Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 141

Here, F is a function that represents the carrying capacity of the tumor, a concept
adapted from ecology, in that it represents that ability of the environment to sustain
the tumor. The Greek letters are constant parameters of the model. For instance,
δ

represents the rate of natural death of T-cells. For our purposes the exact model
structure is not important, and details can be found in [ 1 ]. For instance, the term
− β

x 2 represents the observation that tumors suppress the activity of the immune
system.
The control objective here is to reduce the tumor volume through the use of a
chemotherapeutic agent. We will refer to this as the “treatment.” We implement this
control with a term

represents the control input, namely the
amount of the chemotherapeutic agent administered. The factor x takes account of the
assumption that the effect of chemotherapy on the tumor is proportional to the tumor
volume. Thus, the first equation becomes

− xu , where u = u ( t )

dx / dt = μ C F ( x ) − γ xy − κ xu .

represents the carrying capacity of the tumor, as above. The parameter
μ C captures the growth rate of the tumor, with dimension cells / day . The parameter

Here, F ( x )

γ
denotes the rate at which cancer cells are eliminated as a result of activity by T-cells, so
that the term xy in the equation models the beneficial effect of the immune reaction

on the tumor volume. Finally, we assume that the elimination terms are proportional
to the tumor volume. We therefore subtract a term

γ

xu in the x -equation.
What kind of drug regime is appropriate for a given cancer patient depends in
part on the particular state of this patient's disease. Our goal now is to optimize the
influence of the control variable u , that is, the treatment, with respect to several factors.
On the one hand we want to shrink the tumor through the administration of a maximal
dose of the agent, and on the other hand we need to take into account the toxic side
effects of the medication. Also, the treatment should be as short as possible. All this
can be combined into the cost function

κ

T

J ( u ) = ax ( T ) − by ( T ) + ( cu ( t ) + d ) dt .

0

This equation represents the cost of applying treatment schedule u for a length of
time T . The optimal control problem now is to find a control u that minimizes this cost
function. Solving such control problems is an active area of research, and algorithms
have been developed to find u [ 2 ]. See also [ 3 ] for optimal control problems in systems
biology.
Modeling biological systems through differential equations has its limitations,
however. Inmany cases, the processes involvedmight be fundamentally discrete rather
than continuous. For instance, in the case of a predator-prey relationship between two
species inside an ecosystem, both populations are comprised of discrete individuals
that engage in typically binary discrete interactions. Thus, it is not immediately clear
whether one can apply methods such as differential equations, which assume that the
quantities modeled vary continuously. In molecular biology, when we study regula-
tory relationships between genes inside a cell, these relationships are based on the