Math 4600, Homework 12
1. Derive diffusion equation for the spherically symmetric case (considered
in class) by performing the change of variables.
2. 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.
3. Growth inhibitor model. Find (by making c(r) plots for different R
values) R∗ (the maximum growth radius) of the spherical piece of tissue with
sensitivity level c1 = 0.2. How high the sensitivity level needs to be for the
tissue to continue to grow endlessly? Use β = 1, λ = 0.8, µ = 1.6, D = 1.
4. Suggest a combined model of spherical cancer growth with both immune
system interaction and growth inhibitor. This porblem does not have a unique
solution. Make sure you explain and justify every term in your model.
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