A Mathematical Model for DC Vaccine Treatment of Type I Diabetes

Validation of a Mathematical Model Describing the Dynamics of Chemotherapy for Chronic Lymphocytic Leukemia In Vivo

In recent years, mathematical models have developed into an important tool for cancer research, combining quantitative analysis and natural processes. We have focused on Chronic Lymphocytic Leukemia (CLL), since it is one of the most common adult leukemias, which remains incurable. As the first step toward the mathematical prediction of in vivo drug efficacy, we first found that logistic growth best described the proliferation of fluorescently labeled murine A20 leukemic cells injected in immunocompetent Balb/c mice. Then, we tested the cytotoxic efficacy of Ibrutinib (Ibr) and Cytarabine (Cyt) in A20-bearing mice. The results afforded calculation of the killing rate of the A20 cells as a function of therapy. The experimental data were compared with the simulation model to validate the latter’s applicability. On the basis of these results, we developed a new ordinary differential equations (ODEs) model and provided its sensitivity and stability analysis. There was excellent accordance between numerical simulations of the model and results from in vivo experiments. We found that simulations of our model could predict that the combination of Cyt and Ibr would lead to approximately 95% killing of A20 cells. In its current format, the model can be used as a tool for mathematical prediction of in vivo drug efficacy, and could form the basis of software for prediction of personalized chemotherapy.

Simulation Model for Hashimoto Autoimmune Thyroiditis Disease

Hashimoto thyroiditis (HT) is a pathology that often causes a gradual thyroid insufficiency in affected patients due to the autoimmune destruction of this gland. The cellular immune response mediated by T helper lymphocytes TH1 and TH17 can induce the HT disease. In this pathologic condition, there is an imbalance between the TH17 and Treg lymphocytes as well as a gut microbiota dysfunction. The objective of this work was to describe the interactions of the cell subpopulations that participate in HT. To achieve this goal, we generated a mathematical model that allowed the simulation of different scenarios for the dynamic interaction between thyroid cells, the immune system, and the gut microbiota. We used a hypothetical-deductive design of mathematical modeling based on a system of ordinary differential equations, where the state variables are the TH1, TH17, and Treg lymphocytes, the thyrocytes, and the bacteria from gut microbiota. This work generated a compartmental model of the cellular immune response occurring in the thyroid gland. It was observed that TH1 and TH17 lymphocytes could increase the immune cells' activity, as well as activate effector cells directly and trigger the apoptosis and inflammation processes of healthy thyrocytes indirectly. Likewise, the model showed that a reduction in Treg lymphocytes could increase the activity of TH17 lymphocytes when an imbalance of the gut microbiota composition occurred. The numerical results highlight the TH1, TH17, and bacterial balance of the gut microbiota activities as important factors for the development of HT disease.

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

Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8+T cells, but they neglected the role of CD4+T cells. Other models were constructed to study the role of CD4+T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4+T cells, CD8+T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge-Kutta method failed to solve it. Consequently, the "Adams predictor-corrector" method is used. The results reveal that the patient's immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4+T cells can be an alternative to both CD8+T cell therapy and cytokines in some cases. Moreover, CD4+T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4+T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8+T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients' treatment intervention strategies.