Multiscale modeling of gastrointestinal electrophysiology and experimental validation

A novel compartmental approach for modeling stomach motility and gastric emptying

The stomach, a central organ in the Gastrointestinal (GI) tract, regulates the processing of ingested food through gastric motility and emptying. Understanding the stomach function is crucial for treating gastric disorders. Experimental studies in this field often face difficulties due to limitations and invasiveness of available techniques and ethical concerns. To counter this, researchers resort to computational and numerical methods. However, existing computational studies often isolate one aspect of the stomach function while neglecting the rest and employ computationally expensive methods. This paper proposes a novel cost-efficient multi-compartmental model, offering a comprehensive insight into gastric function at an organ level, thus presenting a promising alternative. The proposed approach divides the spatial geometry of the stomach into four compartments: Proximal/Middle/Terminal antrum and Pyloric sphincter. Each compartment is characterized by a set of ordinary differential equations (ODEs) with respect to time to characterize the stomach function. Electrophysiology is represented by simplified equations reflecting the "slow wave behavior" of Interstitial Cells of Cajal (ICC) and Smooth Muscle Cells (SMC) in the stomach wall. An electro-mechanical coupling model translates SMC "slow waves" into smooth muscle contractions. Muscle contractions induce peristalsis, affecting gastric fluid flow velocity and subsequent emptying when the pyloric sphincter is open. Contraction of the pyloric sphincter initiates a retrograde flow jet at the terminal antrum, modeled by a circular liquid jet flow equation. The results from the proposed model for a healthy human stomach were compared with experimental and computational studies on electrophysiology, muscle tissue mechanics, and fluid behavior during gastric emptying. These findings revealed that each "ICC" slow wave corresponded to a muscle contraction due to electro-mechanical coupling behavior. The rate of gastric emptying and mixing efficiency decreased with increasing viscosity of gastric liquid but remained relatively unchanged with gastric liquid density variations. Utilizing different ODE solvers in MATLAB, the model was solved, with ode15s demonstrating the fastest computation time, simulating 180 s of real-time stomach response in just 2.7 s. This multi-compartmental model signifies a promising advancement in understanding gastric function, providing a cost-effective and comprehensive approach to study complex interactions within the stomach and test innovative therapies like neuromodulation for treating gastric disorders.

Computational models of autonomic regulation in gastric motility: Progress, challenges, and future directions

The stomach is extensively innervated by the vagus nerve and the enteric nervous system. The mechanisms through which this innervation affects gastric motility are being unraveled, motivating the first concerted steps towards the incorporation autonomic regulation into computational models of gastric motility. Computational modeling has been valuable in advancing clinical treatment of other organs, such as the heart. However, to date, computational models of gastric motility have made simplifying assumptions about the link between gastric electrophysiology and motility. Advances in experimental neuroscience mean that these assumptions can be reviewed, and detailed models of autonomic regulation can be incorporated into computational models. This review covers these advances, as well as a vision for the utility of computational models of gastric motility. Diseases of the nervous system, such as Parkinson’s disease, can originate from the brain-gut axis and result in pathological gastric motility. Computational models are a valuable tool for understanding the mechanisms of disease and how treatment may affect gastric motility. This review also covers recent advances in experimental neuroscience that are fundamental to the development of physiology-driven computational models. A vision for the future of computational modeling of gastric motility is proposed and modeling approaches employed for existing mathematical models of autonomic regulation of other gastrointestinal organs and other organ systems are discussed.

Efficient time splitting schemes for the monodomain equation in cardiac electrophysiology

Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE-ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy, and efficiency. Simulations were performed for a benchmark problem with the Beeler-Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this configuration, we provide a reference solution for the transmembrane potential. We found a semi-implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi-ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time.

Simulating human gastrointestinal motility in dynamic in vitro models

The application of dynamic in vitro gastrointestinal (GI) models has grown in popularity to understand the impact of food structure and composition on human health. Given that GI motility is integral to digestion and absorption, a predictive in vitro model should faithfully replicate the motility patterns and motor functions in vivo. In this review, typical characteristics of gastric and small intestinal motility in humans as well as the biomechanical and hydrodynamic events pertinent to gut motility are summarized. The simulation of GI motility in the presently existing dynamic in vitro models is discussed from an engineering perspective and categorized into hydraulic, piston/probe-driven, roller-driven, pneumatic, and other systems. Each system and its representative models are evaluated in terms of their motility patterns, the key hydrodynamic characteristics concerning gut motility, their performance in simulating the key physiological events, and their ability to establish in vitro-in vivo correlations. Practical Application: The review paper provided useful information in the design of dynamic GI models and the simulation of human gastric and small intestinal motility which are important for understanding food and health.

