Modelling tissue electrophysiology with multiple cell types: applications of the extended bidomain framework

On the generality of the finite element modeling physical fields in biological systems by the multiscale smeared concept (Kojic transport model)

The biomechanical and biochemical processes in the biological systems of living organisms are extremely complex. Advances in understanding these processes are mainly achieved by laboratory and clinical investigations, but in recent decades they are supported by computational modeling. Besides enormous efforts and achievements in this modeling, there still is a need for new methods that can be used in everyday research and medical practice. In this report, we give a view of the generality of the finite element methodology introduced by the first author and supported by his collaborators. It is based on the multiscale smeared physical fields, termed as Kojic Transport Model (KTM), published in several journal papers and summarized in a recent book (Kojic et al., 2022) [1]. We review relevant literature to demonstrate the distinctions and advantages of our methodology and indicate possible further applications. We refer to our published results by a selection of a few examples which include modeling of partitioning, blood flow, molecular transport within the pancreas, multiscale-multiphysics model of coupling electrical field and ion concentration, and a model of convective-diffusive transport within the lung parenchyma. Two new examples include a model of convective-diffusive transport within a growing tumor, and drug release from nanofibers with fiber degradation.

On a coupled electro-chemomechanical model of gastric smooth muscle contraction

The stomach is a central organ in the gastrointestinal tract that performs a variety of functions, in which the spatio-temporal organisation of active smooth muscle contraction in the stomach wall (SW) is highly regulated. In the present study, a three-dimensional model of the gastric smooth muscle contraction is presented, including the mechanical contribution of the mucosal and muscular layer of the SW. Layer-specific and direction-dependent model parameters for the active and passive stress-stretch characteristics of the SW were determined experimentally using porcine smooth muscle strips. The electrical activation of the smooth muscle cells (SMC) due to the pacemaker activity of the interstitial cells of Cajal (ICC) is modelled by using FitzHugh-Nagumo-type equations, which simulate the typical ICC and SMC slow wave behaviour. The calcium dynamic in the SMC depends on the SMC membrane potential via a gaussian function, while the chemo-mechanical coupling in the SMC is modelled via an extended Hai-Murphy model. This cascade is coupled with an additional mechano-electrical feedback-mechanism, taking into account the mechanical response of the ICC and SMC due to stretch of the SW. In this way the relaxation responses of the fundus to accommodate incoming food, as well as the typical peristaltic contraction waves in the antrum for mixing and transport of the chyme, have been well replicated in simulations performed at the whole organ level. STATEMENT OF SIGNIFICANCE: In this article, a novel three-dimensional electro-chemomechanical model of the gastric smooth muscle contraction is presented. The propagating waves of electrical membrane potential in the network ofinterstitial cells of Cajal (ICC) and smooth muscle cells (SMC) lead to a global pattern of change in the calciumdynamics inside the SMC. Taking additionally into account the mechanical response of the ICC and SMC due to stretch of the stomach wall, also referred to as mechanical feedback-mechanism, the result is a complex spatio-temporal regulation of the active contraction and relaxation of the gastric smooth muscle tissue. Being a firstapproach, in future view such a three-dimensional model can give an insight into the complexload transferring system of the stomach wall, as well as into the electro-chemomechanicalcoupling process underlying smooth muscle contraction in health and disease.

A finite element approach for gastrointestinal tissue mechanics

The biomechanical properties of gastrointestinal (GI) tissue play a significant role in the normal functioning of the organ. GI soft tissues exhibit a highly nonlinear rate- and time-dependent stress-strain behaviour. In recent years, many constitutive relations have been proposed to characterize these properties. However, a constitutive relation is not sufficient to analyse the biomechanics at the organ level with complex loading and boundary conditions. Hence, for a refined mechanical analysis, a finite element (FE) implementation of the constitutive relation is needed. Here, we propose an FE implementation of a finite nonlinear hyperviscoelastic model suitable for soft biological tissues. The FE model has been validated at first by comparing its results with the analytical solutions of a standard linear solid, and then it has been used to recreate experimental observations performed on tissue strips obtained from different animals. We have also proposed a method, in this work, to construct a residually stressed FE model so that the consequences of residual stresses on GI mechanics can be examined. Our FE formulation was able to capture the nonlinear soft tissue properties and also demonstrated that the addition of residual stresses reduces stress concentrations and the stress gradient in the GI wall.

