# Program Information

## Estimating Tumor Growth Rate After Single Fraction of Dosage

### Y Watanabe*, E Dahlman, University of Minnesota, Minneapolis, MN

SU-E-T-309 Sunday 3:00PM - 6:00PM Room: Exhibit HallPurpose: To develop a mathematical relationship between the cell proliferation probability after single irradiation and the cell survival fraction used for the linear-quadratic (LQ) model.

Methods: A set of differential equations were derived for the numbers of dividing cells and non-dividing cells by introducing the probabilities of mitosis by the end of one cell cycle in a model. The cell survival fraction at the N-th cell cycle obtained from this formulation was then related to the cell survival fraction commonly used in radiation biology as expressed by using two parameters alpha and beta along with the strength of single dosage for the LQ model.

Results: The probability of a cell to divide (or cell proliferation probability, p) after single dosage of d can be expressed by a function of alpha, beta, d, TD, and T. Here TD is the doubling time of tumor volume and T is the time at which the cell survival data are collected during experiments. Normally T is equal to 5 to 10 times the cell cycle time. Using the probability p, then we obtained a formula for the modified growth rate which included the effect of irradiation. This formula indicates that the irradiated tumor keeps growing if the dosage is smaller than a certain threshold dose. The threshold dose increases as the alpha/beta ratio increases. The formula is different from one commonly used in literature; but it can represents actual experimental results more realistically.

Conclusions: The formula of the modified growth rate including the radiation effect can be easily incorporated in a governing equation for tumor growth. The solutions should be able to reproduce experimental and observational data better than using a traditional formulation.

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