- 1 Problem Statement Physiological Systems Are Often Modeled By Dividing Them Into Distinct Functional Units Or Compartm 1 (100.15 KiB) Viewed 18 times
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DIFFERENTIAL EQUATIONS
An Introduction to Modern Methods and Applications
T H I R D E D I T I O N
James R. Brannan
Clemson University
William E. Boyce
Rensselaer Polytechnic Institute
with contributions by
Mark A. McKibben
West Chester University
1. Problem Statement Physiological systems are often modeled by dividing them into distinct functional units or compartments. A simple two-compartment model used to describe the evolution in time of a single intravenous drug dose (or a chemical tracer) is shown in Figure. The central compartment, consisting of blood and extracellular water, is rapidly diffused with the drug. 121 Central compartment Tissue compartment x2 k12 kou Figure: A Two-Compartment Open Model of a Physiological System. The second compartment, known as the tissue compartment, contains tissues that equilibriate more slowly with the drug. If x1 is the concentration of drug in the blood and X2 is its concentration in the tissue, the compartment model is described by the following system: x = -(ko1 + k21)x2 +k12X2 (1) x = k2131 – kızXz or x' = Kx. [You are required to write matrix form for (1) and that will be your Eq. (2)] x' = kx (2) Here, the rate constant kzi is the fraction per unit time of drug in the blood compartment transferred to the tissue compartment; kız is the fraction per unit time of drug in the tissue compartment transferred to the blood; and koi is the fraction per unit time of drug eliminated from the system. In this project, we illustrate a method for estimating the rate constants by using time dependent measurements of concentrations to estimate the eigenvalues and eigenvectors of the rate matrix K in Eq. (2) from which estimates of all rate constants can be computed. 2. To Address You are required to address the following questions from both cases: 11
Assume that all the rate constants in Eq. (1) are positive. a. Show that the eigenvalues of the matrix K are real, distinct, and negative. Hint: Show that the discriminant of the characteristic polynomial of K is positive. b. If 14 and 12 are the eigenvalues of K, show that 1+2 = -(ko1 +k12 + kzı) and Apdz = kızkou- c. Denote the eigenvectors of d, and az by v=C). V = respectively. The solution of Eq. (1) can be expressed as x(t) = aedity,+Beiztva, (i) where a and B, assumed to be nonzero, depend on initial conditions. Compute the eigenvectors corresponding to the found eigenvalues. d. Assume that the eigenvalues i, and iz and corresponding eigenvectors v, and v2 of K are known. Show that the entries of the matrix K must satisfy the following systems of equations: (ii) V11 \u21 V21 Ku V22/K 12 (2111 = and V11 V21 (K2 V21 (x 022(K22 (iii) 9= V21 0 or, using matrix notation, KV = V8, where v=C3972). =) 0 After addressing the given scenario's analytically. You are required to: a. Solve the problem using MATLAB. b. Provide graphs of solutions from MATLAB on a single plot. c. Compare your analytical and MATLAB solution. 3. Program Requirement • On every run, the program must display name of your software house, and your team along with student ids. • At this stage, a message should be displayed to press any key to continue. • On press of a key, the program must ask the user whether they want to provide initial/boundary values, or the program should run with default values. Default values are provided in the problem statement. • After the input from user, the program must solve the problem and display the results. • At this stage, the program must ask the user if they wish to run another query or terminate the program. Based on user input, program must act accordingly.