Metastatic Breast
Cancer in the Lungs
(breast cancer)
Project completed by: Brad Davis, Scott Feldhaus, Patrick
Dolan, Haley Santilli
Brief overview of Metastatic
Breast cancer
 Type of breast cancer that spreads to other organs
 A complication of primary breast cancer
 Process of cancer spreading is called metastasis
 Same type of cancer cells as normal cancer
Data table for graphs
**The table presents the weight of the tumor (p) in mg
at time (t) (days)with the growth rate given by g(p) in
mg/day.
Percent mass change of the tumor per day:
Tumor Growth Model:
Density of tumor (mg) per day
Tumor Growth Model for “fitted” graph:
Tumor Regression Code
Tumor Kuznetsov Model Code
Tumor Kuznetsov Model Code (cont.)
Tumor Fit Data Code
Tumor Fit Data Code (cont.)
Methodology
 To model the tumor growth and interaction with the
immune system we used the Kuznetsov version of a
predator‐prey model:
which is
Methodology
 Taking x(t) as the population of tumor cells and y(t) as the population of
immune cells, we generated a linear regression of x’/x to find the
parameters a and b in the Kuznetsov model.
 For the other parameters, we used those given in panel (a) on page 10 of
“Interactions Between the Immune System and Cancer: A Brief Review of
Non‐spatial Mathematical Models” by R. Eftimie, et al. These parameters,
however, did not generate an appropriate curve, because all the
Kuznetsov models in the Eftimie paper yielded either decreasing, logistic,
or periodic changes in the tumor cell population, while our data appeared
to have an exponential growth rate.
 To improve the model, we altered the parameter n to reduce the effect of
the immune cells on the cancer population, and achieved an exponential
growth curve closer to an unbounded Gompertz model, which fit the tumor
data rather nicely.
 This affected the immune cells causing a rapid collapse in the effector cell
population. We then used the Python optimization code to generate a
tumor growth curve with a better fit.
Works cited page
Article Website:
http://wwwrohan.sdsu.edu/~jmahaffy/courses/s00a/math121/labs/labk/q5v1.htm
Metastatic tumor Information:
http://www.cancer.gov/cancertopics/factsheet/SitesTypes/metastatic