A global phenomenological model of ischemic stroke with stress on spreading depressions

A Data-Informed Mathematical Model of Microglial Cell Dynamics During Ischemic Stroke in the Middle Cerebral Artery

Neuroinflammation immediately follows the onset of ischemic stroke in the middle cerebral artery. During this process, microglial cells are activated in and recruited to the penumbra. Microglial cells can be activated into two different phenotypes: M1, which can worsen brain injury; or M2, which can aid in long-term recovery. In this study, we contribute a summary of experimental data on microglial cell counts in the penumbra following ischemic stroke induced by middle cerebral artery occlusion (MCAO) in mice and compile available data sets into a single set suitable for time series analysis. Further, we formulate a mathematical model of microglial cells in the penumbra during ischemic stroke due to MCAO. Through use of global sensitivity analysis and Markov Chain Monte Carlo (MCMC)-based parameter estimation, we analyze the effects of the model parameters on the number of M1 and M2 cells in the penumbra and fit identifiable parameters to the compiled experimental data set. We utilize results from MCMC parameter estimation to ascertain uncertainty bounds and forward predictions for the number of M1 and M2 microglial cells over time. Results demonstrate the significance of parameters related to M1 and M2 activation on the number of M1 and M2 microglial cells. Simulations further suggest that potential outliers in the observed data may be omitted and forecast predictions suggest a lingering inflammatory response.

Towards a Systemic Concept of the Brain Ishemia Stroke: Monte Carlo Driven in silico Model

BACKGROUND

The work proposes a new mathematical model of dynamic processes of a typical spatially heterogeneous biological system, and sets and solves a mathematical problem of modeling the dynamics of the system of neurovascular units of the brain in conditions of ischemic stroke. There is a description of only a small number of mathematical models of stroke in the literature. This model is being studied and a numerical and software implementation of the corresponding mathematical problem is proposed.

METHODS

This work is the first attempt ever aiming to employ a Monte Carlo computational approach for In Silico simulation of the most critical parameters in molecular and cellular pathogenesis of the brain ischemic stroke. In this work, a new mathematical model of the development of ischemic stroke is proposed in the form of a discrete model based on neurovascular units (NVU) as elements.

RESULTS

As a result of testing the program with the assignment of empirically selected coefficients, data were obtained on the evolution of the states of the lattice of the cellular automaton of the model for the spread of stroke in a region of the brain tissue. A resulting new theoretical model of the particular pathologically altered biosystem might be taken as a promising tool for further studies in neurology; general pathology and cell biology.

CONCLUSION

For the first time, a mathematical model has been constructed that allows us to represent the spatial dynamics of the development of the affected area in ischemic stroke of the brain, taking into account neurovascular units as single morphofunctional structures.

Spreading Depressions and Periinfarct Spreading Depolarizations in the Context of Cortical Plasticity

Studies of cortical function-recovery require a comparison between normal and post-stroke conditions that lead to changes in cortical metaplasticity. Focal cortical stroke impairs experience-dependent plasticity in the neighboring somatosensory cortex and usually evokes periinfarct depolarizations (PiDs) - spreading depression-like waves. Experimentally induced spreading depressions (SDs) affect gene expression and some of these changes persist for at least 30 days. In this study we compare the effects of non-stroke depolarizations that impair cortical experience-dependent plasticity to the effects of stroke, by inducing experience-dependent plasticity in rats with SDs or PiDs by a month of contralateral partial whiskers deprivation. We found that whiskers' deprivation after SDs resulted in normal cortical representation enlargement suggesting that SDs and PiDs depolarization have no influence on experience-dependent plasticity cortical map reorganization. PiDs and the MMP-9, -3, -2 or COX-2 proteins, which are assumed to influence metaplasticity in rats after stroke were compared between SDs induced by high osmolarity KCl solution and the PiDs that followed cortical photothrombotic stroke (PtS). We found that none of these factors directly caused cortical post-stroke metaplasticity changes. The only significant difference between stoke and induced SD was a greater imbalance in interhemispheric activity equilibrium after stroke. The interhemispheric interactions that were modified by stroke may therefore be promising targets for future studies of post-stroke experience-dependent plasticity and of recuperation studies.

