A mixed radiotherapy and chemotherapy model for treatment of cancer with metastasis

Abstract

In this paper, a mathematical model of cancer treatment, in the form of a system of ordinary differential equations, by chemotherapy and radiotherapy where there is metastasis from a primary to a secondary site has been proposed and analyzed. The interaction between immune cells and cancer cells has been examined, and the chemotherapy agent has been considered as a predator on both normal and cancer cells. The metastasis may be time delayed. For better investigation of the treatment process and based on physical investigation, the immanent effects of inputs on cancer dynamic have been investigated. It is supposed that the interaction between NK cells and tumor cells changes during the chemotherapy. Thisnovel approach is useful not only to gain a broad understanding of the specific system dynamics but also to guide the development of combination therapies. The analysis is carried out both analytically (where possible) and numerically. By considering such immanent effects, the tumor-free equilibrium point will be stable at the end of treatment, and the tumor can not recur again, and the patient will totally recover. So, the present analysis suggests that a proper treatment method should change the dynamics of the cancer instead of only reducing the population of cancer cells.

Mathematical Modelling for the Role of CD4+T Cells in Tumor-Immune Interactions

“…where N(t) is the natural killer cell population, L(t) is the CD8 + T cell population, Y(t) is the CD4 + T cell population, C(t) is the circulating lymphocyte cell population not including natural killer cells and active CD8 + Tand CD4 + Tcells [50,51], I(t) is the concentration of the IL-2 cytokine in the body, and D is defined as…”

“…is growth term is tied to the overall immune health levels as measured by the population of circulating lymphocytes [54][55][56]. − fN represents natural killers' turnover [51]. e term (g T 2 /(h + T 2 )) N represents the recruitment of natural killer cells induced due to the existence of the tumor [42,54,57].…”

“…e term (g T 2 /(h + T 2 )) N represents the recruitment of natural killer cells induced due to the existence of the tumor [42,54,57]. e next term − p N T is natural killer death by exhaustion of tumor-killing resources [40,42,51]. In equation (3), − m L represents the inactivation of CD8 + T cells consisting only of natural death rate since no CD8 + T cells are assumed to be present in the absence of tumor cells [40,42].…”

“…e term (j D 2 T 2 /(k + D 2 T 2 ))L represents the recruitment in CD8 + T cell population due to the presence of the tumor [40,42,58]. e next term − qLT is the CD8 + T cell death by exhaustion of tumor-killing resources [40,42,51]. e term r 1 NT represents CD8 + Tcell stimulation by natural killerlysed tumor cell debris, and the term r 2 CT represents the activation of native CD8 + T cells in the general lymphocyte population [42,51].…”

Optimal chemotherapy and immunotherapy schedules for a cancer‐obesity model with Caputo time fractional derivative

“…Moreover, four optimal control problems (OCPs) for chemotherapy schedule are investigated by Engelhart et al, 11 and different choices of the objective functions in the framework of chemotherapy are compared. The combination of chemotherapy and radiotherapy to eradicate the cancer with metastasis is discussed by Ghaffari et al 12 An adaptive robust control is proposed to adjust the drug dosages with an extended Kalman filter observer by Rokhforoz et al 13 An optimal control strategy based on a linear time-varying approximation technique is proposed by Itik et al 14 On the other hand, an OCP with a free final time is solved for tumor-immune interactions with the aim of minimizing not only the tumor population but also the treatment period by Alkama et al 15 A model-free method for chemotherapy based on reinforcement learning is proposed by Padmanabhan et al 16 using the closed-loop control. In particular, they developed an optimal controller using Q-learning algorithm for cancer chemotherapy treatment.…”