number of molecules, m is the mass of one molecule, T is absolute temperature, kB is Boltzmann s constant and u, v, and w are Cartesian velocities respectively. Suppose that the u velocity component is perpendicular to the interface. Integration of Equation (2.25) over v and w velocity components yields: dn S m = 2 k T n B 1/ 2 mu 2 exp 2k T du . B (2.26) 32 Only molecules that that lie within a distance u t, where t is a time interval, will pass through the control surface. The number of such molecules per unit time, per unit area is: 1 1 u t , dj S = dn S S t dx (2.27) where dx is a unit of distance, S is a unit of area and dnS is given by equation (2.26). Integration of Equation (2.27) over all possible values of u yields the total rate at which molecules pass through the control surface per unit area: n 8k BT jS = 4V m 1/ 2 , (2.28) where V is a unit of volume. Combination with the ideal gas law yields an expression for the molecular flux in the terms of local pressure and absolute temperature: M jS = 2 R g 1/ 2 P , mT 1 / 2 (2.29) where M is the molecular weight of vapor, Rg is the universal gas constant and P is local pressure. If there is no net vapor flux through the control surface, dynamic equilibrium between molecular fluxes from the liquid to the vapor phase and back is established. To offset the equilibrium and initiate vaporization the vaporization flux from liquid must exceed condensation flux. 33 It should be noted that Equation (2.29) predicts molecular flux in a stationary gas (u, v, and w are the velocities of random movement and do not result in net motion). However, due to vaporization the vapor above the control surface acquires a bulk velocity u0 in the direction normal to the phase interface. An analysis similar to this shows that the flux in the presence of a net motion is [Scarge, 1953]: jS M = ( a ) 2 R g 1/ 2 P , mT 1 / 2 (2.30) where + or - sign indicates the flux in the direction of the net motion or against it, respectively. The correction factor is: ( a ) = exp(a 2 ) a 1 / 2 [1 erf (a )] , where (2.31) a = u0 M 2Rg T 1/ 2 (2.32) where u0 is the velocity of the vapor in the vicinity of the control surface. In the case of water vaporization the combination of Equations (2.29) for the vaporization flux with Equation (2.30) taken with negative sign for the condensation flux and their evaluation at liquid and vapor local pressure and absolute temperature, respectively, yields total molecule flux through the control surface: 34 M j = 2 R g 1/ 2 P 1 Pl 1 / 2 1 / v2 . T m l Tv (2.33) The total mass flux is respectively: M m '' = 2 R g 1/ 2 Pl P 1 / 2 1 / v2 . T Tv l (2.34) And, finally, the surface vaporization heat flux, q , is: M q = h fg 2 R g '' 1/ 2 Pl P 1 / 2 1 / v2 T Tv l (2.35) where hfg is the latent heat of vaporization. Because the vaporization rate is determined by the heat flux, the resulting velocity of the vapor in the vicinity of the control surface is: q '' , u0 = v h fg (2.36) where v is the vapor density. Thus, Equations (2.35) and (2.36) establish an implicit relation between the vaporization heat flux and the bulk vapor velocity. In addition to the bulk vapor motion effect, the actual vaporization flux is reduced by so-called accommodation effect [van Carey, p.116]. Only a fraction, av, of the molecules cross the control surface due to vaporization. The remaining fraction, 35 1- av, represents reflection of vapor molecules that strike the phase-change interface but do not condense. The same effect occurs for molecules crossing the control surface from the vapor side. The fraction of molecules crossing the interface entirely due to vaporization or condensation is called the vaporization or condensation coefficient: av or ac, respectively. Sometimes they are called evaporation, transmission [Jones, 1992, p. 10] or accommodation [van Carey, 1992, p.116] coefficients. To avoid confusion, the term vaporization coefficient will be used in this study. It is usually assumed that these coefficients are equal even in the dynamic vaporization case. Taking into account the accommodation effect, Equations (2.34) and (2.35) can be rewritten as: M m '' = a v 2 R g 1/ 2 Pl P 1 / 2 1 / v2 T Tv l (2.37) and M q '' = a v h fg 2 R g 1/ 2 Pl P 1 / 2 1 / v2 , T Tv l (2.38) respectively, where av is the accommodation coefficient. 2.5.2 Interstitial Water Vapor Nucleation A liquid heated at constant pressure above its equilibrium vaporization temperature becomes metastable. In such a state, small local fluctuations of molecular density will produce embryo vapor bubbles within the bulk of the liquid. If the radius of an embryo bubble exceeds a critical value it will grow, if not, it will collapse. To initiate homogeneous vapor nucleation, i.e. rapid generation and growth of vapor 36 bubbles in bulk water, some energy is required to overcome the surface tension at the vapor-water interface. This energy is supplied from the liquid overheat. In pure water at atmospheric pressure the necessary energy is rather high and nucleation is initiated only at temperatures of about 300 C [van Carey, 1992, p.153]. However, few natural systems are free of dissolved gases and even weak concentrations of such contaminants will greatly reduce the nucleation threshold [Cole, 1979, p. 83]. In a more common situation when the heat is supplied to the liquid through the solid wall of some containing structure nucleation begins from the preexisting gas phase trapped in microcavities on the heated surface. In this case the phase interface exists already, very little additional energy is required to overcome the surface tension and the nucleation threshold corresponds to temperatures exceeding saturation limit by only a few degrees [Cole, 1979, p.121]. Nucleation from the preexisting phase interface is called heterogeneous . It can be initiated at the solid-liquid and liquid-liquid interfaces as well, though in these cases the required threshold energy or superheat is higher. Tissue liquid contained within the tissue structures, in particular within the muscles, can hardly be approximated as non-contaminated water and assumption of a pure homogenous nucleation in this case will lead to significant overestimation of the required superheat. But, though a normal tissue does not contain micro pockets of gas that can act as heterogeneous nucleation sites, it does contain significant amounts of nitrogen, oxygen and other gas molecules dissolved in tissue fluids and bound to various substances under normal conditions. It is highly likely, that due to the temperature rise and changes in the pressure the bounded gas molecules will dissolve into the water and act as an energy reservoir for homogeneous nucleation. Moreover, a large variety of tissue components, such as collagen fibers, cell membranes etc., 37 may provide numerous sites for heterogeneous nucleation and, in combination with the dissolved gases, may significantly reduce the required superheat. 2.5.2.1 Homogeneous Nucleation In order to create a vapor phase in a metastable liquid a sufficient number of molecules must acquire energies considerably greater than the average. The probability of activating enough of such molecules at the same region of space at about the same time is negligibly small, therefore, it is suggested that nucleation occurs by a stepwise collision process [Cole, 1979, p. 74]: * * An + A1 An +1 (2.39) where An* represents an activated cluster of x such molecules, A1 is a single inactivated molecule and An+1* is the activated cluster of n+1 molecules resulting from the collision of An* and A1. Generally this step-by-step, or bimolecular, process is assumed reversible until the size of the cluster reaches an equilibrium value. This equilibrium is unstable so that addition or removal of one more molecule will result in spontaneous growth of a vapor bubble or its collapse, respectively. The radius of the equilibrium-activated cluster is called the critical radius and the difference between the average molecular energy and the cluster energy is called the activation energy. The latter is equal to the maximum (reversible) work required to bring the system at constant liquid pressure and temperature to its equilibrium state, and in the case of nucleation from a single phase (homogeneous nucleation) is given by Equation (2.39): A = mv [g v g l + (Pl Pv )v v ] + 4 rc2 s , (2.40) 38 where mv is the mass of molecules in the nucleus, vv is the vapor specific volume, s is the vapor-liquid surface tension, rc is the nucleus radius and gv, gl, Pv and Pl are the vapor- and liquid-specific Gibbs functions and pressures respectively. The first term in this equation represents the work necessary to create a vapor volume and the second represents the work to create a spherical phase interface. At thermodynamic equilibrium the specific Gibbs functions of pure substances for liquid and vapor phases are equal and cancel out. The Young-Laplace condition for the bubble mechanical at equilibrium is: Pv Pl = 2 s . rc (2.41) The pressure Pv inside the bubble is correlated with the corresponding vapor equilibrium pressure Pve through [Van Carey, 1992, p.149]: v (P Pve ) Pv = Pve exp l l = Pve , R g Tl (2.42) where vl is the liquid specific volume, Tl is the liquid temperature and is called correction factor . The mass of molecules in the nucleus can be expressed as: 4 1 mv = rc3 3 vv (2.43) Combining Equations from (2.41) to (2.43) the change in the availability function can be rewritten as: 39 A = 3( Pve Pl ) 16 s3 2 . (2.44) The probability to find in a unit of volume Nn activated clusters that contain n molecules of vapor is: Nn A = exp , N kT ( n ) = (2.45) where N is the number of molecules in the unit of volume of liquid. Integration over all clusters that pass from n to n+1 molecules yields the steady-state rate of formation of vapor nuclei per unit volume and unit time [Cole, 1979, p. 79]: 16 s3 . exp 3kT ( P P )2 ve l 3 J = N S m 1/ 2 (2.46) From Equations (2.40) and (2.41) the critical radius of the nuclei is given by: 2 s rc = v ( P Pve ) Pve exp l l Pl R g Tl . (2.47) It should be noted that the equilibrium of these nuclei is unstable [Cole, 1979, p. 77]. Once formed they are bound either to collapse or to grow. In the presence of the external heat source, as is the case with electrosurgery, they have to grow. 40 2.5.2.2 Effect of Dissolved Gases In the presence of dissolved gas the change in availability function is given by [Cole, 1979, p. 82]: 16 s3 3 ' Pve + Pl X X Pl S 2 A = , (2.48) where X and XS are the solute-to-solvent mole ratios in the liquid phase and in a liquid saturated with the gas across a flat interface. The correction factor in front of the vapor equilibrium pressure is: v (P Pve ) ' = exp l l X . R g Tl (2.49) Assuming that the vapor-gas mixture inside the bubble behaves as an ideal gas, the bubble radius can be expressed as [Cole, 1979, p. 82]: 2 s rc = v (P Pve ) Pve exp l l X + Pl X X Pl S R g Tl . (2.50) The expression for the nucleation rate in the presence of dissolved gas is derived from the same probability condition (2.45) and has form similar to Equation (2.44) but for the factor in front of the exponent, which has not been clearly described yet. However, calculations show that the argument of the exponent has a determining effect on the onset of nucleation. In consequence, Equation (2.48) demonstrates that 41 the presence of dissolved gas lowers the availability function and, therefore, lowers required superheat. Equation (2.50) demonstrates that in the presence of dissolved gas the radius of the vapor nucleus will be smaller as well. Ward, et al. [Ward, et al., 1970] have shown that in the absence of the mass exchange, i.e. in a closed volume, the maximum of the availability function, Equation (2.48), corresponds to a stable equilibrium. They also demonstrated that in a closed volume the radiuses of several equilibrium bubbles are inversely proportional to their concentration, and can be as small as several microns each. 2.5.2.3 Heterogeneous Nucleation If a vapor bubble is forming at a liquid-solid interface a portion of its surface area will be provided by the interface. The availability function in this case will be written as [Van Carey, 1992, p.172]: 16 s3 f A = 3( Pve Pl ) 2 , (2.51) where f = 1 2 + 3 cos cos 3 4 ( ) (2.52) and is the contact angle between the liquid-vapor phase and the solid surface. Similarly, the nucleation rate will be: J=N 3/ 2 1 + cos 3 S 2 mf 1/ 2 16 s3 f exp 2 3kT ( P P ) . ve l (2.53) 42 Comparison of equations (2.53) and (2.46) reveals that, at low contact angles the superheat required for homogeneous nucleation is lower than heterogeneous nucleation. At contact angles of approximately 68 , the two modes are equally probable. For contact angles grater than 68 , the heterogeneous mechanism requires lower superheat and at the angles approaching 180 (complete non-wetting) no superheat is possible. 2.5.2.4 Effect of Interfacial Tension Gradient If nucleation occurs at the interface between two liquids with the surface tensions Sa and Sb and interfacial tension Sab the availability function is given by [Van Carey, 1992, p.175]: 3 16 Sa F A = 3( Pve Pl ) 2 , (2.54) where 3 Sa F = 1 3 3 3 Sa 2 3Z a + Z a + Sb 2 3Z b + Z b3 , 4 2 2 2 Sa + Sab Sb , Za = 2 Sa Sab [ ( ) ( )] (2.55a) (2.55b) 2 2 2 Sb + Sab Sa , Zb = 2 Sb Sab (2.55c) 43 The nucleation rate is given by: 3 16 sa F . exp 3kT ( P P )2 ve l J=N 2/3 a 1 Z a 3 Sa 2 mF 1/ 2 (2.56) It is shown [Jarvis, et al., 1975] that nucleation is possible if any of the following conditions is satisfied: Sab > Sa Sb or Sab > Sb Sa , (2.57) otherwise homogeneous nucleation in only one of the liquids is possible. Equations (2.51) and (2.54) demonstrate that heterogeneous nucleation can occur at superheat levels much lower than required for the homogeneous case. 2.5.3 Vapor Bubble Growth When the spherical vapor nucleus is formed in a superheated liquid the molecules at the interface begin to vaporize into the interior consuming the liquid superheat in the process and the bubbles begin to grow. Initially, the temperature inside the bubble and the pressure at the phase interface are equal to the temperature of the superheated liquid and corresponding saturation pressure respectively. The expansion of the bubble results in the pressure drop to the ambient value and temperature increase to the corresponding saturation value. Just after the nucleus is formed and its size is extremely small the heat transfer to the interface is very fast and does not limit the vaporization rate. At this stage only interaction with the surrounding liquid limits bubble growth. The increase of the bubble radius with time 44 is determined only by the value of superheat, latent heat of vaporization and the densities of vapor and liquid respectively. It can be determined from the Rayleigh equation [Van Carey, 1992, p.195] in the form: 2 T Tsat (Pinf ) h fg v R(t ) = l 3 Tsat (Pinf ) l 1/ 2 t, (2.58) where Tl is the temperature of the superheated liquid, Tsat(Pinf) is the liquid saturation pressure at the ambient pressure, hfg is the vaporization heat and v and l are the vapor and liquid densities respectively. The growth rate of the bubble is given by a simple differential of Equation (2.58): 2 T Tsat ( Pinf ) h fg v VR = l 3 Tsat (Pinf ) l 1/ 2 (2.59) and is constant. 2.6 Electric Discharge in Gases Gas discharge is a process of ionization of the gas by the applied electric field and flow of electric current through the ionized gas. If the gas is ionized to a sufficient degree it emits light. There are several types of gas discharges classified by their volt-ampere characteristics, frequency of external electro-magnetic field and mechanism of electron emission. In the present section two discharge types are described briefly. The first is a spark discharge. It may either occur during electrosurgical cutting or have some common properties with the discharge that actually takes place. The second is an arc discharge. Though in electrosurgical practice the electric sparks that are often observed in procedure sometimes are 45 refereed to as arcing , a discharge commonly known as the arc is unlikely to occur in electrosurgical procedures. 2.6.1 Electric Breakdown and Spark Discharge An electric field applied to a volume of gas begins to accelerate seed electrons that are always present due to cosmic rays or other dissociating mechanisms. If a field is strong enough, and the mean free path of an electron is long enough, a seed electron will pick up kinetic energy sufficient to ionize a gas molecule. The result is two slow electrons. They are accelerated again producing two more electrons each, and so forth. The resulting process of ionization build-up is called electron avalanche. Ionization transforms a non-conducting gas into conducting; thus, electric breakdown occurs. If ionization reaches high enough values the breakdown is accompanied with a light flash known as a spark . However, it should be noted, that in gas discharge physics the term spark is used for a specific type of discharge occurring at atmospheric or higher pressures in gaps longer than several centimeters. A spark discharge develops from a thin ionized channel, known as a streamer, which is formed from the primary avalanche and grows from the anode to the cathode [Rees, 1973 p.54]. If a gap is short, the breakdown develops from the avalanche in the entire gap volume and may be accompanied with a visible light flash. I will refer to such an event as a spark as well and will differentiate when necessary. Figure 2.2 shows the so-called Paschen curve: the breakdown voltage Vt in an atmospheric air gap as a function of the product of air pressure p and gap width d [Raizer, 1991, pp.128-134]. There exists a minimum breakdown voltage for a discharge gap Vmin 300 V at (pd)min 0.83 Torr cm. At pd higher than (pd)min 0.83 Torr cm the electrons 46 Figure 2.2 Breakdown voltage in air as a function of product of pressure and discharge gap width [Raizer, 1991]. Vmin = 300 V, (pd)min = 0.83 Torr cm. avalanche and, therefore, the degree of ionization is limited by electron energy losses due to collisions with neutral molecules and attachment in electronegative gases. At lower pd the ionization process is limited by loss of electrons due to diffusion out of the electric field. The threshold voltage, Vt, of the breakdown is given by a phenomenological expression: B( pd ) , C + ln ( pd ) Vt = (2.60) 47 where p is local pressure in the breakdown gap (Torr), d is brekdown gap width (cm), B and C are constants depending on the type of gas in the breakdown channel (for air B=365 and C=1.18) and Vt is given in Volts. In the established spark a bell-type or Gaussian-type distribution of electrons across the discharge channel is developed. A time dependent two-dimensional electron density profile is given [Raizer, 1991, p. 328]: ( z v d t )2 + r 2 exp + ( b )v d t , (2.61) 4 De t ne ( z , r , t ) = (4 De t ) 3/ 2 where z and r are the coordinates along and across the discharge channel respectively, De is the electron diffusion coefficient, vd is electrode drift velocity, and b are ionization and recombination coefficient respectively. From Equation (2.64) the radius of the spark channel, rD, can be deduced as a function of time [Raizer, 1991, p. 328]: rD = 4 De t . (2.62) Total electric current in the spark discharge channel can be estimated as a function of time [Raizer, 1991, p. 331]: i (t ) = ev d exp[( b )v d t ] , d (2.63) where e is electron charge and d is total length of the discharge gap. 48 After electric current is developed the associated temperature rise leads to an increase in electron-molecule collisions, diffusion of electrons out of the discharge and, ultimately, to discharge collapse. However, if the electric field is still applied to the gas, secondary breakdown is possible and, as electron-ion recombination time is much longer than breakdown time [Raizer, 1991, p.128], secondary breakdown can take place at a lower electric field. To prevent breakdown from collapsing and to establish a self-sustaining discharge it is necessary to limit the discharge current using, for instance, ballast resistor. In electrosurgery a spark discharge probably occurs in the fulguration mode. Typical peak fulguration voltages are above 1 kV (r.m.s.) and separation between the electrode and the tissue is from several millimeters to a centimeter. Such conditions appear to be sufficient to breakdown and form a spark discharge, Figure 2.2. A volume breakdown, or a spark, is most likely to occur during electrosurgical cutting. Indeed, the typical time for a spark to develop is from 10-9 to 10-8 seconds [Raizer, 1991, p. 132] short enough for even several sparks to occur during one half cycle of the ESU waveform. Further, typical peak ESU voltages used in electrosurgical cutting are higher than the minimum breakdown voltage of 300 Volts in air at room temperature [Raizer, 1991, p. 134]. From the other side, peak voltages of several hundred volts imply that, at atmospheric pressure, the breakdown distance is on the order from 10 to 100 m, Figure 2.2, on the order of the diameter of a typical muscle cell. 2.6.2 Arc Discharge An arc discharge is a self-sustaining discharge that has cathode thermionic emission as a source of electrons rather than ionization in the gas volume. Thermionic emission occurs if the cathode or a portion of it is heated up to several 49 thousand degrees by, for example, a short circuit current. When an arc covers only a small spot on the cathode such a spot appears and disappears and moves rapidly and randomly on the cathode surface. It often happens that in the process the cathode is melting, eroding and, eventually, is destroyed. An arc discharge burns at voltages as low as several volts and allows the passage of current from several to several thousand amperes [Raizer, 1991, pp. 245-249]. It is unlikely that in electrosurgical cutting procedure temperatures sufficient for thermionic emission and for ignition of an arc discharge can be reached. It was observed by the author that if the tip of a standard electrosurgical electrode is submerged into physiological saline to a depth of 3-5 mm and RF power of at least 100 Watts is applied to it an electric discharge along the electrode s submerged edge is initiated and after several seconds the tip is eroded. Tip erosion might indicate the presence of an arc discharge. However, in the described experiment a discharge was covering the entire submerged perimeter of the electrode in use, while under typical clinical conditions, a discharge occurs only along the electrode leading edge. In this case, heat conduction can prevent the rise of electrode temperature and the formation of arcs. 50 Chapter 3: Thermal Ablation Model 3.1 Abstract In this chapter a qualitative model of the ablation mechanism is suggested for electrosurgical resection of skeletal muscle. The anatomy of skeletal muscle is outlined briefly. Histological samples of the lesions produced by the electrosurgical resectors are examined and a hypothesis of tissue ablation due to water vapor generation is proposed. Simplified numerical analysis of nucleation rate and expansion of vapor bubbles in muscle fiber is conducted and conclusions on the ablation mechanism caused by the interstitial water phase change and its consequences are made. 3.2 Anatomy of Skeletal Muscle Skeletal muscle is made up of thousands of cylindrical muscle fibers (single cells) often running all the way from origin to insertion. The fibers are bound together and attached to bones and skin by a fibrous connective tissue through which run blood vessels and nerves. A fiber is about 50 m in diameter. Each muscle fiber contains an array of myofibrils that run the entire length of the fiber, mitochondria, an endoplasmic reticulum and numerous nuclei. Each myofibril is about 1 m in diameter and made up of arrays of parallel thick and thin filaments. The thick filaments have a diameter of about 15 nm. They are composed of the protein myosin. The thin filaments have a diameter of about 5 nm. They are composed chiefly of the protein actin along with smaller amounts of two other proteins: troponin and tropomyosin [Ganong, 1995]. 51 Seen from the side under the microscope, skeletal muscle fibers show a striated pattern created by alternating dark A bands bisected by the H zone and light I bands bisected by the Z line, Figure 3.1. The entire array of thick and thin filaments between the Z lines is called a sarcomere. At rest a sarcomere is about 2 m in length and contracts to around 70% of this length. Shortening of the sarcomeres in a myofibril produces the shortening of the myofibril and, in turn, contraction of the muscle. A single muscle twitch lasts for approximately 25-50 msec. Figure 3.1: A sarcomere. The thick filaments produce the dark A (Anisotropic) band. The thin filaments extend in each direction from the Z line. Where they do not overlap the thick filaments, they create the light I (Isotropic) band. The H zone is that portion of the A band where the thick and thin filaments do not overlap. The chemical composition of skeletal muscle is approximately 70% water, 7% lipids (fat), and 22% protein [Ganong, 1995]. Water is held by the protein fraction, and a small change in the ability of protein to bind water has a huge effect on the water-holding capacity of the muscle. 52 3.3 Histological Evaluation of Electrosurgical Thermal Damage The degree and the nature of electrosurgical thermal damage to the tissue depends on the energy distribution and exposure time. The distribution of the thermal energy in tissue is determined primarily by the RF energy density, heat conduction and water phase transition processes at the tissue-scalpel interface. Figure 3.2 shows a histological crossection of a typical lesion obtained by applying an electrosurgical resection loop used in the Transurethral Resection of Prostate (TURP) procedure to skeletal muscles in vitro. The thermal damage in this lesion may be differentiated into three zones: 1. a layer of melted tissue with small (diameter 5 - 50 m) vapor bubbles 2. a layer with large (diameter 0.2 - 0.5 mm) vapor vacuoles 3. a layer with lost birefringence. The TURP utilizes the highest power level among all electrosurgical procedures. Consequently, the resulting thermal damage is most extensive and well defined. In the given sample the width of successive layers is about 0.1, 0.5 and 1.5 mm respectively. Due to lower power levels the other procedures may result in narrower zones of damage (especially the bubble and vacuole zone) and smaller and less numerous bubbles and vacuoles. Nevertheless, thermal damage due to vapor formation can be identified on almost all histological samples. Also, it should be noted that observed samples demonstrate lateral cross-section of the cut and do not show the portion of the tissue immediately in front of the scalpel where power density is the highest. Therefore, even insignificant presence of the vapor bubbles along the sides of the cut may indirectly point out that at the front of the scalpel more intense nucleation occurs. 