The evolution of mathematical immunology

COVID-19 Adaptive Humoral Immunity Models: Weakly Neutralizing Versus Antibody-Disease Enhancement Scenarios

The interplay between the virus, infected cells and immune responses to SARS-CoV-2 is still under debate. By extending the basic model of viral dynamics, we propose here a formal approach to describe neutralisation versus weak (or non-)neutralisation scenarios and compare them with the possible effects of antibody-dependent enhancement (ADE). The theoretical model is consistent with the data available in the literature; we show that both weakly neutralising antibodies and ADE can result in final viral clearance or disease progression, but that the immunodynamics are different in each case. As a significant proportion of the world’s population is already naturally immune or vaccinated, we also discuss the implications for secondary infections after vaccination or in the presence of immune system dysfunctions.

Integrative lymph node-mimicking models created with biomaterials and computational tools to study the immune system

The lymph node (LN) is a vital organ of the lymphatic and immune system that enables timely detection, response, and clearance of harmful substances from the body. Each LN comprises of distinct substructures, which host a plethora of immune cell types working in tandem to coordinate complex innate and adaptive immune responses. An improved understanding of LN biology could facilitate treatment in LN-associated pathologies and immunotherapeutic interventions, yet at present, animal models, which often have poor physiological relevance, are the most popular experimental platforms. Emerging biomaterial engineering offers powerful alternatives, with the potential to circumvent limitations of animal models, for in-depth characterization and engineering of the lymphatic and adaptive immune system. In addition, mathematical and computational approaches, particularly in the current age of big data research, are reliable tools to verify and complement biomaterial works. In this review, we first discuss the importance of lymph node in immunity protection followed by recent advances using biomaterials to create in vitro/vivo LN-mimicking models to recreate the lymphoid tissue microstructure and microenvironment, as well as to describe the related immuno-functionality for biological investigation. We also explore the great potential of mathematical and computational models to serve as in silico supports. Furthermore, we suggest how both in vitro/vivo and in silico approaches can be integrated to strengthen basic patho-biological research, translational drug screening and clinical personalized therapies. We hope that this review will promote synergistic collaborations to accelerate progress of LN-mimicking systems to enhance understanding of immuno-complexity.

A mathematical model of combined CD8 T cell costimulation by 4-1BB (CD137) and OX40 (CD134) receptors

Combined agonist stimulation of the TNFR costimulatory receptors 4-1BB (CD137) and OX40(CD134) has been shown to generate supereffector CD8 T cells that clonally expand to greater levels, survive longer, and produce a greater quantity of cytokines compared to T cells stimulated with an agonist of either costimulatory receptor individually. In order to understand the mechanisms for this effect, we have created a mathematical model for the activation of the CD8 T cell intracellular signaling network by mono- or dual-costimulation. We show that supereffector status is generated via downstream interacting pathways that are activated upon engagement of both receptors, and in silico simulations of the model are supported by published experimental results. The model can thus be used to identify critical molecular targets of T cell dual-costimulation in the context of cancer immunotherapy.

Quantitative Mechanistic Modeling in Support of Pharmacological Therapeutics Development in Immuno-Oncology

Following the approval, in recent years, of the first immune checkpoint inhibitor, there has been an explosion in the development of immuno-modulating pharmacological modalities for the treatment of various cancers. From the discovery phase to late-stage clinical testing and regulatory approval, challenges in the development of immuno-oncology (IO) drugs are multi-fold and complex. In the preclinical setting, the multiplicity of potential drug targets around immune checkpoints, the growing list of immuno-modulatory molecular and cellular forces in the tumor microenvironment-with additional opportunities for IO drug targets, the emergence of exploratory biomarkers, and the unleashed potential of modality combinations all have necessitated the development of quantitative, mechanistically-oriented systems models which incorporate key biology and patho-physiology aspects of immuno-oncology and the pharmacokinetics of IO-modulating agents. In the clinical setting, the qualification of surrogate biomarkers predictive of IO treatment efficacy or outcome, and the corresponding optimization of IO trial design have become major challenges. This mini-review focuses on the evolution and state-of-the-art of quantitative systems models describing the tumor vs. immune system interplay, and their merging with quantitative pharmacology models of IO-modulating agents, as companion tools to support the addressing of these challenges.

