Mechanistic Modeling of Tumor Repopulation during Radiation Treatments

A mechanistic mathematical model for radiotherapy is developed to study the effect of tumor repopulation on radiation treatment outcome. The Two-Lesion Kinetic (TLK) model is adopted to describe the interaction between ionizing radiation and malignant cells, and extended in this work to take into account tumor proliferation. The investigated kinetics for tumor cells growth includes exponential, logistic and Gompertz laws, while standard fractionation treatments are considered as case studies. Scaling analysis is used for non-dimensionalizing the system of ordinary differential equations describing the process under investigation. Numerical solution of the proposed model is shown as a function of the dimensionless variables and groups generated by the scaling procedure. The latter allows minimizing the number of parameters needed to describe the radiotherapy process when DNA damage formation and repair, and tumor repopulation phenomena take place simultaneously. The effectiveness in tumor eradication is mapped as a function of two dimensionless groups that take into account the radiation dose rate as well as the DNA lesions repair and malignant cells repopulation rates. The present study illustrates that outcomes of the radiotherapy treatment are strongly affected by the relative rates of phenomena simultaneously occurring. In particular, it appears that in order to achieve an effective treatment by keeping as low as possible the total dose to be administered, the fractionation scheme should be optimized on the base of the ratio between the DNA repair mechanisms and tumor cell growth rates. Model results also show that exponential and logistic kinetics yield similar results in terms of treatments outcomes. By comparison, model simulations with the Gompertz law indicate that repopulation described by this growth kinetics result in a significantly poorer prognosis for tumor eradication than either exponential or logistic models