A reduced mathematical model of the acute inflammatory response II. Capturing scenarios of repeated endotoxin administration

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 inflammatory map: linking individual stress to societal dysfunction

As populations worldwide show increasing levels of stress, understanding emerging links among stress, inflammation, cognition, and behavior is vital to human and planetary health. We hypothesize that inflammation is a multiscale driver connecting stressors that affect individuals to large-scale societal dysfunction and, ultimately, to planetary-scale environmental impacts. We propose a 'central inflammation map' hypothesis to explain how the brain regulates inflammation and how inflammation impairs cognition, emotion, and action. According to our hypothesis, these interdependent inflammatory and neural processes, and the inter-individual transmission of environmental, infectious, and behavioral stressors - amplified via high-throughput digital global communications - can culminate in a multiscale, runaway, feed-forward process that could detrimentally affect human decision-making and behavior at scale, ultimately impairing the ability to address these same stressors. This perspective could provide non-intuitive explanations for behaviors and relationships among cells, organisms, and communities of organisms, potentially including population-level responses to stressors as diverse as global climate change, conflicts, and the COVID-19 pandemic. To illustrate our hypothesis and elucidate its mechanistic underpinnings, we present a mathematical model applicable to the individual and societal levels to test the links among stress, inflammation, control, and healing, including the implications of transmission, intervention (e.g., via lifestyle modification or medication), and resilience. Future research is needed to validate the model's assumptions, expand the factors/variables employed, and validate it against empirical benchmarks. Our model illustrates the need for multilayered, multiscale stress mitigation interventions, including lifestyle measures, precision therapeutics, and human ecosystem design. Our analysis shows the need for a coordinated, interdisciplinary, international research effort to understand the multiscale nature of stress. Doing so would inform the creation of interventions that improve individuals' lives and communities' resilience to stress and mitigate its adverse effects on the world.

Mathematical models of cystic fibrosis as a systemic disease

Cystic fibrosis (CF) is widely known as a disease of the lung, even though it is in truth a systemic disease, whose symptoms typically manifest in gastrointestinal dysfunction first. CF ultimately impairs not only the pancreas and intestine but also the lungs, gonads, liver, kidneys, bones, and the cardiovascular system. It is caused by one of several mutations in the gene of the epithelial ion channel protein CFTR. Intense research and improved antimicrobial treatments during the past eight decades have steadily increased the predicted life expectancy of a person with CF (pwCF) from a few weeks to over 50 years. Moreover, several drugs ameliorating the sequelae of the disease have become available in recent years, and notable treatments of the root cause of the disease have recently generated substantial improvements in health for some but not all pwCF. Yet, numerous fundamental questions remain unanswered. Complicating CF, for instance in the lung, is the fact that the associated insufficient chloride secretion typically perturbs the electrochemical balance across epithelia and, in the airways, leads to the accumulation of thick, viscous mucus and mucus plaques that cannot be cleared effectively and provide a rich breeding ground for a spectrum of bacterial and fungal communities. The subsequent infections often become chronic and respond poorly to antibiotic treatments, with outcomes sometimes only weakly correlated with the drug susceptibility of the target pathogen. Furthermore, in contrast to rapidly resolved acute infections with a single target pathogen, chronic infections commonly involve multi-species bacterial communities, called "infection microbiomes," that develop their own ecological and evolutionary dynamics. It is presently impossible to devise mathematical models of CF in its entirety, but it is feasible to design models for many of the distinct drivers of the disease. Building upon these growing yet isolated modeling efforts, we discuss in the following the feasibility of a multi-scale modeling framework, known as template-and-anchor modeling, that allows the gradual integration of refined sub-models with different granularity. The article first reviews the most important biomedical aspects of CF and subsequently describes mathematical modeling approaches that already exist or have the potential to deepen our understanding of the multitude aspects of the disease and their interrelationships. The conceptual ideas behind the approaches proposed here do not only pertain to CF but are translatable to other systemic diseases. This article is categorized under: Congenital Diseases > Computational Models.

A mathematical model of SARS-CoV-2 immunity predicts paxlovid rebound

Nirmatrelvir/ritonavir (Paxlovid), an oral antiviral medication targeting SARS-CoV-2, remains an important treatment for COVID-19. Initial studies of nirmatrelvir/ritonavir were performed in SARS-CoV-2 unvaccinated patients without prior confirmed SARS-CoV-2 infection; however, most individuals have now either been vaccinated and/or have experienced SARS-CoV-2 infection. After nirmatrelvir/ritonavir became widely available, reports surfaced of "Paxlovid rebound," a phenomenon in which symptoms (and SARS-CoV-2 test positivity) would initially resolve, but after finishing treatment, symptoms and test positivity would return. We used a previously described parsimonious mathematical model of immunity to SARS-CoV-2 infection to model the effect of nirmatrelvir/ritonavir treatment in unvaccinated and vaccinated patients. Model simulations show that viral rebound after treatment occurs only in vaccinated patients, while unvaccinated (SARS-COV-2 naïve) patients treated with nirmatrelvir/ritonavir do not experience any rebound in viral load. This work suggests that an approach combining parsimonious models of the immune system could be used to gain important insights in the context of emerging pathogens.

Mathematical Model of Triple-Negative Breast Cancer in Response to Combination Chemotherapies

Triple-negative breast cancer (TNBC) is a heterogenous disease that is defined by its lack of targetable receptors, thus limiting treatment options and resulting in higher rates of metastasis and recurrence. Combination chemotherapy treatments, which inhibit tumor cell proliferation and regeneration, are a major component of standard-of-care treatment of TNBC. In this manuscript, we build a coupled ordinary differential equation model of TNBC with compartments that represent tumor proliferation, necrosis, apoptosis, and immune response to computationally describe the biological tumor affect to a combination of chemotherapies, doxorubicin (DRB) and paclitaxel (PTX). This model is parameterized using longitudinal [18F]-fluorothymidine positron emission tomography (FLT-PET) imaging data which allows for a noninvasive molecular imaging approach to quantify the tumor proliferation and tumor volume measurements for two murine models of TNBC. Animal models include a human cell line xenograft model, MDA-MB-231, and a syngeneic 4T1 mammary carcinoma model. The mathematical models are parameterized and the percent necrosis at the end time point is predicted and validated using histological hematoxylin and eosin (H&E) data. Global Sobol' sensitivity analysis is conducted to further understand the role each parameter plays in the model's goodness of fit to the data. In both the MDA-MB-231 and the 4T1 tumor models, the designed mathematical model can accurately describe both tumor volume changes and final necrosis volume. This can give insight into the ordering, dosing, and timing of DRB and PTX treatment. More importantly, this model can also give insight into future novel combinations of therapies and how the immune system plays a role in therapeutic response to TNBC, due to its calibration to two types of TNBC murine models.

