Radiotherapy cancer treatment model with fractional derivative coupled with linear-quadratic model

Abstract

A mathematical model that simulates a radiotherapy cancer treatment process is presented in this thesis. The model takes two important radiobiological factors into consideration, which are repair and repopulation of cells. The model is used to simulate the fractionated radiotherapy treatment processes of six patients. The results give the population changes in the cells and the final volumes occupied by the normal and cancer cells. The model is formulated by integrating the Caputo fractional derivative with the previous cancer treatment model. Thereafter, the linear quadratic with the repopulation model is coupled into the model to account for the cells’ population decay due to radiation. The treatment processes are then simulated in MATLAB with numerical variables, numerical parameters, and radiation parameters. The numerical parameters which include the proliferation coefficients of cells, competition coefficients of cells, and the perturbation constant of the normal cells are obtained from a previous research. The radiation parameters are obtained from another previous research that reported clinical data of six patients treated with radiotherapy. From the reported clinical data, the patients had tumor volumes of 24.1cm 3, 17.4cm 3, 28.4cm 3 , 18.8cm 3, 3°.6cm3, and 12.6cm 3 and were treated with fractionated doses of 2.0 Gy for the first two patients and 1.8 Gy for the other four. Next, the integrity of the formulated model is established with the proof of the existence of unique solutions, the stability analysis, the sensitivity analysis, the bifurcation analysis, and the comparative analysis. Also, 96 radiation protocols are simulated by using the biologically effective dose formula. All these protocols are then used to obtain regression equations connecting the value of the Caputo fractional derivative with the fractionated radiation dose, and these regression equations are used to simulate various radiotherapy treatments in four different categories. The final tumor volumes, from the results of the simulations, are 3.58cm3 , 8.61cm3 , 5.68cm3 , 4.36cm3 , 5.75cm3 , and 6.12cm3. Meanwhile the volumes occupied by the normal cells are 23.87cm3, 17.29cm3, 28.1lcm3, 18.68cm3, 30.33cm3 , and 12.55cm3. The stability analysis shows that the model is asymptotically and exponentially stable. Also, the solutions of the simulations are unique and stable even there are changes in initial values. The sensitivity analysis shows that the most sensitive controllable model factor is the value of the Caputo fractional derivative and this model factor has bifurcation values. Furthermore, the comparative analysis shows that the fractional derivative model encompasses the memory effect of the radiotherapy process. The predicted simulated final tumor volumes obtained with the regression equations are then compared with the corresponding reported clinical final tumor volumes. The results of these comparisons show that the predictions have minimal errors, hence they are acceptable. Finally, optimal and complete treatment solutions are simulated and predicted.