Mathematical Modelling of Immune Response in Tissues

A besyian regularisation neural network approach for hepatitis B virus spread prediction and immune system therapy model

The primary aim of the article is to analyze the response of the human immune system when it encounters the hepatitis B virus. This is done using a mathematical system of differential equations. The differential equation system has six components, likely representing various aspects of the immune response or virus dynamics. A Bayesian regularization neural network has been presented in the process of training. These networks are employed to find solutions for different categories or scenarios related to hepatitis B infection. The Adams method is used to generate reference data sets. The back-propagated artificial neural network, based on Bayesian regularization, is trained and validated using the generated data. The data is divided into three sets: 90% for training and 5% each for testing and validation. The correctness and effectiveness of the proposed neural network model have been assessed using various evaluation metrics. The metrics have been used in this study are Mean Square Error (MSE), histogram errors, and regression plots. These measures provide support to the neural network to approximate the immune response to the hepatitis B virus.

Characterization of differences in immune responses during bolus and continuous infusion endotoxin challenges using mathematical modelling

Endotoxin administration is commonly used to study the inflammatory response, and though traditionally given as a bolus injection, it can be administered as a continuous infusion over multiple hours. Several studies hypothesize that the latter better represents the prolonged and pronounced inflammation observed in conditions like sepsis. Yet very few experimental studies have administered endotoxin using both strategies, leaving significant gaps in determining the underlying mechanisms responsible for their differing immune responses. We used mathematical modelling to analyse cytokine data from two studies administering a 2 ng kg-1 dose of endotoxin, one as a bolus and the other as a continuous infusion over 4 h. Using our model, we simulated the dynamics of mean and subject-specific cytokine responses as well as the response to long-term endotoxin administration. Cytokine measurements revealed that the bolus injection led to significantly higher peaks for interleukin (IL)-8, while IL-10 reaches higher peaks during continuous administration. Moreover, the peak timing of all measured cytokines occurred later with continuous infusion. We identified three model parameters that significantly differed between the two administration methods. Monocyte activation of IL-10 was greater during the continuous infusion, while tumour necrosis factor α $ {\alpha} $ and IL-8 recovery rates were faster for the bolus injection. This suggests that a continuous infusion elicits a stronger, longer-lasting systemic reaction through increased stimulation of monocyte anti-inflammatory mediator production and decreased recovery of pro-inflammatory catalysts. Furthermore, the continuous infusion model exhibited prolonged inflammation with recurrent peaks resolving within 2 days during long-term (20-32 h) endotoxin administration.

A multiscale hybrid model for exploring the effect of Resolvin D1 on macrophage polarization during acute inflammation

Dysregulated inflammation underlies various diseases. Specialized pro-resolving mediators (SPMs) like Resolvin D1 (RvD1) have been shown to resolve inflammation and halt disease progression. Macrophages, key immune cells that drive inflammation, respond to the presence of RvD1 by polarizing to an anti-inflammatory type (M₂). However, RvD1's mechanisms, roles, and utility are not fully understood. This paper introduces a gene-regulatory network (GRN) model that contains pathways for RvD1 and other SPMs and proinflammatory molecules like lipopolysaccharides. We couple this GRN model to a partial differential equation - agent-based hybrid model using a multiscale framework to simulate an acute inflammatory response with and without the presence of RvD1. We calibrate and validate the model using experimental data from two animal models. The model reproduces the dynamics of key immune components and the effects of RvD1 during acute inflammation. Our results suggest RvD1 can drive macrophage polarization through the G protein-coupled receptor 32 (GRP32) pathway. The presence of RvD1 leads to an earlier and increased M₂ polarization, reduced neutrophil recruitment, and faster apoptotic neutrophil clearance. These results support a body of literature that suggests that RvD1 is a promising candidate for promoting the resolution of acute inflammation. We conclude that once calibrated and validated on human data, the model can identify critical sources of uncertainty, which could be further elucidated in biological experiments and assessed for clinical use.

A model-informed approach to assess the risk of immune checkpoint inhibitor-induced autoimmune myocarditis

Immune checkpoint inhibitors (ICIs), as a novel immunotherapy, are designed to modulate the immune system to attack malignancies. Despite their promising benefits, immune-related adverse events (IRAEs) may occur, and incidences are bound to increase with surging demand of this class of drugs in treating cancer. Myocarditis, although rare compared to other IRAEs, has a significantly higher fatal frequency. Due to the overwhelming complexity of the immune system, this condition is not well understood, despite the significant research efforts devoted to it. To better understand the development and progression of autoimmune myocarditis and the roles of ICIs therein, we suggest a new approach: mathematical modelling. Mathematical modelling of myocarditis has enormous potential to determine which parts of the immune system are critical to the development and progression of the disease, and therefore warrant further investigation. We provide the immunological background needed to develop a mathematical model of this disease and review relevant existing models of immunology that serve as the mathematical inspiration needed to develop this field.

