A Mathematical Model of Gene Therapy for the Treatment of Cancer - Mathematical Methods and Models in Biomedicine - page 361

for the purpose of gene network regulation in tumors [ 2 ]. In an additional paper,
Burden and colleagues [ 12 ] explored optimal control methods for determining
the best treatment strategy based on the KP equations. They designed a control
functional to maximize numbers of effector cells and interleukin-2 concentration
while minimizing numbers of tumor cells. In 2008, Banerjee [ 7 ] proposed a delay
version of KP equations where the Eq. (3a) was replaced by a new one:

)
d t =

( ( − τ ) ( − τ )

d E t p 1 E t C t

cT ( t ) − μ 2 E ( t )+ + s 1 (4)

g 1 + C ( t )

with the rest of the KP system [Eq. ( 3 )] remaining the same. The introduction of
a time delay, 0, corresponds to the delay that occurs between the production

of a cytokine production, and its downstream binding and activation action on host
effector cells. In that work, Banerjee analyzed the local stability of the cancer free
equilibrium in the presence of the delay using semi-numerical bifurcation methods.
All of the above applications of dynamical system theory were studied using a
similar approach: investigation of local stability of solutions by linear approximation
(i.e., nonlinear equations are replaced by linear ones in a suitable regions of
phase space). However, non-linear phenomena have a much greater complexity and
require analysis on a global level. Very few generalized methods have been devel-
oped; as such, usually each nonlinear system must be studied individually. The main
difficulty is the presence of free, non-numerical parameters in the system as clearly
exemplified in the KP equations. Parameters (13 of them for KP model) such as
antigenicity, c , or maximal growth ratio, a , are not known experimentally. These
parameters are mostly composite parameters that phenomenologically represent a
set of biological mechanisms in a simple way. To describe different qualitative
scenarios of the model when performing a stability analysis without using numerical
values for the model parameters is a critical task. Thus, we consider the pairing of
numerical and analytical methods as the best approach to gain as much information
as possible. Finally, In 2009, Kirschner and Tsygvintsev [ 29 ] performed a global
analysis of the KP system using the generalized Lyapunov method. They derived
sufficient conditions that guarantee asymptotic convergence as t

τ >

)
to 0. For a “virtual” patient, that would imply complete clearance of cancer once
a corresponding therapy is adopted. Another result of [ 29 ] was to analytically prove
the existence of host self-regulation of cytokine levels that never exceed certain
critical values. See also [ 13 , 20 ] for further discussion.

→ + ∞ of T ( t

3 A Gene Therapy Model

The problem with the use of LAK and TIL cells as described above is that only
about half of the TILs that are typically generated are reactive to tumors [ 50 ]. Thus,
the ability to genetically engineer TIL cells that are directed against tumor-specific
antigens is a key objective. Recently, this was attempted in a small clinical trial [ 43 ]