Page 109 - Control Theory in Biomedical Engineering

Modeling and optimal control of cancer-immune system 95

Fig. 2 shows the impact of chemotherapy treatments (with optimal con-
trol) when we choose the parameter values in an unstable region (σ ¼ 0.2,
ρ ¼ 0.2, and τ ¼ 1.5). The tumor cell population is growing over time in the
absence of chemotherapy, while the presence of treatment helps the IS to
keep the growth of the tumor cells under its control. The figure shows that
the optimal treatment strategies reduce the tumor cell load and increase the
effector cells after a few days of therapy. The numerical simulations show the
rationality of the model presented, which in some degree meets the natural
facts.
4.1 Numerical algorithm

Direct and indirect approaches are usually used to solve the OCPs. In the
direct approach, the OCP is transformed into a nonlinear programming
problem. The algorithm of solving the above OCP is roughly based on
the following steps:
Step 1. Provide the initial guess for the control parameters v 0 and w 0 over
the interval, and declare the parameters.
Step 2. Set the initial conditions for the state variables y 0 (t) with the stored
values of v 0 and u 0 and solve the state system forward in time, using
any DDEs solver.
Step 3. Use the transversality condition λ(t f ) ¼ 0, the stored values v 0 and u 0
and x(t) and solve the adjoint system backward in time.
Step 4. Check the control by entering the new values of the state and the
adjoint state into the characterization of the optimal control.
Step 5. Verify for convergence. If values of the variables in this iteration
and the latest iteration are not negligibly small, output the current
values as solutions. If values are not small, return to Step 2.

5 Conclusion
In this chapter, we provided a simple mathematical model with time-lags
and optimal control variables to describe the dynamics of tumor-immune
interactions in presence of chemotherapy treatments. Control variables
are introduced into the originally uncontrolled model and considered L 2
type objective functional to maximize the concentration of effector cells
and minimize the tumor cells with minimal side effects of the chemotherapy.
We showed that an optimal control exists for this problem. We derived the
necessary optimality conditions as a Pontryagin-type minimum principle.
We estimated the optimality system to determine the optimal control
situation (i.e., the drug strategy), and predict the evolution of the tumor