Infectious disease model generalization based on diffuse perturbations under conditions of body's temperature reaction

The infectious disease mathematical model is generalized based on the influence of diffuse perturbations on the development of the disease under conditions of the body's temperature reaction. The singularly perturbed model problem was reduced with delay to a sequence of problems without delay, for which the corresponding asymptotic expansions of solutions are obtained. The presented results of computer modeling in various situational states illustrate the expected decrease in the growth rate of the number of viral particles as a result of the action of the body's protective temperature reaction. The results of numerical experiments demonstrate the influence of the diffuse effect of "scattering" of forcing factors on the dynamics of a viral disease under conditions of the body's temperature reaction are presented too. It is noted that the decrease of the model amount of antigens in the epicenter of infection to a non-critical level caused by diffuse "scattering" over a relatively short time period makes them further destroyed by immune agents presented in the body, or requires the introduction of an injection solution with a smaller amount of donor antibodies.

Mathematical modelling of autoimmune myocarditis and the effects of immune checkpoint inhibitors

Autoimmune myocarditis is a rare, but frequently fatal, side effect of immune checkpoint inhibitors (ICIs), a class of cancer therapies. Despite extensive experimental work on the causes, development and progression of this disease, much still remains unknown about the importance of the different immunological pathways involved. We present a mathematical model of autoimmune myocarditis and the effects of ICIs on its development and progression to either resolution or chronic inflammation. From this, we gain a better understanding of the role of immune cells, cytokines and other components of the immune system in driving the cardiotoxicity of ICIs. We parameterise the model using existing data from the literature, and show that qualitative model behaviour is consistent with disease characteristics seen in patients in an ICI-free context. The bifurcation structures of the model show how the presence of ICIs increases the risk of developing autoimmune myocarditis. This predictive modelling approach is a first step towards determining treatment regimens that balance the benefits of treating cancer with the risk of developing autoimmune myocarditis.

A Validated Mathematical Model of the Cytokine Release Syndrome in Severe COVID-19

By June 2021, a new contagious disease, the Coronavirus disease 2019 (COVID-19), has infected more than 172 million people worldwide, causing more than 3.7 million deaths. Many aspects related to the interactions of the disease's causative agent, SAR2-CoV-2, and the immune response are not well understood: the multiscale interactions among the various components of the human immune system and the pathogen are very complex. Mathematical and computational tools can help researchers to answer these open questions about the disease. In this work, we present a system of fifteen ordinary differential equations that models the immune response to SARS-CoV-2. The model is used to investigate the hypothesis that the SARS-CoV-2 infects immune cells and, for this reason, induces high-level productions of inflammatory cytokines. Simulation results support this hypothesis and further explain why survivors have lower levels of cytokines levels than non-survivors.

Hybrid models of chemotaxis with application to leukocyte migration

Many chemical and biological systems involve reacting species with vastly different numbers of molecules/agents. Hybrid simulations model such phenomena by combining discrete (e.g., agent-based) and continuous (e.g., partial differential equation- or PDE-based) descriptors of the dynamics of reactants with small and large numbers of molecules/agents, respectively. We present a stochastic hybrid algorithm to model a stage of the immune response to inflammation, during which leukocytes reach a pathogen via chemotaxis. While large numbers of chemoattractant molecules justify the use of a PDE-based model to describe the spatiotemporal evolution of its concentration, relatively small numbers of leukocytes and bacteria involved in the process undermine the veracity of their continuum treatment by masking the effects of stochasticity and have to be treated discretely. Motility and interactions between leukocytes and bacteria are modeled via random walk and a stochastic simulation algorithm, respectively. Since the latter assumes the reacting species to be well mixed, the discrete component of our hybrid algorithm deploys stochastic operator splitting, in which the sequence of the diffusion and reaction operations is determined autonomously during each simulation step. We conduct a series of numerical experiments to ascertain the accuracy and computational efficiency of our hybrid simulations and, then, to demonstrate the importance of randomness for predicting leukocyte migration and fate during the immune response to inflammation.

