Transient behavior of a chemotaxis system modelling certain types of tissue inflammation

An in silico Model of T Cell Infiltration Dynamics Based on an Advanced in vitro System to Enhance Preclinical Decision Making in Cancer Immunotherapy

Cancer immunotherapy often involves the use of engineered molecules to selectively bind and activate T cells located within tumour tissue. Fundamental to the success of such treatments is the presence or recruitment of T cells localised within the tumour microenvironment. Advanced organ-on-a-chip systems provide an in vitro setting in which to investigate how novel molecules influence the spatiotemporal dynamics of T cell infiltration into tissue, both in the context of anti-tumour efficacy and off-tumour toxicity. While highly promising, the complexity of these systems is such that mathematical modelling plays a crucial role in the quantitative evaluation of experimental results and maximising the mechanistic insight derived. We develop a mechanistic, mathematical model of a novel microphysiological in vitro platform that recapitulates T cell infiltration into epithelial tissue, which may be normal or transformed. The mathematical model describes the spatiotemporal dynamics of infiltrating T cells in response to chemotactic cytokine signalling. We integrate the model with dynamic imaging data to optimise key model parameters. The mathematical model demonstrates a good fit to the observed experimental data and accurately describes the distribution of infiltrating T cells. This model is designed to complement the in vitro system; with the potential to elucidate complex biological mechanisms, including the mode of action of novel therapies and the drivers of safety events, and, ultimately, improve the efficacy-safety profile of T cell-targeted cancer immunotherapies.

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.

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.

Analysis of Lauffenburger-Kennedy bacterial infection model for tissue inflammation dynamics

In this paper, we analyze a mathematical model for an inflammatory response to bacterial infection of homogeneous tissues. Specifically, we provide a detailed analysis of the Lauffenburger-Kennedy bacterial infection model and show that the model exhibits three possible equilibria corresponding to a bacteria-free and two endemic compromised steady states. Asymptotic results of the steady states along with the existences of saddle-node connection Hopf bifurcations are shown under certain conditions of the parameters. Within the biological ranges of the parameter values, we observe that the system can exhibit both forward and backward bifurcation. In addition, in both cases, the larger compromise bacterial infection steady state can either approach an equilibrium or can oscillate around it via Hopf bifurcation depending on the value of the ratio of leukocyte mortality to phagocytosis rates. Numerical results are used to provide illustrative examples of these different dynamical patterns observed in the model.

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.

Development of a Computational Model of Abscess Formation

In some bacterial infections, the immune system cannot eliminate the invading pathogen. In these cases, the invading pathogen is successful in establishing a favorable environment to survive and persist in the host organism. For example, S. aureus bacteria survive in organ tissues employing a set of mechanisms that work in a coordinated and highly regulated way allowing: (1) efficient impairment of the immune response; and (2) protection from the immune cells and molecules. S. aureus secretes several proteins including coagulases and toxins that drive abscess formation and persistence. Unless staphylococcal abscesses are surgically drained and treated with antibiotics, disseminated infection and septicemia produce a lethal outcome. Within this context, this paper develops a simple mathematical model of abscess formation incorporating characteristics that we judge important for an abscess to be formed. Our aim is to build a mathematical model that reproduces some characteristics and behaviors that are observed in the process of abscess formation.

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.

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.

From data patterns to mechanistic models in acute critical illness

The complexity of the physiologic and inflammatory response in acute critical illness has stymied the accurate diagnosis and development of therapies. The Society for Complex Acute Illness was formed a decade ago with the goal of leveraging multiple complex systems approaches to address this unmet need. Two main paths of development have characterized the society's approach: (i) data pattern analysis, either defining the diagnostic/prognostic utility of complexity metrics of physiologic signals or multivariate analyses of molecular and genetic data and (ii) mechanistic mathematical and computational modeling, all being performed with an explicit translational goal. Here, we summarize the progress to date on each of these approaches, along with pitfalls inherent in the use of each approach alone. We suggest that the next decade holds the potential to merge these approaches, connecting patient diagnosis to treatment via mechanism-based dynamical system modeling and feedback control and allowing extrapolation from physiologic signals to biomarkers to novel drug candidates. As a predicate example, we focus on the role of data-driven and mechanistic models in neuroscience and the impact that merging these modeling approaches can have on general anesthesia.

