Page 102 - Control Theory in Biomedical Engineering

Modeling and optimal control of cancer-immune system 89

ρ
Corollary 1. If μ, then the solutions (E, T) for model (2) are nonneg-
η + T
ative for any nonnegative initial condition. However, if ρ < μ, then there exist
η + T
nonnegative initial conditions such that E(t) becomes negative in a finite time interval.

3 Model with chemotherapy and control
We aim to design an efficient treatment protocol, where we employ the
tools of optimal control theory. The formulation as an OCP allows us to:
(i) investigate the dynamical system of interacting cell populations being
affected by the treatments;
(ii) optimize the application of the control such that the quantity of the
treatments is optimized; and
(iii) minimize the tumor size at some of end-time.
This demonstrates how immunotherapy and chemotherapy might be com-
binedformoreeffectivetreatmentandtoprotectthepatientfromopportunistic
infection, as well as fighting the cancer itself. Unlike chemotherapy, immuno-
therapy does not kill tumor cells directly, but it activates and stimulates the
growth of immune cells, most importantly T cells, and NK cells, which are
capableofdestroyingcancercellsdirectly.Therefore,themaingoalofcombin-
ing immuno-chemotherapy treatment is to eradicate the tumor cells, with
minimum side effect, while maintaining adequate amounts of healthy tissues.
To include external chemotherapy in model (2), we should consider
extra two variables namely amount of chemotherapy u(t) and normal cells
N(t) with two control variables v(t) and w(t) (see Fig. 1). We also assume
a homogeneity of the tumor cells and asynchronous tumor-drug interaction
(Sinek et al., 2004). The modified model is

dEðtÞ ρEðt τÞTðt τÞ
¼ wðtÞσ + μEðt τÞTðt τÞ δEðtÞ F 1 ðuÞEðtÞ,
dt η + Tðt τÞ
dTðtÞ
¼ r 2 TðtÞð1 β TðtÞÞ nEðtÞTðtÞ c 1 NðtÞTðtÞ F 2 ðuÞTðtÞ,
1
dt
dNðtÞ
¼ r 3 NðtÞð1 β NðtÞÞ c 2 TðtÞNðtÞ F 3 ðuÞNðtÞ,
2
dt
duðtÞ
¼ vðtÞ d 1 uðtÞ: ð3Þ
dt
The drug kills all types of cells, with different killing rates for each type of
u
cell: F i (u) ¼ a i (1 e ) is the fraction cell kill for a given amount of drug,