Modelling the electrical activity of skeletal muscle tissue using a multi-domain approach

Electromyography (EMG) can be used to study the behaviour of the motor neurons and thus provides insights into the physiology of the central nervous system. However, due to the high complexity of neuromuscular control, EMG signals are challenging to interpret. While the exact knowledge of the excitation patterns of a specific muscle within an in vivo experimental setting remains elusive, simulations allow to systematically investigate EMG signals in a controlled environment. Within this context, simulations can provide virtual EMG data, which, for example, can be used to validate and optimise signal analysis methods that aim to estimate the relationship between EMG signals and the output of motor neuron pools. However, since existing methods, which are employed to compute EMG signals, exhibit deficiencies with respect to the physical model itself as well as with respect to numerical aspects, we propose a novel homogenised continuum model that closely resolves the electro-physiological behaviour of skeletal muscle tissue. The proposed model is based on an extension of the well-established bidomain model and includes a biophysically detailed description of the electrical activity within the tissue, which is due to the depolarisation of the muscle fibre membranes. In contrast to all other published EMG models, which assume that the electrical potential field for each muscle fibre can be calculated independently, the proposed model assumes that the electrical potential in the muscle fibres is coupled to the electrical potential in the extracellular space. We show that the newly proposed model is able to simulate realistic EMG signals and demonstrate the potential to employ the predicted virtual EMG signal in order to evaluate the goodness of automated decomposition algorithms.

Extended Bidomain Modeling of Defibrillation: Quantifying Virtual Electrode Strengths in Fibrotic Myocardium

Defibrillation is a well-established therapy for atrial and ventricular arrhythmia. Here, we shed light on defibrillation in the fibrotic heart. Using the extended bidomain model of electrical conduction in cardiac tissue, we assessed the influence of fibrosis on the strength of virtual electrodes caused by extracellular electrical current. We created one-dimensional models of rabbit ventricular tissue with a central patch of fibrosis. The fibrosis was incorporated by altering volume fractions for extracellular, myocyte and fibroblast domains. In our prior work, we calculated these volume fractions from microscopic images at the infarct border zone of rabbit hearts. An average and a large degree of fibrosis were modeled. We simulated defibrillation by application of an extracellular current for a short duration (5 ms). We explored the effects of myocyte-fibroblast coupling, intra-fibroblast conductivity and patch length on the strength of the virtual electrodes present at the borders of the normal and fibrotic tissue. We discriminated between effects on myocyte and fibroblast membranes at both borders of the patch. Similarly, we studied defibrillation in two-dimensional models of fibrotic tissue. Square and disk-like patches of fibrotic tissue were embedded in control tissue. We quantified the influence of the geometry and fibrosis composition on virtual electrode strength. We compared the results obtained with a square and disk shape of the fibrotic patch with results from the one-dimensional simulations. Both, one- and two-dimensional simulations indicate that extracellular current application causes virtual electrodes at boundaries of fibrotic patches. A higher degree of fibrosis and larger patch size were associated with an increased strength of the virtual electrodes. Also, patch geometry affected the strength of the virtual electrodes. Our simulations suggest that increased fibroblast-myocyte coupling and intra-fibroblast conductivity reduce virtual electrode strength. However, experimental data to constrain these modeling parameters are limited and thus pinpointing the magnitude of the reduction will require further understanding of electrical coupling of fibroblasts in native cardiac tissues. We propose that the findings from our computational studies are important for development of patient-specific protocols for internal defibrillators.

Smeared multiscale finite element model for electrophysiology and ionic transport in biological tissue

Basic functions of living organisms are governed by the nervous system through bidirectional signals transmitted from the brain to neural networks. These signals are similar to electrical waves. In electrophysiology the goal is to study the electrical properties of biological cells and tissues, and the transmission of signals. From a physics perspective, there exists a field of electrical potential within the living body, the nervous system, extracellular space and cells. Electrophysiological problems can be investigated experimentally and also theoretically by developing appropriate mathematical or computational models. Due to the enormous complexity of biological systems, it would be almost impossible to establish a detailed computational model of the electrical field, even for only a single organ (e.g. heart), including the entirety of cells comprising the neural network. In order to make computational models feasible for practical applications, we here introduce the concept of smeared fields, which represents a generalization of the previously formulated multiscale smeared methodology for mass transport in blood vessels, lymph, and tissue. We demonstrate the accuracy of the smeared finite element computational models for the electric field in numerical examples. The electrical field is further coupled with ionic mass transport within tissue composed of interstitial spaces extracellularly and by cytoplasm and organelles intracellularly. The proposed methodology, which couples electrophysiology and molecular ionic transport, is applicable to a variety of biological systems.

