Tumor Growth and Radio Therapy

1. Tumor Growth and Radio Therapy Bettina Greese Biomathematics, University of Greifswald Nuha Jabakhanji Bioinformatics , University of Alberta

2. Cancer: Background • A cancer tumor is a mass of cells with uncontrolled cell proliferation as a result of defective cell cycle control mechanisms. • How it results: point mutations, DNA rearrangements, gene amplification, translocation, mutations in tumor suppressor genes. • Cancer cells are genetically unstable and are able to become “worse” by accumulating mutations. • Why it’s important: “An estimated 149,000 new cases of cancer and 69,500 deaths from cancer will occur in Canada in 2005.” National Cancer Institute of Canada

3. Mathematical Model: Simple Model • Two populations: Healthy and cancerous cells. • Logistic growth with intrinsic growth rate. • Competition for resources and space. • Initial conditions: 100 healthy cells, 1 cancerous cell.

4. Mathematical Model: Simple Model • Two populations: Healthy and cancerous cells. • Logistic growth with intrinsic growth rate. • Competition for resources and space. • Natural mutations (healthy to cancerous cell).

5. Mathematical Model: Simple Model • Two populations: Healthy and cancerous cells. • Logistic growth with intrinsic growth rate. • Competition for resources and space. • Mutations: natural and radiation induced. • Radiation induced death for cancerous and healthy cells.

6. Simple Model: Analysis • Calculated the steady states and their stability. • Plotted the phase plane for different parameter sets. Extinction Coexistence

7. Simple Model: Analysis • Plots of healthy (red) and cancerous (green) cells versus time. Extinction Coexistence

8. Simple Model: Analysis • Thin lines: with effects of radiation and/or mutation. • Left: The effect of natural mutation on the populations. • Right: Cell deaths caused by constant radiation.

9. Simple Model: Analysis • Left: Cell deaths caused by constant high radiation. • Right: Cell deaths caused by pulsed radiation.

10. Mathematical Model: Final Model • Three populations: Healthy, cancerous and aggressive cancerous cells. • Logistic growth, competition, mutations, radiation induced death are as before. • Initial conditions: 100 healthy, 1 cancerous and 0 aggressive.

11. Final Model: Analysis • Plot of healthy (red), cancerous (green) and aggressive cancerous (blue) cells versus time. • Thin lines: effect of natural mutation.

12. Final Model: Analysis • Plot of healthy (red), cancerous (green) and aggressive cancerous (blue) cells versus time. • Thin lines: effect of radiation induced mutations and death.

13. Final Model: Analysis • Thick lines: natural mutations included. • Thin lines: effect of radiation induced mutations and death in addition to natural mutations.

14. Conclusions and Limitations • Biologically meaningful parameters result in extinction of healthy cells. • Natural mutation accelerates extinction of healthy cells. • Radiation delays extinction. • High doses of radiation are needed to maintain a level of healthy cells above cancer cells. • Pulsed radiation allows higher doses of radiation, thus a higher level of healthy cells is maintained for a significantly longer time. • Pulsed radiation includes breaks in radiation that result in the recovery of the cells.

15. Conclusions and Limitations • In the final model, the cancerous cells are driven to extinction by the aggressive cancerous cells when natural mutation is included. • Also, radiation does not change the qualitative behavior but results in lower levels of cell populations. • We used same mutation rates and carrying capacities for healthy and cancerous cells. • Pulsed radiation was not included in the final model. • Initial cell populations were low. • Further insight can be gained by varying parameters within biological reason.

16. THANK YOU