ExercisesSolutions

B solve the differential equation in part 3a and

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(b) Solve the differential equation in part 3a, and compare the result to exponential tumor growth by plotting solution curves for the two types of growth, using the same initial conditions. (c) Add a growth-limiting term to the differential equation in part 3a, analagous to the overcrowding term in the logistic growth equation. Again, compare numerical simulations of the two systems for, using the same initial conditions, the same proliferation rate, and the same initial conditions. Solution: (a) Since the only proliferating cells are those on the surface of the sphere, and that the number of cells T is proportional to the volume V of the sphere, we find T = kV , where k is a constant. Now since the number of proliferating cells, P is proportional to the surface area of the sphere, we get P = k 2 4 πR ( t ) 2 where k 2 is a constant and R ( t ) is the radius of the sphere as a function of time, t . Recalling that the volume of a sphere is given by V = 4 3 πR ( t ) 3 , rearrange for R ( t ) and substitute into proliferating cells equation to find that P = k 2 4 π ( 3 4 π V ) ( 2 3 ). The differential equation for T ( t ), assuming that only the cells on the surface of the sphere proliferate, can now be written as dT dt = kT 2 3 . (b) Solving the differential equation in part 3a, we find that T ( t ) = parenleftbigg kt 3 + C 1 parenrightbigg 3 where C 1 is a constant of integration. Figure 8 plots this solution (thin line) and compares it with the solution curve for the exponential tumor growth (thick line), with k = 1. (c) The addition of a growth limiting term to the differential equation in part 3a, analogous to the overcrowding term in the logistic growth equation, finds that the differential equation now becomes dT dt = kT 2 3 (1 - bT ) . Figure 9 plots the numerical solution to this equation (thin line) and compares it to the solution curve of the logistic growth equation (thick line), with k = 1 and b = 1 100 . 7
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0 20 40 60 80 100 1 2 3 4 5 6 7 t Figure 8: The solution curves for Exercise 3b. 0 20 40 60 80 100 10 20 30 40 50 60 70 t Figure 9: The solution curves for Exercise 3c. 8
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4. Purpose: Compare the tumor-immune model using Von Bertalanffy growth to the one presented in class using a qualitative analysis. See Exercise 3 Exercise: Use the Von Bertalanffy model of tumor growth in the absence of an immune response: dT dt = aT 2 / 3 (1 - bT ) . Add the immune population, with competition and response terms identical to those used in the model presented in class. Draw nullclines, determine the number of possible equilibria and their stability. Does the qualitative behavior differ from that of the model presented in class? If so, how? Recall the system of equations presented in class: T denotes tumor cells, and E denotes effector (cytotoxic immune) cells. The equations are dE dt = s + pET g + T - dE - mET dT dt = aT (1 - bT ) - cET Solution: The Von Bertalanffy model of tumor growth in the presence of an immune population, competi- tion and response terms identical to the those used in class finds the system of equations to be dE dt = s + pET g + T - dE - mET dT dt = aT 2 / 3 (1 - bT ) - cET where s, p, g, d, m, a, b, c are constants, T
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