Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 307

of the vessel network. In this way, a solid tumor deploys a sophisticated strategy to
control its own growth. Folkman suggested that inhibiting the development of the
tumoral vessel network could be a powerful way to control, in turn, the neoplastic
growth via the reduction of nutrients supply. He termed this new kind of therapy
anti-angiogenic therapy .
Angiogenic inhibitors are commonly classified [ 22 ]as direct inhibitors which act
on the endothelial cells and inhibit their proliferation and migration or induce their
apoptosis, or as indirect inhibitors that block the production of angiogenic factors by
malignant cells, or as mixed agents that target both endothelial and malignant cells.
Most angiogenic inhibitors are cytostatic inhibiting the formation of new blood
vessels. Some of the direct inhibitors have a cytotoxic action that induce a rapid
destruction of existing blood vessels. Various anti-angiogenic drugs have undergone
clinical development in recent years, and some of them have led to improvement
in overall survival or disease-free survival in various clinical scenarios. Since the
therapy targets healthy cells, namely the endothelial cells forming blood vessels,
that are far more genetically stable than tumor cells, anti-angiogenic agents are far
less subject to drug resistance [ 20 ]. Per se , this way of controlling the tumor burden
appears intriguing and there is evidence from experimental work that inhibiting
angiogenesis may induce tumor regression and sometimes cure [ 50 ].
Modeling the interplay between tumor growth and the development of its
vascular network, as well as the action of angiogenic inhibitors, is an important
step that could help to plan effective anti-angiogenic therapies and a large number
of mathematical models have been proposed, e.g., [ 1 , 2 , 16 , 34 , 42 , 43 , 47 ]. Quite
interestingly, Folkman himself and his coworkers formulated a simple, but largely
influential mathematical model in [ 16 ] that describes the vascular phase of tumor
growth assuming that this growth is strictly controlled by the dynamics of the
vascular network and that the vascular dynamics is the result of the opposite
influence of pro-angiogenic and anti-angiogenic factors produced by the tumor
itself. This model provides a framework to portray the effects of anti-angiogenic
therapies, and it was successful in fitting experimental data on the growth and
response to different anti-angiogenic drugs for Lewis lung carcinomas implanted
in mice.
The appreciation of the role of angiogenesis in tumor development has led
Folkman and his coworkers to introduce the concept of a varying carrying capacity,
q

, defined as the tumor size potentially sustainable by the existing vascular
network at a given time [ 16 ]. This carrying capacity may be assumed proportional
to the extent of the actual tumor vasculature. Making the carrying capacity in Eq.
( 8 ) variable, and following [ 16 ] in modeling, the sophisticated, tightly controlled
strategy for the production of vessels reduces to the following dynamical system for
tumor size and carrying capacity under chemotherapy:

( t )

p ln p
q

= − ξ − ϕ ,

p cp (14)

dp 3 q

q = bp − − μ q . (15)