# Gompertz Equation

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

Here

• tumor growth rate

• retardation constant

• initial growth or regression rate.
The sign of determines where the tumor grows or regresses. The solution is given by

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:

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

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)

This leads to the difference equation

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