53 Figure 3.2 Transmission Polarizing Microscopy (TPM) cross-section of the electrosurgical lesions in beef skeletal muscle produced by the loop urological resectors: top - wire loop (Storz, Inc.), bottom - Max Blade (C. R. Bard, Inc.). Cutting was performed under deionized water with the output power of the ESU at 200 Watts and cutting speed 2 mm/sec. Observations of the vapor bubbles in the first zone of the thermal damage (layer of melted tissue) reveal that: a) almost all bubbles are located inside of muscle fibers, b) maximum diameter of the bubble does not exceed the diameter of the fiber, 54 c) many bubbles have an opening (burst) into the cut. Based on these observations the hypothesis suggesting interstitial vaporization as the prime cause of tissue ablation [Pearce, 1986], [LeCarpenter et al., 1993] is adopted for the present study. This hypothesis states that when tissue temperature reaches certain vaporization threshold intracellular nucleation and subsequent expansion of vapor bubbles causes the loss of tissue mechanical cohesion and allows the scalpel to advance [Pearce, 1986]. 3.4 Ablation model 3.4.1 Phenomenological Formulation of Ablation Process Figure 3.3 demonstrate a simplified model of the ETR process in skeletal muscle and schematically shows a cross-section of the electrosurgical cut and advancing scalpel. Cutting was performed across the muscle fibers. In the model the fiber is replaced by a rectangular cell with square cross-section in the plane of the cut of 50x50 m2 and length in the direction perpendicular to that plane much longer than the width of the scalpel. Initially, the scalpel is in mechanical and electric contact with tissue, electric current is heating the tissue up and, at this point, only vaporization of tissue water into the atmosphere takes place at the scalpel-tissue interface, Figure 3.3 a). When the temperature in the cell immediately in front of the scalpel reaches certain nucleation threshold a vapor bubble will appear, Figure 3.3 b), and grow until its volume will be equal to the crossectional cell volume. The latter is defined as the volume of a cube 50x50x50 m3. At this point the cell will burst open and the fiber will be considered cut through or ablated, Figure 3.3 c). In order for the scalpel to advance all cell-fibers in front of it must be respectively cut. It is possible that simultaneously with the 55 appearance and growth of the first vapor bubble more bubbles will be generated in the same cross-sectional cell volume. In this case ablation occurs when cumulative volume of all bubbles becomes equal to the cell crossectional volume. Figure 3.3: Model of the ETR process. Upper and lower rows are side and top views of the cut. Scalpel is in electric contact with tissue (a). Vapor bubbles appear in the muscle fibers in front of the scalpel (b). When they have grown to the size of the cell (c) the cell bursts open and scalpel advances forward (d). At the same time smaller bubbles are growing further away from the scalpel (c); they are filled with water after the temperature dropped (d). 56 If the energy supplied to the tissue is sufficiently high, the nucleation temperature can be reached not only in front of the scalpel but also further away, at the distances from the tissue-scalpel interface exceeding the diameter of the muscle fiber. In this case, the vapor bubbles will be formed and grow while the energy supply allows. It is assumed that after the growth of the interstitial bubble ceases, it is filled with water, Figure 3.3 d). 3.4.2 Functional Analysis of Ablation Process At the present time, there is not sufficient information on the values and temperature dependencies of some material properties of the muscle cell (such as interfacial tension and content of dissolved gases) to allow precise calculation of the nucleation rate and the size of the vapor bubbles. However, a simplified analysis of the nucleation and bubble growth processes based on the known range of the cell s material properties provides a set of assumptions sufficient for a quantitative ablation model. 3.4.2.1 Estimation of Nucleation Threshold and Nucleus Size Vapor nucleation in liquid occurs when molecules contained in a certain volume acquire energy sufficient to overcome the liquid-vapor surface tension. It is reasonable to assume that in the liquid content of a muscle cell vapor nucleation requires much less energy than in pure water. In the muscle cell low surface tension at the intracellular interfaces (such as the interfaces between individual myofibrils and cell membrane) and dissolved gases (primarily nitrogen) provide sufficient nucleation enhancement. Table 3.1 shows values of the surface tension and nitrogen content at atmospheric pressure and room temperature for water and components of muscle cell. 57 Table 3.1: Nucleation parameters of water and muscle cell. Surface tension, s, mN/m 72 0.04-3 N2 content, X, m3/m3 0.0164 0.075 Water Cell membrane b Lipid c a b a [Incropera, De Witt, 1990] [Zachar, 1971] c calculated from data [Van Liew et al., 1995]. Figures 3.4 and 3.5 show, respectively, nucleation rate and average radius of the nuclei as a function of temperature. The calculations were performed for pure water (thin solid lines), water with dissolved nitrogen (thick solid lines), water near an interface with low interfacial tension (thin dashed lines) and for a combination of the two last cases (thick dashed lines). The intracellular nitrogen content was assumed equal to the content in lipids: six times higher than in water [Van Liew et al., 1995]. The low value of interfacial tension was assumed equal to the typical value for a phospholipid membrane of about 0.5 mN/m [Schurch et al., 1971] at room temperature and was assumed to follow the same temperature dependence as in the case of water. In the case of pure water Equations (2.44) and (2.47) were used to calculate the availability change and radius of the nuclei, respectively. In the case of nitrogen contaminated water Equations (2.48) and (2.50) were used, respectively. In the calculations involving low surface tension the same sets of equations were used but the water-vapor surface tension was replaced by the corresponding value for a phospholipid membrane. 58 Figure 3.4 Nucleation rate. Traces correspond to a) pure water, b) gas-contaminated water, c) low surface tension and d) combination of gas contamination and low surface tension respectively. A level of 1.6 1015 nuclei m-3 s-1 corresponds to appearance of one nucleus inside a 50 m cross-sectional volume during one ms time interval. It is useful to define the nucleation threshold as the rate of appearance of one nucleus in a cell cross-sectional volume during a control time interval. For purposes of discussion, the critical time interval, c, can be defined as a time to cut through a fiber: c = 2rc , Uc (3.1) where Uc is anticipated average cutting speed. For a cutting speed of 1 cm/s and 50 m muscle fiber radius the critical time is one ms. 59 Analysis of the functional behavior of the nucleation rate shows that dissolved gas alone does not significantly reduce the nucleation threshold. From the other side, low surface tension has a considerable effect on the threshold but it is a combination of both factors that can allow nucleation at the lowest temperatures. Also, all four models demonstrate rapid increase of the nucleation rate with only slight increase in temperature. At the nucleation threshold a one-degree temperature increase results in at eight thousand times increase in the nucleation rate. Figure 3.5 shows that the average radius of the vapor nucleus decreases with an increase in temperature. In the cases of water without any dissolved gas the radius approaches infinity when the temperature approaches 100 C. Physically, this corresponds to equilibrium vaporization from a flat surface at the atmospheric pressure. A small amount of dissolved gas allows a stable nucleus to exist even below water equilibrium vaporization temperature. The presence of such nuclei in water saturated with nitrogen at room temperature has been demonstrated theoretically [Ward et al., 1970] and proven experimentally [Ward et al., 1982]. The presence of the gas-containing nuclei in the muscle tissue and blood has been postulated in a number of decompression models [Kislyakov, 1988], [Van Liew , 1991], though the experimental proof has to be obtained. Two levels indicated on figure 3.5 correspond to the radii of a myofibril and muscle fiber. The nuclei radii at the temperatures corresponding to the nucleation threshold levels as indicated on figure 3.4 are from 1 to 100 nm in all; much less than the size of any cellular organelle and on the order of a size of a muscle protein filament. 60 Figure 3.5: Nucleus radius. Traces correspond to a) pure water, b) low surface tension, c) gas-contaminated water and d) combination of gas contamination and low surface tension. Levels of 50 and 1 m correspond to the radiuses of myofibril and muscle fiber respectively. 3.4.2.2 Vapor Bubble Growth and Additional Nucleation Given a sufficient supply of energy, once formed inside the cell crosssectional volume, the bubble nucleus will expand due to the vaporization at the vaporliquid interface until the cell bursts open. In the same time the vaporization process will consume a portion of the supplied energy. Depending on the energy density, temperature, expansion velocity and nucleation rate more bubbles can be formed inside the volume. A simplified functional analysis shows that during expansion time additional bubbles can be formed only at relatively high power densities and if nucleation begins at relatively low temperatures. 61 In the beginning of the expansion the bubble growth is inertia-controlled and assuming constant temperature and neglecting the initial radius of the bubble the expansion time (time to expand to the cross-sectional volume), e, can be determined from Equation (2.58) as: 2 T Tsat (Pinf ) h fg (Tl ) v (Tl ) e = l l (Tl ) 3 Tsat (Pinf ) 1 / 2 rc , (3.2) where rc is taken equal to the radius of muscle fiber. Neglecting the energy loss on heat transfer and vaporization into the bubble the temperature rise inside the crosssectional volume can be estimated using relation (1.4) as: T = P e l (Tl )C p l (Tl ) (3.3) where P is power density in the cross-sectional volume. In Equations (3.2) and (3.3) the temperature of the liquid, Tl, was assumed equal to the nucleation temperature, Tn. Finally, the number of created nuclei, Ne, can be calculated by integration of the nucleation rate, J(T), over the expansion time: N e ( e ) = Vc J (T )dt , (3.4) 0 where Vc is cross-sectional volume. Table 3.2 shows the number of additional bubbles formed during expansion of the initial bubble in a 50x50x50 m3 cell volume as predicted by the four nucleation 62 models discussed. Calculations were performed for power densities at the scalpeltissue interface of 1010, 1011 and 1012 W/m3 from the range predicted by the semicylindrical and hemispherical source models, Figure 1.1. Table 3.2 Number of bubbles formed in 50 m cell crossectional volume during expansion of initial bubble. Nucleation threshold, Tn, C 101 176 301 306 Expansion time, e, s 31 1.2 0.3 0.28 Number of bubbles formed, Nc a P=1010 1 10-2 3 10-4 7.1 10-5 7.5 10-5 P=1011 0.2 5 10-3 7.3 10-5 7.8 10-5 P=1012 5 7 10-4 1 10-4 1.2 10-4 Nucleation model Dissolved N2 and low s Low s Dissolved N2 Pure water a Number of bubbles 2 10-2 means that during the expansion time the probability to form one bubble in the cross-sectional volume equal 2 10-2, number like 5 means that at the same conditions the probability to form 5 bubbles equal one. The assumption Tl=Tn adopted for the calculations of the expansion time, e, Equation (3.2), and temperature rise, T, Equation (3.3), appears to be valid for T less than 10 C and neither expansion velocity nor tissue material properties change significantly in this interval. Neglecting the vaporization energy loss in the calculation of the temperature rise, Equation (3.3), is not valid for all temperature and power density ranges. The energy deposited into the cross-sectional volume during the expansion time can be estimated as: E c = P e (Tl )Vc (3.5) and the energy consumed by vaporization during bubble growth is: Ev = h fg (Tl ) v (Tl )Vc (3.6) 63 Figure 3.6 shows the ratio of the deposited and vaporization energies as a function of temperature for power densities of 1010, 1011 and 1012 W/m3. Under the condition Ev>Ec, the temperature of the liquid will remain equal to the nucleation temperature, Tn, and the bubble expansion time, e, will be determined by the power density: h fg (Tl ) v (Tl ) P e (Tl = const ) = (3.7) In the range of power and temperatures discussed e is less than the critical time, c, Equation (3.1), of 1 ms, which means that no additional nuclei will appear in this case as well. Figure 3.6 Deposited, Ec, vs vaporization, Ev, energies. Traces correspond to the power densities of a) 1010, b) 1011 and c) 1012 W/m3 respectively. 64 3.4.3 Limitations of Ablation Model Simplified functional analysis of the interstitial nucleation and growth of vapor bubbles has demonstrated that the most critical parameter required for the modeling of this process is the temperature of the nucleation threshold, Tn. Due to a high uncertainty in definition of the material properties of the muscle cell the value of Tn can not be determined exactly; instead it can be postulated in the numerical model of ETR and predicted results can be compared with experiment. The postulation of the threshold temperature is equivalent to the choice of a set of at least two important for nucleation factors: surface tension and content of dissolved gas. Obviously, one nucleation threshold corresponds to a range of values of material properties. However, once the threshold temperature is postulated, the material properties will be required primarily to describe an additional nucleation in the cell control volume. It has been shown that in the range of anticipated interstitial power densities at the scalpel-tissue interface additional nucleation will take place only at low and moderate values and only a single vapor bubble will be responsible for ablation of each muscle fiber. Therefore, it is expected that the ETR model based on the assumption of threshold temperature will be sufficiently accurate at low and moderate settings of the ESU cutting power. Also, it is expected that such a model will predict vapor damage in the tissue depth were power density will be sufficient to reach nucleation threshold. 3.5 Summary In this chapter a model of tissue electrosurgical thermal ablation was introduced. The model based on numerous studies of sections of tissue samples subjected to electrosurgical cutting. The studies are revealing a large number of small vapor bubbles adjacent to the side of the cut and smaller number of larger vacuoles 65 deeper in tissue. It is suggested that these bubbles are the products of interstitial vapor nucleation and that they are ultimately responsible for tissue mechanical destruction in the electrosurgical cutting procedure. Functional analysis of the nucleation process and vapor bubble growth was performed. The nucleation rate and size of the nuclei were studied as a function of temperature and water-vapor interfacial tension and water contamination. It was shown that the values of the surface tension and gas contamination realistically expected in the muscle cell might result in significant reduction of the nucleation threshold temperature as compared with pure water. However, it was also shown that slight variations in the surface tension and contamination would result in significant shifts of the nucleation threshold. Therefore, a parametric numerical study of the ETR process is recommended to obtain realistic estimates of the value of the nucleation threshold temperature. Analysis of the growth of a generated vapor bubble was conducted. The rate of growth and the energy consumed in the growth process were evaluated. It was found that for the most realistic levels of electrosurgical power dissipated in the tissue and for most of the realistic cutting speeds only one or a few bubbles would be created in a single muscle cell before the volume of created vapor will exceed the volume of the cell. A generation of enough vapor to fill a sphere with diameter equal or larger than the diameter of the muscle cell is suggested as a condition of destruction of a cell adjacent to the electrosurgical scalpel. At the same time, it was assumed that vapor bubble formed deep in the tissue could grow as long as it is supplied with energy. 66 Chapter 4: Evaluation of Water Vaporization Rate from Muscle Tissue Surface 4.1 Abstract Numerous studies of water vaporization at temperatures below the equilibrium vaporization level have indicated that the vaporization rate from an open surface is highly dependent on the accommodation coefficient of the surface, particularly for rough or irregular surfaces. The vaporization of water in temperature range of 25 95 C from a water pan and beef muscle was analyzed experimentally and numerically with the intent of validating the kinetic equation for vaporization flux and calibrating it for the use in the numerical model of the ETR process. The rate of mass loss from tap water, 0.9% saline and beef muscle samples heated to a constant temperature was measured to document the surface vaporization rate and to determine the vaporization coefficients for open water and muscle surfaces, respectively. Vaporization mass flux from a flat open water surface was calculated with the assumption of the vaporization coefficient equal unity. The mass flux predicted by kinetic theory was compared with the experimentally documented fluxes for water and muscle and the values of the vaporization coefficients were determined. The vaporization coefficient, av, introduced in Chapter 2 for calibration of the kinetic equation (2.37), were determined to be equal: for water (1.5 0.3) x 10-4, for saline (1.3 0.1) x 10-4 and for beef muscle (0.14 0.02) x 10-4 respectively. The vaporization rates from muscle tissue predicted by the calibrated kinetic equation were in reasonably good agreement with experiment. 67 4.2 Introduction Accurate prediction of the rate of water vaporization from the tissue surface into the atmosphere is essential to many surgical techniques involving thermal treatment of tissues. The vaporizing water acts as a heat sink reducing the surface temperature [Torres et al., 1993] and resulting in subsurface temperature build up responsible for explosive ablation [LeCarpenter et al., 1993]. In electrosurgical procedures one important tissue physical property dependant on the water content is electric conductivity. Decrease in electric conductivity resulting from the water loss [Ryan et al., 1997] may result in complete cessation of electric current and premature termination of the procedure [Motamedi and Protsenko, 1998]. Previous models of surface vaporization during tissue thermal treatment have been developed primarily for laser applications and are based on Snelling s formula, Equation (2.23) [Torres 1994, Pearce 2000] or on the balance of energy, momentum and mass at the vaporization front [Zweig and Weber 1987, Hu et al., 2001]. Snelling s formula was derived from experimental measurement of vaporization rates from solar ponds under natural environmental conditions in the temperature range from about 20 to 40 C [Brutsaert, 1982]. The vaporization front balance approach has been adopted from analysis of laser drilling and vaporization of solid materials [Krokhin, 1971] and requires that the vaporization is essentially a quasi-equilibrium process and vapor expands into a vacuum [Zweig 1987] or, alternatively, that the liquid pressure is equal to the vapor pressure and, at the same time, much higher than atmospheric pressure. Under natural conditions these assumptions are not satisfied until the vapor temperature reaches 100 C and about 180 C respectively. In Chapter 2 the net vaporization flux from the water surface was described using kinetic theory as a balance between fluxes of molecules leaving the liquid phase 68 and entering it from the vapor phase due to vaporization and condensation respectively, Equations (2.29, 2.30). The vaporization from a liquid surface is driven by the gradient of vapor pressure across the liquid-vapor interface. Since the vapor pressure at the liquid side of the interface is equal to the saturation pressure, therefore, in order to define the vaporization rate, the vapor pressure at the liquid side of the interface has to be defined. In the steady-state process of vaporization from water surface a relatively thick layer of almost saturated vapor is formed above the surface. Yen and Galea [Yen and Galea, 1969] reported that at a water surface temperature of 24 - 25 C the relative humidity at 2 mm above the surface is 97%. Shiba and Ueda, [Shiba and Ueda, 1965] reported relative humidity of more than 87% at 2 cm above the water surface maintained at 26 C. Therefore, in a steady-state conditions, it is reasonable to assume that local vapor pressure, Pv(Tv), in the kinetic vaporization equation (2.34) can be replaced by vapor saturation pressure, Psat(Tv). However, direct application of the kinetic equation will result in significant overestimation of the vaporization rate [Jones, 1992, p.8]. Actual vaporization flux is reduced by the reflection of molecules from the liquid-vapor interface [Jones, 1992, p. 25-43] and by resistance to the vaporization due to diffusion in the gas above the liquid surface [Palmer, 1978]. In this study a vaporization coefficient, av, will be defined, following Hickman and Maa, [Hickman and Maa, 1972], as the ratio of the vaporization fluxes experimentally determined and predicted by the kinetic theory: av = '' mexp '' mkin = '' qexp '' q kin . (4.1) In literature this ratio is also called a transmission, condensation, evaporation or accommodation coefficient [van Carey, 1992, p.116], [Jones, 1992, p. 10]. Its value 69 depends on the roughness of the surface [Brutstaet, 1965], contamination [Jarvis, 1962], salinity or ionic concentration [Salhotra, et al., 1985] and other factors. Experimental determination of the vaporization rate and value of the vaporization coefficient of water has been substantial [Jones, 1992, p. 30]. However, most of the results were obtained for relatively pure water and under specialized conditions [Wyllie, 1949] and could be hardly applicable for tissue fluids. Clinical studies of the vaporization rates from biological tissues surfaces has been conducted for vaporization of sweat from skin [Ayling, 1986], tears from cornea [Herold, 1987], [Mathers, 1993] and tissue fluids from sites of burn injury [Gwosdow, et al., 1993]. Though experimental data are numerous they correspond to a narrow range of skin and body temperatures, therefore additional study of the water vaporization from the tissue surface was required for this research. The primary objective of this study was to confirm the validity of the kinetic vaporization model for prediction of water vaporization rate from the surface of muscle tissue in the temperature range from 22.5 to 95 C. In order to validate the experimental technique, the vaporization rate from open water surface was measured and the results were compared with those reported in literature and predicted by the kinetic model. The vaporization rate of a 0.9% saline solution was measured as well. Then the vaporization rate of water from the surface of beef muscle sample was measured and corresponding vaporization coefficient was calculated. 70 4.3 Methods 4.3.1 Experimental Methods Figure 4.1 schematically shows experimental set-up used for the measurements of water and saline vaporization rates. A cylindrical glass beaker with a diameter of 8 cm and a height of 10 cm was filled with tap water and immersed into a temperature controlled water bath (Constant Temperature Circulator, Model 126762). Circulating water in the water bath elevated the water in the beaker to the desired temperature level. Three thermocouples (T-type, Omega Engineering) were placed immediately above and two thermocouples below the water-vapor interface to measure temperature in the vapor and at the water surface, respectively. The thermocouples were positioned at the center of the beaker within one centimeter of each other and at the distance less than one millimeter from the interface using a micropositioner (not shown). The beaker was filled with water or saline, placed in the water bath and heated to the desired temperature. When the temperatures at the water surface stabilized within 0.5 C the beaker was periodically removed from the water bath and weighted on a digital scale (Type E 5500 S, Sartorius, GmbH). Similar experiments were performed for the 0.9 % physiological saline solution. For the measurements of the vaporization rates from the surface of beef muscle a Styrofoam lid with 5.5x5.5 cm2 opening in the center was placed in the beaker on top of the water surface. A sample of beef muscle with a thickness of 1 cm was inserted into the opening completely covering water surface. One thermocouple was placed at the bottom of the sample, two at the top and two within one millimeter of the top surface. To prevent vaporization before the measurements had begun the surface of the sample was covered by a plastic film. The sample was heated to the desired temperature until the difference between top and bottom surfaces was within 71 1 C and the top surface temperature stabilized within 0.5 C range. The plastic film was removed from the sample and the weight of the beaker was measured periodically. Figure 4.1. Experimental setup for measurement of water vaporization rate from water and saline surfaces. For the measurement of vaporization rate from muscle tissue a tissue sample fixed to a Styrofoam lid was placed on the water surface in the beaker. Weight measurements were performed at ten levels of the surface temperature in the intervals from 22.5 C to 83 C in the cases of water and saline and to 97 C in the case of beef muscle. In the case of liquids the upper limit of the temperature was limited by a strong circulation in the water bath caused by the onset of nucleate boiling on the surface of the heating element. Six weight measurements were performed at each temperature level. In the case of the liquids the time intervals between measurements were chosen to allow the vaporization of between 0.5 grams and 5 grams of water at the lowest and highest temperature levels, and were from 60 minutes to 5 minutes long respectively. In the case of beef muscle, duration of the 72 time intervals was chosen to prevent significant drying of the tissue top layer. The intervals were from 30 minutes to 2 minutes long at the lowest and highest temperature, respectively, and 0.25 gram to 1 gram of water vaporized between each measurement. 