Computational Models of Vascularization and Therapy in Tumor Growth - Mechanical and Chemical Signaling in Angiogenesis - page 224

Fig. 1 Illustration of the modified-Gompertz model that was used to describe untreated and
treated xenograft tumors in animal models. Cycling cells become damaged as a function of the
chemotherapy concentration. Damaged cells finally die with a time delay represented by a transit
compartment. Adapted from [ 10 ]

The proposed Gompertz model was written as follows:

w ðÞ¼ w 0 e a 1 e at

ð Þ

where w denotes the number of tumor cells, w 0 the initial number of tumor cells
also called baseline tumor size, and A and a two positive constants.
In 1970, the Gompertz model was successfully used to describe tumor growth
in animal models [ 5 ]. The Gompertz model was successively applied to clinical
data in 1972 with IgG multiple myeloma patients [ 6 ], and in 1976, Norton and
Simon generalized the use of the Gompertz law for several ranges of solid tumors
[ 7 , 8 ]. The proposed model was at the basis of the so-called Norton-Simon
hypothesis according to which increasing the dose intensity might lead to the
optimization of treatment effects by reducing tumor regrowth between chemo-
therapy cycles. In 2003, this hypothesis reached validation in clinical trials [ 9 ].
In 2004, a model was proposed to integrate, with high accuracy, the effect of
chemotherapy into the Gompertz modeling framework and to analyze the time-
course of untreated and treated tumors in mice models [ 10 ]. In this study, a mod-
ification of the Gompertz model is proposed to remove the final plateau phase of the
Gompertz equation. The equation for the so-called modified-Gompertz model:

dw ðÞ
dt

k 0 w ðÞ

¼

w

w

k 0

1 þ

k 1 w ðÞ

where w ð t Þ denotes the tumor mass, and k 0 and k 1 are two positive constants
regulating the growth in the initial exponential growth and in the linear phase
respectively. w is a constant parameter fixed to a high value (typically w ¼ 20). In
doing so, when w is small, the model reproduces an exponential law, i.e. d dt k 0 w ;
while the growth tends to be linear, i.e.

dw

dt k 1 ; while the tumor mass gets larger

k 1

than the ratio

k 0 :
The previous tumor growth model was successfully integrated into a PK/PD
framework based on transit effect compartments to reproduce both untreated and
treated tumors with several therapeutic agents such as Irinotecan, 5FU and pac-
litaxel. Figure 1 shows an illustration of the model integrating the transit effect
compartments.