pb K bp K e The determinant is det A K bp K e The discriminant is D K pb K bp K

Pb k bp k e the determinant is det a k bp k e the

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pb + K bp + K e ) < 0 The determinant is det | A | = K bp K e > 0 The discriminant is D = ( K pb + K bp + K e ) 2 - 4 K bp K e > 0 These facts prove the eigenvalues are negative and real Since λ 1 < λ 2 < 0, this model has a stable node at the origin Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (15/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example Pharmokinetic Problem 4 Eigenvalues satisfy det - ( K pb + K e ) - λ K bp K pb - K bp - λ = 0 , which gives the characteristic equation λ 2 + ( K pb + K bp + K e ) λ + K bp K e = 0 so λ = 0 . 5 - ( K pb + K bp + K e ) ± q ( K pb + K bp + K e ) 2 - 4 K bp K e This produces the negative, real eigenvalues This model has a stable node at the origin Want to find parameters to fit data Data often only from the Plasma compartment Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (16/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example LSD Example 1 LSD Example: In the early 1960’s 5 healthy male subjects were given LSD (lysergic acid diethylamide) in an experiment to determine its effect on brain function 1 Below is a table averaging the data over the 5 subjects Time (hr) 0.0833 0.25 0.5 1 2 4 8 Plasma (ng/ml) 9.54 7.24 6.44 5.38 4.18 2.825 1 Score (%) 68.6 44.6 29 33.2 38.4 58.8 79.4 Want to fit our Drug Model to these data Have information on Plasma compartment , but must infer levels in Brain compartment Examine correlation between LSD levels and Test performance 1 Aghajanian, G. K. and O. H. L. Bing. 1964. Persistence of lysergic acid diethylamide in the plasma of human subjects . Clinical Pharmacology and Therapeutics. 5 : 611-614. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (17/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example LSD Example 2 LSD Model: From before we have the model ˙ d 1 ˙ d 2 = - ( K pb + K e ) K bp K pb - K bp d 1 d 2 Can only directly fit solution d 1 ( t ) to the plasma data Modify interpretation of model so d 1 and d 2 are masses in their respective compartments Perform a nonlinear least squares fit of d 1 (0) and the kinetic parameters, K pb , K e , and K bp to the LSD plasma data Graph solution and compare to the data for the test scores Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (18/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example LSD Example 3 MatLab Code for finding best parameters Though this linear model could be solved, we’ll fit the numerical solution to the data 1 function Lp = LSD(t,L,Kpb,Kbp,Ke) 2
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