4.3.2 Numerical Methods Water vaporization rate, me'' , was determined as the weight loss during the time between measurements: mt + dt mt , Sdt me'' = (4.3) where mt+dt and mt is weight of the beaker in two consecutive measurements, dt is time interval between measurements and S is the area of vaporization surface. Water vaporization mass flux was estimated from the Snelling s formula: ' m S' = AS w (Ts )[ Psat (Ts ) Psat (Tair )] , (4.4) and from the kinetic theory equation with vaporization coefficient equal unity: Psat (TS ) Psat (TV ) , T TV S '' mkin = M 2 R g (4.5) respectively, where w is water density, Psat is water vapor saturation pressure, Ts is surface temperature, Tair is air (room) temperature, TV is temperature of vapor above the surface, is a salinity factor, is air (room) humidity and As=7.31-11 m/m2sPa is experimental factor, M is the molecular weight of vapor, Rg is the universal gas 73 constant, (assumed to be equal to the vapor temperature) and is the Schagre s factor [Scharge, 1953]. The surface and vapor temperatures were determined as average reading of the thermocouples located at and above the surface, respectively. Statistical evaluation of the experimental data was performed using ANOVA test with the level set to 0.05. The following null hypotheses were tested: a) at room temperature the measured vaporization rates of water, saline and muscle tissue came from a population with the same mean, b) in the temperature range investigated the vaporization coefficients of water, saline and muscle tissue each came from populations with the same mean. 4.4 Results 4.4.1 Temperature Measurements The temperatures in the vapor and at the surface were determined as average values between the reading of the thermocouples placed on and above the surface respectively. Generally, it took about 10% of the time between the weight measurements for the temperatures in the vapor and at the surface to stabilize. The difference in temperature readings between adjacent thermocouples located at the surface did not exceed 0.5 C in all experiments. Above the surface the difference increased approximately from 0.5 C at the lowest to 2.5 C at the highest temperature levels respectively. 74 Figure 4.2. Difference between surface and vapor temperatures as a function of the surface temperature. Figure 4.2 shows the difference between the vapor and surface temperatures as a function of the surface temperature for water, saline and muscle tissue combined. The data points represent the average of six values obtained from the measurements at a given surface temperature level. The error bars indicated the standard deviation of temperature difference measurements for the experimental means. Standard deviation of the surface temperature was within 0.5 C for all measurements below 60 C, and it gradually increased to about 1 C between 60 C and 100 C. At the lowest surface temperature, 22.5 C, the average of all measurements of the surface-vapor temperature difference was 0.5 0.5 C. The line is a least squares fit to the combination of data set of the experiments with water, saline and muscle tissue. The least squares fit function is: 75 T (Ts ) = ATs + B , (4.6) where Ts is surface temperature in C, A = 0.13 C-1 and B = - 0.55 C respectively. The relative error of the least squares fit function demonstrates a gross inaccuracy at the room temperature (deviation of 400%), relatively reasonable fit in the temperature range from 30 C to 75 C (deviation within 20%) and slightly higher but still reasonable fit above 75 C (deviation within 35%). 4.4.2 Water Vaporization Rates Figure 4.3 shows water vaporization rates on a logarithmic scale from the surfaces of water, saline solution and muscle tissue as a function of surface temperature. The points represent the average of five experimental values taken at a given surface temperature. The curves above the data points correspond for water and saline show the vaporization rate as predicted by the Snelling s formula, Equation (4.4). The curve plotted through the data points corresponding to the muscle tissue show the vaporization rate as predicted by kinetic theory where the vaporization coefficient was determined in this study and the vapor temperature was determined from a least squares fit, Equation (4.6). The error bars indicate standard deviation for the experimental values. At room temperature, 22.5 C, and relative humidity of 55% the vaporization rates of water and saline solution were measured to be 0.07 0.02 g/m2s and 0.06 0.02 g/m2s, respectively. The rates predicted by Snelling s formula are 0.086 g/m2s and 0.061 g/m2s respectively. The vaporization rates of water and saline at the highest surface temperature were 3.4 0.2 g/m2s at 85 1 C and 2.7 0.2 g/m2s at 84 1 C, respectively. The corresponding rates predicted by Snelling s formula are 3.06 g/m2s and 3.65 g/m2s, respectively. In the case of the saline solution the calculations 76 were performed with the salinity factor in Snelling s formula assumed to be equal 0.87 [Salhotra, et al., 1985]. At room temperature, 22.5 C, and relative humidity of 55% the measured vaporization rate of water from the surface of the muscle tissue was 0.03 0.01 g/m2s. At the highest surface temperature level of 97 1 C the vaporization rate was 1.10 0.05 g/m2s. Figure 4.3. Vaporization rates from water, saline and muscle tissue surfaces. The points represent experimental data. The lines show vaporization rates for water and saline predicted by the Snelling s formula and for muscle tissue predicted by the kinetic theory based on results of this study. 77 Statistical comparison of the vaporization rates from water, saline and tissue surfaces obtained at room temperature reveals the calculated F values equal 5.6, 3.9 and 4.1, respectively. The critical value, Fcrit, was equal 3.29. Therefore, the null hypothesis is rejected and the vaporization rates of water, saline and muscle tissue at room temperature are statistically different. 4.4.3 Vaporization Coefficients The values of the vaporization coefficients of water, saline solution and muscle tissue calculated from the experimental data and kinetic theory are shown in Figure 4.4. In all experiments, Schagre s factor, , determined through the vapor velocity in the vicinity of the surface, Equations (2.31), (2.32) and (2.39), was found to be close to unity. The data points represent the average of five values obtained from measurements at a given surface temperature level. The lines show least squares fit to the data sets corresponding to water, saline and muscle tissue vaporization coefficients, respectively. The mean values of the vaporization coefficients, av, introduced in Chapter 2, Equation (2.37), calculated from the experimental measurements were (1.5 0.3)x10-4, (1.3 0.2)x10-4 and (0.14 0.03)x10-4 for water, saline solution and beef muscle tissue respectively. Statistical comparison of vaporization coefficients of water, saline and muscle tissue obtained at different levels of the surface temperature reveals the calculated F values equal 1.6, 2.3 and 1.7 respectively. The critical values, Fcrit, were 4.32, 3.21 and 2.18 respectively. Therefore, the null hypothesis is rejected and in the investigated temperature range the vaporization rates of water, saline and muscle tissue are statistically different. 78 Figure 4.4. Vaporization coefficients of water, saline and muscle tissue. The points represent values calculated from experimental data and kinetic theory prediction. The lines show least squares fit. 4.5 Discussion 4.5.1 Temperature in the Vapor Surface Layer To predict the vaporization rate at a given surface temperature by the kinetic model under the assumption of 100% humidity in the vapor surface layer it is required to specify the temperature of this layer. Based on the results of the study it is suggested to approximate the temperature gradient between the surface and the vapor layer by a linear function of the surface temperature. Linear fit of the experimental data defines such an approximation in the surface temperature region form 22.5 C to 79 97 C: Figure 4.2, Equation (4.6). At low surface temperatures the suggested approach is grossly inaccurate. Since at temperatures close to room temperature the pressure difference across the liquid-vapor interface is small, high variation in prediction of the vapor temperature and, therefore, vapor saturation pressure, will result in even higher error in the vaporization rate. However, as the surface temperature increases, the accuracy of the linear fit prediction increases as well. At the same time, the temperature difference, and, therefore the pressure difference, increases and error in the vapor temperature affects the vaporization rate less. Overall, the kinetic model with the vapor temperature calculated by the linear fit of Equation (4.6) gives a reasonably good prediction of the vaporization rate from the muscle tissue in the temperature range from about 35 C to 97 C. 4.5.2 The Accuracy of Snelling s Formula Previous studies have assumed that the vaporization of water from tissue surface into the atmosphere, particularly in the temperature range from 60 C to 100 C [Torres 1994, Pearce 2000], can be approximated by Snelling s formula, Equation (4.4). Snelling s formula gives a phenomenological description of the vaporization as a process driven by the pressure gradient between the vapor pressure immediately at the surface and in the atmosphere, far from the surface. This approach ignores the effect of the almost saturated vapor layer above the surface and, subsequently, overestimates the pressure gradient and vaporization rate. Figure 4.3 demonstrates that above room temperature the vaporization rate from water and saline predicted by the Snelling s formula overestimates the mean experimental values for all but two data points. At room temperature the high uncertainty in the prediction and experimental measurement of the vaporization rate does not allow a definite conclusion. 80 4.5.3 Vaporization Rates from Water and Muscle Tissue It was observed that in the temperature range investigated the vaporization rate of water exceeds the vaporization rate of saline solution, which in turn exceeds the rate of vaporization from muscle. Statistical comparison of the vaporization rates from water, saline and muscle tissue at room temperature rejects the null hypothesis and demonstrates that the mean values of respective vaporization rates belong to different populations. Table 4.1. Vaporization rates from water and tissues and corresponding surface temperature and room humidity levels. Vaporization rate, g / m2s 0.13 0.01 0.05 0.01 0.07 0.02 0.015 0.006 0.076 0.003 0.056 Surface Temperature, C 31 31 32 23 room room Humidity, % 55 55 50 29.5 room room Reference this study this study Herold, 1987 Mathers, 1993 Gwosdow,1993 Ayling, 1984 Tissue Water Beef muscle Cornea Cornea Burn site Skin A relatively high concentration of ions in saline is responsible for its reduced saturation pressure and, therefore, for its lower vaporization rate in comparison with water [Salhotra, et al., 1985]. Relative similarity in the ionic concentration and composition of the 0.9 % physiological saline and seawater [Ganong, 1995] justifies the use of the salinity factor for seawater in Snelling s formula for prediction of the saline vaporization rate. Indeed, the calculated saline vaporization rate exhibits the same type of agreement with the experimental data as for the case of water, Figure 4.3. Further reduction of the vaporization rate from tissue can be likely due to the roughness and complex chemical composition of its surface. In the temperature region from 22.5 C (room temperature) to 31 C (normal skin temperature) the rate of vaporization from muscle tissue was from 0.03 0.01 g/m2s to 0.05 0.01 g/m2s. 81 This is in reasonably good agreement with experimental data on vaporization rates from cornea, skin and burn injury sites as summarized in Table 4.1. 4.5.4 Vaporization Coefficients for Water, Saline and Muscle Tissue As with the vaporization rates, the vaporization coefficient for water exceeds the vaporization coefficient of saline solution, which in turn exceeds the vaporization coefficient of muscle tissue. Statistical evaluation of the average values of the vaporization coefficients for water, saline and muscle tissue measured at all surface temperatures investigated demonstrates that each coefficient can be assumed independent of the surface temperature. At the lowest surface temperature a high uncertainty in the pressure gradient is responsible for the high uncertainty in determination of the vaporization rates and, consequently, the vaporization coefficients. The relative errors of the vaporization coefficients for water, saline and muscle tissue for all data are 20%, 15% and 21% respectively, but, after rejection of the data obtained at room temperature the error drops to 17%, 7% and 14% respectively. In this case the mean values of the vaporization coefficients, av, introduced in Chapter 2, Equation (2.37), becomes equal for water: (1.5 0.3)x10-4, for saline: (1.3 0.1)x10-4 and for muscle: (0.14 0.02)x10-4, respectively. Therefore, the vaporization rate from the surface of beef muscle can be predicted with reasonable accuracy by kinetic theory. Analysis of the vapor temperature, Figure 4.2, reveals that with reasonable accuracy the temperature in the vapor layer can be predicted from the surface temperature using least squares fit, Equation (4.6). At the same time the pressure in the vapor layer can be evaluated as saturation pressure at this temperature. 82 4.6 Conclusion Experimental methods were used to increase understanding of the processes of water vaporization in temperature range from 22.5 C to 95 C and investigate the validity of kinetic theory for prediction of the rate of vaporization from the surface of muscle tissue. Measurements of the mass loss of the samples were used to determine vaporization rates. It was found that at the surface temperatures above 50 C water vaporizes slower than predicted by Snelling s formula. This discrepancy is likely due to the fact that Snelling s formula suggests evaluation of the vapor pressure at the temperature of ambient air, whereas the actual vapor temperature in the vicinity of the phase interface is higher. The experimental data from this study, and from literature, demonstrate that in the investigated temperature range the vaporization rate of water from the surface of muscle tissue is lower than the vaporization rate from relatively pure water. It is suggested that vaporization reduction is due to the morphology, electrolyte and chemical composition of the tissue surface. Statistical analysis of the vaporization coefficient for muscle tissue surface shows that in the investigated temperature range it is constant and equal (0.14 0.02)x10-4. Therefore, according to the definition of the vaporization coefficient, the kinetic theory can be used to predict the rate of water vaporization from the surface of skeletal muscle tissue. 83 Chapter 5: Numerical Study of the Electric Sparks in the ETR Process 5.1 Abstract A novel thermo-electrical model of the electrosurgical tissue resection (ETR) that takes into account variations in electrical and thermal properties with temperature and water content, thermal and electrical processes at the tissue-scalpel interface as well as surface and interstitial water evaporation has been developed. The model predicts measurable parameters of the electric circuit (tissue impedance, ESU output RMS voltage and current) and tissue cutting rate. Results of numerical simulations suggest that high circuit impedance during electrosurgical cutting can result not only from tissue dehydration but from the configuration of the electric field as well. It appears that the area of tissue-scalpel electric contact is significantly smaller than the area of the scalpel itself but is large enough to rule out electric sparks as a major mechanism of electrosurgical cutting. 5.2 Introduction Numerical comparison [Honig, 1975, Pearce 1986, pp. 72-79] of the semicylindrical and hemispherical source models of the electrosurgical cutting have provided indications that the cutting rates of several centimeters per second can be achieved only if the area of scalpel-tissue electric contact is significantly smaller than the front area of the scalpel. To account for the necessary reduction of the contact area the hypotheses of sparks striking in the gap between the scalpel and tissue was introduced [Pearce 1986]. It was claimed that the small cross sectional area of the electric spark is responsible for the high power density at the point of impact and, ultimately, for the faster cutting, Equations (1.1) and (1.2) and Figures 1.2 and 1.3. However, the spark model as described by Pearce [Pearce, 1986] contains several important shortcomings. 84 First, the original example calculations comparing the semicylindrical and hemispherical source models contain several computational errors. For the tissue and electrode parameters introduced in the model the time to reach vaporization temperature at the surface of the cylindrical electrode was estimated as 0.15 seconds [Pearce, 1986 p. 78]. However, calculation using Equations (1.1) and (1.4) and same parameters used by Pearce gives the time of 0.036 seconds. Similarly, the time to reach vaporization temperature in the spark (hemispherical source) model was estimated as 6.86 10-8 s [Pearce, 1986 p. 78], while recalculation by Equations (1.2) and (1.4) gives 3.33 10-6 s. Second, in the case of the hemispherical source, the calculations were performed for a single source but, in order for the scalpel to advance, the sparks must strike along its entire length. Therefore, ignoring the heat transfer effect, the time required for rising temperature along the entire scalpel can be estimated as 3.33 10-6 s times the ratio of the scalpel length to the spark diameter. For the given scalpel length of 7 mm and spark diameter of 0.17 mm the time to reach vaporization temperature increases to 1.4 10-4 s. Third, the cutting rate in the semicylindrical and hemispherical source models is compared using the same value of the total current. However, this comparison is not entirely justifiable because the total power in these two cases is different. The total output power of the ESU can be estimated as the power dissipated in the tissue: P = Rt I 02 , (5.1) where Rt is resistance of a tissue sample and I0 total current. In the case of the semicylindrical source model the resistance of a tissue sample can be estimated as: 85 Rc = r 1 ln t , L rc (5.2) where L is length of an electrode in contact with the tissue, is tissue electric conductivity, rt is distance from the scalpel to the return electrode and rc is distance from the center of the cylindrical scalpel. The resistance in the case of the hemispherical source model is respectively: Rs = 1 2 1 1 , r r t s (5.3) where rs is distance from the center of the spark strike site. Equations (5.1) (5.3) Figure 5.1. Total current as a function of distance from the tissue-scalpel interface to the return electrode for the semicylindrical and hemispherical source models. Total current in the semicylindrical model is 200 mA and total power is the same for both models. 86 show, that for the same total power, total current in the hemispherical source model must be less than in the semicylindrical model. Assuming that total current in the semicylindrical model is 200 mA and total power is the same for both models the difference between total currents in two models as a function of distance from the scalpel to the return electrode is demonstrated on Figure 5.1. Taking into account necessary reduction of the total current in the hemispherical source model in comparison with the semicylindrical source and assuming the distance from the tissue-scalpel interface to the return electrode equal 20 centimeters, the time to reach vaporization temperature further increases to 8.5 10-4 s. Finally, the comparison of the cutting speeds in the source models [Honig, 1975, Pearce 1986, pp. 72-79] based on the energy requirement to rise the temperature in front of the scalpel from 37 C to 100 C. In fact, such a temperature rise is required only for initiation of the cutting but not for the steady state cutting regime. In the steady state cutting the temperature immediately in front of the scalpel must be close or equal to water vaporization temperature and the cutting speed is determined by the energy requirements to maintain this temperature: to balance the diffusion and vaporization heat losses. The geometry of the semicylindrical source provides for lower heat diffusion losses. Also, it is responsible for constant temperature along the scalpel while in the hemispherical case in locations away from site of the spark impact temperature drops. The results of comparison between the two models reduce the difference in the cutting speed from six to less than one and a half orders of magnitude. For the relatively simplified such analysis a difference makes the estimation of the cutting speed in two models almost undistinguishable and does not allow any definite conclusion on the role of electric sparks in the ETR process. To confirm or reject the hypotheses of electric sparks as a primary mechanism of the ETR a more elaborate 87 model accounting for the heat diffusion and vaporization effects as well as material properties of tissue is developed. In addition to clarifying the role of electric sparks, the model introduced here assists in further understanding of the ETR mechanism and used to provide guidelines for development of more elaborate model involving the thermal damage description. 5.3 Methods In this section, basic finite difference representations of the thermo-electric events are summarized while algorithms of the striking of electric sparks at the scalpel-tissue interface and tissue ablation are described in greater detail. 5.3.1 Tissue and Cutting Process Representation A homogeneous rectangular tissue sample was modeled with thermal and electrical parameters as summarized in Table 5.1. Temperature dependent water parameters: vapor saturation pressure, phase change enthalpy and surface tension were determined using 5th order polynomial of the steam table data as explained in Appendix I. Muscle tissue was modeled as a 3D rectangular matrix 5x5x5 mm3 where actual muscle cells are represented as cubical cells with 50 m sides filled with liquid and separated from each other by infinitesimally thin collagen membranes. It was assumed that in order to rupture a muscle cell it is sufficient to form a vapor bubble inside of volume equal to the cell volume. It is further assumed that in order to advance the scalpel a whole line of cells adjacent to it must be destroyed. As the scalpel advances, a tissue layer with the initial parameters is added to the bottom of the tissue space so the total height of the tissue remains constant. 88 Table 5.1. Temperature independent model parameters. Temperature dependent parameters are given in Appendix I. 0.6 1000 0.595 4.22 Initial electric conductivity, S/m Conductivity change rate, %/ C Tissue initial water content, % Surface vaporization coefficient, 0.25 1.5 100 0.14 x10-4 Thermal conductivity, W/Km2 Water density, kg/m3 Vapor density, kg/m3 Specific heat kJ/K kg The scalpel-tissue interface is located in the middle of initially uniform tissue, as shown on Figure 5.2. A given portion of the scalpel is in electric contact with tissue. Generally, the contact area is not uniform and occupies only a certain portion of the interface. The electric contact at the scalpel-tissue interface was modeled as a two-dimensional voltage distribution with a Gaussian profile: x2 y2 V ( x, y ) = V0 exp 2 2 R Ry x (5.4) where V0 is the output ESU voltage and Rx and Ry are the radiuses of the voltage profile in the directions along and across the scalpel respectively. The hemispherical source, corresponding to a hypothesis of an electric spark striking in the scalpel-tissue gap, is approximated by a circular distribution, Rx=Ry. For the semicylindrical source two possibilities for the voltage distributions corresponding to hypotheses of a continuous and interrupted electric contact along the scalpel are considered: a line profile with a Gaussian voltage across the scalpel, Rx = and an elliptical Gaussian profile with the major axis oriented along the scalpel, Rx > Ry respectively. 89 Figure 5.2 Tissue sample and contact spot representation. Muscle tissue is modeled as a uniform matrix constructed from rectangular elements representing muscle fibers (cells). The central axis of the electrosurgical scalpel is located in XZ plane and the scalpel advances in Z direction. It is assumed that application of RF power creates certain voltage profile at the scalpel-tissue interface (XY plane). The algorithm of the ETR model is summarized in the flow chart in Figure 5.3. The output power of the ESU is fixed and the output voltage is determined from calculated tissue impedance. A random site of scalpel tissue electric contact is chosen (in the case of a line voltage distribution it is the entire scalpel) and the heat source in the lowest half of the tissue and temperature field in the entire sample is calculated. If tissue temperature exceeds the nucleation threshold the volume of vapor 90 Figure 5.3 Flow chart of the algorithm used in numerical modeling of ETR. 91 formed and remaining water content are calculated. Tissue electrical and thermal parameters are updated according to changes in temperature and water content. Circuit impedance is calculated and the scalpel voltage is changed accordingly. Then, the heat source and the temperature field are calculated again. If tissue water under the contact area is vaporized completely a new contact site along the scalpel is chosen randomly and the process continues until all cells under the scalpel are ruptured. After that the scalpel advances to the next layer of tissue cells. 5.3.2 Variable Step Grid To reduce running time of the program, simplify iteration and provide sufficient spatial resolution in the vicinity of electrosurgical scalpel a variable step 3D rectangular grid was used in the model. Figure 5.4 shows the grid representing an initially homogenous and isotropic rectangular tissue sample and relative position of the scalpel. A constant grid step along the scalpel (x direction) and variable steps both across the scalpel (y direction) and in direction of the scalpel advance (z direction) were chosen. The grid steps in y and z directions are determined from a geometric progression: dy1 = dy0 , dyiy = dy0 ryiy 2 , iy = 2,3.....Ny 1 (5.5a) dz1 = dz 0 , dz iz = dz 0 rziz 2 , iz = 2,3.....Nz 1 , (5.5b) where dy0 and dz0 are the initial grid steps (choice of equal first and second grid elements in the y and z directions is made for convenience of the boundary condition representation), ry and rz are ratios of the geometrical progression (ry, rz >1) and Ny and Nz are the number of grid elements in each direction. It is convenient to choose: 92 dx0 =dy0=dz0 and ry=rz (5.6) Figure 5.4 Numerical grid. Grid step is constant in X direction and variable in Y and Z directions. 