Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 309

6 Combinations of Chemo- and Anti-angiogenic Therapies

of Vascularized Tumors

Anti-angiogenic therapy is an indirect approach that only limits the tumor's support
mechanism without actually killing the cancer cells. Therefore it is only natural,
and this has been observed consistently, that therapeutic effects are only temporary
and, in the absence of further treatment, the tumor will grow back once treatment is
halted. Thus tumor anti-angiogenesis is not efficient as a stand-alone or monother-
apy treatment, but it needs to be combined with other mechanisms like traditional
chemotherapy or radiotherapy treatments that kill cancer cells. In this context,
it is worth noting that tumors differ from normal tissues also in density, topology
and functionality of their vessel network. Tumor vasculature is characterized by a
remarkable degree of intricacy as well as by a variety of disfunctionalities. Since
the neovessel network that brings nutrients to the tumor is also the route to deliver
chemotherapeutic drugs, R.K. Jain hypothesized that the preliminary delivery of
a vessel disruptive anti-angiogenic agent, by “pruning” the vessel network, may
regularize it with beneficial consequences for the successive delivery of cytotoxic
chemotherapeutic agents [ 18 , 19 ]. If treatment schedules are optimized to minimize
the tumor volume, such a structure of protocols is confirmed as optimal [ 48 ] and our
results support this hypothesis.
Experimental studies on mouse models and clinical trials [ 6 , 7 ] showed that some
cell cytotoxic agents (e.g., cyclophosphamide) also have significant anti-angiogenic
effects. In [ 40 ], this effect was modelled for a chemotherapeutic monotherapy that
also has a vessel disruptive action. On the other hand, here we are interested to
fully assess the effects of a combined therapy when three classes of drugs are
co-present: (a) an anti-angiogenic agent u having effects of vessel disruption, (b)
a chemotherapeutic agent v which may or may not have effects of vessel disruption
or inhibition, and (c) an anti-angiogenic agent w inhibiting the proliferation of the
tumor vessels. Generalizing the model in [ 48 ], these effects are included in the
following equations:

pF q
p

p = − ϕ vp , (18)

q

q
p

v

q = θ ( w , v ) · β − I ( p ) − μ − γ u − η . (19)

θ = θ (

, ) [ , ]

Here w v is a function that takes values in the interval 0 1 and is decreasing

≤ η ≤ ϕ

in both variables. We also assume that 0 since, for biological reasons, the

log-kill effect on the carrying capacity, if it exists, is not the prevalent one. However,
this assumption has no consequences on the asymptotic behavior of the solutions of
the proposed system. The model considered in [ 48 ] was the special case

b
ρ

β ( ρ )=

so that q β ( q / p )= bp .