VeVaPy, a Python Platform for Efficient Verification and Validation of Systems Biology Models with Demonstrations Using Hypothalamic-Pituitary-Adrenal Axis Models

In order for mathematical models to make credible contributions, it is essential for them to be verified and validated. Currently, verification and validation (V&V) of these models does not meet the expectations of the system biology and systems pharmacology communities. Partially as a result of this shortfall, systemic V&V of existing models currently requires a lot of time and effort. In order to facilitate systemic V&V of chosen hypothalamic-pituitary-adrenal (HPA) axis models, we have developed a computational framework named VeVaPy-taking care to follow the recommended best practices regarding the development of mathematical models. VeVaPy includes four functional modules coded in Python, and the source code is publicly available. We demonstrate that VeVaPy can help us efficiently verify and validate the five HPA axis models we have chosen. Supplied with new and independent data, VeVaPy outputs objective V&V benchmarks for each model. We believe that VeVaPy will help future researchers with basic modeling and programming experience to efficiently verify and validate mathematical models from the fields of systems biology and systems pharmacology.

Divergent COVID-19 Disease Trajectories Predicted by a DAMP-Centered Immune Network Model

COVID-19 presentations range from mild to moderate through severe disease but also manifest with persistent illness or viral recrudescence. We hypothesized that the spectrum of COVID-19 disease manifestations was a consequence of SARS-CoV-2-mediated delay in the pathogen-associated molecular pattern (PAMP) response, including dampened type I interferon signaling, thereby shifting the balance of the immune response to be dominated by damage-associated molecular pattern (DAMP) signaling. To test the hypothesis, we constructed a parsimonious mechanistic mathematical model. After calibration of the model for initial viral load and then by varying a few key parameters, we show that the core model generates four distinct viral load, immune response and associated disease trajectories termed “patient archetypes”, whose temporal dynamics are reflected in clinical data from hospitalized COVID-19 patients. The model also accounts for responses to corticosteroid therapy and predicts that vaccine-induced neutralizing antibodies and cellular memory will be protective, including from severe COVID-19 disease. This generalizable modeling framework could be used to analyze protective and pathogenic immune responses to diverse viral infections.

Modeling Neuroimmune Interactions in Human Subjects and Animal Models to Predict Subtype-Specific Multidrug Treatments for Gulf War Illness

Gulf War Illness (GWI) is a persistent chronic neuroinflammatory illness exacerbated by external stressors and characterized by fatigue, musculoskeletal pain, cognitive, and neurological problems linked to underlying immunological dysfunction for which there is no known treatment. As the immune system and the brain communicate through several signaling pathways, including the hypothalamic–pituitary–adrenal (HPA) axis, it underlies many of the behavioral and physiological responses to stressors via blood-borne mediators, such as cytokines, chemokines, and hormones. Signaling by these molecules is mediated by the semipermeable blood–brain barrier (BBB) made up of a monocellular layer forming an integral part of the neuroimmune axis. BBB permeability can be altered and even diminished by both external factors (e.g., chemical agents) and internal conditions (e.g., acute or chronic stress, or cross-signaling from the hypothalamic–pituitary–gonadal (HPG) axis). Such a complex network of regulatory interactions that possess feed-forward and feedback connections can have multiple response dynamics that may include several stable homeostatic states beyond normal health. Here we compare immune and hormone measures in the blood of human clinical samples and mouse models of Gulf War Illness (GWI) subtyped by exposure to traumatic stress for subtyping this complex illness. We do this via constructing a detailed logic model of HPA–HPG–Immune regulatory behavior that also considers signaling pathways across the BBB to neuronal–glial interactions within the brain. We apply conditional interactions to model the effects of changes in BBB permeability. Several stable states are identified in the system beyond typical health. Following alignment of the human and mouse blood profiles in the context of the model, mouse brain sample measures were used to infer the neuroinflammatory state in human GWI and perform treatment simulations using a genetic algorithm to optimize the Monte Carlo simulations of the putative treatment strategies aimed at returning the ill system back to health. We identify several ideal multi-intervention strategies and potential drug candidates that may be used to treat chronic neuroinflammation in GWI.

A Turing mechanism in order to explain the patchy nature of Crohn's disease

Crohn's disease is an inflammatory bowel disease (IBD) that is not well understood. In particular, unlike other IBDs, the inflamed parts of the intestine compromise deep layers of the tissue and are not continuous but separated and distributed through the whole gastrointestinal tract, displaying a patchy inflammatory pattern. In the present paper, we introduce a toy-model which might explain the appearance of such patterns. We consider a reaction-diffusion system involving bacteria and phagocyte and prove that, under certain conditions, this system might reproduce an activator-inhibitor dynamic leading to the occurrence of Turing-type instabilities. In other words, we prove the existence of stable stationary solutions that are spatially periodic and do not vanish in time. We also propose a set of parameters for which the system exhibits such phenomena and compare it with realistic parameters found in the literature. This is the first time, as far as we know, that a Turing pattern is investigated in inflammatory models.

Bistable Mathematical Model of Neutrophil Migratory Patterns After LPS-Induced Epigenetic Reprogramming

The highly controlled migration of neutrophils toward the site of an infection can be altered when they are trained with lipopolysaccharides (LPS), with high dose LPS enhancing neutrophil migratory pattern toward the bacterial derived source signal and super-low dose LPS inducing either migration toward an intermediary signal or dysregulation and oscillatory movement. Empirical studies that use microfluidic chemotaxis-chip devices with two opposing chemoattractants showed differential neutrophil migration after challenge with different LPS doses. The epigenetic alterations responsible for changes in neutrophil migratory behavior are unknown. We developed two mathematical models that evaluate the mechanistic interactions responsible for neutrophil migratory decision-making when exposed to competing chemoattractants and challenged with LPS. The first model, which considers the interactions between the receptor densities of two competing chemoattractants, their kinases, and LPS, displayed bistability between high and low ratios of primary to intermediary chemoattractant receptor densities. In particular, at equilibrium, we observe equal receptor densities for low LPS (< 15ng/mL); and dominance of receptors for the primary chemoattractant for high LPS (> 15ng/mL). The second model, which included additional interactions with an extracellular signal-regulated kinase in both phosphorylated and non-phosphorylated forms, has an additional dynamic outcome, oscillatory dynamics for both receptors, as seen in the data. In particular, it found equal receptor densities in the absence of oscillation for super-low and high LPS challenge (< 0.4 and 1.1 376 ng/mL). Predicting the mechanisms and the type of external LPS challenge responsible for neutrophils migration toward pro-inflammatory chemoattractants, migration toward pro-tolerant chemoattractants, or oscillatory movement is necessary knowledge in designing interventions against immune diseases, such as sepsis.

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.