5.3.3 Governing Differential Equations in Variable Step Grid In the variable step grid the first and second derivatives of, for instance, electric potential V in the y-direction at the point iy can be estimated as: dy iy 1 dy iy dy iy 1 dy iy V Viy +1 + Viy Viy 1 = dy iy dy iy 1 dy iy 1 (dy iy + dy iy 1 ) y dy iy (dy iy + dy iy 1 ) (5.7) and 93 2V 2 2 2 Viy +1 Viy + Viy 1 = 2 dy iy (dy iy + dy iy 1 ) dy iy dy iy 1 dy iy 1 (dy iy + dy iy 1 ) y (5.8) respectively [Hildebrand, 1968] Substitution of 5.6 into 5.7 and 5.8 yields, after simplification, the following representation for the first and second derivatives of electric potential in the grid points iy = 2,3, ..., Ny-1: V 1 = y dy 0 ryiy 2 r2 Viy +1 1 + Viy (ry 1) Viy 1 y ry + 1 ry + 1 2 ry 2ry2 Viy +1 Viy 2ry + Viy 1 ry + 1 ry + 1 (5.9) 2V 1 = 2 y (dy 0 ryiy 2 ) 2 (5.10) 5.3.3.1 Heat Source In the 3D Cartesian coordinate system the continuity equation (2.5) becomes: V 2V V 2V V 2V + 2 + + 2 + + 2 = 0 x x y y z z x y z (5.11) The finite difference solution of the continuity equation (5.11) can be written as a linear combination: Vix ,iy ,iz = AxVix +1 + B xVix 1 + AyViy +1 + B yViy 1 + AzViz +1 + B zViz 1 (5.12) 94 where Vix,iy,iz is the electric potential in the grid element with the coordinates (ix,iy,iz), Vix 1, Viy 1 and Viz 1 are electric potentials in the corresponding neighboring grid elements. In the chosen variable step grid with equal initial grid steps dx0=dy0=dz0 the coefficients of the combination are: 1 1 Ax = (ix, iy, iz ) + ix 4 C 1 1 B x = (ix, iy, iz ) ix 4 C (5.13a) (5.13b) where ix = (ix + 1, iy, iz ) (ix 1, iy, iz ) (5.13c) 2 ry 1 1 1 Ay = (ix, iy, iz ) + iy 2 iy 2 2 ry + 1 (ry + 1) (ry ) C 2 2 ry 2ry 1 1 B y = (ix, iy, iz ) iy 2 iy 2 2 ry + 1 (ry + 1) (ry ) C (5.14a) (5.14b) where iy = (ix, iy + 1, iz ) + (ix, iy, iz )(ry 1) (ix, iy 1, iz )ry2 2 (5.14c) Due to the symmetry of the model the coefficients Az and Bz are similar to Ay and By respectively and 95 iy ry 1 ry r 1 rz (5.15) iz 2iz 2 z C = 2 (ix, iy, iz ) 1 + iy 2 2 + iz 2 2 iy 2 2 (r (r ) (rz ) ) ry + 1 (rz ) rz + 1 y y There are two stability criteria for the solution (5.12) of the finite difference Equation (5.11): Ax + B x + Ay + B y + Az + B z = 1 (5.16) and Ax , B x , Ay , B y , Az , B z 0 (5.17) It is easy to demonstrate that all coefficients A and B always satisfy condition (5.16). However, the second condition (5.17) in the variable step grid has a quite complicated form. In a simple case of the constant step grid (ry = rz = 1) it becomes: i 4 (5.18) where index i stands for ix, iy or iz respectively. It means that in order for the solution (5.12) to be stable the difference of electric conductivity between two grid cells adjacent to any chosen cell must be no more then four times higher then the value of electric conductivity in that chosen cell. It should be noted that stability condition (5.17) does not explicitly depend on the grid geometry. This makes control of the stability of the solution (5.12) somehow difficult and requires adjustments of the grid step during the calculations. These 96 adjustments are done based on projected rate of the change in tissue electric conductivity. The component of the electric field vector (2.7) in grid element (ix, iy, iz) can be written as: Vix +1 Vix 1 2dx0 Ex = (5.19a) Ey = 1 dy 0 ryiy 2 r2 Viy +1 1 + Viy (ry 1) Viy 1 y ry + 1 ry + 1 (5.19b) and Ez = 1 dz 0 rziz 2 r2 1 Viz +1 + Viz (rz 1) Viz 1 z rz + 1 rz + 1 (5.19c) The magnitude of electric field E in the grid element (ix,iy,iz) is absolute value of vector sum of all components: E= (E 2 x 2 + E y + E z2 ) (5.20) The heat source is (2.11): q = E 2 (5.21) 97 where = (W,T) is tissue electric conductivity, Equation (2.3), function of water content and temperature. 5.3.3.2 Temperature Field Neglecting spatial variation in tissue thermal conductivity the diffusion term in the heat diffusion Equation (2.13) becomes: 2T 2T 2T (k T ) = k 2 + 2 + 2 x y z (5.21) Substitution of equation (3.21) with the finite difference representation of the second derivatives (3.8) and conditions (3.6) yields the finite difference equation for the temperature in the grid element with the coordinates (ix,iy,iz): r ( Tiy + Tiz ) + [1 2 FoD]Tt + dt Tt + dt = Fo Tix + r +1 c p dm h fg (5.22) q dt where Fo is the Fourier number: Fo = k dt 2 c p dx0 (5.23) D = 1+ 1 r 2 iy 5 + 1 r 2 iz 5 (5.24) Tix = T (ix + 1, iy, iz ) + T (ix 1, iy, iz ) (5.25a) 98 Tiy = 2 r 2 iy 4 (T (ix, iy + 1, iz ) + rT (ix, iy 1, iz )) (T (ix, iy, iz + 1) + rT (ix, iy, iz 1) ) (5.25b) Tiz = 2 r 2 iz 4 (5.25c) It is easy to see that for any increment of the grid step r >1 the stability criterion for the solution (3.22) is: 1 6 Fo > 0 (5.26) and depends on the value of the minimum grid step. 5.3.4 Water Vaporization The processes of water vaporization considered in the present study are separated into two modes: vaporization from a flat surface into the atmosphere and vapor bubble nucleation and growth. Vaporization into the atmosphere takes place at the scalpel-tissue interface. It is a relatively slow process and, in the absence of ablative effects, will lead to complete dehydration of the tissue-scalpel contact region and subsequent collapse of the RF current. Vapor nucleation occurs in the tissue at depth if and where the temperature exceeds the nucleation threshold. The growth of vapor bubbles in the layer immediately beneath the scalpel is responsible for steady state ablation. More deeply in the tissue only isolated vapor nuclei appear and grow into relatively large vacuoles. 99 5.3.4.1 Surface Vaporization During the steady state cutting process the cellular membrane of the muscle fibers contacting the scalpel is damaged and their water content is exposed to the atmosphere. In this layer the kinetic theory Equations (2.37) and (2.38) are used to define the vaporization mass and heat fluxes, respectively. In these equations local temperature, Tl, and pressure, Pl, on the liquid side of the liquid-vapor interface are defined as the temperature of the matrix element, T(ix, iy, iz=0), and vapor saturation pressure is calculated at this temperature. Therefore, to calculate the vaporization fluxes local temperature, TV, and pressure, PV, at the vapor side of the liquid-vapor interface must be defined. The electrosurgical scalpel advancing through tissue remains confined to a narrow channel containing products of tissue vaporization, mostly water vapor. It is reasonable to assume that under steady-state cutting conditions the vapor in the narrow gap between tissue and scalpel is almost saturated and maintained at a temperature close to the tissue surface temperature. Under these assumptions the vapor temperature in the scalpel-tissue gap above the grid element (ix, iy, iz=0) is defined from the results of Chapter 4, as: T (ix, iy, iz = 0)V = T (ix, iy, iz = 0) + T (T (ix, iy, iz = 0)) , (5.27) where T is defined by Equation (4.6) at the surface temperature TS=T(ix,iy,iz=0) and the vapor pressure is equal to the saturation pressure at this temperature. The vaporization mass loss from the same grid element in the interval, dt, is determined from kinetic theory, Equation (2.37), and given as: 100 m S = aV S (ix, iy )dt M 2 R g Psat (TS ) Psat (TV ) , T TV S (5.28) where av is the muscle tissue vaporization coefficient, determined in Chapter 4, S(ix,iy, iz=0) is the top surface area of the grid element, and the factor is assumed equal unity. 5.3.4.2 Interstitial Vapor Nucleation The energy balance at the scalpel-tissue interface during surface vaporization can be represented as: T dm = h fg z dt k (5.29) where the z-axis is in the direction of the scalpel advance. The equation demonstrates that a positive vaporization mass flux requires a positive temperature gradient and a net heat flux to the surface of the tissue. Therefore, the temperature immediately under the interface must be elevated with respect to the surface vaporization temperature [Rastegar 1987]. Temperature elevation inside the tissue above the equilibrium vaporization point suggests interstitial vapor nucleation. In Chapter 3 an ablation model based on the interstitial vapor nucleation and bubble growth was introduced for the present study. The model describes the process of electrosurgical cutting as a sequence of micro-explosions in front of the scalpel caused by the nucleation and growth of the vapor bubbles. It is assumed that in order destroy a muscle cell located at the scalpel-tissue interface it is sufficient to form 101 inside it a vapor bubble of equal size. It is further assumed that in order for the electrosurgical scalpel to advance all muscle cells in front of it must be destroyed. The rate of the vapor nucleation in pure liquid can be expressed in a general form: dN * A = BN exp , dt kT J= (5.30) where N* is the concentration of activated nuclei, k is Boltzmann constant, T is absolute temperature and the BN product and A have the physical meaning of frequency factor and activation energy respectively. The most critical factor for the calculation of the nucleation rate at a given temperature is the activation energy, A, which is temperature dependent itself and not easy to define. However, functional analysis of the nucleation rate and vapor bubble growth conducted in Chapter 3 demonstrates that at the realistically expected power densities and cutting speeds only one or a few nuclei would be formed inside the cell before it is destroyed. Therefore, in the present study, it is reasonable to treat the activation energy and, therefore, nucleation threshold temperature, as a parameter. It is assumed that at a given nucleation threshold temperature, Tn, during a control time interval, c, Equation (3.1), only one vapor nucleus is formed inside the muscle cell. The corresponding nucleation rate is calculated and, from it, the activation energy is calculated using Equations (2.46) and (5.30). Then, assuming that during the time of bubble growth the activation energy remains constant, further nucleation is estimated. 102 Once a vapor nucleus is formed inside the cell, the rate of its growth, VR, is calculated from the relation of Equation (2.59). In the equation temperature of the surrounding liquid, Tsat(Pinf), is replaced by the average temperature in the surrounding cells. The radius of the bubble at time, t, is calculated as: R(t ) = R(t dt ) + VR dt , (5.31) were R(t-dt) is the bubble radius in the previous iteration. If a vapor bubble is formed in a cell located at the tissue-scalpel interface the cell is assumed destroyed when the diameter of the vapor bubble becomes equal to the diameter of the cell segment, i.e. 50 m. If a bubble is formed in a cell located deep in the tissue it will grow until its radius becomes equal to the distance from the interface or until sufficient energy is provided. 5.3.4.3 Tissue Material Properties as a Function of Water Content Material properties are largely determined by the properties of the tissue components, and, respectively, can be expressed as a function of a weighted sum of components mass fractions, [Cooper and Trezek, 1971]. Muscle tissue contains a significant amount of water, and it is not surprising that in large part the water content determines its properties. Besides water the tissue contains proteins and fats, and with water loss, these components become dominant in defining the material properties. In this study two modes of vaporization or water loss are considered: surface vaporization from tissue cells located at the scalpel-tissue interface, and formation and growth of vapor bubbles. In the case of surface vaporization the mass fraction of water in the cell is reduced and the material properties of a tissue cell are changed 103 accordingly. In the case of interstitial nucleation the conversion of water into vapor occurs inside a tissue cell. Therefore, the volumetric fraction of vapor, rather than mass fraction of water, will be dominant in defining the material properties of a tissue cell. For simplicity reasons, in this study only electric conductivity is assumed to be dependent on water content. Equation (2.3) and (2.4) give the combined temperature and water content dependence of electric conductivity of the tissue cell. Corresponding mass and volumetric fractions are calculated as: Wm = mS m0 , (5.32) and 4 R(t ) 3 WV = , 3 V0 (5.32) where m0 and V0 initial water mass and volume of a tissue cell, mS is combined water mass vaporized through the surface, calculated using Equation (5.28) and R(t) radius of the vapor bubble given by Equation (5.31). 5.3.5 Macroscopic Parameters of Electrosurgical Circuit Tissue impedance Ztis, scalpel voltage, V0 and total current, Itot, were determined from condition of total power, Ptot, dissipated in the tissue sample equal output power of the ESU, PESU. Total dissipated power was calculated is the sum of 104 power dissipated in each grid element multiplied by factor of two due to the symmetry of the model: Ptot = 2 q (ix, iy, iz )V (ix, iy, iz ) , (5.34) where q(ix,iy,iz) and V(ix, iy, iz) are receptively power density (heat source) in and volume of grid element (ix,iy,iz). Tissue impedance was calculated as: Z tis (V ) = Ptot t 2 0 , (5.35) where Vt0 is maximum voltage at the scalpel-tissue interface that was set during the iteration, Equation (5.4). Total current and maximum voltage, Vt+dt0, for next iteration are: I tot = and V0t + dt = PESU Z tis , respectively. PESU , Z tis (5.36) (5.37) 5.4 Results Numerical simulations of the ETR process described in this section were performed for three types of the heat source classified by the geometry of the scalpeltissue contact spot: circular, elliptical and line. Input parameters were: ESU output 105 power, size of the contact spot and nucleation threshold temperature. Their ranges of variation are given in Table 5.2. Table 5.2 Simulation input parameters. Minimum Value Maximum Value Step 1 1.5 0.1 50 25 500 ( for line spot) 50 500 25 25 300 25, 50 105, 107.5, 110, 112.5, 125, 150, 175 Parameter r, increment of grid step Rx, m Ry, m PESU, W Tn, C To evaluate the validity of the hypotheses formulated in the beginning of this study the following simulation data are discussed: 1. histories of the output parameters: a) total electric current b) scalpel voltage amplitude c) tissue impedance d) maximum temperature at the scalpel-tissue interface e) vaporization rate f) total volume of vapor created; 2. distributions of a) heat source b) temperature c) electric conductivity d) water content 106 recorded in the planes across the cut (x,y,z=Nz/2) and along the cut (x, y=0, z) at the steady-state cutting regime; 3. functional dependencies of a) total current b) tissue impedance c) cutting speed from the interface power density and nucleation temperature; 4. established average values standard deviations of a) total electric current b) tissue impedance c) maximum temperature at the scalpel-tissue interface d) average temperature along the scalpel e) cutting speed. 5.4.1 Typical Simulation Results In this section typical features of the simulation results common to all three types of contact spots as well as specific to each particular type are presented. Their origins in the modeled physical processes, model parameters, and assumptions are explained briefly. Results of simulations with the contact spots types: circle radius of 100 m, ellipse radiuses of 50 m and 250 m and line width of 50 m are presented. In all three cases the ESU output power and nucleation threshold temperature are 100 Watts and 112.5 C, respectively. 107 5.4.1.1 Dynamics of the Simulated ETR Process Figures 5.5 to 5.7 show histories of the simulation output parameters for the circular, elliptical and line heat sources respectively, recorded during advance of the scalpel through first five layers of the muscle cells. The histories of total electric current, scalpel voltage amplitude, tissue impedance, maximum temperature at the scalpel-tissue interface, vaporization and total volume of created vapor are presented. In the two last histories the units are the number of vaporized 50x50x50 m3 grid cells in one time step, dt, and number of vaporized grid cells respectively. Figure 5.6 corresponds to the elliptical contact and spot demonstrates the features common to all simulations most clearly. Initial heating of the tissue layer at the scalpel-tissue interface takes place during approximately the first 50 s. The interface temperature rises from the initial 25 C to some value exceeding the nucleation temperature (in these cases Tn=112.5 C). During this time, due to an increase in temperature, the electric conductivity of the layers near the interface rises and the total tissue impedance drops. Since the output ESU power was set constant (in the presented cases PESU=100 W), total electric current rises and scalpel voltage drops respectively. Initially, water vaporization has no apparent effect on the electric conductivity and, respectively, on the tissue impedance, total current and scalpel voltage. But as the heating continues the water loss (due to both surface vaporization and interstitial vapor generation) begins to dominate electric conductivity. First, up to about 150 s it slows down the fall of the impedance and, respectively, the rise and fall in current and voltage. Then, in the time interval from 150 s to about 300 s rapid decline in the electric conductivity, due to the developing interface dehydration, causes a sharp rise in the impedance and, respectively, fall and rise in the total current and scalpel 108 voltage. Toward the end of this time interval the maximum temperature and, as a result, vaporization rate begin to increase. Their rise is likely due to increase in the local power density caused by complete dehydration of the tissue along the scalpel s centerline, which results in: a) sharp gradient in the local electric field (so-called edge-effect ) and b) an increase in the local current density due to decrease in conductive area. At 300 s the first tissue layer is cut through and the scalpel advances to the next. At this moment temperature and material values at the scalpel-tissue interface take values that correspond to the values in the second tissue layer during the previous iteration. Since dehydration in the second layer is limited the impedance drops instantly and the current rises and voltage drops to the values corresponding to the new impedance. From 300 s to about 500 s events in the second tissue layer closely resemble those in the first during the interval from 150 s to 300 s. Dehydration proceeds at the same pace, the impedance increases and, respectively the current decreases and voltage increase. Finally, toward the end of cutting through the second layer, when dehydration becomes extensive, the increase in local power density causes temperature and vaporization rate to increase sharply until the layer is destroyed. 109 Figure 5.5 Histories of output parameters for the simulation with the circular electrode tissue contact spot. Cutting through first five 50 m muscle layers is reproduced. The indicated features are present, though might not be clearly visible, on all six reproduced histories. All indicated features, but the sparks , are common to all three contact spot types. The sparks are organic for the circular and elliptical types only. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx = Ry =100 m. 110 Figure 5.6 Histories of output parameters for the simulation with the elliptical electrode tissue contact spot. Cutting through first five 50 m muscle layers is reproduced. The indicated features are present, though might not be clearly visible, on all six reproduced histories. All indicated features, but the sparks , are common to all three contact spot types. The sparks are organic for the circular and elliptical types only. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx=250 m, Ry = 50 m. 111 Figure 5.7 Histories of output parameters for the simulation with the line electrode tissue contact spot. Cutting through first five 50 m muscle layers is reproduced. The indicated features are present, though might not be clearly visible, on all six reproduced histories. All indicated features, but the sparks , are common to all three contact spot types. The sparks are organic for the circular and elliptical types only. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx= m, Ry = 50 m. 112 The process repeats from layer to layer and after cutting through two or three layers a distinctive periodic pattern is established. While cutting in the surface muscle layer is proceeding the layer immediately beneath it is preheated and partially vaporized (by interstitial vapor generation). When cutting begins in this layer the maximum temperature and, respectively, vaporization rate are not significantly different from the temperature level established in the previous tissue layer immediately before dehydration effects caused its sharp rise. That makes the steadystate cutting period, 50 m, defined as the time to cut trough a 50 m layer, less than the time to cut trough the first muscle layer. 5.4.1.2 The Sparks and Computational Noise Besides the high amplitude, slow and periodic oscillations (period of 50 m) caused by the layer-by-layer cutting representation all data traces on Figures 5.5 through 5.7 demonstrate low amplitude, fast and relatively random oscillations. Figures 5.5 and 5.6, corresponding to the circular and elliptical contact spots respectively, show that these oscillations are present during all of the simulation time except during a brief initial period when electrode just came in contact with tissue and no cutting had occurred yet. In Figure 5.7, (line contact spot) the oscillations appear only near the end of cutting through each muscle layer. Figures 5.8 and 5.9 demonstrate the fast oscillations in the simulations with the elliptical and the line contact spots respectively, recorded in the time interval from the end of the cutting through one layer to the beginning of the cutting through the next layer. The histories of the maximum temperature, vaporization rate, total current and total impedance (as in Figures 5.6 and 5.7), respectively, are shown. 113 Figure 5.8 Histories of output parameters for the simulation with the elliptical electrode tissue contact spot. Events occurring during cutting through fiber are reproduced. All indicated features are common to all three contact spot types. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx=250 m, Ry = 50 m. 114 Figure 5.9 Histories of output parameters for the simulation with the line electrode tissue contact spot. Events occurring during cutting through fiber are reproduced. All indicated features are common to all three contact spot types. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx= m, Ry = 50 m. 115 In the simulations with the circular and elliptical contact spots the fast oscillations are primarily the manifestation of the random shifts of the contact spot along the scalpel from one just-destroyed grid cell to another cell. These shifts simulate electric sparks and nonuniform tissue-scalpel contact. The other contributor to the fast oscillations is stabilization noise that arises from the model requirements to maintain constant vaporization rate at a given temperature and constant power. The oscillations determined by these stabilization processes are the most visible at the end of the cutting trough a layer, just before the shift to the next layer. In the case of the line contact spot, Figure 5.9, (where the random shift oscillations are absent) only stabilization noise is present. 5.4.2 Heat Source and Temperature Distribution 5.4.2.1 Two-Dimensional Distribution Figures 5.10 and 5.11 show two-dimensional distributions of the heat source and temperature recorded in the simulations with the circular and elliptical contact spots, respectively. The records were made in a steady-state cutting regime after the contact spot remained in the same place for several time steps and just a few time steps before the scalpel was through a tissue layer. The distributions were recorded in the planes parallel to the scalpel centerline and co-planar with it. The planes are indicated as the XY plain at Z= Lz/2 plane perpendicular to the cut and parallel to the scalpel, and the XZ plane at Y= 0 plane parallel to the cut and parallel to the scalpel respectively. The heat source is plotted on a logarithmic scale. The heat source distributions are plotted for the entire simulated area of 1 cm2. The temperature distributions are plotted for of the simulated area as indicated on the heat source plots. In the data shown the spot with maximum temperature corresponds to the maximum heat source. However, during most of the simulation time this is not so. The locations and values of the maximum heat source and temperature are 116 indicated. Since the analogous distributions for the line source are symmetrical they can be deduced from the following Figures from 5.12 b) to 5.15 b), respectively. Figure 5.10 Distributions of heat source and temperature in XY (along the scalpel, across the cut) and XZ (along the scalpel, along the cut) planes recorded in simulation with circular contact spot. The temperature distributions are plotted for the square areas as indicated on the heat source plots. The distributions were recorded at the moment just before tissue layer was destroyed. Approximate locations of the grid elements with maximum heat source and temperature and corresponding maximum values are indicated. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx=250 m, Ry = 50 m. 117 Figure 5.11 Distributions of heat source and temperature in XY (along the scalpel, across the cut) and XZ (along the scalpel, along the cut) planes recorded in simulation with elliptical contact spot. The temperature distributions are plotted for the square areas as indicated on the heat source plots. The distributions were recorded at the moment just before tissue layer was destroyed. Approximate locations of the grid elements with maximum heat source and temperature and corresponding maximum values are indicated. The input parameters are ESU output power, PESU = 100 Watts, Nucleation threshold temperature, Tn = 112.5 C, Contact spot radiuses Rx = Ry =100 m. 118 5.4.2.2 Comparison with Theoretical Prediction Figures 5.12 and 5.13 show simulated and theoretically predicted heat source and temperature as a function of the distance from the scalpel s centerline. The simulated distributions in the directions of the cut are shown on Figures 5.12 a) and 5.13 a). The simulated distributions in the directions across the cut are shown on Figures 5.12 b) and 5.13 b) respectively. The distributions corresponding to the maximum values of the heat source and temperature are demonstrated. Theoretical distributions of the heat source and temperature, shown in Figures 5.12 and 5.13 respectively, were calculated using Honig and Pearce current source models [Pearce, 1986], Equations (1.1) and (1.2) respectively. The theoretical distributions were calculated to compare the predicted values for the semicylindrical and hemispherical model with the simulation results for the line and circular contact spots. In the models, total currents, I, were set equal to average values of the total currents in the simulations with the line and circular contact spots: IC = 300 mA for the semicylindrical and IS = 200 mA for the hemispherical models respectively. The radii of the cylindrical electrode and the spark were set equal to the width of the line and the radius of the circular contact spots respectively: rC = 50 m and rS = 100 m. The length of the electrode in Equation (1.1) was equal to the simulated electrode length Lx = 1 cm and tissue electric conductivity was set to = 0.25 S/m. The theoretical temperature distribution was calculated by integrating Equation (1.3). The upper integration limit was determined as the time when the temperature at the radii of the cylindrical electrode, rC, and the spark , rS became equal to the maximum temperature obtained in the simulations with the line and circular contact spots: 125 C and 140 C respectively. The time limit was calculated from Equation (1.4). 119 Figure 5.12 Simulated and analytically predicted heat source as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest heat source made in direction along the scalpel a) and across the scalpel b) shown. 120 Figure 5.13 Simulated and analytically predicted temperature as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest temperature made in direction along the scalpel a) and across the scalpel b) shown. 121 Figure 5.14 Simulated electric conductivity as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest temperature made in direction along the scalpel a) and across the scalpel b) shown. 