# Applications of Calculus: Modeling Tumor Growth

*Deriving the Gompertz function.*

*This post is part of the series Connecting Calculus to the Real World.*

Calculus can help us model the growth of tumors.

Tumors appear to grow exponentially early in their lifecycles, which means that if a tumorâ€™s volume at time $t$ is given by $V(t)$ and its growth rate is a constant $r,$ then

However, tumors do not grow exponentially forever: it has been observed that after some time, tumor growth slows down.

To incorporate this into our model, instead of setting the growth rate to a constant r, we can set it to an exponential decay function given by

Then the full model is

We can separate variables and integrate to solve for $V:$

We can solve for one of the constants in terms of the initial volume and the other constant:

When we plug the constant back in, it cancels out the other constant to yield a final formula:

This is called the Gompertz function, and it has been used to model tumor growth and measure the effectiveness of tumor-killing treatments.

*This post is part of the series Connecting Calculus to the Real World.*