Chapter 1, Problem 16P

Cancer cells grow exponentially with a doubling time of 20 h when they have an unlimited nutrient supply. However, as the cells start to form a solid spherical tumor without a blood supply, growth at the center of the tumor becomes limited, and eventually cells start to die.

(a) Exponential growth of cell number N can be expressed as shown, where $\mu$ is the growth rate of the cells. For cancer cells, find the value of $\mu$.

$\frac{dN}{dt}=\mu N$

(b) Write an equation that will describe the rate of change of tumor volume during exponential growth given that the diameter of an individual cell is 20 microns.

(c) After a particular type of tumor exceeds 500 microns in diameter, the cells at the center of the tumor die (but continue to take up space in the tumor). Determine how long it will take for the tumor to exceed this critical size.