In silico modeling: methods and applications to trauma and sepsis

OBJECTIVES

To familiarize clinicians with advances in computational disease modeling applied to trauma and sepsis.

DATA SOURCES

PubMed search and review of relevant medical literature.

SUMMARY

Definitions, key methods, and applications of computational modeling to trauma and sepsis are reviewed.

CONCLUSIONS

Computational modeling of inflammation and organ dysfunction at the cellular, organ, whole-organism, and population levels has suggested a positive feedback cycle of inflammation → damage → inflammation that manifests via organ-specific inflammatory switching networks. This structure may manifest as multicompartment "tipping points" that drive multiple organ dysfunction. This process may be amenable to rational inflammation reprogramming.

In Silico Augmentation of the Drug Development Pipeline: Examples from the study of Acute Inflammation

The clinical translation of promising basic biomedical findings, whether derived from reductionist studies in academic laboratories or as the product of extensive high-throughput and -content screens in the biotechnology and pharmaceutical industries, has reached a period of stagnation in which ever higher research and development costs are yielding ever fewer new drugs. Systems biology and computational modeling have been touted as potential avenues by which to break through this logjam. However, few mechanistic computational approaches are utilized in a manner that is fully cognizant of the inherent clinical realities in which the drugs developed through this ostensibly rational process will be ultimately used. In this article, we present a Translational Systems Biology approach to inflammation. This approach is based on the use of mechanistic computational modeling centered on inherent clinical applicability, namely that a unified suite of models can be applied to generate in silico clinical trials, individualized computational models as tools for personalized medicine, and rational drug and device design based on disease mechanism.

Translational systems approaches to the biology of inflammation and healing

Inflammation is a complex, non-linear process central to many of the diseases that affect both developed and emerging nations. A systems-based understanding of inflammation, coupled to translational applications, is therefore necessary for efficient development of drugs and devices, for streamlining analyses at the level of populations, and for the implementation of personalized medicine. We have carried out an iterative and ongoing program of literature analysis, generation of prospective data, data analysis, and computational modeling in various experimental and clinical inflammatory disease settings. These simulations have been used to gain basic insights into the inflammatory response under baseline, gene-knockout, and drug-treated experimental animals for in silico studies associated with the clinical settings of sepsis, trauma, acute liver failure, and wound healing to create patient-specific simulations in polytrauma, traumatic brain injury, and vocal fold inflammation; and to gain insight into host-pathogen interactions in malaria, necrotizing enterocolitis, and sepsis. These simulations have converged with other systems biology approaches (e.g., functional genomics) to aid in the design of new drugs or devices geared towards modulating inflammation. Since they include both circulating and tissue-level inflammatory mediators, these simulations transcend typical cytokine networks by associating inflammatory processes with tissue/organ impacts via tissue damage/dysfunction. This framework has now allowed us to suggest how to modulate acute inflammation in a rational, individually optimized fashion. This plethora of computational and intertwined experimental/engineering approaches is the cornerstone of Translational Systems Biology approaches for inflammatory diseases.

Mechanistic simulations of inflammation: current state and future prospects

Inflammation is a normal, robust physiological process. It can also be viewed as a complex system that senses and attempts to resolve homeostatic perturbations initiated from within the body (for example, in autoimmune disease) or from the outside (for example, in infections). Virtually all acute and chronic diseases are either driven or modulated by inflammation. The complex interplay between beneficial and harmful arms of the inflammatory response may underlie the lack of fully effective therapies for many diseases. Mathematical modeling is emerging as a frontline tool for understanding the complexity of the inflammatory response. A series of articles in this issue highlights various modeling approaches to inflammation in the larger context of health and disease, from intracellular signaling to whole-animal physiology. Here we discuss the state of this emerging field. We note several common features of inflammation models, as well as challenges and prospects for future studies.

