Mathematical modeling of radiotherapy cancer treatment using caputo fractional derivative

Abstract

Background: This paper presents a mathematical model that simulates a radiotherapy cancer treatment process. The model takes into consideration two important radiobiological factors, which are repair and repopulation of cells. The model was used to simulate the fractionated treatment process of six patients. The results gave the population changes in the cells and the final volumes of the normal and cancer cells. Method: The model was formulated by integrating the Caputo fractional derivative with the previous cancer treatment model. Thereafter, the linear-quadratic with the repopulation model was coupled into the model to account for the cells’ population decay due to radiation. The treatment process was then simulated with numerical variables, numerical parameters, and radiation parameters. The numerical parameters which included the proliferation coefficients of the cells, competition coefficients of the cells, and the perturbation constant of the normal cells were obtained from previous literature. The radiation and numerical parameters were obtained from reported clinical data of six patients treated with radiotherapy. The patients had tumor volumes of 24.1cm3, 17.4cm3, 28.4cm3, 18.8cm3, 30.6cm3, and 12.6cm3 with fractionated doses of 2 Gy for the first two patients and 1.8 Gy for the other four. The initial tumor volumes were used to obtain initial populations of cells after which the treatment process was simulated in MATLAB. Subsequently, a global sensitivity analysis was done to corroborate the model with clinical data. Finally, 96 radiation protocols were simulated by using the biologically effective dose formula. These protocols were used to obtain a regression equation connecting the value of the Caputo fractional derivative with the fractionated dose. Results: The final tumor volumes, from the results of the simulations, were 3.58cm3, 8.61cm3, 5.68cm3, 4.36cm3, 5.75cm3, and 6.12cm3, while those of the normal cells were 23.87cm3, 17.29cm3, 28.17cm3, 18.68cm3, 30.33cm3, and 12.55cm3. The sensitivity analysis showed that the most sensitive model factors were the value of the Caputo fractional derivative and the proliferation coefficient of the cancer cells. Lastly, the obtained regression equation accounted for 99.14% of the prediction. Conclusion: The model can simulate a cancer treatment process and predict the results of other radiation protocols.