Dynamic analysis of the role of innate immunity in SEIS epidemic model

Consideration of every important aspect while modeling a disease makes the model more precise and the disease eradication strategy more powerful. In the present paper, we analyze the importance of innate immunity on SEIS modeling. We propose an SEIS model with Holling type II and type III functions representing innate immunity. We find the existence and stability conditions for the equilibria. When innate immunity is in the form of Holling type II function, the disease-free equilibrium exists for reproduction number less than unity and is locally asymptotically stable, and supercritical transcritical (forward) as well as subcritical transcritical (backward) bifurcation may occur where the contact rate β = β ∗ acts as the bifurcation parameter. Hence, disease-free equilibrium need not be globally stable. For reproduction number greater than unity unique endemic equilibrium exists which is locally asymptotically stable. The global stability conditions for the same are deduced with the help of Lozinski i ˘ measure. When innate immunity is considered a Holling type III function, the disease-free equilibrium point exists for reproduction number less than unity and is locally as well as globally stable. The existence of either unique or multiple endemic equilibria is found when reproduction number is greater than unity, and there exists at least one locally asymptotically stable equilibrium point and bistability can also be encountered. The conditions for the existence of Andronov-Hopf bifurcation are deduced for both cases. Moreover, we observe that ignoring innate immunity annihilates the possibility of Andronov-Hopf bifurcation. Numerical simulation is performed to validate the mathematical findings. Comparing the obtained results to the case when innate immunity is ignored, it is deduced that ignoring it ends the possibility of backward bifurcation, Andronov-Hopf bifurcation as well as the existence of multiple equilibria, and it also leads to the prediction of higher infection than the actual which may deflect the accuracy of the model to a high extent. This would further lead to false predictions and inefficient disease control strategies which in turn would make disease eradication a difficult and more expensive task.

Dynamical analysis of SEIS model with nonlinear innate immunity and saturated treatment

In this paper, we develop an SEIS model with Holling type II function representing the innate immunity as well as the saturated treatment. We obtain the existence and stability criteria for the equilibrium points. We observe that when the reproduction number is less than unity, the disease-free equilibrium always exists and is locally asymptotically stable. The multiple endemic equilibrium points can exist independent of the basic reproduction number, and the system may experience bistability. We find that the system can encounter backward or forward bifurcation at R 0 = 1 , where the contact rate β = β 0 is the bifurcation parameter. Therefore, the disease-free equilibrium may not be globally stable. We deduce the criteria for the presence of Hopf bifurcation where the parameter γ = γ ∗ acts as the bifurcation parameter and the system is a neutrally stable center. We also observe with the aid of a numerical example that a slight perturbation disrupts the neutral stability and the trajectories become either converging or diverging from the equilibrium point. Numerical simulation is performed with the help of MATLAB to justify the findings. We study the effect of nonlinearity of immunity function and the treatment rate on the dynamics of the disease spread. We find that when both are linear, the reproduction number is the same, but the system has a unique endemic equilibrium point that exists for reproduction number greater than unity. We find that there is neither backward bifurcation nor Hopf bifurcation. We also observe that the saturation in treatment enlarges the domain of backward bifurcation making disease eradication an extremely difficult task. The endemic equilibria in the case of saturated treatment may exist far more to the left of the bifurcation parameter β = β 0 . Hence, the nonlinearity of immunity function and treatment function affects the dynamics of an SEIS model highly; therefore, one must be precautious to choose an appropriate function for both while modeling.

Quantifying immunoregulation by autoantigen-specific T-regulatory type 1 cells in mice with simultaneous hepatic and extra-hepatic autoimmune disorders

Nanoparticles (NPs) displaying autoimmune disease-relevant peptide-major histocompatibility complex class II molecules (pMHCII-NPs) trigger cognate T-regulatory type 1 (Tr1)-cell formation and expansion, capable of reversing organ-specific autoimmune responses. These pMHCII-NPs that display epitopes from mitochondrial protein can blunt the progression of both autoimmune hepatitis (AIH) and experimental autoimmune encephalomyelitis (EAE) in mice carrying either disease. However, with co-morbid mice having both diseases, these pMHCII-NPs selectively treat AIH. In contrast, pMHCII-NPs displaying central nervous system (CNS)-specific epitopes can efficiently treat CNS autoimmunity, both in the absence and presence of AIH, without having any effects on the progression of the latter. Here, we develop a compartmentalized population model of T-cells in co-morbid mice to identify the mechanisms by which Tr1 cells mediate organ-specific immunoregulation. We perform time-series simulations and bifurcation analyses to study how varying physiological parameters, including local cognate antigenic load and rates of Tr1-cell recruitment and retention, affect T-cell allocation and Tr1-mediated immunoregulation. Various regimes of behaviour, including 'competitive autoimmunity' where pMHCII-NP-treatment fails against both diseases, are identified and compared with experimental observations. Our results reveal that a transient delay in Tr1-cell recruitment to the CNS, resulting from inflammation-dependent Tr1-cell allocation, accounts for the liver-centric effects of AIH-specific pMHCII-NPs in co-morbid mice as compared with mice exclusively having EAE. They also suggest that cognate autoantigen expression and local Tr1-cell retention are key determinants of effective regulatory-cell function. These results thus provide new insights into the rules that govern Tr1-cell recruitment and their autoregulatory function.