122 Figure 5.15 Simulated water content as a function of distance from the scalpel s centerline. Simulated distribution through the point of highest temperature made in direction along the scalpel a) and across the scalpel b) shown. 123 5.4.3 Electric Conductivity and Water Content Figures 5.14 and 5.15 show tissue electric conductivity and water content, respectively, as a function of the distance from the scalpel centerline obtained in the simulations with the circular, elliptical and line contact spots. The water content is shown in water fraction remain in the tissue. The data were obtained from the records made in a steady-state cutting regime immediately after the scalpel passed through a tissue layer. The distributions in the XY plane at Z= Lz/2 plane perpendicular to the cut and parallel to the scalpel are shown on Figures 5.14 a) and 5.15 a). The distributions in the XZ plane at Y= 0 plane parallel to the cut and parallel to the scalpel are shown on Figures 5.14 b) and 5.15 b). In the cases of the circular and elliptical contact spots, each data point represents the data average taken along the scalpel. For these cases the standard deviation is indicated. 5.4.4 Total Current, Tissue Impedance and Cutting Speed 5.4.4.1 Effect of the Interface Power Density To present the simulation data more visibly and to illustrate the trends observed in the data the interface power density, Pint, defined as the output power of the ESU, PESU, divided by the area of the contact spot is used. The areas of the contact spot are calculated as Rx2, RxRy and LxRy for the circular, elliptical and line contact spots respectively. Figures 5.16, 5.17 and 5.18 show the total current, tissue impedance and the cutting speed as a functions of the interface power density for the simulations with the circular, elliptical and line contact spots. The simulations were performed for a nucleation temperature of 112.5 C. Each data point corresponds to a unique 124 simulation with particular ESU power and contact spot sizes. The total current, tissue impedance and cutting speed were determined as average values in the steady-state process of cutting through 10 consecutive layers of muscle cells. The standard deviations are plotted for each data point. To cover the same range of interface power density for all three types of contact spots, and to demonstrate a general trend in the data, simulations with unrealistically high power for the line contact spot and large radius for the circular spot had to be performed. The ranges of the data corresponding to the unrealistic simulations are indicated on the plots. Figure 5.16 Tissue impedance as a function of the interface power density. Simulations were performed for all range of power and contact spot sizes as indicated in Table 5.3. The nucleation temperature was 112.5 C in all cases. Data ranges corresponding to unrealistically large circular spot area or, in the case of the line spot, unrealistically high ESU power are indicated. 125 The data presented on all three figures demonstrate distinctive asymptotic behavior. Figure 5.16 demonstrates that in the cases of all three types of the contact spots the total current, Itot, decreases with increase in the surface power density. At the same time, it appears that with increase in Pint, the total current in all three cases approaches a horizontal asymptote. With decrease in Pint, the currents corresponding to the line and circular spots approach two vertical asymptotes. The asymptote of the line spot is shifted to the right, i.e. at the same Pint the Itot is higher for the line spot than for the Figure 5.17 Total current as a function of the interface power density. Simulations were performed for all range of power and contact spot sizes as indicated in Table 5.3. The nucleation temperature was 112.5 C in all cases. Data ranges corresponding to unrealistically large circular spot area or, in the case of the line spot, unrealistically high ESU power are indicated. 126 circular and elliptical spots. At the low and high Pint, the current for the elliptical spot asymptotically approaches the currents for the line and circular spots respectively. Figure 5.17 demonstrates a similar but inverse pattern. The tissue impedance, Ztis, increases with Pint in all three cases. At high Pint the impedance in all three cases approaches a vertical asymptote. At low Pint, the impedance of the line and circular spot approach two horizontal asymptotes. The asymptotic impedance of the circular spot is higher. The impedance for the elliptical spot approaches the line and circular spot impedance at low and high Pint respectively. Figure 5.18 Cutting speed as a function of the interface power density. Simulations were performed for all range of power and contact spot sizes as indicated in Table 5.3. The nucleation temperature was 112.5 C in all cases. Data ranges corresponding to unrealistically large circular spot area or, in the case of the line spot, unrealistically high ESU power are indicated. 127 The asymptotic behavior of the cutting speed is remarkably different in the current and impedance cases. Figure 5.18 demonstrates that the in the case of the circular contact spot the cutting speed, Vcut, increases with increase in Pint and approaches a horizontal asymptote. However, for the cases of the line and elliptical spots the cutting speeds are almost identical over the entire range of Pint. In these two cases the cutting speed increases at low Pint, then it appears to reach saturation level which serves as an asymptote for the Vcut in the case of the circle. 5.4.4.2 Effect of Nucleation Temperature Figures 5.19 and 5.20 show the tissue impedance and cutting speed, respectively, as a function of the nucleation temperature in the simulations of the circular and line contact spots. The simulations were performed in the interval of the interface power densities from 5x107 to 5x109 W/m2. The dimensions of the contact spots, ESU powers and corresponding interfacial power densities are summarized in Table 5.3. Each data point corresponds to a unique simulation with a given ESU power and contact spot size. The tissue impedance and cutting speed were determined as average values in the steady-state process of cutting through 10 consecutive layers of muscle cells. The standard deviations are plotted for each data point. Figure 5.19 demonstrates that the tissue impedance increases with increase in the nucleation threshold temperature. In the investigated range of the input parameters the increase is relatively monotonic and it appears that its character depends neither on the size of the contact spot and the ESU power nor on the type of contact spot. Higher levels of interface power density, Pint, produce higher tissue impedance. A general trend in the dependence of the total current on nucleation temperature can be easily deduced from the tissue impedance dependence: higher levels of interface power density, Pint, produce lower total currents. 128 Figure 5.19 Tissue impedance as a function of nucleation temperature for the circular a) and line b) contact spots. Each data point corresponds to a given ESU power and contact spot sizes and represents an average value obtained during a steady-state cutting through 10 consecutive layers. The simulations were performed in the interval of the interface power density, Pint, from 5x107 to 5x109 W/m2. Arrow in the right portion of the plots shows general increase in interface power density. 129 Figure 5.20 Cutting speed as a function of nucleation temperature for the circular a) and line b) contact spots. Each data point corresponds to a given ESU power and contact spot sizes and represents an average value obtained during a steady-state cutting through 10 consecutive layers. The simulations were performed in the interval of the interface power density, Pint, from 5x107 to 5x109 W/m2. Arrow in the right portion of the plots shows general increase in interface power density. 130 Figure 5.20 demonstrates that in the temperature range investigated the cutting speed generally an increase with increase in nucleation temperature, though the increase is relatively weak. Higher levels of the interface power density produce higher cutting speeds. As in the case of the tissue impedance, it appears that the character of the dependence is influenced neither by the size of the contact spot and the ESU power nor by the type of the contact spot. For all plots the difference between average values of the cutting speeds at the lowest and the highest temperature level, 105 C and 175 C respectively, is well within the standard deviation limits. 5.4.5 Comparison with Data from Literature Table 5.3. Simulation results. Average value standard deviation for the cutting speed, maximum and average temperatures, tissue impedance and total current. Circular contact spot radius: Rx =50 m, elliptical contact spot radiuses: Rx = 500 m, Ry = 50 m, line contact spot width: Ry=50 m, The nucleation temperature Tn=112.5 C. Typical experimental data taken from [Pearce, 1986]. Circle 7.5 0.5 107 8 10 3 50 10 Ellipse 25 1.8 0.1 104 1 1.3 0.3 150 10 100 25 3 114 5 8.6 0.5 150 10 11 0.3 107 1 2.0 0.7 230 30 12 0.3 106 1 0.9 0.2 330 20 0.6 0.1 600 30 Line 1.7 0.1 103 1 0.8 0.1 175 10 Experiment 43 25 Heat source model ESU power (W) Cutting speed (cm/s) Max. temperature ( C) Total impedance (k ) Total current (mA) ESU power (W) Cutting speed (cm/s) Max. temperature ( C) Total impedance (k ) Total current (mA) 1.1 0.8 243 90 208 15 131 Table 5.3 shows average values for the cutting speed, maximum and average temperatures at the scalpel-tissue interface, tissue impedance and total current for the circular, elliptical and line contact spots as well as average values for the tissue impedance and total current density typically obtained in electrosurgical procedures as presented in [Pearce, 1986]. The simulation data were determined as the average values in the steady-state process of cutting through 10 consecutive layers of muscle cells. 5.5 Discussion 5.5.1 Effect of Tissue and Cutting Process Representation In the present study a sample of skeletal muscle tissue is modeled as a periodic matrix structure where each matrix element represents a muscle fiber (cell). The matrix elements are identical and arranged in regular layers in the direction perpendicular to the direction of cutting. The cutting process is represented as the generation of a given amount of vapor inside the layer of muscle fiber and instantaneous advance of the electrosurgical scalpel to the next layer after the threshold vapor volume was reached. Combination of the assumptions of the periodic fiber pattern and instantaneous scalpel movement is ultimately responsible for the periodic pattern observed in the histories of the electric and thermodynamic parameters simulated in the model and presented on Figures from 5.5 to 5.9. The distinctive periodic pattern observed in the simulations, Figures 5.5 to 5.7, is not identifiable in the experimental record of the electrosurgical power, current and voltage obtained during cutting of skeletal muscle. The most obvious reason for this is that the sizes and material properties of muscle fibers are different from each other and that they are arranged relatively randomly as compared to the regular matrix adopted in the present model. However, the variation range of the simulated 132 parameters is likely determined not by the periodic matrix of the fiber layers but by the process of destruction of a single layer. High amplitude variations in the simulated parameters are likely to be caused by variations in the local power density as a result of dynamic variation in the water content and electric conductivity. Therefore, assuming that the model describes the vaporization dynamic in a single layer with relative accuracy, the simulated average values can be compared with experimental observation. 5.5.2 Effect of Local Power Density In the steady-state cutting regime when the scalpel comes in contact with a new tissue layer the vaporization of water immediately begins in this layer. If the temperature is below the nucleation threshold only surface vaporization takes place, but, very soon, the temperature rises to or above the nucleation threshold and more intense interstitial vapor generation is initiated. The fast temperature rise to and above the nucleation threshold is made possible by the high power density reached at the scalpel-tissue interface. Figure 5.12 demonstrates that at an ESU power level of 100 W the local power density at the scalpel-tissue interface is as high as 1013 W/m3 for all types of contact spot. The reduction of the ESU power level to 25 W still results in a power density as high as 1012 W/m3. Equations (3.5) and (3.6) and Figure (3.6) demonstrate that at such power densities the energy deposited in tissue is significantly higher than the energy removed by the vaporization process. Therefore, enough energy remains to contribute to the temperature rise by heat transfer. The high local power density obtained in the simulations, results not only from assumed high ESU power level or small area of the contact spot but from the dehydration of the tissue layer as well. As the heating of the layer continues, more water is removed from the tissue and/or replaced by the vapor. However, the water loss is not uniform it is the highest along the scalpel centerline and (in the cases of 133 the circular and line contact spots) at the spot of the electric contact. The resulting gradient in the water content (Figure 5.15 shows distribution of water content) leads to a gradient in electric conductivity (Figure 5.14 shows distribution of electric conductivity), which is responsible for an increase in local power density in two ways. First, the gradient in the electric conductivity results in the increase in the electric field (the so-called edge effect ). Second, the loss of conductivity through a portion of the contact spot leads to the effective reduction of the contact spot area and, respectively, given constant total power of the ESU, an increase in electric current passing through this area. The effect of the gradient in electric conductivity on the local power density is most noticeable near the end of cutting through a layer, when most of the tissue water is gone and the gradient is the highest. Figures 5.5 to 5.9 demonstrate that (at this time) changes in electric and thermodynamic effects are the fastest and most dramatic. The result of the electric conductivity gradient is also demonstrated in Figure 5.12 b) and Figure 5.13 b); showing the distribution of the power density and temperature respectively, recorded near the end of the cutting through a layer. For all types of contact spots the highest power density is not at the centerline but at a point away from it, near the edge of the dehydrated tissue region. In the case of the circular spot the temperature is the highest at the point of the highest power density, but in two other cases the highest temperature is at the centerline. However, all three cases demonstrate relatively high temperature near the centerline, which is likely due to the same conductivity gradient and resulting increase in the power density. It is hard to separate relative contributions of the edge effect and reduction of the effective spot size especially as they are likely to be linked together and amplify each other. Figure 5.8 demonstrates that the oscillations in the temperature near the end of cutting through a layer are relatively higher than in the beginning. These differences in the oscillation amplitude are likely due to the mutual 134 amplification of the two processes caused by the conductivity gradient. Indeed, when the region at the scalpel s centerline is almost completely dehydrated the only region remaining open for current flow is near the edge of the dehydrated zone. But, in this region the gradient in the electric conductivity is the highest and the increase in the electric field caused by the rise in the local current density will be added to the increase caused by the edge effect . 5.5.3 Effect of Contact Spot Type The prime reason behind the development of the numerical model presented in this study is validation of two hypotheses describing the interstitial heat source and explaining the mechanism of the ETR process proposed respectively by Honig [Honig, 1975] and Pearce [Pearce, 1986, pp. 72-79]. The line contact type of spot corresponds to the hypotheses suggested by Honig and simulates a cylindrical electrode in electric contact along its entire length (the semicylindrical heat source). The circular contact spot type corresponds to the hypotheses suggested by Pearce and simulates an electric discharge striking between the electrode and tissue (hemispherical heat source). The elliptical contact spot type simulates a non-uniform electrode-tissue contact. 5.5.3.1 The Heat Source and Temperature Distributions In this study, the distributions of heat source and temperature obtained in the simulations are compared with the distributions predicted by the two hypotheses calculated using the total current and the radiuses corresponding to those in the simulations. Figure 5.12 a) shows that in the ZX plane the heat source for all three types of contact spot is between the two levels predicted by the hypothesis and decreases in a manner similar to the analytical cases up to a distance of about 0.5 mm from the scalpel. The rapid decrease of the simulated heat sources further away from 135 the scalpel can be explain by the effect of the voltage boundary conditions: zero voltage at a fixed distance instead of infinity, as in analytical cases). However, in the XY plane the behavior of the heat sources is remarkably different from the analytically predicted, Figure 5.12 b). For all three types of contact spots the simulated values of the heat source demonstrate a clear maximum 50 m away from the scalpel. Moreover, beyond the maximum the simulated heat sources do not decrease as rapidly as analytically predicted and, in the cases of the circular and elliptical spots, even exceed the hemispherical level. The resulting exposure to the heating of a larger tissue volume is one of the effects that leads to higher temperatures deeper in tissue in the simulated cases, Figures 5.13 a) and b). Another reason for the increased subsurface temperature in the simulated cases is heat diffusion from the interfacial tissue layer during its relatively long destruction process. The profiles of the heat source and temperature distributions obtained in the simulations and their differences from those analytically predicted, Figures 5.12 and 5.13, can be explained from by distributions of the water content and electric conductivity, Figures 5.14 and 5.15 respectively. As has been pointed out in the previous section, extensive dehydration of the tissue near the scalpel centerline results in a local gradient of electric conductivity in the tissue region at a some distance away from the scalpel center, which, in turn, leads to a rise in local power density. The analytical temperature distributions were obtained assuming that the temperature at the interface is equal to the temperature obtained in the corresponding simulation. In the rest of the tissue the simulated temperatures clearly exceed those predicted from both hypotheses. Comparison of the temperature distributions predicted for the hemispherical and semicylindrical heat sources shows that in the hemispherical case the temperature at the interface is higher (see Figure 1.3 as well) but deeper in the tissue the situation is reversed and temperature in the 136 semicylindrical case is higher. In the simulated cases the temperature at the interface obtained for the circular contact spot (analog of the hemispherical heat source) is higher than obtained for the line (analog of the semicylindrical heat source) spot as well. But, remarkably, deeper in tissue the temperature in the case of the circular spot continues to exceed the temperature of the line and elliptical spots (the last two temperatures are similar). Such excess can be explained from the difference in the distributions of the water content and electric conductivity, Figures 5.14 and 5.15. In the case of the circular spot the edge of the dehydration zone is shifted further away from the scalpel centerline than in the case of the line and elliptical spots. Such a shift is visible in both XY and XZ planes. Therefore in the case of the circular spot more deep tissue layers are exposed for the heating for the longer time. In the case of the circular spot a deeper extent of the dehydration is likely due to high instantaneous power density. 5.5.4 Current, Voltage and Impedance Average values for the cutting speed, maximum temperature, tissue impedance and total current for the circular, elliptical and line heat sources as well as average values for the tissue impedance and total current typically obtained in electrosurgical procedures as presented in [Pearce, 1986] are summarized in Table 5.5. For all simulations average cutting speed is near experimentally observed several centimeters per second and increases proportionally to an increase in total ESU power. However, it should be noted, that the value of the cutting speed strongly depends on assumptions made in the water evaporation model and the criteria for tissue destruction. In the simulations performed with a total ESU power of 25 Watts the cutting speed obtained for the circular heat source is about four times higher that for the elliptical and line sources. But in the simulations performed with a total power of 100 Watts the difference decreases to about two times. 137 Moreover, it appear, that high impedance and low current rule out an electric discharge as the primary mechanism of electrosurgical cutting. Indeed, in order to decrease impedance and increase current, the area of the heat source should be increased. However, evaluation of porosity damage to the coating of electrosurgical blades produced by electric sparks [Konesky, 1998] demonstrates that the average pore radius is about 50 m and there are no pores with radius higher than 100 m. Obviously electric discharge plays a role in the tissue damage process but it is unlikely that it contributes to the cutting process in any significant way. 5.5.5 Effect of Interface Power Density Evaluation of the total current, tissue impedance and cutting speed as functions of the interface power density demonstrates that the geometry of the scalpel-tissue interface established in steady-state cutting is ultimately responsible for the electrical characteristics of the electrosurgical procedure and, as a consequence, for the physical results. Figure 5.17 shows that as the interface power density, Pint, increases the tissue impedance, Ztis, increases for all types of contact spots increases and approaches a vertical asymptote. Increase in the Pint is caused either by a decrease in the area of the contact spot or increase in the ESU output power, PESU. But increase in PESU causes more intense vaporization and as, a result, decrease in effective contact area. Therefore, increase in Pint causes decrease in effective contact area for all types of the contact spots and corresponding increase in Ztis. From the other side, decrease in Pint produces two different asymptotic levels of Ztis for the circular and line contact spots respectively. These two levels reflect the difference in geometry and therefore impedance between circle and line contacts. Strictly speaking, vertical asymptotes for the line and circular spots should be different and their position should depend on the 138 actual sizes of the contact spots. Again, at low Pint two horizontal asymptotes should be different for different radiuses and width of the circular and line contact spots respectively and independent from PESU. The smaller the size the higher the impedance. However, it is likely that under the assumed conditions these differences could not be resolved in the present study. General behavior of the total current manifested in approaching a low horizontal asymptote at the high levels of Pint and in the difference between currents produced by the circular and line spot at the low levels of Pint follows the same reasoning as applied to behavior of the impedance. However, the apparent asymptotic behavior at low Pint is an artifact of this study caused by the limit of PESU = 25 W and lack of spatial resolution in the model. Obviously, further decrease in Pint must result in decrease of the current caused either by reduction of the power or increase in tissue impedance due to complete dehydration. The elliptical contact spot represents an intermediate case between the line and circular spots. An increase in Pint corresponds either to actual or effective decrease in the contact area and results in asymptotic approach to the circular spot behavior. Decrease in Pint corresponds primarily to elongation of the ellipse along the scalpel and asymptotic approach to line spot behavior. With a decrease in Pint the cutting velocity, Vcut, obviously approaches zero for all contact spots. Increase in Pint results in saturation of the cutting speed: faster in the cases of the line and elliptical spots than in the case of the circular spot. The saturation of Vcut, is likely due to vaporization dynamics that leads to the steady-state shape of a dehydrated zone and corresponding electric conductivity zone as demonstrated on Figures 5.14 and 5.15. It is clear that the dehydrated zone is wider across the cut than down under the scalpel. Increase in Pint, and, therefore, increase in local power density results in more intense vaporization at the sides of the cut but not 139 under the scalpel. Therefore, it is reasonable to conclude that increase in PESU will result in increase in width of the thermal damage zone but not in the cutting velocity. 5.5.6 Effect of Nucleation Temperature Parametric study of the effect of the nucleation temperature, Tn, on the simulation results demonstrates the increase in the tissue impedance and subsequent decrease in the total current and increase in scalpel voltage with increase in Tn, Figure 5.19. Increase in impedance likely results from extensive dehydration of the surface tissue layer through the surface vaporization. At low Tn surface vaporization contributes little to the water loss as compared to the process of interstitial nucleation and growth of the vapor bubbles. Increase in the level of Tn prolongs the available time as well as increases the rate of surface vaporization. Slight increase in the cutting speed with increase in Tn, Figure 5.20, is likely a result of two competing processes. The first process is a prolonged time of heating to Tn. However, the histories of the maximum temperatures, Figures from 5.5 to 5.9 show that the time to heat tissue from T0 = 25 C to Tn is relatively short. The second process that increases the cutting speed is the acceleration of bubble growth at higher temperature levels, Equation (2.62) resulting in faster layer destruction. However, such acceleration might not be dramatic enough, as the velocity of the bubble growth depends not on the temperature of the immediate surrounding of the bubble but on the temperature difference between the bubble boundary and surrounding liquid. Figure 5.13 demonstrates that the temperature gradient in the vicinity of the vaporizing region is not especially dramatic, therefore the acceleration of the bubble growth with the rise of Tn might be insignificant. 