Mathematical modelling of trastuzumab-induced immune response in an in vivo murine model of HER2+ breast cancer

The goal of this study is to develop an integrated, mathematical-experimental approach for understanding the interactions between the immune system and the effects of trastuzumab on breast cancer that overexpresses the human epidermal growth factor receptor 2 (HER2+). A system of coupled, ordinary differential equations was constructed to describe the temporal changes in tumour growth, along with intratumoural changes in the immune response, vascularity, necrosis and hypoxia. The mathematical model is calibrated with serially acquired experimental data of tumour volume, vascularity, necrosis and hypoxia obtained from either imaging or histology from a murine model of HER2+ breast cancer. Sensitivity analysis shows that model components are sensitive for 12 of 13 parameters, but accounting for uncertainty in the parameter values, model simulations still agree with the experimental data. Given theinitial conditions, the mathematical model predicts an increase in the immune infiltrates over time in the treated animals. Immunofluorescent staining results are presented that validate this prediction by showing an increased co-staining of CD11c and F4/80 (proteins expressed by dendritic cells and/or macrophages) in the total tissue for the treated tumours compared to the controls ($p < 0.03$). We posit that the proposed mathematical-experimental approach can be used to elucidate driving interactions between the trastuzumab-induced responses in the tumour and the immune system that drive the stabilization of vasculature while simultaneously decreasing tumour growth-conclusions revealed by the mathematical model that were not deducible from the experimental data alone.

The ups and downs of S. aureus nasal carriage

Staphylococcus aureus infections are a growing concern worldwide due to the increasing number of strains that exhibit antibiotic resistance. Recent studies have indicated that some percentage of people carry the bacteria in the nasal cavity and therefore are at a higher risk of subsequent, and more serious, infections in other parts of the body. However, individuals carrying the infection can be classified as only intermittent carriers versus persistent carriers, being able to eliminate the bacteria and later colonized again. Using a model of bacterial colonization of the anterior nares, we investigate oscillatory patterns related to intermittent carriage of S. aureus. Following several studies using global sensitivity analysis techniques, various insights into the model's behaviour were made including interacting effects of the bacteria's growth rate and movement in the mucus, suggesting parameter connections associated with biofilm-like behaviour. Here the bacterial growth rate and bacterial movement are explicitly connected, leading to expanded oscillatory behaviour in the model. We suggest possible implications that this oscillatory behaviour can have on the definition of intermittent carriage and discuss differences in the bacterial virulence dependent upon individual host health. Furthermore, we show that connecting the bacterial growth and movement also expands the region of the parameter space for which the bacteria are able to survive and persist.

Dynamical modeling of pro- and anti-inflammatory cytokines in the early stage of septic shock

A dynamical model of the pathophysiological behaviors of IL18 and IL10 cytokines with their receptors is tested against data for the case of early sepsis. The proposed approach considers the surroundings (organs and bone marrow) and the different subsystems (cells and cyctokines). The interactions between blood cells, cytokines and the surroundings are described via mass balances. Cytokines are adsorbed onto associated receptors at the cell surface. The adsorption is described by the Langmuir model and gives rise to the production of more cytokines and associated receptors inside the cell. The quantities of pro and anti-inflammatory cytokines present in the body are combined to give global information via an inflammation level function which describes the patient’s state. Data for parameter estimation comes from the Sepsis 48 H database. Comparisons between patient data and simulations are presented and are in good agreement. For the IL18/IL10 cytokine pair, 5 key parameters have been found. They are linked to pro-inflammatory IL18 cytokine and show that the early sepsis is driven by components of inflammatory character.

Feature Engineering for ICU Mortality Prediction Based on Hourly to Bi-Hourly Measurements

Mortality prediction for intensive care unit (ICU) patients is a challenging problem that requires extracting discriminative and informative features. This study presents a proof of concept for exploring features that can provide clinical insight. Through a feature engineering approach, it is attempted to improve ICU mortality prediction in field conditions with low frequently measured data (i.e., hourly to bi-hourly). Features are explored by investigating the vital signs measurements of ICU patients, labelled with mortality or survival at discharge. The vital signs of interest in this study are heart and respiration rate, oxygen saturation and blood pressure. The latter comprises systolic, diastolic and mean arterial pressure. In the feature exploration process, it is aimed to extract simple and interpretable features that can provide clinical insight. For this purpose, a classifier is required that maximises the margin between the two classes (i.e., survival and mortality) with minimum tolerance to misclassification errors. Moreover, it preferably has to provide a linear decision surface in the original feature space without mapping to an unlimited dimensionality feature space. Therefore, a linear hard margin support vector machine (SVM) classifier is suggested. The extracted features are grouped in three categories: statistical, dynamic and physiological. Each category plays an important role in enhancing classification error performance. After extracting several features within the three categories, a manual feature fine-tuning is applied to consider only the most efficient features. The final classification, considering mortality as the positive class, resulted in an accuracy of 91.56 % , sensitivity of 90.59 % , precision of 86.52 % and F 1 -score of 88.50 % . The obtained results show that the proposed feature engineering approach and the extracted features are valid to be considered and further enhanced for the mortality prediction purpose. Moreover, the proposed feature engineering approach moved the modelling methodology from black-box modelling to grey-box modelling in combination with the powerful classifier of SVMs.

A mathematical model of the use of supplemental oxygen to combat surgical site infection

Infections are a common complication of any surgery, often requiring a recovery period in hospital. Supplemental oxygen therapy administered during and immediately after surgery is thought to enhance the immune response to bacterial contamination. However, aerobic bacteria thrive in oxygen-rich environments, and so it is unclear whether oxygen has a net positive effect on recovery. Here, we develop a mathematical model of post-surgery infection to investigate the efficacy of supplemental oxygen therapy on surgical-site infections. A 4-species, coupled, set of non-linear partial differential equations that describes the space-time dependence of neutrophils, bacteria, chemoattractant and oxygen is developed and analysed to determine its underlying properties. Through numerical solutions, we quantify the efficacy of different supplemental oxygen regimes on the treatment of surgical site infections in wounds of different initial bacterial load. A sensitivity analysis is performed to investigate the robustness of the predictions to changes in the model parameters. The numerical results are in good agreement with analyses of the associated well-mixed model. Our model findings provide insight into how the nature of the contaminant and its initial density influence bacterial infection dynamics in the surgical wound.

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.

Inflammation and Disease: Modelling and Modulation of the Inflammatory Response to Alleviate Critical Illness

Critical illness, a constellation of interrelated inflammatory and physiological derangements occurring subsequent to severe infection or injury, affects a large number of individuals in both developed and developing countries. The prototypical complex system embodied in critical illness has largely defied therapy beyond supportive care. We have focused on the utility of data-driven and mechanistic computational modelling to help address the complexity of critical illness and provide pathways towards discovering potential therapeutic options and combinations. Herein, we review recent progress in this field, with a focus on both animal and computational models of critical illness. We suggest that therapy for critical illness can be posed as a model-based dynamic control problem, and discuss novel theoretical and experimental approaches involving biohybrid devices aimed at reprogramming inflammation dynamically. Together, these advances offer the potential for Model-based Precision Medicine for critical illness.

