Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 306

in both DNA strands caused by a single particle and a quadratic term that accounts
for two adjacent breaks on different chromosomes caused by two different particles.
This is the so-called linear-quadratic (LQ) model [ 12 , 66 , 68 ] that has become the
accepted model in radiotherapy. The damage of radiation to the tumor can then be
modelled in the form

w

t

− ( )

ϕ + β

( ) ( − ρ ( − ) ) ( ) ,

p t w s exp t s d s t (10)

0

where w represents the radiation dose rate and ϕ , β and ρ are positive constants

with the tumor repair rate.

A small tumor repair rate implies a larger influence of the integral term that describes
the secondary, i.e., quadratic effects, and thus a greater effectiveness of the therapy,
while large repair rates imply that the integral can be neglected. Note that the integral
term in parenthesis in Eq. ( 10 ) is simply the solution to the first-order linear equation

ϕ and β related to the tumor LQ parameters and ρ

r = − ρ r + w , r ( 0 )= 0 . (11)

Mathematically, the structure of the overall model becomes more transparent if we
replace the integral by this differential equation. Briefly, in case of a constant dose
rate w ( t )= W , the eradication condition becomes

W 2

R ( 0 ) < ϕ W +( β / ρ ) , (12)

whereas for periodically delivered therapies it is

R ( 0 ) < ϕ w ( t ) + β w ( t ) r ( t ) . (13)

5 Tumor Angiogenesis and Solid Vascularized Tumors

Chemotherapy targets the main characteristic of tumor cells, their proliferative
derangement. However, as already mentioned, tumor cells show a vast array of
microscopic and macroscopic interactions with other cellular populations. As a
consequence, the study of these phenomena may open the way for the creation
of new therapies. Folkman [ 9 , 10 ] already stressed in the early seventies that the
development of a vascular network inside the tumor mass becomes necessary to
support tumor growth. Indeed, primary solid tumors and metastases require the
formation of new blood vessels in order to grow beyond 1-2 mm 3 . Folkman named
this process neo-angiogenesis. It is sustained by various mechanisms—tumors may
coopt existing vessels, may induce the formation of new vessels from preexisting
ones or may exploit endothelial precursors originating from the bone marrow [ 11 ].
Tumor angiogenesis is a complex process driven by pro-angiogenic factors that are
being released as the tumor cells lack a full level of nutrients [ 70 ]. Interestingly
enough, tumor cells also release anti-angiogenic chemicals that modulate the growth