140 5.5.7 Computational Considerations The computational requirements for this study were significant, and may be the limiting factor in extending the model s precision by decreasing the space and time steps. All simulations were performed on an Optiplex Dell PC computer. The longest simulations required approximately 70 hours of processor time and generated up to 10 megabytes of data. Improvements in memory size and processor speed will facilitate simulations of larger tissue regions and allow the specific geometry of the electrosurgical scalpels to be represented with increased accuracy. 5.6 Conclusion In this study, a thermo-electric model of the ETR process, which allows arbitrary specification of electrode-tissue contact area, has been developed and implemented. By introducing an ablation mechanism based on interstitial vaporization of water in the numerical model, this study has taken a significant step towards understanding the mechanism of electrosurgical cutting procedures. The results of this study have provided a unique perspective on the effects of the geometry of electrode-tissue contact on macroscopic electrical parameters of the electrosurgical circuit and on the ETR process. Simulations performed with the circular contact spot of 100 50 m in radius significantly overestimated the impedance of electrosurgical circuit and cutting speed while impedance and cutting speed predicted in the simulations performed with the line contact spot predicted are in close agreement with experimentally observed values. Therefore, the hypotheses of an electric discharge as a prime mechanism of electrosurgical cutting must be rejected. 141 The model demonstrated the important role that dehydration plays in the cutting process. It was shown that water loss from the centerline of a tissue-electrode interface results in a strong gradient of electric conductivity, which, in turn, is responsible for significant elevation of the heat source and temperature in the vicinity of the electrode. In the case of the linear contact spot, such an elevation results in the cutting speed that exceeds the value estimated from a simplified analytical model and agrees with experimentally observed. Future studies of techniques and parameters used in ETR procedures (such as investigation of the effect of the electrode geometry on thermal damage in next Chapter) will help to clarify the realism of suggested model of interstitial vaporization in simulations of electrosurgical cutting, and may lead to significant improvements in treatment. 142 Chapter 6 Numerical Study of thermal Damage in the ETR Process 6.1 Abstract A numerical study of electrosurgical tissue resection (ETR) has called attention to the importance of dynamic spatial distribution of water vaporization and its influence on electrical and thermal processes at the tissue-scalpel interface. The development of vaporization and coagulation thermal damage during electrosurgical cutting of skeletal muscle tissue was studied numerically and experimentally with the intent of validating the suggested model of interstitial vaporization, as well as investigating the influence of scalpel geometry and output power of ESU on electrosurgical cutting rate and thermal damage. A two-dimensional thermo-electrical model that incorporates dynamic changes in tissue electric conductivity with temperature and water content was developed. The model calculates interstitial heat source from voltage and electric conductivity distributions, temperature, surface and interstitial vaporization and vaporization and coagulation thermal damage. Experiments on electrosurgical cutting of beef muscle samples were performed to validate the developed model in-vitro. Vaporization and coagulation thermal damage and cutting rate were analyzed and compared with predicted. The model showed good functional agreement with experimental results. It was demonstrated numerically, and confirmed experimentally, that with increase in ESU output power relative increase in vaporization damage is stronger than increase in coagulation. At the same time, an increase in width of electrosurgical scalpel results in stronger relative increase in coagulation than vaporization damage. However, moderate discrepancies between absolute values of vaporization and coagulation thermal damage as computed and observed experimentally may have 143 resulted from inaccuracy in definition of tissue material properties used in the model. Particularly, values of tissue parameters critical for interstitial vapor formation are not known with reasonable accuracy at present. 6.2 Introduction In spite of relative maturity of clinical ETR techniques theoretical description of the physical principles involved in the procedure as well as predictions of the resulting thermal damage have remained largely unexplored. Histological evaluation of electrosurgical lesions produced by different types of electrodes at different settings of ESU power have demonstrated, as shown for instance in [Sebben, 1989] and [Wolf et al., 1997], significant differences in the extent of thermal damage. It is clear that in order to understand the physics of ETR, evaluate effects of electrosurgical power and electrode geometry on tissue thermal damage and plan electrosurgical treatment accordingly, a numerical model of the cutting process would be highly useful. The numerical study of ETR discussed in the previous chapters resulted in several important findings. First, the study demonstrated that generation of a nonuniform dehydration region localized in the immediate vicinity of electrosurgical scalpel is highly essential to electrosurgical cutting. The profiles of the heat source and temperature near electrode and, therefore, of thermal damage are shaped by the profile of the dehydrated zone. At the same time, the dehydration resulting from surface and interstitial vaporization is defined by the heat source and temperature distributions. Therefore, to predict results of an electrosurgical procedure and plan clinical treatment a precise understanding of dehydration process and parameters affecting it is required. 144 Second, tis Chapter is confirming the semicylindrical model and, respectively, dismissing the hemispherical model of electrosurgical cutting. Comparison of cutting speeds and electrosurgical circuit parameters such as tissue impedance and total current obtained in the simulations with typical values reported in literature have proven that electric discharges cannot be responsible for electrosurgical cutting. Comparison of the simulation results (cutting speeds and circuit parameters as well as the distributions in tissue of the heat source, temperature and water loss) obtained for the line and elliptical contact spots demonstrated their equivalence. This means that a two-dimensional model can replace a three-dimensional model without significant loss of accuracy. Third, the numerical model introduced in the previous Chapter utilizes a rectangular grid for skeletal muscle. However simple in implementing and reasonably compatible with periodic pattern of muscle fibers, this type of grid lacks either precision or speed of calculation for description of dehydration process around the electrode. The model demonstrated that the shape of the dehydrated zone formed around an electrosurgical cut and, respectively, shapes of the heat source and temperature distributions have distinctive axial simmetry, which better fits polar coordinates. Moreover, most electrosurgical scalpels employed in typical electrosurgical procedures can be approximated as an elliptical or circular cylinder, which favors polar coordinates, as well. Summarizing results of numerical study described in previous chapter, it is possible to conclude that a two-dimensional model describing electrical, thermal and material parameters relevant to the ETR process in elliptical grid might be a reasonably useful tool for prediction of the results of electrosurgical cutting. Development and verification of such a model is the subject of this chapter. 145 6.3 Methods 6.3.1 Experimental Methods 6.3.1.1 Tissue Samples Fresh bovine and porcine muscle (butcher shop) were used as the tissue samples. The tissue was stored in a refrigerator at 7 C from 1 to 5 hours prior to the experiments. Tissue samples of dimensions 5 5 2 cm were prepared. Each sample was wrapped in plastic film to prevent significant drying and kept at room temperature for 30 minutes to equilibrate. Tissue samples were placed on top of a conducting plate electrode and oriented so that muscle fibers were parallel to the top sample surface. 6.3.1.2 Experimental Procedure Figure 6.1 shows the general arrangement of the experiments and configuration of the electric circuit for measuring voltage, current and phase shift. A flat plate electrode with the tissue sample on top was connected by wire to the patient plate input of a Force 30 ESU (Valylab Inc.). A series of cuts were performed on the beef muscle. The cuts were performed in the general direction perpendicular to the muscle fibers at maximum possible speed. The experiments were performed at 40, 60, 80 and 100 Watts of ESU output power. The cutting was recorded on a VCR and the cutting speed was measured from the records. The experiments were performed for two types of electrosurgical scalpels: standard blade electrodes and cylindrical wire electrodes. Diameters of the blade 146 Figure 6.1 Schematic diagram of measurement connections in experiments. The voltage probe is connected to a capacitive voltage divider with the probe capacitance. Electrode voltage is calculated from the measured divider ratio (determined from an open circuit measurement of probe and actual voltage). electrodes were approximated as ellipses with maximum and minimum diameters: a) E1 = 0.45 mm and 2.35 mm and b) E2 = 0.55 mm and 2.85 mm, respectively. Diameters of wire electrodes were a) C1 = 0.45 mm, b) C2 = 0.7 mm and c) C3 = 1.1 mm. Five cuts were performed for each electrode at each power level. Return current was measured with an isolated high frequency current probe attached to the wire connecting a flat plate electrode to the patient plate input of the ESU. In the experiments voltage was measured directly between the cutting/coagulating and return electrodes. Surgical RMS current, RMS voltage and phase angle were measured during one sweep of the oscilloscope. A Tektronix model TDS 350 digital oscilloscope was used for RF current and voltage measurements. The current probe was a Model A6302 (Tektronix) and voltage probe was a Model P6111B (Tektronix). The RF current probe was amplified and calibrated by an AM 503B current amplifier (Tektronix). The output power of the ESU and load impedance were calculated from the voltage, current and phase shift measurements. 147 Immediately after each cut was completed two samples of the lesion crosssection were dissected and retained in 10% neutral-buffered formalin solution for histologic examination. Pathology samples were prepared by the pathology lab at *Pathology Inc.* at Austin, Texas. Trichrome and H&E stains were used. A total of 80 samples were examined: two samples for each type of electrode at each power setting prepared in two stains. The samples were examined using a polarization Figure 6.2 Evaluation of electrosurgical lesions. Three types of thermal damage can be identified on each lesion: melted proteins, vapor bubbles and birefringence loss. The depth of thermal damage was measured from the edge of the cut at six sites, from A to F, approximately located as shown. The damage zones were determined from the edge of the cut; D1 -vapor bubble (water loss) zone, D2 birefringence loss zone. 148 microscope (Nikon Optiphot) and the width of the vaporization zone and the width of the birefringence loss zone were measured for each sample. The measurements were performed at six sites around the circumference of the lesions, as shown on Figure 6.2. The water loss zone was determined as the distance from the lesion edge to the limit of the vapor vacuole layer. The birefringence loss zone was determined as the distance from the lesion edge to the limit of the dark zone. The average size of vapor bubbles and their number per unit length were estimated as well. 6.3.1.3 Data Statistical Analysis The influence of two factors; ESU output power and radius of electrosurgical electrode, on the sizes of birefringence and water loss zones were analyzed for blade and wire electrodes, respectively. The analysis was performed for all four levels of the output power and two and four sizes of the blade and wire electrodes, respectively. Two sets of three null hypotheses each were formulated for the width of the water loss zone and birefringence loss zone respectively. The hypotheses are as follows: Hypotheses tested for the water loss zone: H1.10: All mean values of the width of the water loss zone are the same for all scalpel radiuses. H1.20: All mean values of the width of the water loss zone are the same for all ESU output powers. H1.30: There is no significant interaction effect between all mean values of the width of the water loss zone. 149 Hypotheses tested for the birefringence loss zone: H2.10: All mean values of the width of the birefringence loss zone are the same for all scalpel radiuses. H2.20: All mean values of the width of the birefringence loss zone are the same for all ESU output powers. H2.30: There is no significant interaction effect between all mean values of the width of the birefringence loss zone. The set of alternative hypotheses HA represents a respective negation of the null hypotheses and states that not all means for different scalpel sizes and ESU power levels are the same and there is significant interaction between the means. To perform data analysis two-factor ANOVA with a replication tool was used in Microsoft Excel. The level was set to 0.05. 6.3.2 Numerical Methods In the present section a two-dimensional thermo-electric model of the ETR process is described. The model was developed from the three-dimensional model described in the previous Chapter and utilizes many features from it. Only modifications the model dealing with elliptical and rectangular grids and estimation of coagulation damage are discussed here. Elements of the model that are related to water vaporization and calculation of circuit macroscopic parameters were described in sections 5.3.4 and 5.3.5 in the Chapter 5. 150 6.3.2.1 Tissue and Cutting Process Representation Assuming that the electrosurgical scalpel can be represented as an elliptical cylinder and that the vaporization thermal damage to skeletal muscle tissue displays an axially symmetric pattern as well the voltage, electric field and heat source were calculated in elliptical cylindrical coordinate grid (r, ). Since muscle can be approximated as a matrix of rectangular layers and the cutting mechanism is defined by the vaporization inside each layer rectangular coordinate grid (z,y) was used for calculation of temperature, electric conductivity and water content. Figure 6.3 shows relative position of two coordinate grids. Figure 6.3 Elliptical and rectangular grids. Actual sizes of grid steps, dy0, dz0 and d 0, are much smaller in relation to the scalpel than shown. The cutting direction is Z. Scalpel radiuses Ry = Rz correspond to a polar grid. The material properties of tissue, water and vapor utilized in this study are those given in Tables 5.1 and 5.2. The process of electrosurgical cutting was represented as a destruction of consecutive layers of muscle fibers. Muscle fibers were modeled as rectangular cells of 50x50 m2 crossectional area and infinite length 151 running in direction perpendicular to the scalpel and cutting direction. In order to cut through a layer a vapor bubble filling a fiber, i. e. a bubble with diameter equal to 50 m has to be formed inside. When a layer is cut all tissue is moving upwards with respect to the scalpel. A fresh tissue layer is added to the bottom of the sample and the top layer is removed. The basic differences of the algorithm employed in the model from the algorithm described in Chapter 5 are: a) omission of the random contact spot and b) introduction of two grid systems: one for description of the heat source and the other for calculation of temperature and material properties. Once the heat source was calculated in the cylindrical grid it was interpolated for the rectangular grid and changes in temperature, water content and electric conductivity were determined. Then, the electric conductivity was interpolated back into the cylindrical grid and voltage, electric field and heat source were determined there. 6.3.2.2 Rectangular and Elliptical Grids In both the rectangular and elliptical grids the origin of coordinates was chosen as the center of the scalpel. In the rectangular coordinate system the negative direction on the Z-axis was chosen for cutting and the positive direction is opposite to the cutting. Line Y = 0 is a line of symmetry. In the elliptical coordinate system angles = and = - correspond to the positive and negative directions of Z-axis, respectively. The scalpel is approximated as an elliptical cylinder and orientated with its longest axis in the cutting direction. Location of a grid element (iy, i ) in the elliptical coordinate system is given as: 152 riy ,i = 1 2 cos 2 ( d 0 i ) (R y + dy 0 iy ) 2 , (6.1) where Ry is the smallest scalpel radius, d 0 is angular step, is the eccentricity of the ellipse representing the scalpel: = 1 2 Ry R z2 (6.2) and dy0 is a unit step of rectangular grid along Y-axis: dy 0 = dz 0 Ry Rz , (6.3) where Rz is the largest scalpel radius and dz0 is a unit step of rectangular grid along the Z-axis. In elliptical coordinates the distances between (iy, i ) grid point and adjacent (iy 1, i ) and (iy, i 1) grid points are: Lriy 1,i = riy ,i riy 1,i (6.4) and, approximately: 1 L iy ,i 1 = (R y + dy 0 iy ) 1 2 sin 2 d 0 i d 0 , (6.5) 2 respectively. The perimeter or circumference of the ellipse is calculated by integration of Equation (6.5) over 360 153 6.3.2.3 Heat Source Calculation in Elliptical Grid A convenient method of heat source calculation in elliptical coordinates is to employ integral form of the continuity equation (2.4) represented in the form of Kirchhoff s current law (KCL) written for each grid element [Gandi, et al., 1984]. Summation of currents entering and exiting grid element (iy, i ), Figure 6.4, yields: Viy +1,i Viy ,i Z iy +1,i Viy ,i +1 Viy ,i Z iy ,i +1 Viy ,i Viy 1,i Z iy 1,i Viy ,i Viy ,i 1 Z iy ,i 1 + = + (6.6) and, after rearrangement, the voltage in grid element (iy, ) and be defined as: Viy +1,i Viy ,i +1 Viy 1,i Viy ,i 1 1 Viy ,i = + + + Z Z iy ,i +1 Z iy 1,i Z iy ,i 1 G iy +1,i (6.7) where G is the sum of orthogonal conductivities (inverse of orthogonal impedance) and Ziy 1,i and Ziy,i 1 are the orthogonal impedances defined as: Z iy 1,i = Lriy 1,i iy 1,i L iy 1,i Ls (6.8) and L iy ,i 1 Z iy ,i 1 = iy ,i 1 Lriy ,i 1 Ls , (6.9) where Ls is the length of the scalpel and electric conductivities iy 1,i and iy,i 1 are calculated as the average of values in the two adjacent grid elements. 154 The radial and angular components of the electric field vector are: Viy +1,i Viy ,i Lriy +1,i Viy ,i Viy 1,i Lriy 1,i Eriy ,i = + (6.10) and Viy ,i +1 Viy ,i Z iy ,i +1 Viy ,i Viy ,i 1 Z iy ,i 1 E iy ,i = + , (6.11) The magnitude of the electric field, |E|, in the grid element is absolute value of vector sum of all components: E= (Er 2 iy ,i 2 + E iy ,i ) (6.12) According to Equation (2.11) the heat source, q, is: q = E 2 (6.13) where = (W,T) is tissue electric conductivity, Equation (2.3), function of water content and temperature. 6.3.2.4 Interpolation of the Heat Source into Rectangular Grid Once the heat source is calculated in the elliptical grid its values in the rectangular grid are determined using a two-dimensional Bessel interpolation algorithm [Korn and Korn, 1961, p. 596]. The heat source in element (iy, iz) of the 155 rectangular grid is calculated from its four neighboring values in the elliptical grid, Figure 6.4 (a): Figure 6.4 Arrangement for Bessel interpolation: a) from elliptical into rectangular and b) from rectangular into elliptical grids respectively. In the case of elliptical to rectangualr interpolation unit steps in radial, R, and angular, , direction are taken as shown. 1 1 1 1 1 1 1 q1 + u q 2 + v q3 + u v q 4 , 4 2 2 2 2 2 2 q iy ,iz = (6.14) where u= (R y + dy 0 iy ) y 0 dy 0 (6.15a) and 156 v= and (Rz + dz 0 iz ) z 0 dz 0 , (6.15b) q1 = q 0, 0 + q 0,1 + q1, 0 + q1,1 , q 2 = q 0,0 q 0,1 + q1, 0 + q1,1 , (6.16a) (6.16b) q 4 = q 0,0 + q 0,1 q1,0 + q1,1 , q 4 = q 0, 0 q 0,1 q1,0 + q1,1 , (6.16c) (6.16d) 6.3.2.5 The Temperature Field Neglecting spatial variations in tissue thermal conductivity, the diffusion term in the heat diffusion equation (2.13) becomes: 2T 2T (k T ) = k 2 + 2 y z (6.17) Keeping in mind that the rectangular grid steps along the Y- and Z-axes are not equal, the finite difference equation for the temperature in the grid element with the coordinates (iy,iz) is: Tt + dt = Fo y Tiy + Fo z Tiz + 1 2 Fo y 2 Fo z Tt + [ ] dt dm h fg q c p dt (6.18) 157 where Foy and Fox are Fourier numbers: Fo y = k dt 2 c p dy 0 (6.19a) Fo z = k dt 2 c p dz 0 (6.19b) and Tiy = (T (iy + 1, iz ) + T (iy 1, iz ) ) (6.20a) Tiz = (T (iy, iz + 1) + T (iy, iz 1) ) , (6.20b) respectively. 6.3.2.6 Estimation of Coagulation Damage The extent of coagulation damage in tissue around the cut was estimated using the Arrhenius relation, Equation (2.22): ( ) = A e Ea RgT (t ) dt . (6.21) 0 The frequency factor, A, and the activation energy, Ea, were assumed equal to the values determined for cardiac muscle [Labonte, 1994]: A = 2.39x1092 s-1 and Ea = 5.87x105 J/mol. By definition >= 1 represented irreversible damage. 158 6.3.2.7 Interpolation of Electric Conductivity into Elliptical Grid Once tissue electric conductivity is calculated in the rectangular grid the corresponding values in the elliptical grid are obtained using the Bessel interpolation algorithm. Conductivity values in each grid element (iy, i ) are calculated from four neighboring conductivity values in rectangular grid, Figure 6.4 b), by Equation (6.14) where parameters u and v are given as: u= and v= respectively. (R y + dy 0 iy ) y 0 dy 0 (6.22a) (Rz + dz 0 iz ) z 0 dz 0 , (6.22b) 6.4 Results 6.4.1 Experimental Results The prime goal of this experimental study was verification of the numerical model of the ETR process. Experiments were performed to determine cutting speed, electrosurgical circuit parameters (impedance, current, voltage and phase shift) vaporization and coagulation thermal damage to beef muscle obtained at different ESU power settings and with different electrosurgical scalpels. 159 6.4.1.1 Circuit Parameters and Cutting Speed Table 6.1 summarizes the overall values of cutting speed, voltage, current and phase shift measured in the experiments and the output power and load impedance calculated from these measurements. Each value represents a mean one standard deviation of five measurements obtained during five cuts in different places of tissue sample. Circuit voltage, current and phase shift were measured in the experiment, and delivered power and load impedance were calculated from these measurements. Cutting speed was determined from video records of the cutting process. For all types and sizes of electrodes the electrosurgical voltage and current show gradual increase with increase in the ESU power level. At the same power level, a smaller electrode generally displays lower voltage and higher current. The two smallest wire electrodes, of diameters 0.45 and 0.7 mm, demonstrate higher voltage and lower current than both blade electrodes of 0.45 mm x 2.35 mm and 0.55 mm x 2.85 mm respectively. The largest wire electrode 1.1 mm diameter, demonstrates values of voltage and current close to the blade electrodes. Phase shift appears to be of insignificant value in all cases. Power delivered to tissue gradually increases with increase in the ESU power level. There is no apparent correlation between delivered power and type or size of electrode. For all electrodes load impedance is the highest at the lowest ESU power level. At higher power levels impedance does not display a distinctive decrease or increase. Generally, the smaller the size of the electrode the higher the impedance. Cutting speed increases with increase in ESU power level. The wire electrodes of the two smallest sizes provide the fastest cutting. 160 Table 6.1. Scalpel type Electrosurgical circuit parameters and cutting speed. Values in table are expressed as mean one standard deviation. ESU Power Level, W 40 60 80 100 40 60 80 100 40 60 80 100 40 60 80 100 40 60 80 100 Voltage, V 170 10 190 10 230 20 250 20 200 10 210 10 200 20 220 30 170 30 200 30 260 40 260 50 210 30 180 30 220 50 280 50 160 20 220 30 225 30 230 40 Current, mA 150 10 220 20 260 30 280 30 160 10 260 20 320 30 320 30 120 20 180 30 200 40 250 40 140 20 230 30 240 30 250 40 180 10 250 20 280 30 300 30 Phase shift, 6 5 4 2 0 9 6 3 4 0 7 5 4 2 5 2 4 5 12 7 8 5 5 3 0 5 3 4 3 6 3 0 5 3 0 Delivere d Power, W 26 3 42 6 59 12 70 13 32 4 55 7 64 12 71 15 20 7 36 11 51 18 64 23 29 8 41 12 53 19 70 24 29 8 55 12 63 15 70 19 Load Impedance, k 1.1 0.1 0.9 0.1 0.9 0.2 0.9 0.2 1.3 0.1 0.8 0.1 0.6 0.1 0.7 0.2 1.4 0.5 1.1 0.4 1.3 0.5 1.0 0.4 1.5 0.4 0.8 0.2 0.9 0.3 1.1 0.4 0.9 0.2 0.9 0.2 0.8 0.2 0.8 0.2 Cutting Speed, cm/s 1.7 1.1 3.4 1.6 4.5 2.7 5.2 2.1 2.2 1.2 3.1 1.1 6.4 2.2 5.8 2.6 1.9 1.0 4.2 1.6 6.5 2.3 7.2 5.4 1.7 0.8 3.6 1.6 5.6 1.2 7.2 2.4 1.2 0.9 3.1 1.3 4.5 1.4 5.2 1.8 E1 E2 C1 C2 C3 The relative error of all data but phase shift increases slightly with an increase in power level. The two smallest wire electrodes demonstrate the highest relative error. The error of the data corresponding to the blade electrodes and the largest wire electrode is about the same. 6.4.1.2 Lesion Pathology Lesions produced by all types of electrodes at all power settings have common pathological features indicated in Figures 6.5 and 6.6. Three distinct types of damage: melted proteins, vapor bubbles and birefringence loss could be identified. Outer wall of the lesion consists of a melted protein layer, which appears as a dense mass (like the crust on freshly baked bread). The boundary between the melted protein and tissue still preserved some sort of cohesion and is distinctively sharp. Immediately 161 below or, sometimes inside, the melted protein layer numerous vapor bubbles can be observed. With increase of the distance from the outer wall of the lesions these vapor bubbles gradually increase in diameter until they overlap with each other and form a second layer. Relatively seldom, large isolated vapor vacuoles stretched in the general direction of muscle fibers appear beneath the multiple bubble layers. The layer of muscle tissue with lost birefringence overlaps all vapor bubbles and extends beyond them. Figure 6.5 Typical coagulation and vaporization damage obtained in experiment. H&E stain. Damage types of melted proteins, vapor bubble and birefringence loss are visible in all cases. 