Math 4600, Homework 13
1. The diffusion-limited growth model.
a)Find integration constants and plot steady state solution for the nutrient
concentration as a function of distance from the center of the tumor. Clearly
mark on your graph the boundary of the necrotic core. Also from you graph,
find the thickness of the proliferating cells shell. Use c2 = 2, c1 = 1, r2 = 2,
k = 1 and D = 0.5.
b) Now use the same parameters, but r2 = 10. Does the thickness of the
proliferating layer match the large size approximation we found in class?
c) Predict (without much computation) the size of necrotic core for r2 = 19.
2. In the non-dimensionalized version of the G1-checkpoint crossing model
plot phase planes (including nullclines) for small and large α and use them to
explain the mechanism of checkpoint crossing by the cell. You can use parameter
vales κs = 1, κi = 0.1, κa = 0.001, λ = 37, µ = 4.
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