1. Metastatic Breast Cancer in the Lungs (breast cancer) Project completed by: Brad Davis, Scott Feldhaus, Patrick Dolan, Haley Santilli

2. 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

3. **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. Data table for graphs

4. Percent mass change of the tumor per day:

5. Tumor Growth Model: Density of tumor (mg) per day

6. Tumor Growth Model for “fitted” graph:

7. Tumor Regression Code

8. Tumor Kuznetsov Model Code

9. Tumor Kuznetsov Model Code (cont.)

10. Tumor Fit Data Code

11. Tumor Fit Data Code (cont.)

12. Methodology To model the tumor growth and interaction with the immune system we used the Kuznetsov version of a predator ‐ prey model: which is

13. 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.

14. Works cited page Article Website: http://www- rohan.sdsu.edu/~jmahaffy/courses/s00a/math121/labs/labk/q5v1.htm Metastatic tumor Information: http://www.cancer.gov/cancertopics/factsheet/Sites- Types/metastatic