Relevance of Porcine Stroke Models to Bridge the Gap from Pre-Clinical Findings to Clinical Implementation

In the search of animal stroke models providing translational advantages for biomedical research, pigs are large mammals with interesting brain characteristics and wide social acceptance. Compared to rodents, pigs have human-like highly gyrencephalic brains. In addition, increasingly through phylogeny, animals have more sophisticated white matter connectivity; thus, ratios of white-to-gray matter in humans and pigs are higher than in rodents. Swine models provide the opportunity to study the effect of stroke with emphasis on white matter damage and neuroanatomical changes in connectivity, and their pathophysiological correlate. In addition, the subarachnoid space surrounding the swine brain resembles that of humans. This allows the accumulation of blood and clots in subarachnoid hemorrhage models mimicking the clinical condition. The clot accumulation has been reported to mediate pathological mechanisms known to contribute to infarct progression and final damage in stroke patients. Importantly, swine allows trustworthy tracking of brain damage evolution using the same non-invasive multimodal imaging sequences used in the clinical practice. Moreover, several models of comorbidities and pathologies usually found in stroke patients have recently been established in swine. We review here ischemic and hemorrhagic stroke models reported so far in pigs. The advantages and limitations of each model are also discussed.

Nonautonomous dynamics of acute cell injury

Medical conditions due to acute cell injury, such as stroke and heart attack, are of tremendous impact and have attracted huge amounts of research effort. The biomedical research that seeks cures for these conditions has been dominated by a qualitative, inductive mind-set. Although the inductive approach has not been effective in developing medical treatments, it has amassed enough information to allow construction of quantitative, deductive models of acute cell injury. In this work we develop a modeling approach by extending an autonomous nonlinear dynamic theory of acute cell injury that offered new ways to conceptualize cell injury but possessed limitations that decrease its effectiveness. Here we study the global dynamics of the cell injury theory using a nonautonomous formulation. Different from the standard scenario in nonlinear dynamics that is determined by the steady state and fixed points of the model equations, in this nonautonomous model with a trivial fixed point, the system property is dominated by the transient states and the corresponding dynamic processes. The model gives rise to four qualitative types of dynamical patterns that can be mapped to the behavior of cells after clinical acute injuries. The nonautonomous theory predicts the existence of a latent stress response capacity (LSRC) possessed by injured cells. The LSRC provides a theoretical explanation of how therapies, such as hypothermia, can prevent cell death after lethal injuries. The nonautonomous theory of acute cell injury provides an improved quantitative framework for understanding cell death and recovery and lays a foundation for developing effective therapeutics for acute injury.

Modeling cortical spreading depression induced by the hyperactivity of interneurons

Cortical spreading depression (CSD) is a wave of transient intense neuronal firing leading to a long lasting depolarizing block of neuronal activity. It is a proposed pathological mechanism of migraine with aura. Some forms of migraine are associated with a genetic mutation of the Na_v1.1 channel, resulting in its gain of function and implying hyperexcitability of interneurons. This leads to the counterintuitive hypothesis that intense firing of interneurons can cause CSD ignition. To test this hypothesis in silico, we developed a computational model of an E-I pair (a pyramidal cell and an interneuron), in which the coupling between the cells in not just synaptic, but takes into account also the effects of the accumulation of extracellular potassium caused by the activity of the neurons and of the synapses. In the context of this model, we show that the intense firing of the interneuron can lead to CSD. We have investigated the effect of various biophysical parameters on the transition to CSD, including the levels of glutamate or GABA, frequency of the interneuron firing and the efficacy of the KCC2 co-transporter. The key element for CSD ignition in our model was the frequency of interneuron firing and the related accumulation of extracellular potassium, which induced a depolarizing block of the pyramidal cell. This constitutes a new mechanism of CSD ignition.

Clinical neurophysiology of stroke

Stroke constitutes the third most common cause of death and the leading cause of acquired neurologic handicap. During ischemic stroke, very early after the onset of the focal perfusion deficit, excitotoxicity triggers a number of events that can further contribute to tissue death. Such events include peri-infarct depolarizations and spreading depolarizations (SDs) within the ischemic penumbra. SDs spread slowly through continuous gray matter at a typical velocity of 2-5mm/min. SDs exacerbate neuronal injury through prolonged ionic breakdown and SD-related hypoperfusion (spreading ischemia). Scalp EEG alone is not yet sufficient to reliably diagnose SDs. Hyperexcitability occurs in parallel, both in the acute and chronic phases of stroke. Stroke is a common cause of new-onset epileptic seizures after middle age and is the leading cause of symptomatic epilepsy in adults. The last part of this chapter is dedicated to noninvasive neurophysiologic techniques that can be used to promote stroke rehabilitation. These techniques mainly include repetitive transcranial magnetic stimulation and tDCS. These approaches are based on the concept of interhemispheric rivalry and aim at modulating the imbalance of cortical activities between both hemispheres resulting from stroke.

