Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 311

then, independently from the initial burden of the tumor and of the vessels
(i.e.,

2
+

for all initial conditions ( p ( 0 ) , q ( 0 )) ∈ R ), the tumor is eradicated,

0 + , 0 + ) lim t → + ∞ ( . Moreover, under generic time varying therapies, a

sufficient condition for eradication is that

p ( t ) , q ( t )) = (

θ ( w min , v min ) β ( A ( v min )) − ( μ + γ u min + η v min ) ≤ 0 , (25)

where u min ,v min and w min denote the minimal values during therapy.
It is interesting to notice the various implications of condition Eq. ( 24 ) in case of
combinations of a non-null chemotherapy ( V 0) with anti-angiogenic therapies.

We start with the biologically interesting case when no anti-angiogenic agents are
present [ 69 ], i.e., U

>

0. In this case, the model studies the effects of the
inclusion of a varying carrying capacity to a continuous infusion chemotherapy.
In the classical setting, with constant carrying capacity K

= W =

> 0,

pF K
p

p = − ϕ Vp , (26)

it follows that chemotherapy can never eradicate (at least, in theory) the tumor in
case of an unbounded value F . However, with the inclusion of the dynamics

of the vessels, there exists a threshold V ∗ such that the tumor can be eradicated if
V

( ∞ )

V ∗ . It is easily determined from Eq. ( 24 ) with W ≥ = U = 0:

V ∗ ) β ( V ∗ )) − μ − η V ∗ =

θ ( W , A ( 0 . (27)

μ > η =

Note that, provided that 0, the threshold also exists in case 0. Of course,

μ since μ usually is small and satisfies b , such a threshold is very large. On the

μ = η = other hand, if 0, then there cannot be eradication, even if we add a

proliferation inhibiting effects, since θ (

, ) β ( ( )) >

W V A V 0.

7 Beyond Linear Models of Chemotherapy in Vascularized

Tu m o rs

For many solid tumors, a log-kill law to model the effects of cytotoxic drugs might
be oversimplified. Indeed, the efficacy of a blood-borne agent on the tumor cells will
depend on its actual concentration at the cell site, and thus it will be influenced by
the geometry of the vascular network and by the extent of blood flow. The efficacy
of a drug will be higher if vessels are close to each other and sufficiently regular to
permit a fast blood flow; it will be lower if vessels are distanced, or irregular and
tortuous so to hamper the flow. To represent these phenomena in a simple form,
in [ 45 , 46 ] it has been assumed that the drug action to be included in the equation
for p is dependent on the vessel density, i.e., in our model on the ratio ρ = q / p .