Representation of Multiple Cellular Phenotypes Within Tissue-Level Simulations of Cardiac Electrophysiology

Distinct electrophysiological phenotypes are exhibited by biological cells that have differentiated into particular cell types. The usual approach when simulating the cardiac electrophysiology of tissue that includes different cell types is to model the different cell types as occupying spatially distinct yet coupled regions. Instead, we model the electrophysiology of well-mixed cells by using homogenisation to derive an extension to the commonly used monodomain or bidomain equations. These new equations permit spatial variations in the distribution of the different subtypes of cells and will reduce the computational demands of solving the governing equations. We validate the homogenisation computationally, and then use the new model to explain some experimental observations from stem cell-derived cardiomyocyte monolayers.

Calcium movement in smooth muscle and evaluation of graded functional intercellular coupling

Spontaneous activity of vascular smooth muscle is present in small arteries and some venous tissues like the hepatic portal vein. Whereas the ability to generate rhythmic membrane potential changes is expressed in a high number of primary oscillators, the generation of physiological tone and phasic activity requires synchronization of specialized pacemaker activity (Interstitial Cajal-like cells) by intercellular propagation and regeneration of excitation or a strong coupling mechanism of smooth muscle cells. The aim of this study was to deduce oscillator coupling by analyzing the spatiotemporal homogeneity of calcium oscillations within a native tissue preparation. Portal vein tissue was loaded with a calcium-sensitive dye (Fluo-3). By combining confocal microscopy and computation of spatial auto- and cross-correlation of the calcium signals, temporal and spatial coupling between cells was characterized. Spontaneous oscillations of calcium signals were measured at different predefined regions of interest. Cross-correlation analysis of these signals revealed that their damping was very similar in all directions of the investigated z-plane. In single experiments, improved cell-to-cell coupling was seen when noradrenaline (1-10 μM) was added to the bath-solution. With the chosen parameters of frame refresh, the velocity of signal propagation was faster than the maximum detectable velocity, but it could be estimated to exceed 0.1 mm/s. Correlative Network Analysis is a new and very useful tool to determine the functional coupling parameters of quasi-homogenous biological networks and their temporal changes. The action and significance of pharmacological modulators can be well studied on cellular and functional aspects with this newly introduced technique in biological sciences.

Progress in Mathematical Modeling of Gastrointestinal Slow Wave Abnormalities

Gastrointestinal (GI) motility is regulated in part by electrophysiological events called slow waves, which are generated by the interstitial cells of Cajal (ICC). Slow waves propagate by a process of "entrainment," which occurs over a decreasing gradient of intrinsic frequencies in the antegrade direction across much of the GI tract. Abnormal initiation and conduction of slow waves have been demonstrated in, and linked to, a number of GI motility disorders. A range of mathematical models have been developed to study abnormal slow waves and applied to propose novel methods for non-invasive detection and therapy. This review provides a general outline of GI slow wave abnormalities and their recent classification using multi-electrode (high-resolution) mapping methods, with a particular emphasis on the spatial patterns of these abnormal activities. The recently-developed mathematical models are introduced in order of their biophysical scale from cellular to whole-organ levels. The modeling techniques, main findings from the simulations, and potential future directions arising from notable studies are discussed.