A user's guide to PDE models for chemotaxis

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems

When searching for hosts, parasitoids are observed to aggregate in response to chemical signalling cues emitted by plants during host feeding. In this paper we model aggregative parasitoid behaviour in a multi-species host-parasitoid community using a system of reaction-diffusion-chemotaxis equations. The stability properties of the steady-states of the model system are studied using linear stability analysis which highlights the possibility of interesting dynamical behaviour when the chemotactic response is above a certain threshold. We observe quasi-chaotic dynamic heterogeneous spatio-temporal patterns, quasi-stationary heterogeneous patterns and a destabilisation of the steady-states of the system. The generation of heterogeneous spatio-temporal patterns and destabilisation of the steady state are due to parasitoid chemotactic response to hosts. The dynamical behaviour of our system has both mathematical and ecological implications and the concepts of chemotaxis-driven instability and coexistence and ecological change are discussed.

Evidence-based modeling of critical illness: an initial consensus from the Society for Complexity in Acute Illness

INTRODUCTION

Given the complexity of biological systems, understanding their dynamic behaviors, such as the Acute Inflammatory Response (AIR), requires a formal synthetic process. Dynamic Mathematical Modeling (DMM) represents a suite of methods intended for inclusion within the required synthetic framework. The DMM, however, is a relatively novel approach in the practice of biomedical research. The Society for Complexity in Acute Illness (SCAI) was formed in 2004 from the leading research groups using DMM in the study of acute inflammation. This society believes that it is important to offer guidelines for the design, development, and use of DMM in the setting of AIR research to avoid the "garbage in, garbage out" problem. Accordingly, SCAI identified a need for and carried out a critical appraisal of DMM as currently used in the setting of acute illness.

METHODS

The SCAI annual meeting in 2005, the Fourth International Conference on Complexity in Acute Illness (Cologne, Germany), was structured with the intent of developing a consensus statement on the methods and execution of DMM in AIR research. The conference was organized to include a series of interactive breakout sessions that included thought leaders from both the DMM and acute illness fields, the results of which were then presented in summary form to the entire group for discussion and consensus. The information in this article represents the concatenation of those presentations.

RESULTS

The output from the Fourth International Conference on Complexity in Acute Illness involved consensus statements for the following topics: (1) the need for DMM; (2) a suggested approach for the process of establishing a modeling project; (3) the type of "wet" laboratory experiments and data needed to establish a modeling project; (4) general quality measures for data to be input to a modeling project; and (5) a descriptive list of several types of DMM to provide guidance in selection of a method for a project.

CONCLUSION

We believe that the complexity of biological systems requires that DMM needs to be among the methods used to improve understanding and make progress with attempts to characterize and manipulate the AIR. We believe that this consensus statement will help guide the integration, rational implementation, and standardization of DMM into general biomedical research.

IN SILICO MODELS OF ACUTE INFLAMMATION IN ANIMALS

Trauma and hemorrhagic shock elicit an acute inflammatory response, predisposing patients to sepsis, organ dysfunction, and death. Few approved therapies exist for these acute inflammatory states, mainly due to the complex interplay of interacting inflammatory and physiological elements working at multiple levels. Various animal models have been used to simulate these phenomena, but these models often do not replicate the clinical setting of multiple overlapping insults. Mathematical modeling of complex systems is an approach for understanding the interplay among biological interactions. We constructed a mathematical model using ordinary differential equations that encompass the dynamics of cells and cytokines of the acute inflammatory response, as well as global tissue dysfunction. The model was calibrated in C57Bl/6 mice subjected to (1) various doses of lipopolysaccharide (LPS) alone, (2) surgical trauma, and (3) surgery + hemorrhagic shock. We tested the model's predictive ability in scenarios on which it had not been trained, namely, (1) surgery +/- hemorrhagic shock + LPS given at times after the beginning of surgical instrumentation, and (2) surgery + hemorrhagic shock + bilateral femoral fracture. Software was created that facilitated fitting of the mathematical model to experimental data, as well as for simulation of experiments with various inflammatory challenges and associated variations (gene knockouts, inhibition of specific cytokines, etc.). Using this software, the C57Bl/6-specific model was recalibrated for inflammatory analyte data in CD14-/- mice and was used to elucidate altered features of inflammation in these animals. In other experiments, rats were subjected to surgical trauma +/- LPS or to bacterial infection via fibrin clots impregnated with various inocula of Escherichia coli. Mathematical modeling may provide insights into the complex dynamics of acute inflammation in a manner that can be tested in vivo using many fewer animals than has been possible previously.