Global well-posedness and pattern formations of the immune system induced by chemotaxis

This paper studies a reaction-diffusion-advection system describing a directed movement of immune cells toward chemokines during the immune process. We investigate the global solvability of the model based on the bootstrap argument for minimal chemotaxis models. We also examine the stability of nonconstant steady states and the existence of periodic orbits from theoretical aspects of bifurcation analysis. Through numerical simulations, we observe the occurrence of steady or time-periodic pattern formations.

Approximate solutions for HBV infection with stability analysis using LHAM during antiviral therapy

Hepatitis B virus (HBV) is a life-threatening virus that causes very serious liver-related diseases from the family of Hepadnaviridae having very rare qualities resembling retroviruses. In this paper, we analyze the effect of antiviral therapy through mathematical modeling by using Liao's homotopy analysis method (LHAM) that defines the connection between the target liver cells and the HBV. We also examine the basic nonlinear differential equation by LHAM to get a semi-analytical solution. This can be a very straight and direct method which provides the appropriate solution. Moreover, the local and global stability analysis of disease-free and endemic equilibrium is done using Lyapunov function. Mathematica 12 software is used to find out the solutions and graphical representations. We also discuss the numerical simulations up to sixth-order approximation and error analysis using the same software.

Mechanisms and Points of Control in the Spread of Inflammation: A Mathematical Investigation

Understanding the mechanisms that control the body’s response to inflammation is of key importance, due to its involvement in myriad medical conditions, including cancer, arthritis, Alzheimer’s disease and asthma. While resolving inflammation has historically been considered a passive process, since the turn of the century the hunt for novel therapeutic interventions has begun to focus upon active manipulation of constituent mechanisms, particularly involving the roles of apoptosing neutrophils, phagocytosing macrophages and anti-inflammatory mediators. Moreover, there is growing interest in how inflammatory damage can spread spatially due to the motility of inflammatory mediators and immune cells. For example, impaired neutrophil chemotaxis is implicated in causing chronic inflammation under trauma and in ageing, while neutrophil migration is an attractive therapeutic target in ailments such as chronic obstructive pulmonary disease. We extend an existing homogeneous model that captures interactions between inflammatory mediators, neutrophils and macrophages to incorporate spatial behaviour. Through bifurcation analysis and numerical simulation, we show that spatially inhomogeneous outcomes can present close to the switch from bistability to guaranteed resolution in the corresponding homogeneous model. Finally, we show how aberrant spatial mechanisms can play a role in the failure of inflammation to resolve and discuss our results within the broader context of seeking novel inflammatory treatments.

A minimal modeling framework of radiation and immune system synergy to assist radiotherapy planning

Recent evidence indicates the ability of radiotherapy to induce local and systemic tumor-specific immune responses as a result of immunogenic cell death. However, fractionation regimes routinely used in clinical practice typically ignore the synergy between radiation and the immune system, and instead attempt to completely eradicate tumors by the direct lethal effect of radiation on cancer cells. This paradigm is expected to change in the near future due to the potential benefits of considering radiation-induced antitumor immunity during treatment planning. Towards this goal, we propose a minimal modeling framework based on key aspects of the tumor-immune system interplay to simulate the effects of radiation on tumors and the immunological consequences of radiotherapy. The impacts of tumor-associated vasculature and intratumoral oxygen-mediated heterogeneity on treatment outcomes are ininvestigated. The model provides estimates of the minimum radiation doses required for tumor eradication given a certain number of treatment fractions. Moreover, estimates of treatment duration for disease control given predetermined fractional radiation doses can be also obtained. Although theoretical in nature, this study motivates the development and establishment of immune-based decision-support tools in radiotherapy planning.

A personalized computational model of edema formation in myocarditis based on long-axis biventricular MRI images

Background

Myocarditis is defined as the inflammation of the myocardium, i.e. the cardiac muscle. Among the reasons that lead to this disease, we may include infections caused by a virus, bacteria, protozoa, fungus, and others. One of the signs of the inflammation is the formation of edema, which may be a consequence of the interaction between interstitial fluid dynamics and immune response. This complex physiological process was mathematically modeled using a nonlinear system of partial differential equations (PDE) based on porous media approach. By combing a model based on Biot’s poroelasticity theory with a model for the immune response we developed a new hydro-mechanical model for inflammatory edema. To verify this new computational model, T2 parametric mapping obtained by Magnetic Resonance (MR) imaging was used to identify the region of edema in a patient diagnosed with unspecific myocarditis.