A computational analysis of dynamic, multi-organ inflammatory crosstalk induced by endotoxin in mice

Bacterial lipopolysaccharide (LPS) induces an acute inflammatory response across multiple organs, primarily via Toll-like receptor 4 (TLR4). We sought to define novel aspects of the complex spatiotemporal dynamics of LPS-induced inflammation using computational modeling, with a special focus on the timing of pathological systemic spillover. An analysis of principal drivers of LPS-induced inflammation in the heart, gut, lung, liver, spleen, and kidney to assess organ-specific dynamics, as well as in the plasma (as an assessment of systemic spillover), was carried out using data on 20 protein-level inflammatory mediators measured over 0-48h in both C57BL/6 and TLR4-null mice. Using a suite of computational techniques, including a time-interval variant of Principal Component Analysis, we confirm key roles for cytokines such as tumor necrosis factor-α and interleukin-17A, define a temporal hierarchy of organ-localized inflammation, and infer the point at which organ-localized inflammation spills over systemically. Thus, by employing a systems biology approach, we obtain a novel perspective on the time- and organ-specific components in the propagation of acute systemic inflammation.

Gram-negative bacterial lipopolysaccharide (LPS) is both a central mediator of sepsis and a canonical inducer of acute inflammation via Toll-like receptor 4 (TLR4). Sepsis involves the systemic spillover of inflammation that normally remains localized in individual organs. The goal of this study was to gain insights into 1) early vs. later drivers of LPS-induced inflammation in various compartments, and 2) the systemic spillover from affected organs vs. local production of inflammatory mediators in the blood. This study involved a large number of data points on the dynamics of inflammatory mediators at the protein level, data-driven computational modeling of principal characteristics and cross-correlations, and validation of key hypotheses. In addition to verifying key mechanisms in LPS/TLR4-driven acute inflammation, this approach yielded key insights into the progression of inflammation across tissues, and also suggested the presence of TLR4-independent pathways (especially in the gut). This is, to our knowledge, the first study examining the dynamic evolution of some key inflammatory mediators and their interactions with each other in both the systemic circulation and within a number of targeted parenchymal organs in mice.

Modeling the Bistable Dynamics of the Innate Immune System

The size of primary challenge with lipopolysaccharide induces changes in the innate immune cells phenotype between pro-inflammatory and pro-tolerant states when facing a secondary lipopolysaccharide challenge. To determine the molecular mechanisms governing this differential response, we propose a mathematical model for the interaction between three proteins involved in the immune cell decision making: IRAK-1, PI3K, and RelB. The mutual inhibition of IRAK-1 and PI3K in the model leads to bistable dynamics. By using the levels of RelB as indicative of strength of the immune responses, we connect the size of different primary lipopolysaccharide doses to the differential phenotypical outcomes following a secondary challenge. We further predict under what circumstances the primary LPS dose does not influence the response to a secondary challenge. Our results can be used to guide treatments for patients with either autoimmune disease or compromised immune system.

A review of inflammatory mechanism in airway diseases

BACKGROUND

Inflammation in the lung is the body's natural response to injury. It acts to remove harmful stimuli such as pathogens, irritants, and damaged cells and initiate the healing process. Acute and chronic pulmonary inflammation are seen in different respiratory diseases such as; acute respiratory distress syndrome, chronic obstructive pulmonary disease (COPD), asthma, and cystic fibrosis (CF).

FINDINGS

In this review, we found that inflammatory response in COPD is determined by the activation of epithelial cells and macrophages in the respiratory tract. Epithelial cells and macrophages discharge transforming growth factor-β (TGF-β), which trigger fibroblast proliferation and tissue remodeling. Asthma leads to airway hyper-responsiveness, obstruction, mucus hyper-production, and airway-wall remodeling. Cytokines, allergens, chemokines, and infectious agents are the main stimuli that activate signaling pathways in epithelial cells in asthma. Mutation of the CF transmembrane conductance regulator (CFTR) gene results in CF. Mutations in CFTR influence the lung epithelial innate immune function that leads to exaggerated and ineffective airway inflammation that fails to abolish pulmonary pathogens. We present mechanistic computational models (based on ordinary differential equations, partial differential equations and agent-based models) that have been applied in studying the complex physiological and pathological mechanisms of chronic inflammation in different airway diseases.

CONCLUSION

The scope of the present review is to explore the inflammatory mechanism in airway diseases and highlight the influence of aging on airways' inflammation mechanism. The main goal of this review is to encourage research collaborations between experimentalist and modelers to promote our understanding of the physiological and pathological mechanisms that control inflammation in different airway diseases.

Toward a model-free feedback control synthesis for treating acute inflammation

An effective and patient-specific feedback control synthesis for inflammation resolution is still an ongoing research area. A strategy consisting of manipulating a pro and anti-inflammatory mediator is considered here as used in some promising model-based control studies. These earlier studies, unfortunately, suffer from the difficultly of calibration due to the heterogeneity of individual patient responses even under similar initial conditions. We exploit a new model-free control approach and its corresponding "intelligent" controllers for this biomedical problem. A crucial feature of the proposed control problem is as follows: the two most important outputs which must be driven to their respective desired states are sensorless. This difficulty is overcome by assigning suitable reference trajectories to the other two outputs that do have sensors. A mathematical model, via a system of ordinary differential equations, is nevertheless employed as a "virtual" patient for in silico testing. We display several simulation results with respect to the most varied situations, which highlight the effectiveness of our viewpoint.

A System Dynamics Model to Predict the Human Monocyte Response to Endotoxins

System dynamics is a powerful tool that allows modeling of complex and highly networked systems such as those found in the human immune system. We have developed a model that reproduces how the exposure of human monocytes to lipopolysaccharides (LPSs) induces an inflammatory state characterized by high production of tumor necrosis factor alpha (TNFα), which is rapidly modulated to enter into a tolerant state, known as endotoxin tolerance (ET). The model contains two subsystems with a total of six states, seven flows, two auxiliary variables, and 14 parameters that interact through six differential and nine algebraic equations. The parameters were estimated and optimized to obtain a model that fits the experimental data obtained from human monocytes treated with various LPS doses. In contrast to publications on other animal models, stimulation of human monocytes with super-low-dose LPSs did not alter the response to a second LPSs challenge, neither inducing ET, nor enhancing the inflammatory response. Moreover, the model confirms the low production of TNFα and increased levels of C-C motif ligand 2 when monocytes exhibit a tolerant state similar to that of patients with sepsis. At present, the model can help us better understand the ET response and might offer new insights on sepsis diagnostics and prognosis by examining the monocyte response to endotoxins in patients with sepsis.

