Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 308

Here bp models the growth stimulated by the pro-angiogenic factors, dq 2 / 3 is a
variable loss rate “constant” due to the endogenous anti-angiogenic factors produced
autonomously by tumor cells, is the natural loss rate constant of the vasculature,

and c denotes the concentration of the chemotherapeutic agent. This model was
obtained under a series of simplifying assumptions that include spherical symmetry
of the tumor, a fast degradation of pro-angiogenic factors and a slow degradation
of inhibitory factors. The dynamics of anti-angiogenic factors reflects their more
systemic effects and leads to an interaction term between the surface area of the
spheroid and the vasculature of the form dp 3 q , whereas the dynamics of the pro-
angiogenic factors suggests the term bp . A mathematical analysis of this model was
presented in [ 42 ], focusing on the tumor eradication under regimens of continuous
or periodic anti-angiogenic therapy; the problem of determining optimal treatment
schedules for a given amount of inhibitors has been solved in [ 26 ].
By relaxing the assumptions made in [ 16 ], and also by considering more general
laws of tumor growth, the above model was generalized in [ 44 ] assuming that the
specific growth rate of the tumor,

μ

p , and the specific birth rate of vessels depend on
the ratio between the carrying capacity and the tumor size. Since the ratio q p may
be interpreted as proportional to the tumor vessel density, the second assumption
agrees with the model proposed by Agur et al. [ 1 ]. In the absence of therapy, the
model proposed in [ 44 ] takes the form

pF q
p

p = , (16)

q

q
p

q = β − I ( p ) − μ , (17)

where the growth function F : ( 0 , ∞ ) → R is strictly increasing and satisfies

− ∞ ≤ lim

ρ → 0 +

F ( ρ ) < 0 , F ( 1 )= 0 , and 0 < lim

ρ → + ∞

F ( ρ ) ≤ + ∞ .

The stimulation term β : ( 0 , ∞ ) → R is strictly decreasing and satisfies β (+ ∞ )= 0

and β ( 1 ) > μ . It may be unbounded, like in the biologically important case of power

ρ − δ , laws, β ( ρ )= b δ > 0, or bounded such as

β M

β ( ρ )=

n ,

n ≥ 1 .

1 + k ρ

0 + β ( ρ )

In this case, lim is a decreasing Hill function. Also,

combinations of the above two expressions are allowed. The inhibition term
is a strictly increasing function I :

is finite and β

ρ →

[ , ∞ ) → [ , ∞ ) ( )=

0 0 that satisfies I 0 0and

( )=+ ∞

lim p → + ∞ I . Equations ( 16 )and( 17 ) together provide a general mathemat-

ical framework within which the time evolution of solid vascularized tumors can be
analyzed.

p