Spatio-temporal patterns generated by Salmonella typhimurium

We present experimental results on the bacterium Salmonella typhimurium which show that cells of chemotactic strains aggregate in response to gradients of amino acids, attractants that they themselves excrete. Depending on the conditions under which cells are cultured, they form periodic arrays of continuous or perforated rings, which arise sequentially within a spreading bacterial lawn. Based on these experiments, we develop a biologically realistic cell-chemotaxis model to describe the self-organization of bacteria. Numerical and analytical investigations of the model mechanism show how the two types of observed geometric patterns can be generated by the interaction of the cells with chemoattractant they produce.

Finite time blow-up in some models of chemotaxis

We consider a class of models of chemotactic bacterial populations, introduced by Keller-Segel. For those models, we investigate the possibility of chemotactic collapse, in other words, the possibility that in finite time the population of predators aggregates to form a delta-function. To study this phenomenon, we construct self-similar solutions, which may or may not blow-up (in finite time), depending on the relative strength of three mechanisms in competition: (i) the chemotactic attraction of bacteria towards regions of high concentration in substrate (ii) the rate of consumption of the substrate by the bacteria and (iii) (possibly) the diffusion of bacteria. The solutions we construct are radially symmetric, and therefore have no relation with the classical traveling wave solutions. Our scaling can be justified by a dimensional analysis. We give some evidence of numerical stability.

Chemotaxis and chemokinesis in eukaryotic cells: the Keller-Segel equations as an approximation to a detailed model

More than 20 years after its proposal, Keller and Segel's model (1971, J. theor. Biol., 30, 235-248) remains by far the most popular model for chemical control of cell movement. However, before the Keller-Segel equations can be applied to a particular system, appropriate functional forms must be specified for the dependence on chemical concentration of the cell transport coefficients and the chemical degradation rate. In the vast majority of applications, these functional forms have been chosen using simple intuitive criteria. We focus on the particular case of eukaryotic cell movement, and derive an approximation to the detailed model of Sherratt et al. (1993, J. theor. Biol., 162, 23-40). The approximation consists of the Keller-Segel equations, with specific forms predicted for the cell transport coefficients and chemical degradation rate. Moreover, the parameter values in these functional forms can be directly measured experimentally. In the case of the much studied neutrophil-peptide system, we test our approximation using both the Boyden chamber and under-agarose assays. Finally, we show that for other cell-chemical interactions, a simple comparison of time scales provides a rapid check on the validity of our Keller-Segel approximation.

Cell migration in multicell spheroids: swimming against the tide

Multicell spheroids, small spherical clusters of cancer cells, have become an important in vitro model for studying tumour development given the diffusion limited geometry associated with many solid tumour growths. Spheroids expand until they reach a dormant state where they exhibit a grossly static three-layered structure. However, at a cellular level, the spheroid is demonstrably dynamic with constituent cells migrating from the outer well-nourished region of the spheroid toward the necrotic central core. The mechanism that drives the migrating cells in the spheroid is not well understood. In this paper we demonstrate that recent experiments on internationalization can be adequately described by implicating pressure gradients caused by differential cell proliferation and cell death as the primary mechanism. Although chemotaxis plays a role in cell movement, we argue that it acts against the passive movement caused by pressure differences.