Cheater-altruist synergy in public goods games

Much research has focused on the deleterious effects of free-riding in public goods games, and a variety of mechanisms that suppress cheating behavior. Here we argue that under certain conditions cheating can be beneficial to the population. In a public goods game, cheaters do not pay for the cost of the public goods, yet they receive the benefit. Although this free-riding harms the entire population in the long run, the success of cheaters may aid the population when there is a common enemy that antagonizes both cooperators and cheaters. Here we study models of the interactions between tumor cells, which play a public goods game, and the immune system. We investigate three population dynamics models of cancer growth combined with a model of effector cell dynamics. We show that under a public good with a limiting benefit, the presence of cheaters aids the tumor in overcoming immune system suppression, and explore the parameter space wherein it occurs. The mechanism of this phenomenon is that a polymorphism of cheaters and altruists optimizes the average growth rate of the tumor, which is what determines whether or not the immune response is overcome. Our results give support for a possible synergy between cooperators and cheaters in ecological public goods games.

Mathematical modeling of tumor-induced immunosuppression by myeloid-derived suppressor cells: Implications for therapeutic targeting strategies

Myeloid-derived suppressor cells (MDSCs) belong to immature myeloid cells that are generated and accumulated during the tumor development. MDSCs strongly suppress the anti-tumor immunity and provide conditions for tumor progression and metastasis. In this study, we present a mathematical model based on ordinary differential equations (ODE) to describe tumor-induced immunosuppression caused by MDSCs. The model consists of four equations and incorporates tumor cells, cytotoxic T cells (CTLs), natural killer (NK) cells and MDSCs. We also provide simulation models that evaluate or predict the effects of anti-MDSC drugs (e.g., l-arginine and 5-Fluorouracil (5-FU)) on the tumor growth and the restoration of anti-tumor immunity. The simulated results obtained using our model were in good agreement with the corresponding experimental findings on the expansion of splenic MDSCs, immunosuppressive effects of these cells at the tumor site and effectiveness of l-arginine and 5-FU on the re-establishment of antitumor immunity. Regarding this latter issue, our predictive simulation results demonstrated that intermittent therapy with low-dose 5-FU alone could eradicate the tumors irrespective of their origins and types. Furthermore, at the time of tumor eradication, the number of CTLs prevailed over that of cancer cells and the number of splenic MDSCs returned to the normal levels. Finally, our predictive simulation results also showed that the addition of l-arginine supplementation to the intermittent 5-FU therapy reduced the time of the tumor eradication and the number of iterations for 5-FU treatment. Thus, the present mathematical model provides important implications for designing new therapeutic strategies that aim to restore antitumor immunity by targeting MDSCs.

Mathematical Models for Immunology: Current State of the Art and Future Research Directions

The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years.

Computational immuno-biology for organ transplantation and regenerative medicine

Organ transplantation and regenerative medicine are adopted platforms that provide replacement tissues and organs from natural or engineered sources. Acceptance, tolerance and rejection depend greatly on the proper control of the immune response against graft antigens, motivating the development of immunological and genetical therapies that prevent organ failure. They rely on a complete, or partial, understanding of the immune system. Ultimately, they are innovative technologies that ensure permanent graft tolerance and indefinite graft survival through the modulation of the immune system. Computational immunology has arisen as a tool towards a mechanistic understanding of the biological and physicochemical processes surrounding an immune response. It comprehends theoretical and computational frameworks that simulate immuno-biological systems. The challenge is centered on the multi-scale character of the immune system that spans from atomistic scales, during peptide-epitope and protein interactions, to macroscopic scales, for lymph transport and organ-organ reactions. In this paper, we discuss, from an engineering perspective, the biological processes that are involved during the immune response of organ transplantation. Previous computational efforts, including their characteristics and visible limitations, are described. Finally, future perspectives and challenges are listed to motivate further developments.

