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Highlights • This is the first mathematical model of RA which quantitatively describes the progression of the disease. • The model explains the roles of macrophages, inflammatory fibroblasts, and T cells in the degradation of cartilage in a joint. • Example is given how to achieve the same efficacy in stabilization of the cartilage while decreasing negative side-effects. Abstract Rheumatoid arthritis (RA) is a common autoimmune disease that mainly affects the joints. It is characterized by synovial inflammation, which may result in cartilage and bone destruction. The present paper develops a mathematical model of chronic RA. The model is represented by a system of partial differential equations (PDEs) in the synovial fluid, the synovial membrane, and the cartilage. The model characterizes the progression of the disease in terms of the degradation of the cartilage. More precisely, we assume a simplified geometry in which the synovial membrane and the cartilage are planar layers adjacent to each other. We then quantify the state of the disease by how much the cartilage layer has decreased, or, equivalently, how much the synovial layer has increased. The model is used to evaluate treatments of RA by currently used drugs, as well as by experimental drugs. [ABSTRACT FROM AUTHOR]

• A model is proposed based on the key role of Regulatory T cells in chemoimmunotherapy. • The model's fixed points are obtained and the bifurcation diagrams are discussed. • New features of personalizing a cancer mathematical model are presented. • Considering Regulatory T cells leads to identifying critical patient conditions more precisely. Studying the mathematical dynamics of cancer has gained the attention of bioengineers in the past three decades. Different kinds of modelling considering various aspects of treatment have been proposed. In this paper, the key role of Regulatory T cells is discussed and a model in ordinary differential equation (ODE) form is proposed by adding this state to the system dynamics considering chemoimmunotherapy treatment. Regulatory T cells are considered as one of the main tumor cells' tactics to deceive the body's immune system. The improved model is verified mathematically and biologically and fits all criteria in both fields. The results show that entering Regulatory T cells state on cancer mathematical modelling for simulating body cells for chemoimmunotherapy provides a way to identify critical cases more carefully, which a simplified model is unable to accomplish. This point emphasizes the fact that this state must be present in cancer modelling to anticipate immune response more accurately. The advanced system fixed points are obtained by the Newton method and bifurcation diagrams are derived and discussed. New features and remarks are proposed during the journey of developing more accurate models that have the best fit with laboratory data. The sensitivity chart of the model is illustrated and novel aspects of discussions are made with the aim of personalizing a model for a patient and identifying critical conditions based on the chart before any treatment begins. This point enables physicians to determine whether critical conditions have occurred for a patient in a specific treatment or not. [ABSTRACT FROM AUTHOR]

• An individual-based model describing interactions between tumour and CD8+ T cells. • The model takes into account tumour antigen expression and presentation. • The degree of tumour heterogeneity has an impact on immune response efficacy. • Wider ranges of antigens promote immune escape of tumours. • Lower levels of antigen presentation diminish the immune action. Intra-tumour heterogeneity (ITH) has a strong impact on the efficacy of the immune response against solid tumours. The number of sub-populations of cancer cells expressing different antigens and the percentage of immunogenic cells (i.e. tumour cells that are effectively targeted by immune cells) in a tumour are both expressions of ITH. Here, we present a spatially explicit stochastic individual-based model of the interaction dynamics between tumour cells and CD8+ T cells, which makes it possible to dissect out the specific impact of these two expressions of ITH on anti-tumour immune response. The set-up of numerical simulations of the model is defined so as to mimic scenarios considered in previous experimental studies. Moreover, the ability of the model to qualitatively reproduce experimental observations of successful and unsuccessful immune surveillance is demonstrated. First, the results of numerical simulations of this model indicate that the presence of a larger number of sub-populations of tumour cells that express different antigens is associated with a reduced ability of CD8+ T cells to mount an effective anti-tumour immune response. Secondly, the presence of a larger percentage of tumour cells that are not effectively targeted by CD8+ T cells may reduce the effectiveness of anti-tumour immunity. Ultimately, the mathematical model presented in this paper may provide a framework to help biologists and clinicians to better understand the mechanisms that are responsible for the emergence of different outcomes of immunotherapy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we sought to identify the CD4+ T-cell dynamics in the course of HIV infection in response to continuous and intermittent intravenous courses of interleukin-2 (IL-2), the principal cytokine responsible for progression of CD4+ T-lymphocytes from the G1 to the S phase of the cell cycle. Based on multivariate regression models, previous literature has concluded that the increase in survival of CD4+ T-cell appears to be the critical mechanism leading to sustained CD4+ T-cell levels in HIV-infected patients receiving intermittent IL-2 therapy. Underscored by comprehensive mathematical modeling, a major finding of the present work is related to the fact that, rather than due to any increase in survival of CD4+ T-cells, the expressive, selective and sustained CD4+ T-cell expansions following IL-2 administration may be related to the role of IL-2 in modulating the dynamics of Fas-dependent apoptotic pathways, such as activation-induced cell death (AICD) or HIV-specific apoptotic routes triggered by viral proteins. [Copyright &y& Elsevier]