Results

A patient-specific geometrical model was created using MRI images from the patient with myocarditis. With this model, edema formation was simulated using the proposed hydro-mechanical mathematical model in a two-dimensional domain. The computer simulations allowed us to correlate spatiotemporal dynamics of representative cells of the immune systems, such as leucocytes and the pathogen, with fluid accumulation and cardiac tissue deformation.

Conclusions

This study demonstrates that the proposed mathematical model is a very promising tool to better understand edema formation in myocarditis. Simulations obtained from a patient-specific model reproduced important aspects related to the formation of cardiac edema, its area, position, and shape, and how these features are related to immune response.

Game of neutrophils: modeling the balance between apoptosis and necrosis

BACKGROUND

Neutrophils are one of the key players in the human innate immune system (HIIS). In the event of an insult where the body is exposed to inflammation triggering moieties (ITMs), neutrophils are mobilized towards the site of insult and antagonize the inflammation. If the inflammation is cleared, neutrophils go into a programmed death called apoptosis. However, if the insult is intense or persistent, neutrophils take on a violent death pathway called necrosis, which involves the rupture of their cytoplasmic content into the surrounding tissue that causes local tissue damage, thus further aggravating inflammation. This seemingly paradoxical phenomenon fuels the inflammatory process by triggering the recruitment of additional neutrophils to the site of inflammation, aimed to contribute to the complete neutralization of severe inflammation. This delicate balance between the cost and benefit of the neutrophils' choice of death pathway has been optimized during the evolution of the innate immune system. The goal of our work is to understand how the tradeoff between the cost and benefit of the different death pathways of neutrophils, in response to various levels of insults, has been optimized over evolutionary time by using the concepts of evolutionary game theory.

RESULTS

We show that by using evolutionary game theory, we are able to formulate a game that predicts the percentage of necrosis and apoptosis when exposed to various levels of insults.

CONCLUSION

By adopting an evolutionary perspective, we identify the driving mechanisms leading to the delicate balance between apoptosis and necrosis in neutrophils' cell death in response to different insults. Using our simple model, we verify that indeed, the global cost of remaining ITMs is the driving mechanism that reproduces the percentage of necrosis and apoptosis observed in data and neutrophils need sufficient information of the overall inflammation to be able to pick a death pathway that presumably increases the survival of the organism.

A multiscale in silico model of endothelial to mesenchymal transformation in a tumor microenvironment

Endothelial to mesenchymal transformation (EndMT) is a process in which endothelial cells gain a mesenchymal-like phenotype in response to mechanobiological signals that results in the remodeling or repair of underlying tissue. While initially associated with embryonic development, this process has since been shown to occur in adult tissue remodeling including wound healing, fibrosis, and cancer. In an attempt to understand the role of EndMT in cancer progression and metastasis, we present a multiscale, three-dimensional, in silico model. The model couples tissue level phenomena such as extracellular matrix remodeling, cellular level phenomena such as migration and proliferation, and chemical transport in the tumor microenvironment to mimic in vitro tissue models of the cancer microenvironment. The model is used to study the presence of EndMT-derived activated fibroblasts (EDAFs) and varying substrate stiffness on tumor cell migration and proliferation. The simulations accurately model the behavior of tumor cells under given conditions. The presence of EDAFs and/or an increase in substrate stiffness resulted in an increase in tumor cell activity. This model lays the foundation of further studies of EDAFs in a tumor microenvironment on a cellular and subcellular physiological level.

Mathematical modeling of septic shock based on clinical data

BACKGROUND

Mathematical models of diseases may provide a unified approach for establishing effective treatment strategies based on fundamental pathophysiology. However, models that are useful for clinical practice must overcome the massive complexity of human physiology and the diversity of patients' environmental conditions. With the aim of modeling a complex disease, we choose sepsis, which is highly complex, life-threatening systemic disease with high mortality. In particular, we focused on septic shock, a subset of sepsis in which underlying circulatory and cellular/metabolic abnormalities are profound enough to substantially increase mortality. Our model includes cardiovascular, immune, nervous system models and a pharmacological model as submodels and integrates them to create a sepsis model based on pathological facts.

