A dynamical perspective of CTL cross-priming and regulation: implications for cancer immunology

Threshold dynamics of a HCV model with virus to cell transmission in both liver with CTL immune response and the extrahepatic tissue

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number R01 and immune reproduction number R02 are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when R01<1 , the virus will be cleared out eventually and the CTL immune response will also disappear; when R02<1<R01 , the virus persists within the host, but the CTL immune response disappears eventually; when R02>1 , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given.

Mathematical Analysis of a Mathematical Model of Chemovirotherapy: Effect of Drug Infusion Method

A mathematical model for the treatment of cancer using chemovirotherapy is developed with the aim of determining the efficacy of three drug infusion methods: constant, single bolus, and periodic treatments. The model is in the form of ODEs and is further extended into DDEs to account for delays as a result of the infection of tumor cells by the virus and chemotherapeutic drug responses. Analysis of the model is carried out for each of the three drug infusion methods. Analytic solutions are determined where possible and stability analysis of both steady state solutions for the ODEs and DDEs is presented. The results indicate that constant and periodic drug infusion methods are more efficient compared to a single bolus injection. Numerical simulations show that with a large virus burst size, irrespective of the drug infusion method, chemovirotherapy is highly effective compared to either treatments. The simulations further show that both delays increase the period within which a tumor can be cleared from body tissue.

Dynamical analysis on a chronic hepatitis C virus infection model with immune response

A mathematical model for HCV infection is established, in which the effect of dendritic cells (DC) and cytotoxic T lymphocytes (CTL) on HCV infection is considered. The basic reproduction numbers of chronic HCV infection and immune control are found. The obtained results show that the infection dies out finally as the basic reproduction number of HCV infection is less than unity, and the infection becomes chronic as it is greater than unity. In the presence of chronic infection, the existence of immune control equilibrium is discussed completely, which illustrates that the backward bifurcation may occur under certain conditions, and that the two quantities, the sizes of the activated DC and the removed CTL during their average life-terms, play a critical role in controlling chronic HCV infection and immune response. The occurrence of backward bifurcation implies that there may be bistability for the model, i.e., the outcome of infection depends on the initial situation. By choosing the activated rate of non-activated DC or the cross-representation rate of activated DC as bifurcation number, Hopf bifurcation for certain condition shows the existence of periodic solution of the model. Again, numerical simulations suggest the dynamical complexity of the model including the instability of immune control equilibrium and the existence of stable periodic solution.

Understanding the roles of activation threshold and infections in the dynamics of autoimmune disease

Onset and development of autoimmunity have been attributed to a number of factors, including genetic predisposition, age and different environmental factors. In this paper we discuss mathematical models of autoimmunity with an emphasis on two particular aspects of immune dynamics: breakdown of immune tolerance in response to an infection with a pathogen, and interactions between T cells with different activation thresholds. We illustrate how the explicit account of T cells with different activation thresholds provides a viable model of immune dynamics able to reproduce several types of immune behaviour, including normal clearance of infection, emergence of a chronic state, and development of a recurrent infection with autoimmunity. We discuss a number of open research problems that can be addressed within the same modelling framework.

How do tumors actively escape from host immunosurveillance?

The immunological background for the process of tumor growth is still obscure. However, our understanding of what happens could have important consequences, namely in the context of cancer immunotherapy. A tumor is able to grow in the host environment either because it is recognizable as normal tissue and tolerated by host immune cells, or because it can "escape" from host immunosurveillance. According to the second option the mechanisms of tumor recognition and consequent destruction are actively disturbed by such processes as: change of tumor immunogenicity, production of tumor-derived regulatory molecules, and interaction of cancer cells with tumor-infiltrating immune cells. The results of studies devoted to the problem of immunoregulation in the tumor environment seem to support the "escape" hypothesis.

How regulatory CD25+CD4+ T cells impinge on tumor immunobiology: the differential response of tumors to therapies

Aiming to get a better insight on the impact of regulatory CD25(+)CD4(+) T cells in tumor-immunobiology, a simple mathematical model was previously formulated and studied. This model predicts the existence of two alternative modes of uncontrolled tumor growth, which differ on their coupling with the immune system, providing a plausible explanation to the observation that the development of some tumors expand regulatory T cells whereas others do not. We report now the study of how these two tumor classes respond to different therapies, namely vaccination, immune suppression, surgery, and their different combinations. We show 1) how the timing and the dose applied in each particular treatment determine whether the tumor will be rejected, with or without concomitant autoimmunity, or whether it will continue progressing with slower or faster pace; 2) that both regulatory T cell-dependent and independent tumors are equally sensitive to vaccination, although the former are more sensitive to T cell depletion treatments and are unresponsive to partial surgery alone; 3) that surgery, suppression, and vaccination treatments, can synergistically improve their individual effects, when properly combined. Particularly, we predict rational combinations helping to overcome the limitation of these individual treatments on the late stage of tumor development.

Dynamical properties of autoimmune disease models: Tolerance, flare-up, dormancy

The mechanisms of autoimmune disease have remained puzzling for a long time. Here we construct a simple mathematical model for autoimmune disease based on the personal immune response function and the target cell growth function. We show that these two functions are sufficient to capture the essence of autoimmune disease and can explain characteristic symptom phases such as tolerance, repeated flare-ups and dormancy. Our results strongly suggest that a more complete understanding of these two functions will underlie the development of an effective therapy for autoimmune disease.

Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls

We investigate a mathematical model of tumor-immune interactions with chemotherapy, and strategies for optimally administering treatment. In this paper we analyze the dynamics of this model, characterize the optimal controls related to drug therapy, and discuss numerical results of the optimal strategies. The form of the model allows us to test and compare various optimal control strategies, including a quadratic control, a linear control, and a state-constraint. We establish the existence of the optimal control, and solve for the control in both the quadratic and linear case. In the linear control case, we show that we cannot rule out the possibility of a singular control. An interesting aspect of this paper is that we provide a graphical representation of regions on which the singular control is optimal.

Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations

We develop and analyze a mathematical model, in the form of a system of ordinary differential equations (ODEs), governing cancer growth on a cell population level with combination immune, vaccine and chemotherapy treatments. We characterize the ODE system dynamics by locating equilibrium points, determining stability properties, performing a bifurcation analysis, and identifying basins of attraction. These system characteristics are useful not only to gain a broad understanding of the specific system dynamics, but also to help guide the development of combination therapies. Numerical simulations of mixed chemo-immuno and vaccine therapy using both mouse and human parameters are presented. We illustrate situations for which neither chemotherapy nor immunotherapy alone are sufficient to control tumor growth, but in combination the therapies are able to eliminate the entire tumor.