Computer modeling of ischemic stroke

The occlusion of a blood vessel in the brain causes an ischemic stroke. Current treatment relies restoration of blood flow within 3 hours. Substantial research has focused on neuroprotection to spare compromised neural tissue and extend the treatment time window. Despite success with animal models and extensive associated clinical testing, there are still no therapies of this kind. Ischemic stroke is fundamentally a multiscale phenomenon where a cascade of changes triggered by loss of blood flow involves processes at spatial scales from molecular to centimeters with damage occurring in milliseconds to days and recovery into years. Multiscale computational modeling is a technique to assist understanding of the many agents involved in these multitudinous interacting pathways to provide clues for in silico development of multi-target polypharmacy drug cocktails.

Computational modeling of neurostimulation in brain diseases

Neurostimulation as a therapeutic tool has been developed and used for a range of different diseases such as Parkinson's disease, epilepsy, and migraine. However, it is not known why the efficacy of the stimulation varies dramatically across patients or why some patients suffer from severe side effects. This is largely due to the lack of mechanistic understanding of neurostimulation. Hence, theoretical computational approaches to address this issue are in demand. This chapter provides a review of mechanistic computational modeling of brain stimulation. In particular, we will focus on brain diseases, where mechanistic models (e.g., neural population models or detailed neuronal models) have been used to bridge the gap between cellular-level processes of affected neural circuits and the symptomatic expression of disease dynamics. We show how such models have been, and can be, used to investigate the effects of neurostimulation in the diseased brain. We argue that these models are crucial for the mechanistic understanding of the effect of stimulation, allowing for a rational design of stimulation protocols. Based on mechanistic models, we argue that the development of closed-loop stimulation is essential in order to avoid inference with healthy ongoing brain activity. Furthermore, patient-specific data, such as neuroanatomic information and connectivity profiles obtainable from neuroimaging, can be readily incorporated to address the clinical issue of variability in efficacy between subjects. We conclude that mechanistic computational models can and should play a key role in the rational design of effective, fully integrated, patient-specific therapeutic brain stimulation.

A methodological framework for drug development in rare diseases

INTRODUCTION

Developing orphan drugs is challenging because of their severity and the requisite for effective drugs. The small number of patients does not allow conducting adequately powered randomized controlled trials (RCTs). There is a need to develop high quality, ethically investigated, and appropriately authorized medicines, without subjecting patients to unnecessary trials.

AIMS AND OBJECTIVES

The main aim is to develop generalizable framework for choosing the best-performing drug/endpoint/design combinations in orphan drug development using an in silico modeling and trial simulation approach. The two main objectives were (i) to provide a global strategy for each disease to identify the most relevant drugs to be evaluated in specific patients during phase III RCTs, (ii) and select the best design for each drug to be used in future RCTs.

METHODOLOGICAL APPROACH

In silico phase III RCT simulation will be used to find the optimal trial design and was carried out in two steps: (i) statistical analysis of available clinical databases and (ii) integrative modeling that combines mathematical models for diseases with pharmacokinetic-pharmacodynamics models for the selected drug candidates.

CONCLUSION

There is a need to speed up the process of orphan drug development, develop new methods for translational research and personalized medicine, and contribute to European Medicines Agency guidelines. The approach presented here offers many perspectives in clinical trial conception.

Inductive and Deductive Approaches to Acute Cell Injury

Many clinically relevant forms of acute injury, such as stroke, traumatic brain injury, and myocardial infarction, have resisted treatments to prevent cell death following injury. The clinical failures can be linked to the currently used inductive models based on biological specifics of the injury system. Here we contrast the application of inductive and deductive models of acute cell injury. Using brain ischemia as a case study, we discuss limitations in inductive inferences, including the inability to unambiguously assign cell death causality and the lack of a systematic quantitative framework. These limitations follow from an overemphasis on qualitative molecular pathways specific to the injured system. Our recently developed nonlinear dynamical theory of cell injury provides a generic, systematic approach to cell injury in which attractor states and system parameters are used to quantitatively characterize acute injury systems. The theoretical, empirical, and therapeutic implications of shifting to a deductive framework are discussed. We illustrate how a deductive mathematical framework offers tangible advantages over qualitative inductive models for the development of therapeutics of acutely injured biological systems.