Immune therapeutic strategies using optimal controls with L1 and L2 type objectives

Therapeutic strategies to correct an excessive immune response to pathogenic infection is investigated as an optimal control problem. The control problem is formulated around a four dimensional mathematical model describing the inflammatory response to a pathogenic insult with two therapeutic control inputs which have either a direct pro- or anti-inflammatory effect in the given system. We use Pontryagin's maximum principle and discuss necessary optimality conditions. We consider both an L¹ type objective functional as well as an L² type objective. For the former, the presence of singular control will be addressed. For each case, numerical simulations using a nonlinear programming optimization solver to acquire different drug treatment strategies are presented and discussed. The results provide insight for possible treatment strategies and the methods could be a relevant tool for future practice to assist in better prediction of clinical outcomes and subsequently better treatment for patients.

Modeling cytokine regulatory network dynamics driving neuroinflammation in central nervous system disorders

A central goal of pharmacological efforts to treat central nervous system (CNS) diseases is to develop systemic therapeutics that can restore CNS homeostasis. Achieving this goal requires a fundamental understanding of CNS function within the organismal context so as to leverage the mechanistic insights on the molecular basis of cellular and tissue functions towards novel drug target identification. The immune system constitutes a key link between the periphery and CNS, and many neurological disorders and neurodegenerative diseases are characterized by immune dysfunction. We review the salient opportunities for applying computational models to CNS disease research, and summarize relevant approaches from studies of immune function and neuroinflammation. While the accurate prediction of disease-related phenomena is often considered the central goal of modeling studies, we highlight the utility of computational modeling applications beyond making predictions, particularly for drawing counterintuitive insights from model-based analysis of multi-parametric and time series data sets.

Combining Theoretical and Experimental Techniques to Study Murine Heart Transplant Rejection

The quality of life of organ transplant recipients is compromised by complications associated with life-long immunosuppression, such as hypertension, diabetes, opportunistic infections, and cancer. Moreover, the absence of established tolerance to the transplanted tissues causes limited long-term graft survival rates. Thus, there is a great medical need to understand the complex immune system interactions that lead to transplant rejection so that novel and effective strategies of intervention that redirect the system toward transplant acceptance (while preserving overall immune competence) can be identified. This study implements a systems biology approach in which an experimentally based mathematical model is used to predict how alterations in the immune response influence the rejection of mouse heart transplants. Five stages of conventional mouse heart transplantation are modeled using a system of 13 ordinary differential equations that tracks populations of both innate and adaptive immunity as well as proxies for pro- and anti-inflammatory factors within the graft and a representative draining lymph node. The model correctly reproduces known experimental outcomes, such as indefinite survival of the graft in the absence of CD4⁺ T cells and quick rejection in the absence of CD8⁺ T cells. The model predicts that decreasing the translocation rate of effector cells from the lymph node to the graft delays transplant rejection. Increasing the starting number of quiescent regulatory T cells in the model yields a significant but somewhat limited protective effect on graft survival. Surprisingly, the model shows that a delayed appearance of alloreactive T cells has an impact on graft survival that does not correlate linearly with the time delay. This computational model represents one of the first comprehensive approaches toward simulating the many interacting components of the immune system. Despite some limitations, the model provides important suggestions of experimental investigations that could improve the understanding of rejection. Overall, the systems biology approach used here is a first step in predicting treatments and interventions that can induce transplant tolerance while preserving the capacity of the immune system to protect against legitimate pathogens.

Modeling and Hemofiltration Treatment of Acute Inflammation

The body responds to endotoxins by triggering the acute inflammatory response system to eliminate the threat posed by gram-negative bacteria (endotoxin) and restore health. However, an uncontrolled inflammatory response can lead to tissue damage, organ failure, and ultimately death; this is clinically known as sepsis. Mathematical models of acute inflammatory disease have the potential to guide treatment decisions in critically ill patients. In this work, an 8-state (8-D) differential equation model of the acute inflammatory response system to endotoxin challenge was developed. Endotoxin challenges at 3 and 12 mg/kg were administered to rats, and dynamic cytokine data for interleukin (IL)-6, tumor necrosis factor (TNF), and IL-10 were obtained and used to calibrate the model. Evaluation of competing model structures was performed by analyzing model predictions at 3, 6, and 12 mg/kg endotoxin challenges with respect to experimental data from rats. Subsequently, a model predictive control (MPC) algorithm was synthesized to control a hemoadsorption (HA) device, a blood purification treatment for acute inflammation. A particle filter (PF) algorithm was implemented to estimate the full state vector of the endotoxemic rat based on time series cytokine measurements. Treatment simulations show that: (i) the apparent primary mechanism of HA efficacy is white blood cell (WBC) capture, with cytokine capture a secondary benefit; and (ii) differential filtering of cytokines and WBC does not provide substantial improvement in treatment outcomes vs. existing HA devices.

Inflammation Following Traumatic Brain Injury in Humans: Insights from Data-Driven and Mechanistic Models into Survival and Death

Inflammation induced by traumatic brain injury (TBI) is a complex mediator of morbidity and mortality. We have previously demonstrated the utility of both data-driven and mechanistic models in settings of traumatic injury. We hypothesized that differential dynamic inflammation programs characterize TBI survivors vs. non-survivors, and sought to leverage computational modeling to derive novel insights into this life/death bifurcation. Thirteen inflammatory cytokines and chemokines were determined using Luminex™ in serial cerebrospinal fluid (CSF) samples from 31 TBI patients over 5 days. In this cohort, 5 were non-survivors (Glasgow Outcome Scale [GOS] score = 1) and 26 were survivors (GOS > 1). A Pearson correlation analysis of initial injury (Glasgow Coma Scale [GCS]) vs. GOS suggested that survivors and non-survivors had distinct clinical response trajectories to injury. Statistically significant differences in interleukin (IL)-4, IL-5, IL-6, IL-8, IL-13, and tumor necrosis factor-α (TNF-α) were observed between TBI survivors vs. non-survivors over 5 days. Principal Component Analysis and Dynamic Bayesian Network inference suggested differential roles of chemokines, TNF-α, IL-6, and IL-10, based upon which an ordinary differential equation model of TBI was generated. This model was calibrated separately to the time course data of TBI survivors vs. non-survivors as a function of initial GCS. Analysis of parameter values in ensembles of simulations from these models suggested differences in microglial and damage responses in TBI survivors vs. non-survivors. These studies suggest the utility of combined data-driven and mechanistic models in the context of human TBI.