Comparing stochastic differential equations and agent-based modelling and simulation for early-stage cancer

There is great potential to be explored regarding the use of agent-based modelling and simulation as an alternative paradigm to investigate early-stage cancer interactions with the immune system. It does not suffer from some limitations of ordinary differential equation models, such as the lack of stochasticity, representation of individual behaviours rather than aggregates and individual memory. In this paper we investigate the potential contribution of agent-based modelling and simulation when contrasted with stochastic versions of ODE models using early-stage cancer examples. We seek answers to the following questions: (1) Does this new stochastic formulation produce similar results to the agent-based version? (2) Can these methods be used interchangeably? (3) Do agent-based models outcomes reveal any benefit when compared to the Gillespie results? To answer these research questions we investigate three well-established mathematical models describing interactions between tumour cells and immune elements. These case studies were re-conceptualised under an agent-based perspective and also converted to the Gillespie algorithm formulation. Our interest in this work, therefore, is to establish a methodological discussion regarding the usability of different simulation approaches, rather than provide further biological insights into the investigated case studies. Our results show that it is possible to obtain equivalent models that implement the same mechanisms; however, the incapacity of the Gillespie algorithm to retain individual memory of past events affects the similarity of some results. Furthermore, the emergent behaviour of ABMS produces extra patters of behaviour in the system, which was not obtained by the Gillespie algorithm.

Investigating mathematical models of immuno-interactions with early-stage cancer under an agent-based modelling perspective

Many advances in research regarding immuno-interactions with cancer were developed with the help of ordinary differential equation (ODE) models. These models, however, are not effectively capable of representing problems involving individual localisation, memory and emerging properties, which are common characteristics of cells and molecules of the immune system. Agent-based modelling and simulation is an alternative paradigm to ODE models that overcomes these limitations. In this paper we investigate the potential contribution of agent-based modelling and simulation when compared to ODE modelling and simulation. We seek answers to the following questions: Is it possible to obtain an equivalent agent-based model from the ODE formulation? Do the outcomes differ? Are there any benefits of using one method compared to the other? To answer these questions, we have considered three case studies using established mathematical models of immune interactions with early-stage cancer. These case studies were re-conceptualised under an agent-based perspective and the simulation results were then compared with those from the ODE models. Our results show that it is possible to obtain equivalent agent-based models (i.e. implementing the same mechanisms); the simulation output of both types of models however might differ depending on the attributes of the system to be modelled. In some cases, additional insight from using agent-based modelling was obtained. Overall, we can confirm that agent-based modelling is a useful addition to the tool set of immunologists, as it has extra features that allow for simulations with characteristics that are closer to the biological phenomena.

Novel methods for quantifying individual host response to infectious pathogens for genetic analyses

We propose two novel approaches for describing and quantifying the response of individual hosts to pathogen challenge in terms of infection severity and impact on host performance. The first approach is a direct extension of the methodology for estimating group tolerance (the change in performance with respect to changes in pathogen burden in a host population) to the level of individuals. The second approach aims to capture the dynamic aspects of individual resistance and tolerance over the entire time course of infections. In contrast to the first approach, which provides a means to disentangle host resistance from tolerance, the second approach focuses on the combined effects of both characteristics. Both approaches provide new individual phenotypes for subsequent genetic analyses and come with specific data requirements. In particular, both approaches rely on the availability of repeated performance and pathogen burden measurements of individuals over the time course of one or several episodes of infection. Consideration of individual tolerance also highlights some of the assumptions hidden within the concept of group tolerance, indicating where care needs to be taken in trait definition and measurement.

Mathematical modeling of biological systems

Mathematical and computational models are increasingly used to help interpret biomedical data produced by high-throughput genomics and proteomics projects. The application of advanced computer models enabling the simulation of complex biological processes generates hypotheses and suggests experiments. Appropriately interfaced with biomedical databases, models are necessary for rapid access to, and sharing of knowledge through data mining and knowledge discovery approaches.

A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays

Some key features of a mathematical description of an immune response are an estimate of the number of responding cells and the manner in which those cells divide, differentiate, and die. The intracellular dye CFSE is a powerful experimental tool for the analysis of a population of dividing cells, and numerous mathematical treatments have been aimed at using CFSE data to describe an immune response [30,31,32,37,38,42,48,49]. Recently, partial differential equation structured population models, with measured CFSE fluorescence intensity as the structure variable, have been shown to accurately fit histogram data obtained from CFSE flow cytometry experiments [18,19,52,54]. In this report, the population of cells is mathematically organized into compartments, with all cells in a single compartment having undergone the same number of divisions. A system of structured partial differential equations is derived which can be fit directly to CFSE histogram data. From such a model, cell counts (in terms of the number of divisions undergone) can be directly computed and thus key biological parameters such as population doubling time and precursor viability can be determined. Mathematical aspects of this compartmental model are discussed, and the model is fit to a data set. As in [18,19], we find temporal and division dependence in the rates of proliferation and death to be essential features of a structured population model for CFSE data. Variability in cellular autofluorescence is found to play a significant role in the data, as well. Finally, the compartmental model is compared to previous work, and statistical aspects of the experimental data are discussed.

