Page 99 - Control Theory in Biomedical Engineering

86 Control theory in biomedical engineering

bacterial infections and cancerous cells. Several mathematical models have
been suggested to describe the interactions of tumors and the IS over time
(see, e.g., the research papers; Araujo and McElwain, 2004; Bellomo et al.,
2008; Chaplain, 2008; Nagy, 2005; Roose et al., 2007). Most of these papers
describe the interactions between tumor cells and immune cells, tumor cells,
and normal cells alone (Rihan et al., 2012), or consider the interactions of
tumor-IS with chemotherapy treatment (de Pillis and Radunskaya, 2003;
Swan, 1985). In this chapter, we provide a mathematical model of
tumor-immune interactions in presence of chemotherapy treatment and
optimal control variables. The control variables are incorporated to justify
the best strategy of treatment and minimize side effects of the external treat-
ment by reducing the production of new tumor cells while keeping the
number of normal cells above the average of its carrying capacity.
This chapter is organized as follows. In Section 2, we provide a simple
mathematical model with time-delay (time-lag) to represent the interaction
of the IS with tumor cells. Boundedness and nonnegativity of the model
solutions are also discussed. In Section 3, we extend the model to include
control variables of chemotherapy treatment. Existence of optimal controls
are also investigated. Section 4 presents numerical simulations and discussion
to show the effectiveness of the theoretical results.

2 Mathematical models
Mathematical models provide biologists and clinicians with the tools that may
guideeffortstoclarifyfundamentalmechanismsofcancerprogressandimprove
current strategies to stimulate the development of new ones. We first present a
simple model that describes the dynamics of tumor cells, T(t), and activated
effector cells, E(t), such as cytotoxic T cells. The model takes the form

dEðtÞ
¼ σ + FðEðtÞ,TðtÞÞ μEðtÞTðtÞ δEðtÞ, (1a)
dt

dTðtÞ
¼ αTðtÞ 1 βTðtÞÞ nEðtÞTðtÞ, (1b)
ð
dt
with initial conditions: E(0) ¼ E 0 , T(0) ¼ T 0 . Of course, the interaction
between the effector and tumor cells leads to a reduction in the size of
both populations with different rates, which are expressed by μE(t)T(t)
and nE(t)T(t), respectively. As a result of this interaction, the immune
effector cells decrease the population of tumor cells at rate n. Meanwhile,