Anomaly Detection in Host Signaling Pathways for the Early Prognosis of Acute Infection

Clinical diagnosis of acute infectious diseases during the early stages of infection is critical to administering the appropriate treatment to improve the disease outcome. We present a data driven analysis of the human cellular response to respiratory viruses including influenza, respiratory syncytia virus, and human rhinovirus, and compared this with the response to the bacterial endotoxin, Lipopolysaccharides (LPS). Using an anomaly detection framework we identified pathways that clearly distinguish between asymptomatic and symptomatic patients infected with the four different respiratory viruses and that accurately diagnosed patients exposed to a bacterial infection. Connectivity pathway analysis comparing the viral and bacterial diagnostic signatures identified host cellular pathways that were unique to patients exposed to LPS endotoxin indicating this type of analysis could be used to identify host biomarkers that can differentiate clinical etiologies of acute infection. We applied the Multivariate State Estimation Technique (MSET) on two human influenza (H1N1 and H3N2) gene expression data sets to define host networks perturbed in the asymptomatic phase of infection. Our analysis identified pathways in the respiratory virus diagnostic signature as prognostic biomarkers that triggered prior to clinical presentation of acute symptoms. These early warning pathways correctly predicted that almost half of the subjects would become symptomatic in less than forty hours post-infection and that three of the 18 subjects would become symptomatic after only 8 hours. These results provide a proof-of-concept for utility of anomaly detection algorithms to classify host pathway signatures that can identify presymptomatic signatures of acute diseases and differentiate between etiologies of infection. On a global scale, acute respiratory infections cause a significant proportion of human co-morbidities and account for 4.25 million deaths annually. The development of clinical diagnostic tools to distinguish between acute viral and bacterial respiratory infections is critical to improve patient care and limit the overuse of antibiotics in the medical community. The identification of prognostic respiratory virus biomarkers provides an early warning system that is capable of predicting which subjects will become symptomatic to expand our medical diagnostic capabilities and treatment options for acute infectious diseases. The host response to acute infection may be viewed as a deterministic signaling network responsible for maintaining the health of the host organism. We identify pathway signatures that reflect the very earliest perturbations in the host response to acute infection. These pathways provide a monitor the health state of the host using anomaly detection to quantify and predict health outcomes to pathogens.

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.

Deciphering the Counterplay of Aspergillus fumigatus Infection and Host Inflammation by Evolutionary Games on Graphs

Microbial invaders are ubiquitously present and pose the constant risk of infections that are opposed by various defence mechanisms of the human immune system. A tight regulation of the immune response ensures clearance of microbial invaders and concomitantly limits host damage that is crucial for host viability. To investigate the counterplay of infection and inflammation, we simulated the invasion of the human-pathogenic fungus Aspergillus fumigatus in lung alveoli by evolutionary games on graphs. The layered structure of the innate immune system is represented by a sequence of games in the virtual model. We show that the inflammatory cascade of the immune response is essential for microbial clearance and that the inflammation level correlates with the infection-dose. At low infection-doses, corresponding to daily inhalation of conidia, the resident alveolar macrophages may be sufficient to clear infections, however, at higher infection-doses their primary task shifts towards recruitment of neutrophils to infection sites.

A One-Nearest-Neighbor Approach to Identify the Original Time of Infection Using Censored Baboon Sepsis Data

OBJECTIVES

Sepsis therapies have proven to be elusive because of the difficulty of translating biologically sound and effective interventions in animal models to humans. A part of this problem originates from the fact that septic patients present at various times after the onset of sepsis, whereas the exact time of infection is controlled in animal models. We sought to determine whether data mining longitudinal physiologic data in a nonhuman primate model of Escherichia coli-induced sepsis could help inform the time of onset of infection.

DESIGN

A nearest-neighbor approach was used to back cast the time of onset of infection in animal models of sepsis. Animal data were censored to simulate prospective monitoring at any moment along the septic infection. This was compared against an uncensored database to find the most similar animal in order to estimate the infection onset time. Leave-one-out cross-validation was used for validation. Biomarker selection was performed based on the criteria of estimation accuracy and/or ease of measurement.

SETTING

Computational experimental on existing experimental data.

SUBJECTS

Retrospective data from 33 septic baboons (Papio ursinus) subjected to Escherichia coli infusion. Validation was performed using 14 pigs that were subjected to surgically induced fecal peritonitis and 22 pigs that were subjected to lipopolysaccharide infusion.

MEASUREMENTS AND MAIN RESULTS

Longitudinal physiologic and serum markers, time of death. The presence of uniquely changing biomarkers during septic infection enabled the estimation of infection onset time in the datasets. Various combinations of temporal biomarkers, such as WBC, oxygen content, mean arterial pressure, and heart rate, yielded estimation accuracies of up to 97.8%. The use of temporal vital signs and a single measurement of serum biomarkers yielded highly accurate estimates without the need for invasive measurements. Validation in the pig data revealed similar results despite the heterogeneity of multiple experimental cohorts. This suggests that the method may be effective if sufficiently similar subjects are present in the database.

CONCLUSIONS

One nearest-neighbor analysis showed promise in accurately identifying the onset of infection given a database of known infection times and of sufficient breadth. We suggest that this approach is ready for evaluation within the clinical setting using human data.

Reverse Engineering the Inflammatory "Clock": From Computational Modeling to Rational Resetting

Properly-regulated inflammation is central to homeostasis. Traumatic injury, hemorrhagic shock, septic shock, and other injury-related processes such as wound healing are associated with dysregulated inflammation. Like many biological processes, inflammation is a dynamic, complex system whose function, like that of an analog clock, cannot be discerned simply from a laundry list of its parts (data). The advent of multiplexed platforms for gathering biological data, while providing an unprecedented level of detailed information about the inflammatory response, has paradoxically also proven to be overwhelming. This problem is especially acute when the datasets involve time courses, since typical statistical analyses and data-driven modeling are geared towards single time points. Various groups have addressed this problem using dynamic approaches to data-driven and mechanistic computational modeling. These modeling tools can be thought of as the "gears" and "hands" of the "clock," and have led to insights regarding principal drivers, dynamic networks, feedbacks, and regulatory switches that characterize and perhaps regulate the inflammatory response. In parallel, mechanistic computational models have given an abstracted sense of how the inflammatory "clock" works, leading to in silico models of critically ill individuals and populations. Integrating data-driven and mechanistic modeling may point the way to a rational "resetting" of inflammation via model-driven precision medicine.

Insights into the Role of Chemokines, Damage-Associated Molecular Patterns, and Lymphocyte-Derived Mediators from Computational Models of Trauma-Induced Inflammation

SIGNIFICANCE

Traumatic injury elicits a complex, dynamic, multidimensional inflammatory response that is intertwined with complications such as multiple organ dysfunction and nosocomial infection. The complex interplay between inflammation and physiology in critical illness remains a challenge for translational research, including the extrapolation to human disease from animal models.

RECENT ADVANCES

Over the past decade, we and others have attempted to decipher the biocomplexity of inflammation in these settings of acute illness, using computational models to improve clinical translation. In silico modeling has been suggested as a computationally based framework for integrating data derived from basic biology experiments as well as preclinical and clinical studies.

