Stochastic modelling of cancer cell proliferation and death in response to anticancer therapeutics of thymoquinone - Shabana, Tabassum (2023)
Recent studies have revealed the role of Thymoquinone (TQ) as an active ingredient of black seed (Nigella Sativa) in apoptotic activities. TQ induced apoptotic (the program cell death) can modulate cell life and death, hence able to provide therapeutic potential in cancer disease. The biological mechanism of apoptotic induced by TQ is not yet fully understood. Mathematical model is useful in promoting effective knowledge about the effects of TQ in cancer proliferation and apoptotic activities. It provides an insightful way to explore and predict the growth of the cancer as well as the response to therapy. Furthermore, the cancer cell proliferation is subjected to uncontrolled factors, referred as white noise. Stochastic model provides a way to describe the process. Although potentially useful, no stochastic model has been formulated to represent the growth of cancer affected by anticancer therapeutic of TQ and apoptotic activities. This research is aimed to formulate a system of stochastic differential equations (SDEs) for the apoptosis process in signalling pathways of cancer cell proliferation and death in response to TQ. To achieve this objective, the logistic and Gompertz growth laws of population dynamics were included in the prey-predator model to form a deterministic model of ordinary differential equations (ODEs). Therefore, the deterministic form of logistic prey-predator and Gompertz prey-predator was developed to model the cancer cells proliferation (prey) in the presence of TQ. TQ was also recruited by the cancer cells through a Michaelis-Menten law which provided the saturation effect in the predator of the equation. The models were extended to their stochastic counterpart with the inclusion of the Wiener process to the kinetic growth rate parameters of cancer cells and TQ. The qualitative dynamic of the logistic and Gompertz prey-predator models had shown that the model possesses the properties of positive solution. Cancer cells would grow to the equilibrium point of the treatment failure, but under the success of treatment, the cancer cells would shrink to the equilibrium points of the treatment. Deterministic and stochastic models were simulated, and the results were compared with the experimental data of HSC-3 and HSC-4 lines. Laboratory experiment of TQ in response to cancer cell was carried out in International Islamic University Malaysia (IIUM) laboratory and the experimental data were used to validate the model. The simulated results of the deterministic and stochastic models were consistent with the experimental data and low values of root mean square error (RMSE) in SDEs model. This indicated good fit of the SDEs in modelling the proliferation of the cancer cells in the presence of TQ. Modelling of the system was extended to the mechanism of the apoptotic signalling pathway for cancer cells in the presence of TQ. Two pathways, which are the intrinsic mitochondrial pathway that promotes the activation of the Caspase 3 (pathway 1) and the intrinsic mitochondrial pathway that promotes the activation of the Caspase 10 (pathway 2) had been identified. The kinetic reaction of those pathways has been developed and mathematical model of a system of ODEs was constructed based on the biochemical kinetic reactions of the pathways 1 and 2. Then, the perturbation was performed through the kinetic rate parameters of external growth factor rate (EGFR) and apoptosis to form a system of SDEs. In this research, the kinetics parameter was estimated using Markov Chain Monte Carlo Method (MCMC). Numerical method of 4-stage stochastic Runge-Kutta (SRK4) was employed to simulate the solution of SDEs. The results showed that as the TQ reacted with EGFR, the activation of Caspase family for intrinsic pathways 1 and 2, led to the activation of the apoptosis mechanism. It was consistent with the results of the experimentation and modelling of the cancer size under treatment using two equations model, which was apoptosis mechanism (increasing trend in the amount of protein) that vi shrank the size of the cancer cells. The newly developed stochastic model can help oncologists to understand the physical and biological barriers in apoptotic activities of anticancer therapeutic. The model can be used to predict the growth of cancer affected by TQ accurately and subsequently help to plan better treatment strategies for cancer.