Computational Models of Vascularization and Therapy in Tumor Growth - Mechanical and Chemical Signaling in Angiogenesis - page 227

Fig. 2 Illustration of the
tumor growth angiogenesis
model that was used to
describe untreated xenograft
tumors in animal models by
combining tumor size
measurements together with
histological markers such as
the percentage of hypoxic
tissue. Adapted from [ 34 ]

The model is written as follows:

dP
dt ¼ k P P 1 s a

Þ k PH Ps a ð

dH
dt ¼ k PH Ps a þ k H H 1 s a

ð Þ k HN H

dN
dt ¼ k HN H
dK

dt ¼ bP where P ¼ P þ H þ N and s ¼ P K

P denotes the tumor mean diameter as measured in mice and s the hypoxic stress
regulating the logistic growth of the non-hypoxic ð P Þ and hypoxic tissue ð H Þ and
modulating the transfer between these two types of tissue. N denotes the necrotic
tissue. The exponent a was fixed to a low value (\1) for which the logistic
proliferation term tends to the Gompertz equation as used to model tumor growth
in animal models [ 10 , 27 ]. K stands for the carrying capacity as in [ 27 ] and is here
described by a simple growing function, as a function of the tumor size.

4 Multiscale Approach for Modeling Vascular Tumor
Growth

To treat with model complexity, researchers have also put efforts on the development
of 'multiscale' models. The aim of multiscale modeling is to account, in a single
computational model, for several biological processes occurring at different space
and time scales. Conceptually, this is a very promising approach but practically,
many issues must be addressed. In particular, no real methodology has been proposed
to assess validation and qualification aspects of those models. Nevertheless, it is
believed that the development of multiscale models can provide the basis for an
integrative holistic approach to predict drug response and effect [ 35 ].
Most of the mathematical multiscale models developed so far are theoretical
attempts and none of them showed yet benefit for drug development. The reader