Gompertz Equation

The Gompertz growth law has been shown to provide a good fit for the growth data of numerous tumors. Let $  V(t)  $ measure the size of the tumor (e.g. volume, number, etc.). The Gompertz growth law is described by the following system of differential equations:

\[  \frac{dV}{dx} = r(t)V(t), \;\; V(0)=V_0  \]

\[  \frac{dr}{dx} = \alpha r(t), \;\; r(0)=\beta  \]


The sign of $  \beta  $ determines where the tumor grows or regresses. The solution is given by \[  V(t)=V_0 e^{\displaystyle \frac{\beta}{\alpha} 
(1-e^{-\alpha t})} \]

where the constants are determined by the data.

In the serial treatment model discussed in class, we assume that the post-treatment tumor has the same growth characteristics as the original tumor. Specifically, we assume that the ultimate tumor size remains the unchanged which we denote as:

\[   \theta = V_0 e^{\beta/\alpha}  \]

The Gompertz growth model can be written in the following form:

\[  \frac{dV}{dt} = - \alpha V \ln \left(\frac{V}{\theta}\right).  \]

To determine the tumor regression from just before one treatment to just before the next, we integrate the first order differential equation as follows (see figure in the exponential case)

\[  \int_{FV_n}^{V_{n+1}} \frac{dV}{\ln \left(\frac{V}{\theta}\right)}= 
\int_0^T \alpha dt  \]

This leads to the difference equation

\[   \ln \left(\frac{V_{n+1}}{\theta}\right) = 
\ln \left(\frac{F \ V_n}{\theta}\right)e^{-\alpha T}  \]