A Mathematical Model of Gene Therapy for the Treatment of Cancer - Mathematical Methods and Models in Biomedicine - page 357

best suited for each case not known a priori, the choice of which gene to target, the
dose and timing of treatments all must be determined. A recent review summarizes
advances in gene therapy and also highlights opportunities for systems biology and
mathematical modeling to synergize efforts with experimentalists and clinicians to
push cancer research forward [ 23 ].
Mathematical modeling has been instrumental in the past 50 years in helping
decipher different aspects of complex systems in biology. In particular, mathe-
matical modeling has had an impact on our understanding of cancer biology and
treatment (cf. [ 4 , 24 , 25 ] for excellent reviews). We begin by briefly reviewing
existing models designed specifically for capturing tumor-immune dynamics, one
of which forms the basis for our current work. As a first step to exploring the use
of gene therapy on the tumor-immune interaction during cancer, we will apply a
simple mathematical model to explore the dynamics of these different types of gene
therapies, with the goal of predicting optimal combinations of approaches leading
to clearance of a tumor. We present the model and its analysis (both analytical and
numerical) and offer some conclusions.

2 Brief Review of Mathematical Models Describing

Tumor-Immune Dynamics

For the past 40 years, mathematical models have been developed describing many
aspects of cancer from tumor growth dynamics (cf. [ 11 , 22 , 32 ]), angiogenesis, and
vascularization (cf. [ 36 , 41 , 52 ]), to the topic of immune response to tumors. Since
the work herein will be focused solely on tumor-immune dynamics, we briefly
review work in this area.
Tumor-immune models have been around since the early 1990s and have
evolved to capture very complex aspects of the immune response as knowledge
of the molecular dynamics of immunity has increased. For example, an important
aspect of immunity is the recognition of non-self, or foreign antigens. Specialized
antigen presenting cells (known as macrophages and dendritic cells) present foreign
antigen to immune cells, such as T cells, to train them to respond and clear the
foreign material (like bacteria and viruses). Of course, since tumor cells began
as self, or non-foreign host cells, the level of antigenicity of the tumor may
be weak as the human immune system is trained to not kill self. Mathematical
models of tumor-immune interactions that have explored dynamics at this scale are
[ 27 , 34 , 40 , 47 , 51 ]. Recently, Joshi et al. [ 26 ] develop a new mathematical model
to capture immunotherapy that involves the antigen presentation pathway and its
role in tumor-immune dynamics. Other models focus only on therapy as well as
on boosting immunity [ 3 , 6 , 8 - 10 , 16 - 18 , 21 , 30 , 48 ]. Immune competition models