ENDOGENOUS VERSUS EXOGENOUS ORIGINS OF DISEASES

Many illnesses are associated with an alteration of the immune system homeostasis due to a combination of factors, including exogenous bacterial insult, endogenous breakdown (e.g. development of a disease that results in immuno suppression), or an exogenous hit like surgery that simultaneously alters immune responsiveness and provides access to bacteria, or genetic disorder. We conjecture that, as a consequence of the co-evolution of the human immune system with the ecology of pathogens, the homeostasis of the immune system requires the influx of pathogens. This allows the immune system to keep the ever-present pathogens under control and to react and adjust fast to bursts of infections. We construct the simplest and most general system of rate equations which describes the dynamics of five compartments: healthy cells, altered cells, adaptive immune cells, innate immune cells, and pathogens. We study four regimes obtained with or without auto-immune disorder and with or without spontaneous proliferation of infected cells. For each of the four regimes, the phase space is always characterized by four (but not necessary identical) coexisting stationary structurally stable states. Over all four regimes among the 4 × 4 possibilities, there are only seven different states that are naturally described by the model: (i) strong healthy immune system, (ii) healthy organism with evanescent immune cells, (iii) chronic infections, (iv) strong infections, (v) cancer, (vi) critically ill state, and (vii) death. Our description provides a natural framework for describing the relationships and transitions between these seven states. The analysis of stability conditions demonstrates that these seven states depend on the balance between the robustness of the immune system and the influx of pathogens. In particular, the healthy state A is found to exist only under the influence of a sufficiently large pathogen flux, which suggests that health is not the absence of pathogens, but rather a strong ability to find balance by counteracting any pathogen attack.

A new model for the estimation of cell proliferation dynamics using CFSE data

CFSE analysis of a proliferating cell population is a popular tool for the study of cell division and divisionlinked changes in cell behavior. Recently Banks et al. (2011), Luzyanina et al. (2009), Luzyanina et al. (2007), a partial differential equation (PDE) model to describe lymphocyte dynamics in a CFSE proliferation assay was proposed. We present a significant revision of this model which improves the physiological understanding of several parameters. Namely, the parameter used previously as a heuristic explanation for the dilution of CFSE dye by cell division is replaced with a more physical component, cellular autofluorescence. The rate at which label decays is also quantified using a Gompertz decay process. We then demonstrate a revised method of fitting the model to the commonly used histogram representation of the data. It is shown that these improvements result in a model with a strong physiological basis which is fully capable of replicating the behavior observed in the data.

Proteomic approaches to understanding the role of the cytoskeleton in host-defense mechanisms

The cytoskeleton is a cellular scaffolding system whose functions include maintenance of cellular shape, enabling cellular migration, division, intracellular transport, signaling and membrane organization. In addition, in immune cells, the cytoskeleton is essential for phagocytosis. Following the advances in proteomics technology over the past two decades, cytoskeleton proteome analysis in resting and activated immune cells has emerged as a possible powerful approach to expand our understanding of cytoskeletal composition and function. However, so far there have only been a handful of studies of the cytoskeleton proteome in immune cells. This article considers promising proteomics strategies that could augment our understanding of the role of the cytoskeleton in host-defense mechanisms.

Predicting outcomes of prostate cancer immunotherapy by personalized mathematical models

BACKGROUND

Therapeutic vaccination against disseminated prostate cancer (PCa) is partially effective in some PCa patients. We hypothesized that the efficacy of treatment will be enhanced by individualized vaccination regimens tailored by simple mathematical models.