Refractory period modulates the spatiotemporal evolution of cortical spreading depression: a computational study

Cortical spreading depression (CSD) is a pathophysiological phenomenon, which underlies some neurological disorders, such as migraine and stroke, but its mechanisms are still not completely understood. One of the striking facts is that the spatiotemporal evolution of CSD wave is varying. Observations in experiments reveal that a CSD wave may propagate through the entire cortex, or just bypass some areas of the cortex. In this paper, we have applied a 2D reaction-diffusion equation with recovery term to study the spatiotemporal evolution of CSD. By modulating the recovery rate from CSD in the modeled cortex, CSD waves with different spatiotemporal evolutions, either bypassing some areas or propagating slowly in these areas, were present. Moreover, spiral CSD waves could also be induced in case of the transiently altered recovery rate, i.e. block release from the absolute refractory period. These results suggest that the refractory period contributes to the different propagation patterns of CSD, which may help to interpret the mechanisms of CSD propagation.

Spatiotemporal dynamic simulation of acute perfusion/diffusion ischemic stroke lesions evolution: a pilot study derived from longitudinal MR patient data

The spatiotemporal evolution of stroke lesions, from acute injury to final tissue damage, is complex. Diffusion-weighted (DWI) and perfusion-weighted (PWI) imaging is commonly used to detect early ischemic changes and attempts to distinguish between permanently damaged and salvageable tissues. To date, 2D and 3D measures of diffusion/perfusion regions at individual timepoints have been widely used but may underestimate the true lesion spatio-temporal dynamics. Currently there is no spatio-temporal 4D dynamic model that simulates the continuous evolution of ischemic stroke from MR images. We determined whether a 4D current-based diffeomorphic model, developed in the field of statistical modeling for measuring the variability of anatomical surfaces, could estimate patient-specific spatio-temporal continuous evolution for MR PWI (measured as mean transit time, (MTT)) and DWI lesions. In our representative pilot sample, the model fitted the data well. Our dynamic analysis of lesion evolution showed different patterns; for example, some DWI/PWI dynamic changes corresponded with DWI lesion expansion into PWI lesions, but other patterns were much more complex and diverse. There was wide variation in the time when the final tissue damage was reached after stroke for DWI and MTT.

Modeling the contributions of Ca2+ flows to spontaneous Ca2+ oscillations and cortical spreading depression-triggered Ca2+ waves in astrocyte networks

Astrocytes participate in brain functions through Ca(2+) signals, including Ca(2+) waves and Ca(2+) oscillations. Currently the mechanisms of Ca(2+) signals in astrocytes are not fully clear. Here, we present a computational model to specify the relative contributions of different Ca(2+) flows between the extracellular space, the cytoplasm and the endoplasmic reticulum of astrocytes to the generation of spontaneous Ca(2+) oscillations (CASs) and cortical spreading depression (CSD)-triggered Ca(2+) waves (CSDCWs) in a one-dimensional astrocyte network. This model shows that CASs depend primarily on Ca(2+) released from internal stores of astrocytes, and CSDCWs depend mainly on voltage-gated Ca(2+) influx. It predicts that voltage-gated Ca(2+) influx is able to generate Ca(2+) waves during the process of CSD even after depleting internal Ca(2+) stores. Furthermore, the model investigates the interactions between CASs and CSDCWs and shows that the pass of CSDCWs suppresses CASs, whereas CASs do not prevent the generation of CSDCWs. This work quantitatively analyzes the generation of astrocytic Ca(2+) signals and indicates different mechanisms underlying CSDCWs and non-CSDCWs. Research on the different types of Ca(2+) signals might help to understand the ways by which astrocytes participate in information processing in brain functions.

Medical image analysis methods in MR/CT-imaged acute-subacute ischemic stroke lesion: Segmentation, prediction and insights into dynamic evolution simulation models. A critical appraisal

Over the last 15 years, basic thresholding techniques in combination with standard statistical correlation-based data analysis tools have been widely used to investigate different aspects of evolution of acute or subacute to late stage ischemic stroke in both human and animal data. Yet, a wave of biology-dependent and imaging-dependent issues is still untackled pointing towards the key question: "how does an ischemic stroke evolve?" Paving the way for potential answers to this question, both magnetic resonance (MRI) and CT (computed tomography) images have been used to visualize the lesion extent, either with or without spatial distinction between dead and salvageable tissue. Combining diffusion and perfusion imaging modalities may provide the possibility of predicting further tissue recovery or eventual necrosis. Going beyond these basic thresholding techniques, in this critical appraisal, we explore different semi-automatic or fully automatic 2D/3D medical image analysis methods and mathematical models applied to human, animal (rats/rodents) and/or synthetic ischemic stroke to tackle one of the following three problems: (1) segmentation of infarcted and/or salvageable (also called penumbral) tissue, (2) prediction of final ischemic tissue fate (death or recovery) and (3) dynamic simulation of the lesion core and/or penumbra evolution. To highlight the key features in the reviewed segmentation and prediction methods, we propose a common categorization pattern. We also emphasize some key aspects of the methods such as the imaging modalities required to build and test the presented approach, the number of patients/animals or synthetic samples, the use of external user interaction and the methods of assessment (clinical or imaging-based). Furthermore, we investigate how any key difficulties, posed by the evolution of stroke such as swelling or reperfusion, were detected (or not) by each method. In the absence of any imaging-based macroscopic dynamic model applied to ischemic stroke, we have insights into relevant microscopic dynamic models simulating the evolution of brain ischemia in the hope to further promising and challenging 4D imaging-based dynamic models. By depicting the major pitfalls and the advanced aspects of the different reviewed methods, we present an overall critique of their performances and concluded our discussion by suggesting some recommendations for future research work focusing on one or more of the three addressed problems.