Biomaterials for Bioprinting Microvasculature

Microvasculature functions at the tissue and cell level, regulating local mass exchange of oxygen and nutrient-rich blood. While there has been considerable success in the biofabrication of large- and small-vessel replacements, functional microvasculature has been particularly challenging to engineer due to its size and complexity. Recently, three-dimensional bioprinting has expanded the possibilities of fabricating sophisticated microvascular systems by enabling precise spatiotemporal placement of cells and biomaterials based on computer-aided design. However, there are still significant challenges facing the development of printable biomaterials that promote robust formation and controlled 3D organization of microvascular networks. This review provides a thorough examination and critical evaluation of contemporary biomaterials and their specific roles in bioprinting microvasculature. We first provide an overview of bioprinting methods and techniques that enable the fabrication of microvessels. We then offer an in-depth critical analysis on the use of hydrogel bioinks for printing microvascularized constructs within the framework of current bioprinting modalities. We end with a review of recent applications of bioprinted microvasculature for disease modeling, drug testing, and tissue engineering, and conclude with an outlook on the challenges facing the evolution of biomaterials design for bioprinting microvasculature with physiological complexity.

Current applications of mathematical models of the interstitial cells of Cajal in the gastrointestinal tract

The interstitial cells of Cajal (ICC) form interconnected networks throughout the gastrointestinal (GI) tract. ICC act as the pacemaker cells that initiate the rhythmic bioelectrical slow waves and intermediary between the GI musculature and nerves, both of which are critical to GI motility. Disruptions to the number of ICC and the integrity of ICC networks have been identified as a key pathophysiological mechanism in a number of clinically challenging GI disorders. The current analyses of ICC generally rely on either functional recordings taken directly from excised tissue or morphological analysis based on images of labeled ICC, where the structural-functional relationship is investigated in an associative manner rather than mechanistically. On the other hand, computational physiology has played a significant role in facilitating our understanding of a number of physiological systems in both health and disease, and investigations in the GI field are beginning to incorporate several mathematical models of the ICC. The main aim of this review is to present the major modeling advances in GI electrophysiology, in order to introduce a multi-scale framework for mathematically quantifying the functional consequences of ICC degradation at both cellular and tissue scales. The outcomes will inform future investigators utilizing modeling techniques in their studies. This article is categorized under: Metabolic Diseases > Computational Models.

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.

Biomechanical and microstructural characterisation of the porcine stomach wall: Location- and layer-dependent investigations

The mechanical properties of the stomach wall help to explain its function of storing, mixing, and emptying in health and disease. However, much remains unknown about its mechanical properties, especially regarding regional heterogeneities and wall microstructure. Consequently, the present study aimed to assess regional differences in the mechanical properties and microstructure of the stomach wall. In general, the stomach wall and the different tissue layers exhibited a nonlinear stress-stretch relationship. Regional differences were found in the mechanical response and the microstructure. The highest stresses of the entire stomach wall in longitudinal direction were found in the corpus (201.5 kPa), where food is ground followed by the antrum (73.1 kPa) and the fundus (26.6 kPa). In contrast, the maximum stresses in circumferential direction were 39.7 kPa, 26.2 kPa, and 15.7 kPa for the antrum, fundus, and corpus, respectively. Independent of the fibre orientation and with respect to the biaxial loading direction, partially clear anisotropic responses were detected in the intact wall and the muscular layer. In contrast, the innermost mucosal layer featured isotropic mechanical characteristics. Pronounced layers of circumferential and longitudinal muscle fibres were found in the fundus only, whereas corpus and antrum contained almost exclusively circumferential orientated muscle fibres. This specific stomach structure mirrors functional differences in the fundus as well as corpus and antrum. Within this study, the load transfer mechanisms, connected with these wavy layers but also in total with the stomach wall's microstructure, are discussed. STATEMENT OF SIGNIFICANCE: This article examines for the first time the layer-specific mechanical and histological properties of the stomach wall attending to the location of the sample. Moreover, both mechanical behaviour and microstructure were explicitly match identifying the heterogeneous characteristics of the stomach. On the one hand, the results of this study contribute to the understanding of stomach mechanics and thus to their functional understanding of stomach motility. On the other hand, they are relevant to the fields of constitutive formulation of stomach tissue, whole stomach mechanics, and stomach-derived scaffolds i.e., tissue-engineering grafts.

