In this model formulated by Hahnfeldt, Panigrahy, Folkman and Hlatky [3], the growth of
the primary tumor and evolution of the vascular network are presented in a two-dimensional
dynamical system. The volume of the primary tumor and the carrying capacity of the
vascular network are respectively represented by the variables p and q.
A growth function describes the rate of change in the volume of the primary tumor which
is dependent on the carrying capacity of the vascular network q and is chosen as Gompertzian
in the original model. Hence the growth function in this model is represented as
p
p_ = pln (2.1)q
where denotes a tumor growth parameter. The dynamics of the carrying capacity of the
vascular network shows a balance between stimulation and inhibition and is represented as
followed:
q_ = q + S(p;q) I(p;q) Guq: (2.2)
In this model q > 0 represents the death of endothelial cells from natural causes, I and S
are functions of p > 0 and q > 0 which describes inhibition, and stimulation respectively.
The control in this system which represents the dosage rate of angiogenic drug is denoted
by u 2 [0;a],where 0 represents no dosage and a, full dosage. G stands for anti-angiogenic
killing parameter. Let’s also keep in mind that the variable is relatively smaller than the
other factors and in some models can be ignored.
The analysis from [3] that leads to the construction of the I(p;q) and S(p;q) functions in
this model is based on two main results. The …rst one suggests the following: the inhibitory
mechanism a¤ects endothelial cells in a way that grows like volume of cancerous cells to
the power 23. Intuitively, the fraction 23 makes perfect sense because the inhibiting proteins
are released through the surface (2 dimensions) of cancerous cells and target the volume (3
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