162 Figure 6.5 demonstrates typical coagulation and vaporization damage obtained in the experiments. Four tissue samples presented there highlight the difference in thermal damage corresponding to output power levels of 40 and 80 Watts and diameters of wire electrodes of 0.45 and 1.1 mm respectively. The zone of birefringence loss is clearly wider for the larger electrode and vapor bubbles are more numerous and larger at the higher power level. Figure 6.6 demonstrates the difference in the vapor bubble size observed in two samples obtained at power levels of 40 and 80 Watts and with the 0.45 mm wire electrode. High magnification of the image, Figure 6.6, enables one to see a sharp boundary between melted proteins and relatively cohesive myofibrils. On the level of individual muscle fibers this boundary is much more distinctive than the boundary of birefringence loss but on larger scale it is still quite irregular. 6.4.1.3 Lesion Factorial Analysis To investigate the influence of scalpel size and ESU power level on extent of thermal damage a factorial analysis of the width of birefringence and water loss zones was conducted. Thermal damage data were organized into data sets corresponding to particular factor: power level or electrode size. Each data point in a set represents the average value of six measurements (three from each side of the cut) of thermal damage to sample. The power levels of ESU of 40, 60, 80 and 100 Watts correspond to the average values one standard deviation of power dissipated in tissue of 26 8, 44 12, 56 17 and 68 22 Watts respectively. All three levels of the wire diameters and two levels of blade sizes were considered. 163 Figure 6.6 Typical vapor bubbles and melted protein thermal damage. Note a relatively sharp boundary of melted protein layer as indicated by white arrows. 164 Figures 6.7 Results of factorial analysis of vaporization thermal damage obtained with cylindrical wire electrodes. Means Diamonds are shown for each data set. Top and bottom endpoints of a diamond indicate 95% confidence interval of the data set. Two horizontal marks inside indicate 95% overlapping level of the data sets. 165 Figures 6.8 Results of factorial analysis of coagulation thermal damage obtained with cylindrical wire electrodes. Means Diamonds are shown for each data set. Top and bottom endpoints of a diamond indicate 95% confidence interval of the data set. Two horizontal marks inside indicate 95% overlapping level of the data sets. 166 Figures 6.7 and 6.8 are the results of factorial analysis of thermal damage obtained with cylindrical wire electrodes. Means Diamonds are shown for each data set. Top and bottom endpoints of a diamond indicate 95% confidence interval of the data set. Two horizontal marks inside indicate 95% overlapping level of the data sets. To be significantly different, two means must not overlap their overlap marks. The data for the blade electrodes demonstrate similar functional behavior. Summary of the two-factor ANOVA study is given in Tables 6.2 and 6.3 (study of width of water loss and birefringence loss zones respectively for cylindrical wire electrodes) and Tables 6.4 and 6.5 (study for the blade electrodes) respectively. Calculated and corresponding critical F-values are given for each null hypothesis as well as conclusion on their comparison. Table 6.2. Evaluation of width of the water loss zone for the wire electrodes F calculated 1.66 14.79 0.17 F critical 3.19 2.79 2.29 Conclusion accepted rejected accepted Hypotheses / Factor H1.10 /scalpel radius H1.20 /ESU power H1.30 / interaction Table 6.3. Evaluation of width of the birefringence loss zone for the wire electrodes F calculated 26.4 0.41 0.29 F critical 3.19 2.79 2.29 Conclusion rejected accepted accepted Hypotheses / Factor H2.10 /scalpel radius H2.20 /ESU power H2.30 / interaction Table 6.4. Evaluation of width of the water loss zone for the blade electrodes F calculated 2.12 18.45 0.12 F critical 2.76 2.12 1.46 Conclusion accepted rejected accepted Hypotheses / Factor H1.10 /scalpel radius H1.20 /ESU power H1.30 / interaction 167 Table 6.5. Evaluation of width of the birefringence loss zone for the blade electrodes F calculated 47.5 0.93 0.18 F critical 2.76 2.12 1.46 Conclusion rejected accepted accepted Hypotheses / Factor H2.10 /scalpel radius H2.20 /ESU power H2.30 / interaction Figures 6.7 and 6.8 as well as Tables from 6.2 to 6.5 demonstrate that the power level of the ESU has a significant effect on the width of the water loss zone and no apparent effect on the width of the birefringence zone. At the same time, the size of the scalpel plays important role in defining the width of the birefringence loss zone and has no noticeable effect on the water loss. Generally, the higher the value of the factor the great extent of thermal damage. Results presented in Tables 6.2 to 6.5 demonstrate that size of the scalpel and ESU power level are independent factors in defining thermal damage. 6.4.2 Numerical Simulation Results The intention of the numerical study was to verify ETR model by comparing absolute values as well as functional behavior of electrosurgical circuit parameters, cutting speed and thermal damage predicted in simulations with those experimentally observed. The calculations were performed for ESU output powers of 25, 40, 55 and 70 Watts and scalpel sizes: wire diameter of 0.45, 0.7 and 1.1 mm and elliptical cylinders of 0.45 x 0.9, 1.35, 1.8 and 2.25 mm respectively. For simplicity reasons the nucleation temperature was set to 107.5 C in all simulations. 168 6.4.2.1 Typical Distributions of Electric and Thermal Parameters Figures 6.9 and 6.10 show typical steady-state distributions of voltage, heat source, temperature and water content under the scalpel for the wire electrode of 0.45 mm diameter and elliptical electrode of 0.45 x 0.9 mm diameters, respectively. The records were made in a steady-state cutting regime just several time steps before the scalpel was through a tissue layer. The power was set at 25 Watts. In both cases points of maximum heat source and temperature were located in front of the electrodes but slightly of the centerline. Wire and elliptical electrodes demonstrate distinctively different distribution patterns of electric and thermal parameters. Heat source, temperature and dehydration zone in the case of the wire electrode are elongated across the cutting direction while in the case of the elliptical electrode the elongation is in the cutting direction. In the case of the elliptical electrode a relatively large uniform area of high temperature was located in front of the scalpel and extends deep into the tissue. Temperature in front of wire electrode drops much faster but the area of high temperature is wider than in the case of elliptical electrode. Figures 6.11 and 6.12 demonstrate heat source a) and electric conductivity b) as a function of the distance from the scalpel s surface for the wire and elliptical electrode, respectively. Distributions in directions 6 o clock, 4:30 and 3 o clock relative to the origin of coordinates are demonstrated. Both heat source and conductivity distributions display a much more regular shape than similar distributions obtained in the simulations described in previous Chapter 5. With increase in the distance from electrode the heat source demonstrates much more gradual decrease than was obtained in three-dimensional simulations. In fact, far away from electrode, the heat source behaves relatively close to the prediction obtained for the Honig model. 169 Figure 6.9 Distributions of heat source, temperature, electric conductivity and water content in XY plain recorded in simulation with circular electrode. The records correspond to the time just before tissue layer was destroyed. The input parameters are ESU output power, PESU = 25 Watts, Nucleation threshold temperature, Tn = 107.5 C, Electrode diameter 0.45 mm. 170 Figure 6.10 Distributions of heat source, temperature, electric conductivity and water content in XY plain recorded in simulation with elliptical electrode. The records correspond to the time just before tissue layer was destroyed. The input parameters are ESU output power, PESU = 25 Watts, Nucleation threshold temperature, Tn = 107.5 C, Electrode diameters 0.45 x 0.9 mm 171 Figure 6.11 Simulated for circular electrode a) heat source and b) electric conductivity as a function of distance from the scalpel s center. Distributions along the cutting plane (6 o clock), across the cutting plane (3 o clock) and 4:30 are shown. 172 Figure 6.12 Simulated for elliptical electrode a) heat source and b) electric conductivity as a function of distance from the scalpel s center. Distributions along the cutting plane (6 o clock), across the cutting plane (3 o clock) and 4:30 are shown. 173 Conductivity profiles demonstrate relatively smooth character without significant sharp edges. The region of high conductivity in the case of elliptical electrode is significantly wider than in the case of wire electrode but conductivity gradient is sharper in latter case. 6.4.2.2 Circuit Parameters and Cutting Speed Macroscopic parameters of electrosurgical circuit obtained in experiment and predicted in the simulations are compared in Table 6.6. Mean values one standard deviation of experimental data are presented. Simulated values were obtained during steady state cutting. In most of the simulations steady-state cutting had commenced already when no more than ten layers of muscle fibers had been destroyed. Average values one standard deviation correspond to cutting trough ten consecutive layers. Simulated and experimentally observed electrosurgical circuit parameters demonstrate surprisingly accurate agreement within experimental standard deviation ranges. Generally, load impedance obtained in the simulations is slightly lower and the voltage and current are respectively higher and lower than experimentally observed values. Simulated load impedance increases with an increase in ESU power while in the experimental case the load is high at low power and, with increase in power it decreases and then stays constant. 174 Table 6.6 Comparisons of electrosurgical circuit parameters obtained in experiment with the predicted by the numerical model. Mean values one standard deviation are shown for the experimental data. Simulated values were obtained in steady-state cutting. The simulated values are the average values one standard deviation obtained during cutting trough ten consecutive layers of muscle fibers. Scalpel diameters: E1 experiment: 0.45 x 2.35 mm, simulation: 0.45 x 2.25 mm; C1: 0.45 mm, C2: 0.7 mm, C3: 1.1 mm. Power, W 26 3 42 6 70 13 36 11 51 18 41 12 70 24 55 12 70 19 Experiment Voltage, Current, V mA 170 10 150 10 190 10 220 20 250 20 280 30 200 30 260 40 180 30 280 50 220 30 230 40 180 30 200 40 230 30 250 40 250 20 300 30 Load, k 1.1 0.1 0.9 0.1 0.9 0.2 1.1 0.4 1.3 0.5 0.8 0.2 1.1 0.4 0.9 0.2 0.8 0.2 Power, W 25 40 70 40 55 40 70 55 70 Simulation Voltage, Current, V mA 140 10 180 10 170 10 230 20 230 20 300 30 190 20 220 20 180 20 240 20 200 20 230 20 210 20 250 20 220 20 280 20 270 20 300 30 Load , k 0.76 0.01 0.76 0.01 0.78 0.01 0.91 0.01 0.91 0.01 0.83 0.01 0.87 0.01 0.76 0.01 0.78 0.01 Scalpel type E1 C1 C2 C3 Cutting speed as a function of surface power density is shown in Figure 6.13. The surface power density is defined as the power dissipated in tissue divided by half of the surface area of electrode. Note, that the surface power density defined in this chapter is different from the interface power density used in Chapter 5 (defined as the ratio of power and contact spot area). Experimental data are presented as points with standard deviation bars. Linear regression fits for the data sets corresponding to the wire and blade electrodes are shown. Correlation coefficients are Rw = 0.9 and Rb = 0.8 for the wire and blade electrode data sets, respectively. High standard deviation shown for the data obtained for the wire electrode at power density of 2x106 W/m2 is likely due to especially irregular muscle sample. Simulated cutting speeds are shown for circular and elliptical electrodes. Data points correspond to values obtained at particular power setting for particular electrode size. Standard deviations are well within data markers and do not exceed a few percent. 175 Figure 6.13 Cutting speed as a function of surface power density. Simulations were performed for all range of power and electrode sizes as indicated. The nucleation temperature was 107.5 C in all cases. Experimental data and corresponding linear regression fit are shown together with simulation results. In all experimental and simulated cases the cutting speed increases with an increase in surface power density. At the same power density level the simulated cutting speed obtained for the elliptical electrodes is higher than that obtained for the circular electrode. Linear regression fit of the experimental data shows that in the case of the blade electrodes (approximated as elliptical cylinders) the cutting speed obtained is lower than for the wire electrodes. However, this observation cannot be conclusive as the difference between the fit results is well within experimental standard deviation. 176 Figure 6.14 Vaporization thermal damage. Depth of the vaporization zone is determined at 50% out of 100% initial water content. 177 Figure 6.15 Coagulation thermal damage. Depth of the birefringence loss zone is determined at > 1 threshold level. 178 6.4.2.3 Thermal Damage Figures 6.14 and 6.15 show dimensions of vaporization and coagulation thermal damage, respectively, obtained in the simulation with different electrosurgical power dissipated in tissue and different electrode sizes. In the case of the elliptical scalpel the smallest diameter was constant and equal 0.45 mm and its largest diameter (length along the cut) was varied. Width of the water loss zone was defined as the distance of approximately 50% water loss measured from Z-axis in Y direction. Width of the birefringence loss zone was defined as the distance from Zaxis to tissue region with thermal damage parameter < 1 measured in Y direction. The dimensions were recorded when, after cutting through 200 layers of 50 m fibers, the electrosurgical power was set to zero and tissue maximum temperature dropped to 50 C. Values of thermal damage were recorded at 5 mm distance from final position of the scalpel back down the cut. For both types of scalpels the vaporization and coagulation thermal damage display similar dependencies on ESU power and scalpel size as were observed in the experiment. Increase in vaporization depth with increase in ESU power is relatively strong while the vaporization depth is decreasing, but not significant with increase in diameter of a scalpel. Increase in coagulation depth with increase in diameter of a scalpel is significant, but with increase in ESU power it is hardly noticeable, though present. Generally, the elliptical scalpel produces a little less vaporization damage than the circular. The coagulation damage produced by the elliptical scalpel is noticeably less than the circular scalpel. At the same time, in the case of the elliptical scalpel the increase in coagulation depth is lower. 179 6.5 Discussion 6.5.1 General Representation of ETR Process by the Numerical Model The two-dimensional model discussed in this chapter was developed with the intention accounting for the effects of scalpel geometry on degree and distribution of electrosurgical damage and minimizing computational time without the loss of spatial resolution. A typical electrosurgical blade can be approximated as an elliptical cylinder with its major and minor axis corresponding to the blade s width and thickness respectively. Therefore, an elliptical grid was introduced to accommodate the geometry of the blade as well as distribution of electric field, temperature and thermal damage possibly defined by the blade s geometry. Grid definition allows for arbitrary setting of the major and minor axis ratio of the blade. Ratio equal one corresponds to a wire electrode. Comparison with the three-dimensional model introduced in previous chapter demonstrates that in the vicinity of the scalpel the two-dimensional model described in this chapter resolves the distributions of heat source, temperature and thermal damage much better. The comparison can be done best using data for tissue electric conductivity, which combines effects from and, at the same time, influences on heat source, temperature and thermal damage. For example, Figures 5.14 a) and b) and 6.12 b) show the distribution of tissue electric conductivity around the scalpel, which were computed using the three- and two-dimensional models, respectively. The 50 m grid step used in the three-dimensional model is obviously too large for satisfactory description of electrical and thermal processes in the vicinity of tissuescalpel interface. Poor spatial resolution results in a sharp conductivity gradient, which, in turn, leads to overestimation of the local electric field, heat source, temperature, vaporization rate and, as an effective positive feed-back loop 180 mechanism, leads to an even sharper conductivity gradient. The 10 m maximum grid step used in the two-dimensional model together with the grid geometry adapted to the scalpel results in a more monotonic electric conductivity distribution and effective elimination of sharp gradients in electric and thermal parameters around the scalpel. An elliptical geometry results in a more accurate prediction of the electric field and, consequently, heat source distributions further away from the scalpel. Figures 5.12 a) and b) and 6.12 a) show the distributions of the heat source in tissue computed using the three- and two-dimensional models respectively. In the case of the elliptical geometry the behavior of the heat source as a function of distance from the electrode s centerline closely resembles the distribution that was analytically defined by Equation (1.1). In the case of the three-dimensional model the heat source demonstrates more rapid drop with increase in the distance from the scalpel s centerline. Such a rapid drop is a likely effect of the rectangular geometry rather than poor resolution because an increase in the three-dimensional grid resolution does not result in a marked approach of the computed heat source to that predicted analytically. It might be expected that a direct consequence of better spatial resolution and more accurate representation of geometry of tissue-scalpel interface would be the two-dimensional model as compared with the three-dimensional predict a general reduction of the vaporization thermal damage. Indeed, elimination of sharp gradients in electric conductivity an artifact of the three-dimensional model likely results in effective reduction of the electric field, temperature and vaporization rate. Another manifestation of higher accuracy of the elliptical two-dimensional model is the smaller relative error in predicted total current, scalpel voltage and tissue impedance, 181 as Tables 5.5 and 6.6 show. Reduction of stabilization noise increases accuracy in the prediction of electrosurgical circuit parameters. From the other side, systematic error introduced in the three-dimensional model out of necessity to interpolate between elliptical and rectangular numerical grids appears to be compensated by a much more fine spatial resolution of both grids. 6.5.2 Experiment and Model Comparison 6.5.2.1 Electrosurgical Circuit Parameters In experiments on electrosurgical cutting of beef skeletal muscle RMS values of electrosurgical current and voltage between the electrosurgical scalpel and return electrode and phase shift between current and voltage were measured directly. From these measurements values of dissipated power and circuit impedance were calculated. In numerical simulations total power dissipated in tissue was set constant and defined as a sum of power dissipated in each grid element, Equation (5.34). Tissue impedance, total current and scalpel voltage were computed from total power as described by Equations (5.35) to (5.37). The difference in parameter computational algorithms reflects in high variation of experimental power and impedance and simulated current and voltage and, respectively, in low variations of experimental current and voltage and simulated power and impedance, as shown in Table 6.6. It should be noted that while in the experiment variations in circuit parameters were caused primarily by tissue nonuniformity, in simulations the source of variations is impedance changes during cutting through a fiber layer. In the calculation of simulated impedance, voltage and current an assumption of zero phase shift was made. An experimentally observed low value of phase shift, Table 6.1, confirms the validity of this assumption. 182 Table 6.6 demonstrates that simulated and experimentally observed electrosurgical current and voltage agree relatively well in absolute value and functional behavior observed with changes in electrosurgical power and scalpel geometry. As was shown in the previous Chapter, the simulated tissue impedance rises slightly with an increase in ESU power and declines with increase in diameter of the scalpel. Such behavior can be explained by the effect of dehydration at the scalpel-tissue interface. Increase in the interface power density stimulates the vaporization process and a resulting effective decrease in the contact area is responsible for the impedance rise. Experimentally observed impedance decreases with increase of the scalpel size, but an impedance increase with increase in dissipated power is not observable, Table 6.1. On the contrary, in all cases impedance at the lowest power setting is the highest. The lack of a marked change in impedance with increase in delivered power is likely due to relative insignificance of the effect compared to experimental error. Indeed, simulations show that in the investigated power range impedance increases about 1%, while experimental error often exceeds 35%. High impedance at the lowest ESU power settings arises probably from the fact that this power level is barely sufficient for the cutting. At these power levels the role of surface vaporization in dehydration of the tissue-scalpel interface area is likely comparable with the role played by interstitial vapor nucleation. Therefore, dehydration of the interface, leading to an impedance increase, develops without marked muscle fiber destruction. 6.5.2.2 Cutting Speed To compare experimental and numerically predicted cutting performance of electrodes of different size and geometry at different ESU power settings a surface power density parameter was introduced. For the purpose of this study surface power 183 density was defined as power dissipated in the tissue divided by half of scalpel area of circumference (area of front portion). Cutting speed obtained in simulations overestimates experimentally observed values in all investigated ranges of surface power density. The complex nature of the ETR process and significant inaccuracy in determination of material properties of the tissue allows several explanations for this discrepancy. In the simulations the cutting speed depends on the ablation model; i.e. on criteria and mechanisms of nucleation, bubble growth and final fiber destruction, and on amount of energy supplied for the ablation process. Overestimation of predicted cutting speed could be caused, for example, by underestimation of nucleation temperature, overestimation of bubble growth rate, overestimation of tissue electric conductivity at high temperatures, neglecting heat loss through mechanical removal of tissue debris in micro explosions associated with interstitial vaporization, etc. Physical limitation of the operator and manual cutting process itself that did not allow maintaining cutting speed above some limit can be another explanation of the discrepancy between experiment and prediction as well. Experimentally observed and simulated cutting speed increases with an increase in surface power density but demonstrated in the previous Chapter the effect of cutting speed saturation at high interface power densities was not observed. There are two likely explanations for this. First, the surface and interface power densities were defined as ESU power divided by the scalpel front half area and contact spot area, respectively. In the later case the contact spot is an arbitrary parameter introduced to analyze the heat source hypothesis. Its area is not an equivalent to the scalpel area, and is generally a lot smaller. Therefore, it is likely that the investigated range of surface power densities did not cover the equivalent range of interface power densities it fell short, and the saturation effect may exist at a higher ESU power or for a smaller scalpel. Second a possible explanation of absence of the cutting speed 184 saturation is reduction of the edge effect caused by tissue dehydration due to the more refined geometry of the two-dimensional model. It was suggested that cutting speed saturation had been caused by increased vaporization from tissue located at the scalpel s sides rather than from under the scalpel. Reduction of the edge effect in the two-dimensional model might reduce vaporization and changed shape of the dehydrated zone so that vaporization from the sides of the cut did not exceed vaporization from under the scalpel. The effect of the electrode type (wire or blade) on the cutting speed is inconclusive. The experimentally observed wire electrode cutting speed slightly exceeds the speed observed for blade electrodes. However, the difference is well within experimental error. On the contrary, simulations demonstrate that an elliptical electrode produces a higher cutting speed than a wire electrode. This difference can be explained from the shape of the dehydrated zone formed around wire and elliptical scalpels as demonstrated on Figures 6.9 and 6.10 respectively. The wire electrode produces a wider dehydration zone than the elliptical scalpel, therefore, more energy is spent on water vaporization from the sides of the cut than for vaporization under the scalpel and, respectively, the cutting speed is reduced. 6.5.2.3 Thermal Damage Accurate prediction of vaporization and coagulation thermal damage is the ultimate goal of modeling ETR process. In the following section the effects of ESU output power and scalpel size on the predicted and experimentally observed outcomes of electrosurgical cutting are compared and discussed. Factorial analysis of experimentally observed thermal damage was conducted and results are presented in Figures 6.7 and 6.8 and in Tables 6.2 to 6.5 respectively. Predicted thermal damage is plotted on Figures 6.14 and 6.15 in a format similar to the used for experimental data presentation. 185 The general agreement in functional behavior between experimental and simulated thermal damage as functions of ESU power and scalpel size is surprisingly good. Figure 6.7, Tables 6.2 and 6.3 and Figure 6.14 demonstrate that in both experimental and simulated cases and for both wire and elliptical blade electrode types vaporization thermal damage does not strongly depend on scalpel size, but increases with increase in ESU power level. On the contrary, Figure 6.8, Tables 6.4 and 6.5 and Figure 6.15 demonstrate that in both cases, and for both electrode, types coagulation thermal damage (defined as the zone of birefringence loss) increases with increase in scalpel size but does not strongly depend on ESU power level. 