RESULTS

Model validation was done in two steps. First, we established a model for a standard patient in order to confirm the validity of our approach in general aspects. For this, we checked the correspondence between the severity of infection defined in terms of pathogen growth rate and the ease of recovery defined in terms of the intensity of treatment required for recovery. The simulations for a standard patient showed good correspondence. We then applied the same simulations to a patient with heart failure as an underlying disease. The model showed that spontaneous recovery would not occur without treatment, even for a very mild infection. This is consistent with clinical experience. We next validated the model using clinical data of three sepsis patients. The model parameters were tuned for these patients based on the model for the standard patient used in the first part of the validation. In these cases, the simulations agreed well with clinical data. In fact, only a handful parameters need to be tuned for the simulations to match with the data.

CONCLUSIONS

We have constructed a model of septic shock and have shown that it can reproduce well the time courses of treatment and disease progression. Tuning of model parameters for each patient could be easily done. This study demonstrates the feasibility of disease models, suggesting the possibility of clinical use in the prediction of disease progression, decisions on the timing of drug dosages, and the estimation of time of infection.

Mathematical modeling of energy consumption in the acute inflammatory response

When a pathogen invades the body, an acute inflammatory response is activated to eliminate the intruder. In some patients, runaway activation of the immune system may lead to collateral tissue damage and, in the extreme, organ failure and death. Experimental studies have found an association between severe infections and depletion in levels of adenosine triphosphate (ATP), increase in nitric oxide production, and accumulation of lactate, suggesting that tissue energetics is compromised. In this work we present a differential equations model that incorporates the dynamics of ATP, nitric oxide, and lactate accompanying an acute inflammatory response and employ this model to explore their roles in shaping this response. The bifurcation diagram of the model system with respect to the pathogen growth rate reveals three equilibrium states characterizing the health, aseptic and septic conditions. We explore the domains of attraction of these states to inform the instantiation of heterogeneous virtual patient populations utilized in a survival analysis. We then apply the model to study alterations in the inflammatory response and survival outcomes in metabolically altered conditions such as hypoglycemia, hyperglycemia, and hypoxia.

Supplemented Alkaline Phosphatase Supports the Immune Response in Patients Undergoing Cardiac Surgery: Clinical and Computational Evidence

Alkaline phosphatase (AP) is an enzyme that exhibits anti-inflammatory effects by dephosphorylating inflammation triggering moieties (ITMs) like bacterial lipopolysaccharides and extracellular nucleotides. AP administration aims to prevent and treat peri- and post-surgical ischemia reperfusion injury in cardiothoracic surgery patients. Recent studies reported that intravenous bolus administration and continuous infusion of AP in patients undergoing coronary artery bypass grafting with cardiac valve surgery induce an increased release of liver-type “tissue non-specific alkaline phosphatase” (TNAP) into the bloodstream. The release of liver-type TNAP into circulation could be the body's way of strengthening its defense against a massive ischemic insult. However, the underlying mechanism behind the induction of TNAP is still unclear. To obtain a deeper insight into the role of AP during surgery, we developed a mathematical model of systemic inflammation that clarifies the relation between supplemented AP and TNAP and describes a plausible induction mechanism of TNAP in patients undergoing cardiothoracic surgery. The model was validated against clinical data from patients treated with bovine Intestinal AP (bIAP treatment) or without AP (placebo treatment), in addition to standard care procedures. We performed additional in-silico experiments adding a secondary source of ITMs after surgery, as observed in some patients with complications, and predicted the response to different AP treatment regimens. Our results show a strong protective effect of supplemented AP for patients with complications. The model provides evidence of the existence of an induction mechanism of liver-type tissue non-specific alkaline phosphatase, triggered by the supplementation of AP in patients undergoing cardiac surgery. To the best of our knowledge this is the first time that a quantitative and validated numerical model of systemic inflammation under clinical treatment conditions is presented.

Importance of Feedback and Feedforward Loops to Adaptive Immune Response Modeling

The human adaptive immune system is a very complex network of different types of cells, cytokines, and signaling molecules. This complex network makes it difficult to understand the system level regulations. To properly explain the immune system, it is necessary to explicitly investigate the presence of different feedback and feedforward loops (FFLs) and their crosstalks. Considering that these loops increase the complexity of the system, the mathematical modeling has been proved to be an important tool to explain such complex biological systems. This review focuses on these regulatory loops and discusses their importance on systems modeling of the immune system.

Mathematical modeling and its analysis for instability of the immune system induced by chemotaxis

In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathematical model, a reaction-diffusion-advection system, describing a cross-talk between antigens and immune cells via chemokines. We analyze the stability and instability arising in our chemotaxis model and find their conditions for different chemotactic strengths by using energy estimates, spectral analysis, and bootstrap argument. Numerical simulations are also performed to the model, by using the finite volume method in order to deal with the chemotaxis term, and the fractional step methods are used to solve the whole system. From the analytical and numerical results for our model, we explain not only the effective attraction of immune cells toward the site of infection but also hypersensitivity when chemotactic strength is greater than some threshold.