Gastrointestinal Safety Pharmacology in Drug Discovery and Development

Although the basic structure of the gastrointestinal tract (GIT) is similar across species, there are significant differences in the anatomy, physiology, and biochemistry between humans and laboratory animals, which should be taken into account when conducting a gastrointestinal (GI) assessment. Historically, the percentage of cases of drug attrition associated with GI-related adverse effects is small; however, this incidence has increased over the last few years. Drug-related GI effects are very diverse, usually functional in nature, and not limited to a single pharmacological class. The most common GI signs are nausea and vomiting, diarrhea, constipation, and gastric ulceration. Despite being generally not life-threatening, they can greatly affect patient compliance and quality of life. There is therefore a real need for improved and/or more extensive GI screening of candidate drugs in preclinical development, which may help to better predict clinical effects. Models to identify drug effects on GI function cover GI motility, nausea and emesis liability, secretory function (mainly gastric secretion), and absorption aspects. Both in vitro and in vivo assessments are described in this chapter. Drug-induced effects on GI function can be assessed in stand-alone safety pharmacology studies or as endpoints integrated into toxicology studies. In silico approaches are also being developed, such as the gut-on-a-chip model, but await further optimization and validation before routine use in drug development. GI injuries are still in their infancy with regard to biomarkers, probably due to their greater diversity. Nevertheless, several potential blood, stool, and breath biomarkers have been investigated. However, additional validation studies are necessary to assess the relevance of these biomarkers and their predictive value for GI injuries.

Determining the efficient inter-electrode distance for high-resolution mapping using a mathematical model of human gastric dysrhythmias

Motility of the stomach is in part coordinated by an electrophysiological event called slow waves, which are generated by pacemaker cells called the interstitial cells of Cajal (ICC). In functional motility disorders, which can be associated with a reduction of ICC, dynamic slow wave dysrhythmias can occur. In recent years, high-resolution (HR) mapping techniques have been applied to describe both normal and dysrhythmic slow wave patterns. The main aim of this study was to inform gastric HR mapping array design by determining the efficient inter-electrode distance required to accurately capture normal and dysrhythmic gastric slow wave activity. A two-dimensional mathematical model was used to simulate normal activity and four types of reported slow wave dysrhythmias in human patients: ectopic activation, retrograde propagation, slow conduction, conduction block. For each case, the simulated data were re-sampled at 4, 6, 10, 12, 20 and 30mm inter-electrode distances. The accuracy of each distance was compared to a reference set sampled at 2mm inter-electrode distance, in terms of accuracy of velocity, using an ANOVA. Manual groupings were also conducted to test the ability of the human markers to distinguish separate cycles of slow waves as inter-electrode distance increases. The largest interelectrode distance for human gastric slow wave analysis, which produced both accurate grouping and velocity, was 10mm (CI [0.3 2.4]mms(-1); p<;0.05). Therefore an inter-electrode distance of less than 10mm was required to accurately describe the types of baseline and dysrhythmic activities reported in this study. However, it is likely that more spatially complex dysrhythmias, such as re-entry, may require finer inter-electrode distances.

A Multiscale Tridomain Model for Simulating Bioelectric Gastric Pacing

GOAL

Gastric motility disorders have been associated with abnormal slow wave electrical activity (gastric dysrhythmias). Gastric pacing is a potential therapy for gastric dysrhythmias; however, new pacing protocols are required that can effectively modulate motility patterns, while being power efficient. This study presents a novel comprehensive 3-D multiscale modeling framework of the human stomach, including anisotropic conduction, capable of evaluating pacing strategies.

METHODS

A high-resolution anatomically realistic mesh was generated from CT images taken from a human stomach. Principal conduction axes were calculated and embedded within this model based on a modified Laplace-Dirichlet rule-based algorithm. A continuum-based tridomain formulation was implemented and evaluated for performance and used to model the slow-wave propagation, which takes into account the two main cell types present in gastric musculature. Model parameters were found by matching predicted normal slow-wave activity to experimental observation and data. These simulation parameters were applied while modeling an external pacing event to entrain slow-wave patterns.

RESULTS

The proposed formulation was found to be two times more efficient than a previous formulation for a normal slow-wave simulation. Convergence analysis showed that a mesh resolution of [Formula: see text] is required for an accurate solution process.

CONCLUSION

The effect of different pacing frequencies on entrainment demonstrated that the pacing protocols are limited by the frequency of the native propagation and the refractory period of the cellular activity.

SIGNIFICANCE

The model is expected to become an important tool in studying pacing protocols for both efficiency and effectiveness.

Slow wave conduction patterns in the stomach: from Waller's foundations to current challenges

This review provides an overview of our understanding of motility and slow wave propagation in the stomach. It begins by reviewing seminal studies conducted by Walter Cannon and Augustus Waller on in vivo motility and slow wave patterns. Then our current understanding of slow wave patterns in common laboratory animals and humans is presented. The implications of slow wave arrhythmic patterns that have been recorded in animals and patients suffering from gastroparesis are discussed. Finally, current challenges in experimental methods and techniques, slow wave modulation and the use of mathematical models are discussed.