6.5.2.3.1 Vaporization Damage Simulation results show a steady increase of about 200% in vaporization damage with 100% increase in electrosurgical power. Experimental results show an overall increase in vaporization damage of about 25% with 100% increase in power and vaporization damage appears to be saturated at higher power levels. The increase is likely due to a combination of two factors: overall increase in thermal energy available for vaporization, and rise of maximum temperature in tissue. The effect of an increase in deposited energy naturally leads to an increase in the amount of water vaporized and, since vaporization occurs not only in front of the scalpel but also at its sides, an overall vaporization increase results respectively in increase in cutting speed and width of the vaporized zone. If the local energy deposition rate exceeds the rate of energy removal by vaporization, then the local temperature may rise quite significantly. Figure 3.6 shows that for nucleation temperatures exceeding 100 C by a few degrees such a condition is likely to be satisfied at an energy density higher than 1010 W/m3. Figures 6.9 and 6.10 demonstrate that for both types of electrodes the maximum power densities are clearly above this threshold. Temperatures in the 186 vaporization region that exceed the nucleation threshold provide for more vapor bubbles and faster growth and for an increased heat flux out of the vaporization region. Both effects likely contribute to the increase in the vaporization zone. Apparent saturation of the width of the vaporization zone in experiment with an increase in ESU power is likely a manifestation of the same factors as those responsible for inability to achieve high cutting rate predicted in the simulations. In the simulation, increase in deposited energy causes respective increase in vaporization and the cutting speed automatically increases so that contact conditions at the scalpel-tissue interface remain more or less stable. In the case of experiment, reduction of cutting speed might cause a temporary break in scalpel-tissue electric contact, effective reduction of the deposited energy, and subsequent decrease or reduction of the vaporization zone. In contrast with the ESU power settings, the effect of the size of the scalpel on vaporization damage is remarkably small or non existent at all. Experimental data show no statistically significant change in the vaporization zone dimensions with increase in the scalpel diameter. It is likely that the effective size of the scalpel-tissue contact established under a steady-state cutting regime depends more strongly on total deposited energy than on energy density. Simulation results show that the width of vaporization zone slightly decreases with an increase in scalpel diameter. The drop in the simulated data is relatively small: about 20% reduction for 100 % increase in the scalpel diameter. This is probably caused by a drop in power density at the scalpeltissue interface and subsequent reduction of local deposited energy and maximum tissue temperature. But the drop in the vaporization zone is relatively small and could not been detected experimentally. A similar argument regarding the dependence of electric contact on total deposited energy, rather than on energy density, can be used to explain that neither in 187 experiments nor in simulations is there a clear difference between vaporization thermal damage produced by wire and blade (elliptical) electrode types. Simulations generally underestimate the vaporization zone at low ESU power levels and overestimate it at high levels. Overestimation of the vaporization damage at high power levels can be explained using the same reasons suggested for the explanation of overestimation of the cutting speed: that is underestimation of nucleation temperature, overestimation of bubble growth rate, overestimation of tissue electric conductivity at high temperatures, etc. Underestimation of the vaporization damage at low power levels can be explained by overall underestimation of vaporization efficiency. Apparently actual vaporization, primarily formation and expansion of interstitial vapor bubbles, might occur at lower temperatures and/or proceed at faster rates than was postulated in the model. However, such an explanation is clearly at odds with the explanation given for cutting speed behavior. It is likely that more definite explanation of the vaporization thermal damage and cutting speed behavior can be found with a model accounting for the mechanical effects of vapor bubble formation and growth. 6.5.2.3.2 Coagulation (Birefringence Loss) Damage Simulation results show a steady increase in coagulation damage of about 80% and 50% for wire and elliptical electrodes, respectively, with 100% increase in scalpel diameter. Experiments show 120% increase with 100% increase in scalpel diameter. No difference between the wire and elliptical blade electrodes was detected. Increase in coagulation damage with increase in scalpel size is likely due to effective increase of the tissue zone that is maintained near the vaporization temperature. Increase in the volume of this zone allows more energy to accumulate in the tissue after vaporization ceases or is significantly diminished. This energy is transported to tissue depth by heat diffusion, allowing large tissue volumes to be exposed at 188 coagulation temperature for longer time. Apparent difference in the simulation results obtained in the cases of wire and elliptical electrodes can be explained from the characteristic shapes of high temperature zones demonstrated on Figures 6.9 and 6.10 respectively. For the elliptical scalpel the high temperature zone is elongated in the direction of cutting. Therefore, tissue is continuously destroyed as the scalpel advances. In the case of the wire electrode, the high temperature zone is stretched across the path of the scalpel, exposing deeper layers to high temperature and storing more thermal energy in the tissue. No increase whatsoever in coagulation thermal damage is observed with increase in ESU power level in the experiments and only a slight increase (a few percent) was obtained in the simulations. This is likely due to effective consumption of energy in the vaporization process. Most of the energy dissipated in tissue is spent in the relatively small volume where vaporization is taking place. Increase in the energy dumped into this volume will lead to an increase in vaporization rate; therefore, to an increase in cutting speed and vaporization damage. However, since vaporization temperature is determined by the vaporization mechanism, the temperature in the vaporizing volume will stay close to it. The temperature at the vaporization front will stay constant and equal to the vaporization threshold temperature. Though increase in electrosurgical power shifts the vaporization and, respectively, vaporization temperature fronts deeper into the tissue, this shift is relatively less comparatively with the width of coagulation zone and has no significant effect on final temperature distribution in the bulk of tissue. Generally, coagulation thermal damage obtained in simulation overestimates the values observed in experiment. The most reasonable explanation is overestimation of the temperature in simulations. This overestimation might arise from misrepresentation of the temperature distribution. It is possible that in a real situation, due to better electric contact at the front portion of the scalpel, the heat source and 189 temperature distributions resemble those obtained in the simulation with the elliptical electrode; i.e. more elongated in the cutting direction. In this case the tissue exposed to high temperature is destroyed before heat has a chance to diffuse to deeper layers. Another source of temperature overestimation is underestimation of vaporization efficiency; particularly, overestimation of nucleation threshold temperature. As has been mentioned, deeper analysis involving mechanical events in tissue is required to clarify this problem. 6.6 Conclusion The previous study has demonstrated that a model of the ETR process can be developed in a cylindrical geometry, as had been suggested by Honig. A twodimensional thermo-electric finite difference model of the ETR process was developed. Macroscopic electrical parameters of the electrosurgical circuit, cutting rate and thermal damage were studied in experiment and compared with predictions from the model. The model demonstrated reasonably good agreement with experimentally observed macroscopic parameters of the electrosurgical circuit: cutting rate and thermal damage. Results indicate, in general, that the extent of vaporization damage is strongly dependent on output ESU power level and increases with an increase in power. At the same time, vaporization damage is relatively insensitive to the size and shape of electrosurgical scalpel. On the other hand, the extent of coagulation zone increases with an increase in scalpel diameter, but increases in electrosurgical power results in no apparent effect. Overall, the model predicts a wider coagulation zone for the wire electrode than for elliptical one. This prediction was not confirmed experimentally. The results of this study indicate the need for further development of the model, particularly the necessity to introduce mechanical effects and to clarify the 190 values of thermodynamic parameters associated with vapor bubble formation and expansion. Further refinement of the numerical model might resolve conflicting overestimation of the cutting speed and underestimation of vaporization damage and, at the same time, resolve overestimation of coagulation damage. 191 Chapter 7 Summary and Conclusion The studies in this dissertation were performed in order to increase our understanding of the mechanism of the electrosurgical tissue resection (ETR) process, as well as to develop a numerical model capable of predicting extent and nature of the thermal damage produced in ETR. This chapter summarizes the most significant results of this study and contains suggestions regarding future work. 7.1 Thermal Ablation and Water Vaporization Model The foundation of the numerical study performed in this dissertation is a novel model of tissue thermal ablation based on interstitial vapor nucleation and expansion. It had been pointed out that in tissue a combination of a volumetric heat and convective cooling of the surface (or conductive or evaporative cooling for that matter) might result in a significant subsurface temperature rise, formation of interstitial vapor bubbles and explosive tissue ablation, so called popcorn effect [LeCarpentier, et al., 1993]. Earlier, it had been suggested that almost instantaneous expansion of a vapor bubble formed at 100 C inside a tissue cell might be the mechanism responsible for electrosurgical tissue cutting [Pearce 1986, p. 76]. Numerous observations of histological sections of electrosurgical lesions in skeletal muscles have demonstrated that the sizes of vapor bubbles formed in the immediate proximity of the lesion surface are about the same as the diameter of the muscle fiber, and that some of the bubbles are open up into interior of the lesion. Analytical estimates of the rates of vapor nucleation and bubble growth conducted in this study have indicated that under the typical conditions of an electrosurgical procedure only one or a few vapor bubbles can form inside a cell and expand to the size of the muscle fiber. 192 The thermal ablation model introduced in this dissertation postulates that formation of a single vapor bubble inside a fiber adjacent to the electrosurgical scalpel and its expansion to the diameter of the fiber is sufficient for a fiber to be cut . Subsequent destruction of all muscle fibers in front of the scalpel allows the scalpel to advance forward. In its present form, the model describes interstitial vapor nucleation and bubble growth using homogeneous nucleation theory [Van Cary, 1992] with nucleation temperature treated as a threshold parameter and Rayleigh equation [Rayleigh, 1917]. Besides interstitial vapor formation, considerable attention in this dissertation was given to vaporization from the tissue surface directly into the atmosphere. A coefficient of vaporization from muscle tissue surface was determined experimentally and incorporated into numerical model. It was demonstrated that Snelling s formula [Brutsaert, 1982], developed for steady-state vaporization from an open surface of water and routinely used to predict vaporization from tissue surface overestimates experimentally observed vaporization rate from skeletal muscle [Torres 1994], [Pearce 2000]. 7.2 Implementation of Numerical Models Two thermo-electric numerical models of the ETR process were introduced in the present dissertation with the intention to validate the hypotheses of electrosurgical cutting, to study this mechanism in detail and to attempt numerical prediction of electrosurgical thermal damage. Though most of the contemporary numerical models of tissue thermal treatment by optical [London, et al., 1997], ultrasound [Meaney, et al., 1998] or radiofrequency [Tungjitkusolmun, et al., 2001], [Jain, et al., 1998], [Panescu, et al., 1995] methods had been routinely implemented using a more accurate finite element algorithm, the models in this study were implemented using a finite-difference algorithm. The choice of the finite-difference method was 193 determined, paradoxically, by the expected complex geometry of the ETR process. A system of partial differential equations can be solved most effectively and precisely by a finite element method when the geometry of the mesh reflects the geometry of solution. However, since the present study involves complex vaporization processes in tissue that were never modeled before, the geometry of the electric field, heat source, temperature, etc. could not be predicted beforehand. Therefore, the finite difference method with a high spatial grid resolution appeared a satisfactory choice. Initial simulations were conducted using a three-dimensional rectangular grid with variable steps in the cutting direction and across the cut and a constant step along the scalpel. A simplified representation of the skeletal muscle as a structure of periodic layers was used. Each layer was constructed from identical muscle fibers and cutting of all fibers in a layer was required for its destruction. A tissue-scalpel contact spot replaced the electrosurgical scalpel, and its size and geometry were varied in order to study the two hypotheses of electrosurgical cutting. In the course of the study it was established that a two-dimensional elliptical geometry is sufficient for representation of the ETR process and could be used to study the effects of scalpel geometry on electrosurgical thermal damage. Simulations with an elliptical grid were performed using the same layer representation skeletal muscles. The results demonstrated improved accuracy in spatial resolution of the model and general agreement between the predicted and experimentally observed parameters of electrosurgical circuit and thermal damage. 7.3 Geometry of Scalpel-Tissue Electric Contact One of the intentions of this study was a verification of the hypothesis of electrosurgical tissue cutting suggested by Honig, [Honig, 1975]. This hypothesis assumes that tissue is in perfect electric contact with electrosurgical blade and electric current flowing through whole contact area heats tissue above vaporization threshold. 194 The alternative hypothesis suggested by Pearce [Pearce 1986] states that electric contact between tissue and electrosurgical blade is maintained through electric sparks. Numerical simulations conducted in the course of the study have demonstrated that at steady-state an electric contact between the tissue and electrosurgical electrode can exist only along relatively narrow stripe on the scalpel s front surface. In steady-state cutting, rapid dehydration of the area of the scalpeltissue interface is responsible for effective decrease of the interface area. Subsequent increase in local power density is responsible for the ability to achieve realistically high cutting speeds. Thus, Honig s hypothesis can be described as relatively realistic and modified with respect to effective decrease of electrode-tissue contact area. From the other side, simulations of electric sparks striking randomly along the electrosurgical scalpel demonstrated significant overestimation of electrosurgical circuit impedance. Simulations predicted impedance values on the order of 10 k while experimentally observed values rarely exceeded 1 k . Electrosurgical voltage and current are similarly over- and underestimated. Since the impedance overestimation resulted almost exclusively from a small area of spark-tissue contact area rather than from a dehydration effect, the sparks can be ruled out as a major mechanism of electrosurgical cutting. Overall, since a model based on Honig s hypotheses predicts more realistic parameters of the electrosurgical circuit, it is reasonable to use it as a corner stone for future work. 195 7.4 Parameters of Electrosurgical Circuit Numerical simulations of the ETR process were performed under the assumption of constant output power of the ESU. Macroscopic parameters of electrosurgical circuit total tissue impedance, electrode voltage and total electrosurgical current were calculated in simulations and compared with experimental measurements. Time dependencies of the simulated circuit parameters demonstrated distinctive periodical pattern consistent with periodic pattern adopted for skeletal muscle. Though in experiment a periodic behavior of circuit parameters was not observed (quite naturally, as tissue samples are not homogeneous enough) time average values of experimentally measured impedance, voltage and current agree to a relatively high degree of accuracy with predictions. 7.5 Vaporization and Coagulation Thermal Damage The effects of ESU output power and size and geometry of electrosurgical scalpel on extent of vaporization and coagulation thermal damage zones were studied experimentally and numerically. Simulations and experimental observations indicated that the size of the vaporization zone depends strongly on the level of ESU output power and is independent or weakly dependent on the size of the scalpel. At the same time, the size of the coagulation zone (defined as a birefringence loss zone) depends strongly on the size of the scalpel and independent or weakly dependent on the level of ESU output power. Numerical simulation also indicated that an elliptically shaped scalpel produces less coagulation damage than circular wire scalpel. However, no statistically significant difference in performance of the blade (approximated as an elliptical scalpel) and wire electrodes was observed in experiment. 196 Though numerical simulations have shown relatively good functional agreement with experimental observations, predicted absolute values of the width of the thermal damage zones were not in agreement with experimental observations. The simulations underestimated the width of the vaporization damage zone at low ESU power levels and overestimated it at high levels while the coagulation thermal damage obtained in simulation was overestimated. Such a discrepancy between prediction and experiment arises from a poor understanding and oversimplification of vaporization and thermo-electric processes at the scalpel-tissue interface. Accurate definition of tissue material properties responsible for as well as consideration of mechanical events taking place during water phase change and tissue ablation might undoubtedly result in better correlation between prediction and experiment. 7.6 Future Directions The results obtained in presented study highlight a number of issues that require further investigation before a comprehensive model of ETR can be developed. Several areas that needed to be clarified are listed below in descending order of importance. Interstitial vapor nucleation mechanism. To determine a threshold temperature for vapor nucleation values of interfacial surface tension and concentration of dissolved gases in tissue should be determined with sufficient precision. Experimental confirmation of interstitial vapor nucleation is essential as well. Mechanics of tissue ablation. A more comprehensive description of mechanical events of the growth of a vapor bubble considering tissue elastic behavior should be included in the model. This description must describe a simultaneous growth of multiple bubbles in close proximity to each other. Values of appropriate tissue mechanical properties should be obtained. 197 Surface vaporization. Coefficient of water vaporization from tissue surface should be determined for a non steady-state process in broad range of temperatures. Water diffusion from tissue bulk to the surface and vapor diffusion and convection in interior of electrosurgical lesion should be described and included in model. Finite-element model. Better spatial resolution resulting in a more accurate prediction of thermal damage and more realistic overall description of ETR process can be obtained with finite-element numerical algorithm. A finite-element mesh should be developed with regard to geometry of electrosurgical scalpel and shape of water loss zone. Tissue material properties. Temperature - and water content - dependent thermal conductivity, density and heat capacity should be considered in the model. Temperature dependence of electric conductivity at high temperatures should be confirmed. Dielectric properties should be considered in the case of tissues with high fat content. Birefringence loss rate coefficients should be obtained for skeletal muscle. 7.7 Conclusion The results of this dissertation rejected the hypothesis of electric sparks as the primary mechanism of electrosurgical cutting and confirmed validity of hypothesis of scalpel-tissue mechanical contact with modification that the contact exists only along relatively narrow portion of the scalpel. Although the research has not resulted in comprehensive numerical model capable of prediction of the outcome of the ETR procedure with sufficient accuracy, nevertheless it has pointed out a correlation between procedure parameters and thermal damage. 198 The prime result of this investigation should be initiation of a series of new research in the area of electrosurgical tissue resection. It might prove useful for development of numerical models describing particular electrosurgical procedures, for studying the effects of ESU parameters on treatment outcome and for general improvement of electrosurgical procedures. 199 Appendix: Temperature Dependent Material Properties. Temperature dependencies of water saturation pressure, phase change enthalpy and surface tension were approximated from tabulated data of water thermophysical properties [Incropera and de Witt, 1990, p. A22] by a fifth order polynomial: X (T ) = 5 C nT n , (A.1) n =0 where X is saturation pressure in N/m2, phase change enthalpy in J/g or surface tension in N/m, T is absolute temperature and Cn are coefficients of the corresponding polynomial. Approximation was performed in temperature interval from 290 to 450 K (from 17 to 177 C). Polynomial coefficients are given in Table A.1. Table A.1. Coefficients for 5th order polynomial fit of temperature dependant model parameters. The temperature is in K. C0 -1.203 x10-3 0.037 -3.589 x10-10 C1 -0.144 4.476 4.239 x10-8 C2 -10.77 335 3.213 x10-6 C3 0.118 -2.339 6.407 x10-8 C4 -4.338 x10-4 6.019 x10-3 2.689 x10-10 C5 5.341 x10-7 5.478 x10-6 3.332 x10-13 Saturation Pressure, Psat (N/m2) Phase changes enthalpy, Hfg (J/g) Water surface tension, (N/m) Figures A.1 a), A.2 a) and A.3 a) show tabulated data and polynomial fit of water saturation pressure, phase change enthalpy and surface tension. Figures A.1 b), A.2 b) and A.3 b) show relative errors of the polynomial fits, respectively. In the case of the saturation pressure at the temperatures below 40 C the error is higher than one percent but significantly below ten percent. In all other cases the errors are below one percent. 200 a) Figure A.1. b) a) Water saturation pressure as a function of temperature. Data points correspond to tabulated values, solid line represent a fifth order polynomial fit. b) Relative error of polynomial approximation. b) Figure A.2. b) a) Water phase change enthalpy as a function of temperature. Data points correspond to tabulated values, solid line represent a fifth order polynomial fit. b) Relative error of polynomial approximation, 201 c) Figure A.3. b) a) Water phase change enthalpy as a function of temperature. Data points correspond to tabulated values, solid line represent a fifth order polynomial fit. b) Relative error of polynomial approximation. 202 Bibliography Aronson, J. 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Thomsen, S., Pearce, J.A., Cheong, W. F., Changes in birefringence as markers of thermal damage in tissues, IEEE Transactions on Biomedical Engineering, v. 36, n. 12, pp. 1174-1179, 1989. 209 Tsubota, K. and Yamada, M., Tear Evaporation from the Ocular Surface, Investigative Ophthalmology & Visual Science, v. 33, n. 10, pp. 2942-50, 1992. Tungjitkusolmun, S., Vorperian, V. R., Bhavaraju, N.. Cao, H., Tsai, J.Z., Webster, J. G., Guidelines for predicting lesion size at common endocardial locations during radio-frequency ablation, IEEE Transactions on Biomedical Engineering, v. 48, n. 2, pp.194-201, 2001. Van Leeuwen, Ton G., Jansen, E. D., Motamedi, M., Excimer laser ablation of soft tissue: a study of the content of rapidly expanding and collapsing bubbles, IEEE Journal of Quantum Electronics, v. 30, pp. 1339-45, 1995. Van Liew, H, D., Similation of the dynamics of decompression sickness bubbles and the generation of new bubbles, Undersea Biomedical Research, v. 18, n. 4, pp. 333346, 1991. Welch, A J., Gemert, M. J. C., Optical-thermal response of laser-irradiated tissue, New York, 1995 Wicker, P,. Electrosurgery--Part I. The history of diathermy, NATNEWS, v. 27, n. pp. 6-7, 1990. Wolf, J. S. Jr., Rayala, H. J., Humphrey, P. A., Clayman, R. V., In vivo comparison of electrosurgical vaporization electrodes, Journal of Endourology, v. 11, n. 1, pp. 83-87, 1997. Wrobel, L.C., Computational modelling of free and moving boundary problems, Southampton, Boston, Computational Mechanics Publications, 1995. 210 Wurzer, H. at al, Optimization of tissue interaction in electrosurgery, Proc. SPIE, v. 2624, pp. 276-282, 1996 Wurzer, H., Maeckel, R., Lademann, J., Audring, H., Liess, H. D., A spark counter as a control unit of a radio frequency surgery device, IEEE Transactions on Biomedical Engineering, v. 44 n. 9 pp. 831-8, 1997. Zweig, A. D., Weber, H.P., Mechanical and Thermal Parameters in Pulsed Laser Cutting of Tissue, IEEE Journal of Quantum Electronics, v. 23, no 10, p. 1787-1793, 1987. 211 Vita Dmitriy Evgenievich Protsenko was born in Moscow, Russia on June 13, 1970, the son of Evgeniy and Emma Protsenko. He completed primary education in Moscow, Russia, and graduated from Physics and Math High School #542 in 1986. Dmitriy received a Diploma of Engineer-Physicist with specialization in Laser Physics from Moscow Engineering and Physics Institute, Russia in 1993. During the following year he was employed as a Junior Research Associate at General Physics Institute of Russian Academy of Sciences in Moscow, Russia. In 1994 he entered the University of Texas at Austin to pursue a Ph.D. in the Biomedical Engineering Program. His research advisors were Dr. John Pearce and Dr. Massoud Motamedi and his area of specialization was electrosurgery. Permanent Address: 6-2 Proletarskiy Prospekt, #344 Moscow, Russia 115522 This dissertation was typed by the author. 212

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