Spatiotemporal Dynamics of Virus Infection Spreading in Tissues

Virus spreading in tissues is determined by virus transport, virus multiplication in host cells and the virus-induced immune response. Cytotoxic T cells remove infected cells with a rate determined by the infection level. The intensity of the immune response has a bell-shaped dependence on the concentration of virus, i.e., it increases at low and decays at high infection levels. A combination of these effects and a time delay in the immune response determine the development of virus infection in tissues like spleen or lymph nodes. The mathematical model described in this work consists of reaction-diffusion equations with a delay. It shows that the different regimes of infection spreading like the establishment of a low level infection, a high level infection or a transition between both are determined by the initial virus load and by the intensity of the immune response. The dynamics of the model solutions include simple and composed waves, and periodic and aperiodic oscillations. The results of analytical and numerical studies of the model provide a systematic basis for a quantitative understanding and interpretation of the determinants of the infection process in target organs and tissues from the image-derived data as well as of the spatiotemporal mechanisms of viral disease pathogenesis, and have direct implications for a biopsy-based medical testing of the chronic infection processes caused by viruses, e.g. HIV, HCV and HBV.

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.

Cancer therapeutic potential of combinatorial immuno- and vasomodulatory interventions

Currently, most of the basic mechanisms governing tumour-immune system interactions, in combination with modulations of tumour-associated vasculature, are far from being completely understood. Here, we propose a mathematical model of vascularized tumour growth, where the main novelty is the modelling of the interplay between functional tumour vasculature and effector cell recruitment dynamics. Parameters are calibrated on the basis of different in vivo immunocompromised Rag1(-/-) and wild-type (WT) BALB/c murine tumour growth experiments. The model analysis supports that tumour vasculature normalization can be a plausible and effective strategy to treat cancer when combined with appropriate immunostimulations. We find that improved levels of functional tumour vasculature, potentially mediated by normalization or stress alleviation strategies, can provide beneficial outcomes in terms of tumour burden reduction and growth control. Normalization of tumour blood vessels opens a therapeutic window of opportunity to augment the antitumour immune responses, as well as to reduce intratumoral immunosuppression and induced hypoxia due to vascular abnormalities. The potential success of normalizing tumour-associated vasculature closely depends on the effector cell recruitment dynamics and tumour sizes. Furthermore, an arbitrary increase in the initial effector cell concentration does not necessarily imply better tumour control. We evidence the existence of an optimal concentration range of effector cells for tumour shrinkage. Based on these findings, we suggest a theory-driven therapeutic proposal that optimally combines immuno- and vasomodulatory interventions.

The role of the hypothalamic-pituitary-adrenal axis in modulating seasonal changes in immunity

Seasonal changes in environmental conditions are accompanied by significant adjustment of multiple biological processes. In temperate regions, the day fraction, or photoperiod, is a robust environmental cue that synchronizes seasonal variations in neuroendocrine and metabolic function. In this work, we propose a semimechanistic mathematical model that considers the influence of seasonal photoperiod changes as well as cellular and molecular adaptations to investigate the seasonality of immune function. Our model predicts that the circadian rhythms of cortisol, our proinflammatory mediator, and its receptor exhibit seasonal differences in amplitude and phase, oscillating at higher amplitudes in the winter season with peak times occurring later in the day. Furthermore, the reduced photoperiod of winter coupled with seasonal alterations in physiological activity induces a more exacerbated immune response to acute stress, simulated in our studies as the administration of an acute dose of endotoxin. Our findings are therefore in accordance with experimental data that reflect the predominance of a proinflammatory state during the winter months. These changes in circadian rhythm dynamics may play a significant role in the seasonality of disease incidence and regulate the diurnal and seasonal variation of disease symptom severity.

Bayesian integrated testing strategy (ITS) for skin sensitization potency assessment: a decision support system for quantitative weight of evidence and adaptive testing strategy