Abstract: Epstein-Barr virus (EBV) can infect both B cells and epithelial cells. Infection of B cells enables the virus to persist within a host while infection of epithelial cells is suggested to amplify viral output. Data from a recent study have shown that the virus shedding in EBV positive individuals is relatively stable over short periods of time but varies significantly over long periods. The mechanisms underlying the regulation of virus shedding within a host are not fully understood. In this paper, we construct a model of ordinary differential equations to study the dynamics of virus shedding into the saliva of infected hosts. Infection of epithelial cells is further separated into infection by virus released from B cells and virus released from epithelial cells. We use the model to investigate whether the long-term variation and short-term stability of virus shedding can be generated by three possible factors: stochastic variations in the number of epithelial cells susceptible to virus released from infected B cells, to virus released from infected epithelial cells, or random variation in the probability that CD8+ T cells encounter and successfully kill infected cells. The results support all three factors to explain the long-term variation but only the first and third factors to explain the short-term stability of virus shedding into saliva. Our analysis also shows that clearance of virus shedding is possible only when there is no virus reactivation from B cells. [Copyright &y& Elsevier]

Abstract: The strategies of structured treatment interruptions (STIs) of antiretroviral therapies have been proposed for clinical management of HIV infected patients, but clinical studies on STIs failed to achieve a consistent conclusion for this strategy. To evaluate the STI strategies, in particular, CD4+ T cell count-guided STIs, and explain these controversial conclusions from different clinical studies, in this paper we propose to use piecewise HIV virus dynamic models to quantitatively explore the STI strategies and investigate their dynamic behaviors. Our analysis results indicate that CD4+ T cell counts can be maintained above a safe level using the STI with a single threshold or a threshold window. Numerical simulations show that the CD4+ T cell counts either fluctuate or approach a stable level for a patient, depending on the prescribed upper or lower threshold values. In particular, the CD4+ T cell counts can be stabilized at a desired level if the threshold policy control is applied. The durations of drug-on and drug-off are very sensitive to the prescribed upper or lower threshold levels, which possibly explains why the on–off strategy with fixed schedule or an STI strategy with frequent switches are associated with the high rate of failure. Our findings suggest that it is critical to carefully choose the thresholds of CD4+ T cell count and individualize the STIs for each individual patient based on initial CD4+ T cell counts. [Copyright &y& Elsevier]

Abstract: Some self-reactive immature T cells escape negative selection in the thymus and may cause autoimmune diseases later. In the periphery, if T cells are stimulated insufficiently by peptide-major histocompatibility complex, they become inactive and their production of cytokines changes, a phenomenon called “T cell anergy”. In this paper, we explore the hypothesis that T cell anergy may function to reduce the risk of autoimmunity. The underlying logic is as follows: Since those self-reactive T cells that receive strong stimuli from self-antigens are eliminated in the thymus, T cells that receive strong stimuli in the periphery are likely to be non-self-reactive. As a consequence, when a T cell receives a weak stimulus, the likelihood that the cell is self-reactive is higher than in the case that it receives a strong stimulus. Therefore, inactivation of the T cell may reduce the danger of autoimmunity. We consider the formalism in which each T cell chooses its response depending on the strength of stimuli in order to reduce the risk of autoimmune diseases while maintaining its ability to attack non-self-antigens effectively. The optimal T cell responses to a weak and a strong stimulus are obtained both when the cells respond in a deterministic manner and when they respond in a probabilistic manner. We conclude that T cell anergy is the optimal response when a T cell meets with antigen-presenting cells many times in its lifetime, and when the product of the autoimmunity risk and the number of self-reactive T cells has an intermediate value. [Copyright &y& Elsevier]