CRITICAL ISSUES

Extensive studies in cells, mice, and human blunt trauma patients have led us to suggest (i) that while an adequate level of inflammation is required for healing post-trauma, inflammation can be harmful when it becomes self-sustaining via a damage-associated molecular pattern/Toll-like receptor-driven feed-forward circuit; (ii) that chemokines play a central regulatory role in driving either self-resolving or self-maintaining inflammation that drives the early activation of both classical innate and more recently recognized lymphoid pathways; and (iii) the presence of multiple thresholds and feedback loops, which could significantly affect the propagation of inflammation across multiple body compartments.

FUTURE DIRECTIONS

These insights from data-driven models into the primary drivers and interconnected networks of inflammation have been used to generate mechanistic computational models. Together, these models may be used to gain basic insights as well as serving to help define novel biomarkers and therapeutic targets.

Automated parameter estimation for biological models using Bayesian statistical model checking

Background

Probabilistic models have gained widespread acceptance in the systems biology community as a useful way to represent complex biological systems. Such models are developed using existing knowledge of the structure and dynamics of the system, experimental observations, and inferences drawn from statistical analysis of empirical data. A key bottleneck in building such models is that some system variables cannot be measured experimentally. These variables are incorporated into the model as numerical parameters. Determining values of these parameters that justify existing experiments and provide reliable predictions when model simulations are performed is a key research problem.

Domain experts usually estimate the values of these parameters by fitting the model to experimental data. Model fitting is usually expressed as an optimization problem that requires minimizing a cost-function which measures some notion of distance between the model and the data. This optimization problem is often solved by combining local and global search methods that tend to perform well for the specific application domain. When some prior information about parameters is available, methods such as Bayesian inference are commonly used for parameter learning. Choosing the appropriate parameter search technique requires detailed domain knowledge and insight into the underlying system.

Results

Using an agent-based model of the dynamics of acute inflammation, we demonstrate a novel parameter estimation algorithm by discovering the amount and schedule of doses of bacterial lipopolysaccharide that guarantee a set of observed clinical outcomes with high probability. We synthesized values of twenty-eight unknown parameters such that the parameterized model instantiated with these parameter values satisfies four specifications describing the dynamic behavior of the model.

Conclusions

We have developed a new algorithmic technique for discovering parameters in complex stochastic models of biological systems given behavioral specifications written in a formal mathematical logic. Our algorithm uses Bayesian model checking, sequential hypothesis testing, and stochastic optimization to automatically synthesize parameters of probabilistic biological models.

The macrophage and its role in inflammation and tissue repair: mathematical and systems biology approaches

Macrophages are central to the inflammatory response and its ability to resolve effectively. They are complex cells that adopt a range of subtypes depending on the tissue type and stimulus that they find themselves under. This flexibility allows them to play multiple, sometimes opposing, roles in inflammation and tissue repair. Their central role in the inflammatory process is reflected in macrophage dysfunction being implicated in chronic inflammation and poorly healing wounds. In this study, we discuss recent attempts to model mathematically and computationally the macrophage and how it partakes in the complex processes of inflammation and tissue repair. There are increasing data describing the variety of macrophage phenotypes and their underlying transcriptional programs. Dynamic mathematical and computational models are an ideal way to test biological hypotheses against experimental data and could aid in understanding this multi-functional cell and its potential role as an attractive therapeutic target for inflammatory conditions and tissue repair.

Dynamics of the HPA axis and inflammatory cytokines: Insights from mathematical modeling

In the work presented here, a novel mathematical model was developed to explore the bi-directional communication between the hypothalamic-pituitary-adrenal (HPA) axis and inflammatory cytokines in acute inflammation. The dynamic model consists of five delay differential equations 5D for two main pro-inflammatory cytokines (TNF-α and IL-6) and two hormones of the HPA axis (ACTH and cortisol) and LPS endotoxin. The model is an attempt to increase the understanding of the role of primary hormones and cytokines in this complex relationship by demonstrating the influence of different organs and hormones in the regulation of the inflammatory response. The model captures the main qualitative features of cytokine and hormone dynamics when a toxic challenge is introduced. Moreover, in this work a new simple delayed model of the HPA axis is introduced which supports the understanding of the ultradian rhythm of HPA hormones both in normal and infection conditions. Through simulations using the model, the role of key inflammatory cytokines and cortisol in transition from acute to persistent inflammation through stability analysis is investigated. Also, by employing a Markov chain Monte Carlo (MCMC) method, parameter uncertainty and the effects of parameter variations on each other are analyzed. This model confirms the important role of the HPA axis in acute and prolonged inflammation and can be a useful tool in further investigation of the role of stress on the immune response to infectious diseases.

Mathematical Modeling of Early Cellular Innate and Adaptive Immune Responses to Ischemia/Reperfusion Injury and Solid Organ Allotransplantation

A mathematical model of the early inflammatory response in transplantation is formulated with ordinary differential equations. We first consider the inflammatory events associated only with the initial surgical procedure and the subsequent ischemia/reperfusion (I/R) events that cause tissue damage to the host as well as the donor graft. These events release damage-associated molecular pattern molecules (DAMPs), thereby initiating an acute inflammatory response. In simulations of this model, resolution of inflammation depends on the severity of the tissue damage caused by these events and the patient's (co)-morbidities. We augment a portion of a previously published mathematical model of acute inflammation with the inflammatory effects of T cells in the absence of antigenic allograft mismatch (but with DAMP release proportional to the degree of graft damage prior to transplant). Finally, we include the antigenic mismatch of the graft, which leads to the stimulation of potent memory T cell responses, leading to further DAMP release from the graft and concomitant increase in allograft damage. Regulatory mechanisms are also included at the final stage. Our simulations suggest that surgical injury and I/R-induced graft damage can be well-tolerated by the recipient when each is present alone, but that their combination (along with antigenic mismatch) may lead to acute rejection, as seen clinically in a subset of patients. An emergent phenomenon from our simulations is that low-level DAMP release can tolerize the recipient to a mismatched allograft, whereas different restimulation regimens resulted in an exaggerated rejection response, in agreement with published studies. We suggest that mechanistic mathematical models might serve as an adjunct for patient- or sub-group-specific predictions, simulated clinical studies, and rational design of immunosuppression.