METHODOLOGY/PRINCIPAL FINDINGS

We developed a general mathematical model encompassing the basic interactions of a vaccine, immune system and PCa cells, and validated it by the results of a clinical trial testing an allogeneic PCa whole-cell vaccine. For model validation in the absence of any other pertinent marker, we used the clinically measured changes in prostate-specific antigen (PSA) levels as a correlate of tumor burden. Up to 26 PSA levels measured per patient were divided into each patient's training set and his validation set. The training set, used for model personalization, contained the patient's initial sequence of PSA levels; the validation set contained his subsequent PSA data points. Personalized models were simulated to predict changes in tumor burden and PSA levels and predictions were compared to the validation set. The model accurately predicted PSA levels over the entire measured period in 12 of the 15 vaccination-responsive patients (the coefficient of determination between the predicted and observed PSA values was R(2) = 0.972). The model could not account for the inconsistent changes in PSA levels in 3 of the 15 responsive patients at the end of treatment. Each validated personalized model was simulated under many hypothetical immunotherapy protocols to suggest alternative vaccination regimens. Personalized regimens predicted to enhance the effects of therapy differed among the patients.

CONCLUSIONS/SIGNIFICANCE

Using a few initial measurements, we constructed robust patient-specific models of PCa immunotherapy, which were retrospectively validated by clinical trial results. Our results emphasize the potential value and feasibility of individualized model-suggested immunotherapy protocols.

Dynamic models of immune responses: what is the ideal level of detail?

Background

One of the goals of computational immunology is to facilitate the study of infectious diseases. Dynamic modeling is a powerful tool to integrate empirical data from independent sources, make novel predictions, and to foresee the gaps in the current knowledge. Dynamic models constructed to study the interactions between pathogens and hosts' immune responses have revealed key regulatory processes in the infection.

Optimum complexity and dynamic modeling

We discuss the usability of various deterministic dynamic modeling approaches to study the progression of infectious diseases. The complexity of these models is dependent on the number of components and the temporal resolution in the model. We comment on the specific use of simple and complex models in the study of the progression of infectious diseases.

Conclusions

Models of sub-systems or simplified immune response can be used to hypothesize phenomena of host-pathogen interactions and to estimate rates and parameters. Nevertheless, to study the pathogenesis of an infection we need to develop models describing the dynamics of the immune components involved in the progression of the disease. Incorporation of the large number and variety of immune processes involved in pathogenesis requires tradeoffs in modeling.

ImmunoGrid: towards agent-based simulations of the human immune system at a natural scale

The ultimate aim of the EU-funded ImmunoGrid project is to develop a natural-scale model of the human immune system-that is, one that reflects both the diversity and the relative proportions of the molecules and cells that comprise it-together with the grid infrastructure necessary to apply this model to specific applications in the field of immunology. These objectives present the ImmunoGrid Consortium with formidable challenges in terms of complexity of the immune system, our partial understanding about how the immune system works, the lack of reliable data and the scale of computational resources required. In this paper, we explain the key challenges and the approaches adopted to overcome them. We also consider wider implications for the present ambitious plans to develop natural-scale, integrated models of the human body that can make contributions to personalized health care, such as the European Virtual Physiological Human initiative. Finally, we ask a key question: How long will it take us to resolve these challenges and when can we expect to have fully functional models that will deliver health-care benefits in the form of personalized care solutions and improved disease prevention?

ImmunoGrid, an integrative environment for large-scale simulation of the immune system for vaccine discovery, design and optimization

Vaccine research is a combinatorial science requiring computational analysis of vaccine components, formulations and optimization. We have developed a framework that combines computational tools for the study of immune function and vaccine development. This framework, named ImmunoGrid combines conceptual models of the immune system, models of antigen processing and presentation, system-level models of the immune system, Grid computing, and database technology to facilitate discovery, formulation and optimization of vaccines. ImmunoGrid modules share common conceptual models and ontologies. The ImmunoGrid portal offers access to educational simulators where previously defined cases can be displayed, and to research simulators that allow the development of new, or tuning of existing, computational models. The portal is accessible at <igrid-ext.cryst.bbk.ac.uk/immunogrid>.

Immunoinformatics: an overview of computational tools and techniques for understanding immune function

In recent years, there has been a rapid expansion in the application of information technology to biological data. Although the use of information science techniques is less common for the discipline of immunology, this field has seen great strides in recent years. This review addresses why in silico modeling is needed in immunology research, highlights some of the major areas of research and suggests what may be important for the future of immunoinformatics.