Abstract: The adaptive immune system of a vertebrate may attack its own body, causing autoimmune diseases. Regulatory T cells suppress the activity of the autoreactive effector T cells, but they also interrupt normal immune reactions against foreign antigens. In this paper, we discuss the optimal number of regulatory T cells that should be produced. We make the assumptions that some self-reactive immature T cells may fail to interact with their target antigens during the limited training period and later become effector T cells causing autoimmunity, and that regulatory T cells exist that recognize self-antigens. When a regulatory T cell is stimulated by its target self-antigen on an antigen-presenting cell (APC), it stays there and suppresses the activation of other naive T cells on the same APC. Analysis of the benefit and the harm of having regulatory T cells suggests that the optimal number of regulatory T cells depends on the number of self-antigens, the severity of the autoimmunity, the abundance of pathogenic foreign antigens, and the spatial distribution of self-antigens in the body. For multiple types of self-antigen, we discuss the optimal number of regulatory T cells when the self-antigens are localized in different parts of the body and when they are co-localized. We also examine the separate regulation of the abundances of regulatory T cells for different self-antigens, comparing it with the situation in which they are constrained to be equal. [Copyright &y& Elsevier]

Abstract: The immune system of vertebrates may attack its own body and cause autoimmunity diseases. To prevent autoimmunity, regulatory T cells suppress the activity of the autoreactive effector T cells, but they also interrupt normal immune reactions against foreign antigens. In this paper, we discuss the advantage of having some regulatory T cells by considering the host''s ability of coping with foreign antigens and the harm of autoimmunity. Assumptions are as follows: the immature T cells reactive to abundant self-antigens are eliminated, those reactive to rare self-antigen will become regulatory T cells, and those that fail to interact with the antigens to which they are reactive will become effector T cells. Some self-reactive immature T cells may fail to interact with their own target antigens during the limited training period, and will later become effector T cells, causing autoimmunity. Analysis suggests that, having some regulatory T cells can never be advantageous to the host, if activated regulatory T cells suppress effector T cells at any location of the body (global suppression). In contrast, producing some regulatory T cells can be beneficial, if the body is composed of many compartments and regulatory T cells suppress the immune reactions only within the same compartment (localized suppression). This requires regulatory T cells to stop circulating once they are activated by their own target self-antigens. [Copyright &y& Elsevier]

Abstract: Cytotoxic T lymphocytes (CTLs) play a dominant role in the pathogenesis of autoimmune diabetes, commonly denoted Type 1 Diabetes (T1D). These CTLs (notably T cells) recognize and kill insulin-secreting pancreatic cells, reducing their number by . The resulting reduction of insulin secretion causes the defective regulation of glucose metabolism, leading to the characteristic symptoms of diabetes. Recognition of cells as targets by CTLs depends on the interactions between MHC-peptide complexes on the surface of cells and receptors (TCRs) on T cells. Those CTLs with high affinity TCRs (also called high avidity T cells) cause most of the harm, while those with low affinity TCRs (also called low avidity T cells) play a more mysterious role. Recent experimental evidence suggests that low avidity T cells accumulate as memory T cells during the disease and may be protective in NOD mice (a strain prone to developing T1D), delaying disease progression. It has been hypothesized that such low avidity T cells afford disease protection either by crowding the islets of Langerhans, where cells reside, or by killing antigen presenting cells (APCs). In this paper, we explore the hypothesized mechanisms for this protective effect in the context of a series of models for (1) the interactions of low and high avidity T cells, (2) the effect of APCs and (3) the feedback from cell killing to autoantigen-induced T cell proliferation. We analyze properties of these models, noting consistency of predictions with observed behaviour. We then use the models to examine the influence of various treatment strategies on the progression of the disease. The model reveals that progressive accumulation of memory low avidity autoreactive T cells during disease progression makes treatments aimed at expanding these protective T cell types more effective close to, or at the onset of clinical disease. It also provides evidence for the hypothesis that low avidity T cells kill APCs (rather than the alternate hypothesis that they crowd the islets). [Copyright &y& Elsevier]

