1. Modeling Tumor Growth Jessie Xiao, Cindy Zhang, Jessica Zheng

2. ●The model ●Tumor growth ●Lymphocyte growth ●Coupling tumor growth and NK cell growth ●Adding in chemotherapy Overview of Our Project

3. Tumors ●An abnormal proliferation of tissues

4. ● T = Population of tumor cells ● N = Population of NK (natural killer) cells ● C = Population of lymphocytes ● M = chemotherapy drug concentration in the bloodstream Variables

5. ●Growth and death terms ●Cell-cell kill term (Tumor cells only) ●Cell recruitment (NK cells only) ●Inactivation terms (NK cells only) ●Chemotherapy term Explanation of Models

6. The Model

7. ●Logistic growth: ●Carrying capacity at (approximately 10 9 ) ●Tumor population at time t where A is a constant: A Tumor Cell By Itself

9. ● ●Steady state: (≈ 6.25 x 10 10 cells) A Lymphocyte By Itself

10. NK Cells ● ● Setting to 0 to find nullclines: ● For T = : Coupling Tumor and NK cell growth

11. Tumor Cells ● ● Setting to zero to find the nullclines: Coupling Tumor and NK cell growth cont.

12. Tumor and NK Cell Nullclines

13. Simulating Initial Conditions

14. Tumor and NK Nullclines (zoomed in)

15. Simulating Initial Conditions

16. ● M was previously assumed to be zero ●Adding in the effect of chemotherapy ● v M (t) is assumed to be a constant Adding in Chemotherapy

17. ● ●Solving for M gives: ● is a stable steady state ●Assume v M (t) is 5, so M is ≈ 5.56 Adding in Chemotherapy cont.

18. Initial Conditions for M

19. ●Solving for C : ● ● C was 6.25 x 10 10 cells without chemotherapy ●Need to change C in the other equations Chemotherapy and Lymphocytes

20. ●New C * value: ●Setting to 0: ●For : Chemotherapy and NK Cells

21. ● Setting to zero : Chemotherapy and Tumor Cells

22. Chemotherapy and Tumor-NK Nullclines

23. Simulating Initial Conditions with Chemotherapy

24. Simulating a System of Treatment

25. Adding in Chemotherapy

26. Adding Chemotherapy (zoomed in)

27. Removing Chemotherapy

28. Removing Chemotherapy (zoomed in)

29. ●Limitations of our model ●Untreated tumors are unlikely to be eradicated by the immune system alone ●Chemotherapy kills tumor cells very effectively, but also kills immune cells effectively Conclusions and Discussion

30. ●"Medical Diagnosis and Medicinal Plants." Medical Diagnosis and Medicinal Plants. N.p., n.d. Web. 30 July 2015. ●Pillis, L.g. De, W. Gu, and A.e. Radunskaya. "Mixed Immunotherapy and Chemotherapy of Tumors: Modeling, Applications and Biological Interpretations." Journal of Theoretical Biology 238.4 (2006): 841-62. Web. 30 July 2015. ●Pillis L.g. De, A.e. Radunskaya, and C.L. Wiseman. “A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth.” Web. 30 July 2015. ●"Tumor Doubling Defense : PA Cervical Cancer Misdiagnosis Attorneys." Meyers Evans & Associates, LLC. N.p., n.d. Web. 30 July 2015. ●Pillis, L.g. De, and A. Radunskaya. "A Mathematical Model of Immune Response to Tumor Invasion." Computational Fluid and Solid Mechanics 2003 (2003): 1661-668. Web. 30 July 2015. Sources

31. Thank you to: ●Professors Tim Lewis, Sebastian Schreiber, and Bob Guy ●David and William ●Mr. Gonzalez Acknowledgements