Outcome Prediction in Mathematical Models of Immune Response to Infection

Clinicians need to predict patient outcomes with high accuracy as early as possible after disease inception. In this manuscript, we show that patient-to-patient variability sets a fundamental limit on outcome prediction accuracy for a general class of mathematical models for the immune response to infection. However, accuracy can be increased at the expense of delayed prognosis. We investigate several systems of ordinary differential equations (ODEs) that model the host immune response to a pathogen load. Advantages of systems of ODEs for investigating the immune response to infection include the ability to collect data on large numbers of ‘virtual patients’, each with a given set of model parameters, and obtain many time points during the course of the infection. We implement patient-to-patient variability v in the ODE models by randomly selecting the model parameters from distributions with coefficients of variation v that are centered on physiological values. We use logistic regression with one-versus-all classification to predict the discrete steady-state outcomes of the system. We find that the prediction algorithm achieves near 100% accuracy for v = 0, and the accuracy decreases with increasing v for all ODE models studied. The fact that multiple steady-state outcomes can be obtained for a given initial condition, i.e. the basins of attraction overlap in the space of initial conditions, limits the prediction accuracy for v > 0. Increasing the elapsed time of the variables used to train and test the classifier, increases the prediction accuracy, while adding explicit external noise to the ODE models decreases the prediction accuracy. Our results quantify the competition between early prognosis and high prediction accuracy that is frequently encountered by clinicians.

Achieving Remission in Gulf War Illness: A Simulation-Based Approach to Treatment Design

Gulf War Illness (GWI) is a chronic multi-symptom disorder affecting up to one-third of the 700,000 returning veterans of the 1991 Persian Gulf War and for which there is no known cure. GWI symptoms span several of the body’s principal regulatory systems and include debilitating fatigue, severe musculoskeletal pain, cognitive and neurological problems. Using computational models, our group reported previously that GWI might be perpetuated at least in part by natural homeostatic regulation of the neuroendocrine-immune network. In this work, we attempt to harness these regulatory dynamics to identify treatment courses that might produce lasting remission. Towards this we apply a combinatorial optimization scheme to the Monte Carlo simulation of a discrete ternary logic model that represents combined hypothalamic-pituitary-adrenal (HPA), gonadal (HPG), and immune system regulation in males. In this work we found that no single intervention target allowed a robust return to normal homeostatic control. All combined interventions leading to a predicted remission involved an initial inhibition of Th1 inflammatory cytokines (Th1Cyt) followed by a subsequent inhibition of glucocorticoid receptor function (GR). These first two intervention events alone ended in stable and lasting return to the normal regulatory control in 40% of the simulated cases. Applying a second cycle of this combined treatment improved this predicted remission rate to 2 out of 3 simulated subjects (63%). These results suggest that in a complex illness such as GWI, a multi-tiered intervention strategy that formally accounts for regulatory dynamics may be required to reset neuroendocrine-immune homeostasis and support extended remission.

A Boundary Value Approach to Optimization with an Application to Salmonella Competition

We develop a novel optimization framework to study strategies in ecological competition processes. The optimization method uses theory from dynamical systems describing the asymptotic behavior of a bistable system based on initial conditions, which we implement using a numerical boundary value problem. As an application of our method, we develop a model of the competition between Salmonella Typhimurium and the host's native microflora, which constantly and densely inhabit the intestinal lining of most mammals. S. Typhimurium invades the gut in two distinct phenotypic populations, one virulent and one avirulent, though the avirulent bacteria have the ability to activate a virulence factor and thereby "switch" into the virulent population. Counterintuitively, some studies have found that the combined population of S. Typhimurium gains an environmental advantage over the commensal microbiota after the virulent subpopulation provokes the body's inflammatory defenses. Our model represents the competition between the commensal microbiota, the avirulent salmonella, and the virulent salmonella populations and incorporates a simple representation of the immune response. We use our model to predict optimal strategies that would favor salmonella in its competition with the commensal bacteria.

The inflammatory response to influenza A virus (H1N1): An experimental and mathematical study

Mortality from influenza infections continues as a global public health issue, with the host inflammatory response contributing to fatalities related to the primary infection. Based on Ordinary Differential Equation (ODE) formalism, a computational model was developed for the in-host response to influenza A virus, merging inflammatory, innate, adaptive and humoral responses to virus and linking severity of infection, the inflammatory response, and mortality. The model was calibrated using dense cytokine and cell data from adult BALB/c mice infected with the H1N1 influenza strain A/PR/8/34 in sublethal and lethal doses. Uncertainty in model parameters and disease mechanisms was quantified using Bayesian inference and ensemble model methodology that generates probabilistic predictions of survival, defined as viral clearance and recovery of the respiratory epithelium. The ensemble recovers the expected relationship between magnitude of viral exposure and the duration of survival, and suggests mechanisms primarily responsible for survival, which could guide the development of immuno-modulatory interventions as adjuncts to current anti-viral treatments. The model is employed to extrapolate from available data survival curves for the population and their dependence on initial viral aliquot. In addition, the model allows us to illustrate the positive effect of controlled inflammation on influenza survival.

Modeling the effects of systemic mediators on the inflammatory phase of wound healing

The normal wound healing response is characterized by a progression from clot formation, to an inflammatory phase, to a repair phase, and finally, to remodeling. In many chronic wounds there is an extended inflammatory phase that stops this progression. In order to understand the inflammatory phase in more detail, we developed an ordinary differential equation model that accounts for two systemic mediators that are known to modulate this phase, estrogen (a protective hormone during wound healing) and cortisol (a hormone elevated after trauma that slows healing). This model describes the interactions in the wound between wound debris, pathogens, neutrophils and macrophages and the modulation of these interactions by estrogen and cortisol. A collection of parameter sets, which qualitatively match published data on the dynamics of wound healing, was chosen using Latin Hypercube Sampling. This collection of parameter sets represents normal healing in the population as a whole better than one single parameter set. Including the effects of estrogen and cortisol is a necessary step to creating a patient specific model that accounts for gender and trauma. Utilization of math modeling techniques to better understand the wound healing inflammatory phase could lead to new therapeutic strategies for the treatment of chronic wounds. This inflammatory phase model will later become the inflammatory subsystem of our full wound healing model, which includes fibroblast activity, collagen accumulation and remodeling.

Computational modelling of the inflammatory response in trauma, sepsis and wound healing: implications for modelling resilience

Resilience refers to the ability to recover from illness or adversity. At the cell, tissue, organ and whole-organism levels, the response to perturbations such as infections and injury involves the acute inflammatory response, which in turn is connected to and controlled by changes in physiology across all organ systems. When coordinated properly, inflammation can lead to the clearance of infection and healing of damaged tissues. However, when either overly or insufficiently robust, inflammation can drive further cell stress, tissue damage, organ dysfunction and death through a feed-forward process of inflammation → damage → inflammation. To address this complexity, we have obtained extensive datasets regarding the dynamics of inflammation in cells, animals and patients, and created data-driven and mechanistic computational simulations of inflammation and its recursive effects on tissue, organ and whole-organism (patho)physiology. Through this approach, we have discerned key regulatory mechanisms, recapitulated in silico key features of clinical trials for acute inflammation and captured diverse, patient-specific outcomes. These insights may allow for the determination of individual-specific tolerances to illness and adversity, thereby defining the role of inflammation in resilience.