Abstract: Epstein–Barr virus (EBV) is an important human pathogen that establishes a life-long persistent infection and for which no precise animal model exists. In this paper, we describe in detail an agent-based model and computer simulation of EBV infection. Agents representing EBV and sets of B and T lymphocytes move and interact on a three-dimensional grid approximating Waldeyer''s ring, together with abstract compartments for lymph and blood. The simulation allows us to explore the development and resolution of virtual infections in a manner not possible in actual human experiments. Specifically, we identify parameters capable of inducing clearance, persistent infection, or death. [Copyright &y& Elsevier]

Abstract: Mathematical models of HIV therapy have traditionally amalgamated the action of antiretroviral drugs, trading the complexity of the situation in favour of simpler—and hence mathematically tractable—models. However, the effects of ignoring such dynamics remain underexamined. In this paper, the traditional method of dosing (where the dose is modelled implicitly as a proportional inhibition of viral infection and production) is compared to a model that accounts for drug dynamics via explicit compartments. Four limiting cases are examined: frequent dosing of both major classes of drugs, absence of either drug, frequent dosing of one drug alone, or frequent dosing of the other drug alone. When drugs are absent, both models predict that the virus will dominate and the uninfected T cell counts will be low. When reverse transcriptase inhibitors are given frequently, both models predict that the virus will be theoretically eliminated and the uninfected T cell counts will be high; this is true regardless of whether the reverse transcriptase inhibitors act alone or in conjunction with protease inhibitors. However, if protease inhibitors alone are given frequently, then the implicit model predicts that the virus will be eliminated and the uninfected T cell counts will be high, whereas the (more realistic) explicit model predicts that the reverse situation may occur. In the latter case, critically, protease-only regimens may ultimately result in the death of the patient. It follows that the impact of drug regimens consisting only of protease inhibitors must be urgently re-examined, if such outcomes have been based on overly simplistic modelling. [Copyright &y& Elsevier]

Abstract: Mathematical models of T cell population dynamics after infection typically assume that T cells differentiate according to a linear process in which they first become effector cells, and then after some time, differentiate further into memory cells. In this paper, we offer a different mathematical model which can equally well capture T cell dynamics, using data from lymphocytic choriomeningitis (LCMV) infection. Our model assumes that memory cells are intermediates that further differentiate into effector cells only from additional or stronger antigenic stimulation. Our assumption naturally leads to a testable prediction about the generation of T cell memory—that the memory phenotype of T cells should be present in detectable numbers during the expansion phase of the response. We use our model to estimate a rate of differentiation from memory type cells to effectors. We argue that this differentiation assumption, where memory cells are intermediates, captures recent experimental work on T cell differentiation, and hence this new mathematical model could be helpful in doing further studies of T cell population dynamics. We also propose a method of distinguishing the models by examining the ratio of memory T cells detectable long after an infection to the peak numbers of T cells at the end of the expansion phase. [Copyright &y& Elsevier]

Abstract: Antigen recognition by T cells is a key event in the adaptive immune response. T cells scan the surface of antigen-presenting cells (APCs) or target cells for specific peptides bound to MHC molecules. In the physiological setting, a typical APC presents tens of thousands of diverse endogenous self-derived peptides complexed to MHC (pMHC complexes). When ‘foreign’ peptides are presented, they constitute a small fraction of the total surface peptide repertoire. As T cells seem to be capable of discerning minute amounts of ‘foreign’ peptides among a complex background of self-peptides, endogenous peptides are generally assumed to play no role in recognition. However, recent results suggest that these background peptides may alter the sensitivity of T cells to foreign peptides. Current experimental limitations preclude analysis of peptide mixtures approaching physiological complexity, making it difficult to further address the role of complex background peptides. In this paper, we present a computational model to test how complex, varied peptide populations on an APC could potentially modulate a T cell''s ability to detect the presence of small numbers of agonist peptides among a diverse population. We use the model to investigate the notion that under physiological conditions, T cell recognition of foreign peptides is context dependent, that is, T cells process signals gathered from all pMHC interactions, not just from a few agonist peptides while ignoring all others. [Copyright &y& Elsevier]

In this paper, we propose and analyse a mathematical model for chronic myelogenous leukemia (CML), a cancer of the blood. We model the interaction between naive T cells, effector T cells, and CML cancer cells in the body, using a system of ordinary differential equations which gives rates of change of the three cell populations. One of the difficulties in modeling CML is the scarcity of experimental data which can be used to estimate parameters values. To compensate for the resulting uncertainties, we use Latin hypercube sampling (LHS) on large ranges of possible parameter values in our analysis. A major goal of this work is the determination of parameters which play a critical role in remission or clearance of the cancer in the model. Our analysis examines 12 parameters, and identifies two of these, the growth and death rates of CML, as critical to the outcome of the system. Our results indicate that the most promising research avenues for treatments of CML should be those that affect these two significant parameters (CML growth and death rates), while altering the other parameters should have little effect on the outcome. [Copyright &y& Elsevier]

