Numerical simulation of normal and cancer cells' populations with fractional derivative under radiotherapy

Background This paper presents a numerical simulation of normal and cancer cells' population dynamics during radiotherapy. The model used for the simulation was the improved cancer treatment model with radiotherapy. The model simulated the population changes during a fractionated cancer treatment process. The results gave the final populations of the cells, which provided the final volumes of the tumor and normal cells. Method The improved model was obtained by integrating the previous cancer treatment model with the Caputo fractional derivative. In addition, the cells' population decay due to radiation was accounted for by coupling the linear-quadratic model into the improved model. The simulation of the treatment process was done with numerical variables, numerical parameters, and radiation parameters. The numerical variables include the populations of the cells and the time of treatment. The numerical parameters were the model factors which included the proliferation rates of cells, competition coefficients of cells, and perturbation constant for normal cells. The radiation parameters were clinical data based on the treatment procedure. The numerical parameters were obtained from the previous literature while the numerical variables and radiation parameters, which were clinical data, were obtained from reported data of four cancer patients treated with radiotherapy. The four cancer patients had tumor volumes of 28.4 cm3, 18.8 cm3, 30.6 cm3, and 12.6 cm3 and were treated with different treatment plans and a fractionated dose of 1.8 Gy each. The initial populations of cells were obtained by using the tumor volumes. The computer simulations were done with MATLAB. Results The final volumes of the tumors, from the results of the simulations, were 5.67 cm3, 4.36 cm3, 5.74 cm3, and 6.15 cm3 while the normal cells' volumes were 28.17 cm3, 18.68 cm3, 30.34 cm3, and 12.54 cm3. The powers of the derivatives were 0.16774, 0.16557, 0.16835, and 0.16. A variance-based sensitivity analysis was done to corroborate the model with the clinical data. The result showed that the most sensitive factors were the power of the derivative and the cancer cells' proliferation rate. Conclusion The model provided information concerning the status of treatments and can also predict outcomes of other treatment plans.