Self-terminating wave patterns and self-organized pacemakers in a phenomenological model of spreading depression

A simple reaction-diffusion model of spreading depression (SD) is presented. Its local dynamics are governed by two activator and two inhibitor variables that provide an extremely simplified description of the mutual interaction between the neurons and extracellular space. This interaction is realized by the substances in the extracellular space that are increasing excitability of the neurons that have released them and are diffusing to the neighboring neurons, thereby spreading this excitation. Typical dynamic patterns of simulated activity are presented. The focus is laid on the case where response of the extracellular medium is relatively fast, and retracting waves, spiral-shaped waves, and autonomous pacemakers are observed, which is in good agreement with experimental observations. The underlying mechanisms are found to be related to switching between the local bi-stable, excitable, and self-sustained dynamics in the simulated medium. This article is part of a Special Issue entitled: Neural Coding.

In silico study of the influence of intensity and duration of blood flow reduction on cell death through necrosis or apoptosis during acute ischemic stroke

Ischemic stroke involves numerous and complex pathophysiological mechanisms including blood flow reduction, ionic exchanges, spreading depressions and cell death through necrosis or apoptosis. We used a mathematical model based on these phenomena to study the influences of intensity and duration of ischemia on the final size of the infarcted area. This model relies on a set of ordinary and partial differential equations. After a sensibility study, the model was used to carry out in silico experiments in various ischemic conditions. The simulation results show that the proportion of apoptotic cells increases when the intensity of ischemia decreases, which contributes to the model validation. The simulation results also show that the influence of ischemia duration on the infarct size is more complicated. They suggest that reperfusion is beneficial when performed in the early stroke but may be either inefficacious or even deleterious when performed later after the stroke onset. This aggravation could be explained by the depolarisation waves which might continue to spread ischemic damage and by the speeding up of the apoptotic process leading to cell death. The effect of reperfusion on cell death through these two phenomena needs to be further studied in order to develop new therapeutic strategies for stroke patients.

Towards a dynamical network view of brain ischemia and reperfusion. Part IV: additional considerations

The general failure of neuroprotectants in clinical trials of ischemic stroke points to the possibility of a fundamental blind spot in the current conception of ischemic brain injury, the "ischemic cascade". This is the fourth in a series of four papers whose purpose is to work towards a revision of the concept of brain ischemia by applying network concepts to develop a bistable model of brain ischemia. Here we consider additional issues to round out and close out this initial presentation of the bistable network view of brain ischemia. Initial considerations of the network architecture underlying the post-ischemic state space are discussed. Network and differential equation models of brain ischemia are compared. We offer a first look at applying the bistable model to focal cerebral ischemia. The limitations of the present formulation of the bistable model are discussed. This work concludes with a series of questions by which to direct future efforts.

Modelling methodology in physiopathology

Diseases are complex systems. Modelling them, i.e. systems physiopathology, is a quite demanding, complicated, multidimensional, multiscale process. As such, in order to achieve the goal of the model and further to optimise a rather-time and resource-consuming process, a relevant and easy to practice methodology is required. It includes guidance for validation. Also, the model development should be managed as a complicated process, along a strategy which has been elaborated in the beginning. It should be flexible enough to meet every case. A model is a representation of the available knowledge. All available knowledge does not have the same level of evidence and, further, there is a large variability of the values of all parameters (e.g. affinity constant or ionic current) across the literature. In addition, in a complex biological system there are always values lacking for a few or sometimes many parameters. All these three aspects are sources of uncertainty on the range of validity of the models and raise unsolved problems for designing a relevant model. Tools and techniques for integrating the parameter range of experimental values, level of evidence and missing data are needed.