Abstract
•The presented model simulates a radiotherapy cancer treatment process taking into consideration the repair and repopulation of the cells.•The model was obtained by integrating the previous cancer treatment model with the Caputo fractional derivative.•Cells' population decay due to radiation was accounted for by coupling the linear-quadratic with the repopulation model into the improved model. The model was then used to simulate the cancer treatment process of six patients with the use of numerical variables, numerical parameters, and radiation parameters.•The results from the simulation showed the population changes and the final volumes of tumors and normal cells. In addition, a global sensitivity analysis was done to prioritize the model factors.•The biologically effective dose formula was used to simulate 96 protocols from which a regression equation for deriving the value of the Caputo fractional derivative was obtained. Finally, this paper is aimed at contributing to research activities in the field of cancer treatment with radiotherapy.
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.
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.
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.
The model can simulate a cancer treatment process and predict the results of other radiation protocols.