Computational Models of Vascularization and Therapy in Tumor Growth - Mechanical and Chemical Signaling in Angiogenesis - page 226

therapy or radiotherapy. With the integration of more informative data (such as
information for functional imaging in addition to morphological data), the newly
developed models will also become more complex, integrating in processes that
are important to tumor dynamics such as cell cycle regulation and angiogenesis.

3 Models of Tumor Growth and Angiogenesis

An extension of the Gompertz model that includes the process of angiogenesis was
proposed in 1999 [ 27 ]. In this paper, the authors expanded the Gompertz model to
incorporate the process of angiogenesis using a new variable, called ''carrying
capacity'' to account for tumor vascularization. The proposed model is as follows:

dw
dt ¼ k 1 w log

w
k

dk
dt ¼ k 2 k þ bS w ; k

ð Þ dI w ; k ð Þ ekg ;

where w denotes the tumor volume and k denotes the carrying capacity. The first
equation describes the Gompertz growth of the tumor volume. The growth satu-
rates when w reaches k : The evolution of k depends on the process of angiogenesis.
It is regulated by a positive 'pro-angiogenic' term denoted S ; and a negative 'anti-
angiogenic' term denoted by I : Two additional terms account for natural decay (at
rate k 2 ) and eventual degradation due to the effect of anti-angiogenic drugs, where
exposure is denoted g ð t Þ:
The authors showed that the model successfully predicts tumor growth inhibition
by several anti-angiogenic compounds in mice models. Interestingly, this model
has been subjected to mathematical analysis to study its main properties that
highlighting the best strategies to optimize the delivery of anti-angiogenic drugs
[ 28 - 32 ].
In 2011, a more complex model of tumor growth was proposed accounting for
different types of tumor tissue, with a specific focus on hypoxic tissue that is known
to play a crucial role in tumor angiogenesis [ 33 ]. The main innovation of this
model is the integration, together with tumor size, of classical histological bio-
markers such as those commonly retrieved in preclinical studies. This model
integrates three types of tissue (proliferative non-hypoxic, hypoxic, necrotic) and is
based on the following hypothesis: as the tumor grows, oxygen tends to lack within
the spheroid and drives the formation of hypoxic tissue. Hypoxic tissues become
in turn necrotic with a constant transfer rate. Depending on the whole tumor size,
the carrying capacity increases as a result of the process of angiogenesis.
The model shows correct predictions of tumor size progression as well as the
percentages of necrotic and hypoxic tissue in 30 mice that were xenografted with
either HT29 or HCT116 colorectal cancer cell lines [ 34 ]. The diagram of the
model is presented Fig. 2 .