Tumor Growth
The Gompertz growth curve is used to model the growth of tumors. It is assumed that tumor growth starts slow, then speeds up and then slows again as the tumor approaches some maximum size. If the size of the tumor is L(t), the tumor growth rate varies with tumor size. The tumor growth rate is given by a function of L
r(L) = aL ln(K/L)
where a and K are positive constants that are different for different tumor types.
1. Find all values of L for which the rate of change given by the differential equation
dL/dt = r(L)
equals 0.
2. Which of the above values is likely to represent a limiting value for L(t)? Are both equally likely? Explain your answer using a plot of r(L) versus L for L > 0.
3. Show that r(L) is positive for small values of L, and negative for large values of L. At what value of L does r(L) switch from being negative to positive?
4. Verify that the function L(t) = Ke^-ce^-at solves the given differential equation.
5. What is the role played by the constant K?
6. Letting the initial size of the tumor equal L0 > 0, find an expression for the c in terms of K and L0.
7. Setting K = 100, Use your answer to question 6 to plot c as a function of L0. Interpret the meaning of the plot from the point of view of treating the tumor in terms of its initial size L0.
8. Setting L0 = 100, Use your answer to question 6 to plot c as a function of K. Interpret the meaning of the plot from the point of view of treating the tumor in terms of the constant K. (Recall that c relates the rate at which the tumor grows or shrinks)
9. Plot solutions L(t) for L0 > K and L0 < K using question 3, by choosing at least 3 initial values L0 on either side of K. You may assume K = 100, a = 2. What conclusions can you draw about the development of the tumor with time, and its relation to the initial size of the tumor?