Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 140

CHAPTER

5

Agent-Based Models and
Optimal Control in Biology:
A Discrete Approach
Reinhard Laubenbacher ∗ , Franziska Hinkelmann † and Matt Oremland ∗
∗ Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA, USA
† Mathematical Biosciences Institute, The Ohio State University, Columbus, OH, USA

5.1 INTRODUCTION
The need to control complex systems, both engineered and natural, pervades our lives.
From the thermostat that controls the temperature in our homes to the software that
controls flight characteristics of the space shuttle during landing, the vast majority of
engineered systems have built-in control mechanisms. Being able to control certain
biological systems is no less important. For instance, we control ecosystems for
agriculture and wildlife management; we control different parts of the human body to
treat and cure diseases such as hypertension, cancer, or heart disease. And we control
microbes for the efficient production of a vast array of biomaterials. With control
comes the requirement to carry it out in a fashion that is optimal with respect to a
given objective. For instance, we want to devise a schedule for administering radiation
therapy to cancer patients in a way that maximizes the number of cancer cells killed
while minimizing side effects. We apply pesticides to fields in a way that minimizes
environmental damage. Andwe aim to control themetabolismof engineered strains of
microbes so they produce the maximal amount of biofuel. Thus, the need for optimal
control is a problem we face everywhere. This chapter will focus on optimal control
of biological systems.
The most common approach to optimal control is through the use of mathematical
models, often consisting of one or more (ordinary or partial) differential equations.
These equations model key features of the system to be controlled and include one
or more variables that represent control inputs. The following example illustrates this
approach; it is taken from [ 1 ], where more details can be found. The problem we
want to focus on is the optimization of cancer chemotherapy, taking into account
certain immunological activity. The two relevant variables are x , which represents
the volume of the tumor to be treated, and y , which quantifies the density of so-
called immunocompetent cells, capturing various types of T-cells activated during
the immune reaction to the tumor. These two variables are governed by the system of
ordinary differential equations,

dx / dt = μ C F ( x ) − γ xy ,

x 2 dy / dt = μ I ( x − β ) y − δ y + α.