Lab 14 – Differential Equations, Integration, and Tumor Growth
Date: November 29, 2011
Assignment Due Date: December 6, 2011
Goal: In today’s lab we will explore differential equation models of tumor growth. We aim to better
understand differential equations and how to obtain solutions using integration. We will use R to plot
and analyze solutions of differential equations.
Tumor Growth
Cancer cells gain the ability to overcome normal cellular controls and rapidly reproduce through varied
genetic mutations. Tumors are masses of uncontrolled cell growth. Many researchers are currently
trying to understand tumor formation and create mathematical models that will provide insight into
how cancer forms and what treatments might prove effective. Today we will explore two very simplistic
differential equations that we will use as crude models of avascular solid tumor growth. We assume
that over time the growth slows down due to limited resources to supply the enlarging tumor. Often,
the interior of tumors become necrotic as those cells cannot get enough resources to survive. (Cancer
cells overcome this challenge through angiogenesis, formation of new blood vessels, which we will not
discuss today.)
We will let the variable represent the number of cells in the tumor. In both cases we will assume that
we start with just one tumor cell. Therefore our initial condition is
.
Model #1
is measured in
Plot
. Time is measured in days.
in R for the time span of one year. What happens to the growth rate over time?
Use your knowledge of integration to solve for
and plot this in the same figure in R. Don’t forget to
use the initial condition when solving. What happens to the tumor size over time? Does this seem
logical? Why or why not?
Model #2
is measured in
Plot
. Time is measured in days.
in R for the time span of one year. What happens to the growth rate over time?
Use your knowledge of integration to solve for
and plot this in the same figure in R. Don’t forget to
use the initial condition when solving. What happens to the tumor size over time? Does this seem
logical? Why or why not?
Which model of tumor growth seems like a more appropriate model? Why? Plot both solutions for
longer than one year and see what happens in the long run.
Comparing to Data
Assume that a certain type of tumor grows according to the following differential equation,
where
and are parameters that describe the growth.
Suppose you are given the following plot and data which was collected from a growing solid avascular
tumor. The data given is number of cells (*
) collected every 14 days in a year. Use R to experiment
with parameters and find values for and which seem to match the given data well.
0.2172247
0.9719847
3.310323
4.254782
4.480685
5.957431
4.711694
7.057095
6.426778
7.873677
7.368535
8.562457
7.404883
8.466175
7.657692
9.053041
9.445027
9.362287
9.064911
9.963894
9.026285
9.618952
8.93085
10.47605
9.197494
9.139538
9.603312