Fratricide between CD8+ T lymphocytes is known to occur in HTLV-I and possibly HSV-1 and HIV-1 infection. However it is not known what effect, if any, T-cell fratricide has on the course of infection. Here we present simple mathematical techniques to investigate T-cell fratricide with particular reference to HTLV-I infection. Using a general model we predict the qualitative and quantitative effect of fratricide on HTLV-I equilibrium proviral load. We also investigate the effect of fratricide on the probability of viral clearance.We show that, surprisingly, fratricide can lead either to an increase or a decrease in equilibrium proviral load. We derive the conditions necessary for fratricide to cause a decrease in load and deduce that, for the five HTLV-I-positive patients considered here, fratricide has probably caused an increase in equilibrium load. We also estimate the percentage increase in load that is attributable to fratricide and determine the parameters that should be measured in order to improve this estimate. Finally, we show that fratricide reduces the probability of viral clearance.Mathematical modelling of HTLV-I infection, as is often the case in biology, is severely hampered by a lack of experimental data. Consequently it is difficult to know what functional form a model should take. The behaviour of complex nonlinear systems is highly model-dependent. Predictions based on theoretical models are therefore sensitive to the choice of model; this is a very severe problem that undermines and limits the success of the application of mathematics to immunology. In this paper we reduce the model dependency of the results in two ways—by considering (analytically) a general model with a minimal number of assumptions and, where this is not possible, by checking (numerically) that a wide range of models yield the same results. We therefore begin to develop two practical methods for dealing with the problem of robustness in mathematical models of the immune system. [Copyright &y& Elsevier]

Recent experimental results show that even brief stimulation with antigen can cause antigen-specific CD8 T-cells to undergo sustained proliferation followed by differentiation into memory cells. These results show that the dynamics of these immune responses are not governed by constant monitoring of antigen levels, but rather that following stimulation immune cells commit to a “program”. At present relatively little is known about the program which governs CD8 cell proliferation and differentiation. For example, we do not know whether the program is completely specified by the initial encounter of a T cell with antigen, or whether it subsequently can be modified by the amount of antigen present. Nor do we know whether the entire program for T cell proliferation and differentiation resides within the T cell itself, or whether some component(s) of the program are determined by cells or molecules external to the CD8 cell. In this paper we construct simple mathematical models which incorporate antigen-independent proliferation and differentiation of CD8 cells during acute infections. We use these models to determine what characteristics the program must have in order to be consistent with the existing data on the dynamics of CD8 responses, and in particular to answer the questions posed above. Our results suggest that the program is not completely defined by the initial encounter of T cell with antigen but may be augmented by exposure to antigen in a brief window shortly after infection; furthermore, parts of the program may reside external to the T-cells. Finally we examine some of the consequences of the “program” for pathogen–host coevolution. [Copyright &y& Elsevier]

• A mathematical model for combination therapy using engineered T cells and IL-2. • The results provide a TCR T cell dose window for a successful therapy. • Combination therapy does not always provide a better outcome. Adoptive T cell based immunotherapy is gaining significant traction in cancer treatment. Despite its limited efficacy so far in treating solid tumors compared to hematologic cancers, recent advances in T cell engineering render this treatment increasingly more successful in solid tumors, demonstrating its broader therapeutic potential. In this paper we develop a mathematical model to study the efficacy of engineered T cell receptor (TCR) T cell therapy targeting the E7 antigen in cervical cancer cell lines. We consider a dynamical system that follows the population of cancer cells, TCR T cells, and IL-2 treatment concentration. We demonstrate that there exists a TCR T cell dosage window for a successful cancer elimination that can be expressed in terms of the initial tumor size. We obtain the TCR T cell dose for two cervical cancer cell lines: 4050 and CaSki. Finally, a combination therapy of TCR T cell and IL-2 treatment is studied. We show that certain treatment protocols can improve therapy responses in the 4050 cell line, but not in the CaSki cell line. [ABSTRACT FROM AUTHOR]