The presented Bayesian network Integrated Testing Strategy (ITS-3) for skin sensitization potency assessment is a decision support system for a risk assessor that provides quantitative weight of evidence, leading to a mechanistically interpretable potency hypothesis, and formulates adaptive testing strategy for a chemical. The system was constructed with an aim to improve precision and accuracy for predicting LLNA potency beyond ITS-2 (Jaworska et al., J Appl Toxicol 33(11):1353-1364, 2013) by improving representation of chemistry and biology. Among novel elements are corrections for bioavailability both in vivo and in vitro as well as consideration of the individual assays' applicability domains in the prediction process. In ITS-3 structure, three validated alternative assays, DPRA, KeratinoSens and h-CLAT, represent first three key events of the adverse outcome pathway for skin sensitization. The skin sensitization potency prediction is provided as a probability distribution over four potency classes. The probability distribution is converted to Bayes factors to: 1) remove prediction bias introduced by the training set potency distribution and 2) express uncertainty in a quantitative manner, allowing transparent and consistent criteria to accept a prediction. The novel ITS-3 database includes 207 chemicals with a full set of in vivo and in vitro data. The accuracy for predicting LLNA outcomes on the external test set (n = 60) was as follows: hazard (two classes)-100 %, GHS potency classification (three classes)-96 %, potency (four classes)-89 %. This work demonstrates that skin sensitization potency prediction based on data from three key events, and often less, is possible, reliable over broad chemical classes and ready for practical applications.

In silico tumor control induced via alternating immunostimulating and immunosuppressive phases

Despite recent advances in the field of Oncoimmunology, the success potential of immunomodulatory therapies against cancer remains to be elucidated. One of the reasons is the lack of understanding on the complex interplay between tumor growth dynamics and the associated immune system responses. Toward this goal, we consider a mathematical model of vascularized tumor growth and the corresponding effector cell recruitment dynamics. Bifurcation analysis allows for the exploration of model's dynamic behavior and the determination of these parameter regimes that result in immune-mediated tumor control. In this work, we focus on a particular tumor evasion regime that involves tumor and effector cell concentration oscillations of slowly increasing and decreasing amplitude, respectively. Considering a temporal multiscale analysis, we derive an analytically tractable mapping of model solutions onto a weakly negatively damped harmonic oscillator. Based on our analysis, we propose a theory-driven intervention strategy involving immunostimulating and immunosuppressive phases to induce long-term tumor control.

The resolution of inflammation: a mathematical model of neutrophil and macrophage interactions

There is growing interest in inflammation due to its involvement in many diverse medical conditions, including Alzheimer's disease, cancer, arthritis and asthma. The traditional view that resolution of inflammation is a passive process is now being superceded by an alternative hypothesis whereby its resolution is an active, anti-inflammatory process that can be manipulated therapeutically. This shift in mindset has stimulated a resurgence of interest in the biological mechanisms by which inflammation resolves. The anti-inflammatory processes central to the resolution of inflammation revolve around macrophages and are closely related to pro-inflammatory processes mediated by neutrophils and their ability to damage healthy tissue. We develop a spatially averaged model of inflammation centring on its resolution, accounting for populations of neutrophils and macrophages and incorporating both pro- and anti-inflammatory processes. Our ordinary differential equation model exhibits two outcomes that we relate to healthy and unhealthy states. We use bifurcation analysis to investigate how variation in the system parameters affects its outcome. We find that therapeutic manipulation of the rate of macrophage phagocytosis can aid in resolving inflammation but success is critically dependent on the rate of neutrophil apoptosis. Indeed our model predicts that an effective treatment protocol would take a dual approach, targeting macrophage phagocytosis alongside neutrophil apoptosis.

On the coupling of two models of the human immune response to an antigen

The development of mathematical models of the immune response allows a better understanding of the multifaceted mechanisms of the defense system. The main purpose of this work is to present a scheme for coupling distinct models of different scales and aspects of the immune system. As an example, we propose a new model where the local tissue inflammation processes are simulated with partial differential equations (PDEs) whereas a system of ordinary differential equations (ODEs) is used as a model for the systemic response. The simulation of distinct scenarios allows the analysis of the dynamics of various immune cells in the presence of an antigen. Preliminary results of this approach with a sensitivity analysis of the coupled model are shown but further validation is still required.

On the computational modeling of the innate immune system

In recent years, there has been an increasing interest in the mathematical and computational modeling of the human immune system (HIS). Computational models of HIS dynamics may contribute to a better understanding of the relationship between complex phenomena and immune response; in addition, computational models will support the development of new drugs and therapies for different diseases. However, modeling the HIS is an extremely difficult task that demands a huge amount of work to be performed by multidisciplinary teams. In this study, our objective is to model the spatio-temporal dynamics of representative cells and molecules of the HIS during an immune response after the injection of lipopolysaccharide (LPS) into a section of tissue. LPS constitutes the cellular wall of Gram-negative bacteria, and it is a highly immunogenic molecule, which means that it has a remarkable capacity to elicit strong immune responses. We present a descriptive, mechanistic and deterministic model that is based on partial differential equations (PDE). Therefore, this model enables the understanding of how the different complex phenomena interact with structures and elements during an immune response. In addition, the model's parameters reflect physiological features of the system, which makes the model appropriate for general use.