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

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

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.

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).

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.

Simple Model: Analysis  Calculated the steady states and their stability.  Plotted the phase plane for different parameter sets. ExtinctionCoexistence

Simple Model: Analysis ExtinctionCoexistence  Plots of healthy (red) and cancerous (green) cells versus time.

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.

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

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.

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

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.

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

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.

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.

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