Chapter D1.1, Problem 56E

Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is

$\frac{dM}{dt}=-rM\left(t\right)\ln \left(\frac{M\left(t\right)}{K}\right),M\left(0\right)=M_0,$

where r and K are positive constants and 0 < M₀ < K.

a.    Show by substitution that the solution of the initial value problem is

$M\left(t\right)=K{\left(\frac{M_0}{K}\right)}^{\exp \left(-rt\right)}.$

b.    Graph the solution for M₀ = 100 and r = 0.05.

c.    Using the graph in part (b), estimate $\underset{t\to \infty }{\lim }M\left(t\right)$, the limiting size of the tumor.