Unformatted text preview: BE104 Lab 6a The purpose of this lab is to model the reversible binding of antibodies to CMV
infected vascular endothelial cells lining a blood vessel. We want to answer two questions with this model: (1) if the blood vessel begins with no antibodies and the incoming blood brings with it a flood of antibodies , how long does it take before antibody binding to the infected cells reaches a steady
state, and (2) what concentration of antibody is bound to the surface at steay
state? During this lab you should focus on learning the following lessons: • Solving unsteady problems – how to solve transient (time
dependent) unsteady problems with COMSOL. • Avoiding overlapping geometries – if multiple shapes overlap in space, COMSOL is going to end up generating frustrating errors instead of solutions. When drawing complex shapes, always be sure that they do not overlap. • Avoiding Navier
Stokes – when flow is steady and fully
developed through a simple tube or channel, an appropriate flow profile can be assigned to the geometry instead of having to add and solve for N
S. • Reversible surface reactions – COMSOL is capable of modeling a reversible reaction at a surface by using a physics called “Weak Form, Boundary” to link the reaction at a boundary to the fluid in contact with it. Modeling antibody binding to the lining of a blood vessel using COMSOL: (1) Open COMSOL. The Model Navigator window will open. (2) Select Space Dimension > 2D and click OK. (3) Draw an ellipse of any size using the Ellipse/Circle (Centered) button. (4) (5) (6) (7) Double
click on your ellipse and adjust the properties to produce a circle that is 2.0 cm in radius (2.0e
2 m), centered at (0,0). Extrude the circle to make a tube that is 40 cm long. Go back to Geom1 and Extrude the circle again to make another tube that is 4.0 cm long. You should now be in Geom2 displaying the tube. Go to Multiphysics > Model Navigator. Add Application Modes > COMSOL Multiphysics > Convection and Diffusion > Convection and Diffusion > Transient analysis. Now select Application Modes > COMSOL Multiphysics > PDE Modes > Weak Form, Boundary, type “Cs” into the dependent variables box, and then click OK. Cs is going to be our concentration of antibody at the infected surface, and we’re going to use this PDE mode to couple what’s going on at the surface with what’s going on in the fluid. The smaller tube needs to be moved – it represents the part of the vessel that is infected. Select the smaller tube, copy it, and then paste it (with a z
displacement of 0.1). Delete the original smaller tube. Unfortunately, you now have two tubes that overlap, which is going to give you trouble later when solving. Select the smaller tube, copy it, and then paste it (with no displacements). (8) (9) (10) Go to Draw > Create Composite Object. Enter EXT1
EXT3 (the large tube minus the copied and pasted smaller tube). Be sure that Keep Interior Boundaries is checked and click OK. Now you have no overlap between the uninfected and infected vessel subdomains. (11) Go to Options > Constants and enter the following: Dab kr kf atotal vmax Cin 6e
11 0.02 1e8 1e12 0.25 1e
7 (12) Go to Options > Expressions > Boundary Expressions and place the following expression into every boundary that makes up the infected part of the vessel wall: binding, kf*c*(atotal
Cs)
kr*Cs. (13) Select Convection and Diffusion and under Physics > Boundary Settings set the diffusion coefficient to Dab for all subdomains. Convection will only occur in the z
direction, since there is no flow in the r
direction. We’re going to assign fully
developed, parabolic flow to the entire vessel without solving for Navier
Stokes. Theoretically, we know that a straight tube will have a parabolic flow profile. Input the equation above as the z
velocity in all subdomains. Lastly, click the Artificial Diffusion button. Check Isotropic Diffusion and leave the tuning parameter at the default value of 0.5. (14) Go to Physics > Boundary Settings and assign a concentration of Cin to the inlet at the origin, convective flux to the outlet, insulation to the uninfected parts of the wall, and a flux equal to “
binding” at the infected parts of the wall. (15) Select Weak Form, Boundary and go to Physics > Boundary Settings. Select the infected parts of the wall and copy the following into the Weak box: “Dab*(
test(CsTx)*CsTx
test(CsTy)*CsTy)+test(Cs)*(binding
Cst)” This equation will ensure that what’s lost by the fluid subdomain matches what’s bound to the infected surface. For all other uninfected parts of the wall, uncheck Active in this domain. (16) Press the Initialize Mesh button. button, then the Refine Mesh (17) Go to Solver Parameters. Choose Analysis: Transient and enter the 0:1:20. This tells the solver we want a transient solution showing what happens from t = 0 seconds to t = 20 seconds, saving snapshots at 1.0 s intervals. Enter 1e
4 for the Relative tolerance and 1e
10 for the Absolute tolerance. Click OK. (18) Before we solve for convection and diffusion, we want to tell COMSOL to keep track of the antibody bound to the infected vessel wall. Go to Postprocessing > Probe Plot Parameters. Create a new probe plot called “Bound Antibody” (or whatever you prefer) Choose Plot type: Integration and Domain type: Boundary in the dialogue box. Select the infected parts of the wall under Boundary selection and enter “Cs” under expression. (19) Click Solve. Note the automatically generated antibody binding plot. (20) Has the antibody binding reached steady
state in 20 seconds? If not, adjust the solver parameters and resolve to see how long the system takes to reach steady
state. Is the steady
state concentration of antibody bound what you would expect it to be, given the kinetic parameters that you used? ...
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 Spring '11
 TERRY
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