Detection of abnormal cardiac activity using principal component analysis--a theoretical study

Electrogram-guided ablation has been recently developed for allowing better detection and localization of abnormal atrial activity that may be the source of arrhythmogeneity. Nevertheless, no clear indication for the benefit of using electrograms guided ablation over empirical ablation was established thus far, and there is a clear need of improving the localization of cardiac arrhythmogenic targets for ablation. In this paper, we propose a new approach for detection and localization of irregular cardiac activity during ablation procedures that is based on dimension reduction algorithms and principal component analysis (PCA). Using an 8×8 electrode array, our method produces manifolds that allow easy visualization and detection of possible arrhythmogenic ablation targets characterized by irregular conduction. We employ mathematical modeling and computer simulations to demonstrate the feasibility of the new approach for two well established arrhythmogenic sources for irregular conduction--spiral waves and patchy fibrosis. Our results show that the PCA method can differentiate between focal ectopic activity and spiral wave activity, as these two types of activity produce substantially different manifold shapes. Moreover, the technique allows the detection of spiral wave cores and their general meandering and drifting pattern. Fibrotic patches larger than 2 mm(2) could also be visualized using the PCA method, both for quiescent atrial tissue and for tissue exhibiting spiral wave activity. We envision that this method, contingent to further numerical and experimental validation studies in more complex, realistic geometrical configurations and with clinical data, can improve existing atrial ablation mapping capabilities, thus increasing success rates and optimizing arrhythmia management.

Chaste: an open source C++ library for computational physiology and biology

Chaste — Cancer, Heart And Soft Tissue Environment — is an open source C++ library for the computational simulation of mathematical models developed for physiology and biology. Code development has been driven by two initial applications: cardiac electrophysiology and cancer development. A large number of cardiac electrophysiology studies have been enabled and performed, including high-performance computational investigations of defibrillation on realistic human cardiac geometries. New models for the initiation and growth of tumours have been developed. In particular, cell-based simulations have provided novel insight into the role of stem cells in the colorectal crypt. Chaste is constantly evolving and is now being applied to a far wider range of problems. The code provides modules for handling common scientific computing components, such as meshes and solvers for ordinary and partial differential equations (ODEs/PDEs). Re-use of these components avoids the need for researchers to ‘re-invent the wheel’ with each new project, accelerating the rate of progress in new applications. Chaste is developed using industrially-derived techniques, in particular test-driven development, to ensure code quality, re-use and reliability. In this article we provide examples that illustrate the types of problems Chaste can be used to solve, which can be run on a desktop computer. We highlight some scientific studies that have used or are using Chaste, and the insights they have provided. The source code, both for specific releases and the development version, is available to download under an open source Berkeley Software Distribution (BSD) licence at http://www.cs.ox.ac.uk/chaste, together with details of a mailing list and links to documentation and tutorials.

Toward the virtual stomach: progress in multiscale modeling of gastric electrophysiology and motility

Experimental progress in investigating normal and disordered gastric motility is increasingly being complimented by sophisticated multiscale modeling studies. Mathematical modeling has become a valuable tool in this effort, as there is an ever-increasing need to gain an integrative and quantitative understanding of how physiological mechanisms achieve coordinated functions across multiple biophysical scales. These interdisciplinary efforts have been particularly notable in the area of gastric electrophysiology, where they are beginning to yield a comprehensive and integrated in silico organ modeling framework, or 'virtual stomach'. At the cellular level, a number of biophysically based mathematical cell models have been developed, and these are now being applied in areas including investigations of gastric electrical pacemaker mechanisms, smooth muscle electrophysiology, and electromechanical coupling. At the tissue level, micro-structural models are being creatively developed and employed to investigate clinically significant questions, such as the functional effects of ICC degradation on gastrointestinal (GI) electrical activation. At the organ level, high-resolution electrical mapping and modeling studies are combined to provide improved insights into normal and dysrhythmic gastric electrical activation. These efforts are also enabling detailed forward and inverse modeling studies at the 'whole body' level, with implications for diagnostic techniques for gastric dysrhythmias. These recent advances, together with several others highlighted in this review, collectively demonstrate a powerful trend toward applying mathematical models to effectively investigate structure-function relationships and overcome multiscale challenges in basic and clinical GI research.