Optimal control techniques are of great use in developing optimal strategies for chemotherapy [173]. Specifically, a treatment regimen, cast as the control, which will minimize the tumor density and drug side-effects over a given time interval, can be found. This technique was employed by Fister and Panetta in [61]. There, the tumor is assumed to have Gompertzian growth. Several models of chemotherapeutic kill-cell (killing of tumor cells) exist. Three different models are treated in [61]. Here, we examine only one, namely, Skipper’s log-kill hypothesis, which states cell-kill due to chemotherapeutic drugs is proportional to tumor population. Thus, if N(t) is the normalized density of the tumor at time t, we have the model

N ′(t) = rN(t) ln (

1 N(t)

) − u(t)δN(t),

where r is the growth rate of the tumor, δ is the magnitude of the dose, and u(t) describes the pharmacokinetics of the drug, i.e., u(t) = 0 implies no drug effect and u(t) > 0 is the strength of the drug effect. The initial condition is taken to be N(0) = N0, where 0 < N0 < 1, as the tumor cells have been normalized. The objective functional used is quadratic, where the cost of the control, representing possible side-effects, and the tumor density N are minimized over a time interval. Finally, we require u(t) ≥ 0 for all t. So, our problem is

min u

aN(t)2 + u(t)2 dt

subject to N ′(t) = rN(t) ln (

1 N(t)

) − u(t)δN(t), N(0) = N0,

u(t) ≥ 0. Here, a is a positive weight parameter. Enter MATLAB and begin lab5.

First, try the values