Progress in research of functional constipation in children

Constipation is a common symptom in the children population, which can be a manifestation of organic diseases or may be caused by functional factors. More than 90% of cases of constipation in children are functional constipation (FC). As a physical and mental illness in childhood or even a public health problem, FC has multidimensional impacts on children and their families, posing a great healthcare burden on the society. The Rome Ⅳ criteria for diagnosis of functional gastrointestinal disorders were published in 2016, in which a global consensus was formed and modified for the diagnostic criteria of FC in children. However, there are still difficulties in the choice of FC treatments and assessment of its outcome not only because of the distinct understanding and application of the criteria but also because of the incomplete understanding of the pathogenesis of FC. Therefore, the pathophysiological mechanism of FC and its therapeutic efficacy evaluation should be further studied in the future.

Multiscale modeling methods in biomechanics

More and more frequently, computational biomechanics deals with problems where the portion of physical reality to be modeled spans over such a large range of spatial and temporal dimensions, that it is impossible to represent it as a single space-time continuum. We are forced to consider multiple space-time continua, each representing the phenomenon of interest at a characteristic space-time scale. Multiscale models describe a complex process across multiple scales, and account for how quantities transform as we move from one scale to another. This review offers a set of definitions for this emerging field, and provides a brief summary of the most recent developments on multiscale modeling in biomechanics. Of all possible perspectives, we chose that of the modeling intent, which vastly affect the nature and the structure of each research activity. To the purpose we organized all papers reviewed in three categories: 'causal confirmation,' where multiscale models are used as materializations of the causation theories; 'predictive accuracy,' where multiscale modeling is aimed to improve the predictive accuracy; and 'determination of effect,' where multiscale modeling is used to model how a change at one scale manifests in an effect at another radically different space-time scale. Consistent with how the volume of computational biomechanics research is distributed across application targets, we extensively reviewed papers targeting the musculoskeletal and the cardiovascular systems, and covered only a few exemplary papers targeting other organ systems. The review shows a research subdomain still in its infancy, where causal confirmation papers remain the most common. WIREs Syst Biol Med 2017, 9:e1375. doi: 10.1002/wsbm.1375 For further resources related to this article, please visit the WIREs website.

A Theoretical Analysis of Electrogastrography (EGG) Signatures Associated With Gastric Dysrhythmias

Routine screening and accurate diagnosis of chronic gastrointestinal motility disorders represent a significant problem in current clinical practice. The electrogastrography (EGG) provides a noninvasive option for assessing gastric slow waves, as a means of diagnosing gastric dysrhythmias, but its uptake in motility practice has been limited partly due to an incomplete sensitivity and specificity. This paper presents the development of a human whole-organ gastric model to enable virtual (in silico) testing of gastric electrophysiological dispersion in order to improve the diagnostic accuracy of EGG. The model was developed to simulate normal gastric slow wave conduction as well as three types of dysrhythmias identified in recent high-resolution gastric mapping studies: conduction block, re-entry, and ectopic pacemaking. The stomach simulations were then applied in a torso model to identify predicted EGG signatures of normal and dysrhythmic slow wave profiles. The resulting EGG data were compared using percentage differences and correlation coefficients. Virtual EGG channels that demonstrated a percentage difference over 100% and a correlation coefficient less than ±0.2 (threshold relaxed to ±0.5 for the ectopic pacemaker case) were further investigated for their specific distinguishing features. In particular, anatomical locations from the epigastric region and specific channel configurations were identified that could be used to clinically diagnose the three classes of human gastric dysrhythmia. These locations and channels predicted by simulations present a promising methodology for improving the clinical reliability and applications of EGG.

A physiome interoperability roadmap for personalized drug development

The goal of developing therapies and dosage regimes for characterized subgroups of the general population can be facilitated by the use of simulation models able to incorporate information about inter-individual variability in drug disposition (pharmacokinetics), toxicity and response effect (pharmacodynamics). Such observed variability can have multiple causes at various scales, ranging from gross anatomical differences to differences in genome sequence. Relevant data for many of these aspects, particularly related to molecular assays (known as '-omics'), are available in online resources, but identification and assignment to appropriate model variables and parameters is a significant bottleneck in the model development process. Through its efforts to standardize annotation with consequent increase in data usability, the human physiome project has a vital role in improving productivity in model development and, thus, the development of personalized therapy regimes. Here, we review the current status of personalized medicine in clinical practice, outline some of the challenges that must be overcome in order to expand its applicability, and discuss the relevance of personalized medicine to the more widespread challenges being faced in drug discovery and development. We then review some of (i) the key data resources available for use in model development and (ii) the potential areas where advances made within the physiome modelling community could contribute to physiologically based pharmacokinetic and physiologically based pharmacokinetic/pharmacodynamic modelling in support of personalized drug development. We conclude by proposing a roadmap to further guide the physiome community in its on-going efforts to improve data usability, and integration with modelling efforts in the support of personalized medicine development.

The impact of surgical excisions on human gastric slow wave conduction, defined by high-resolution electrical mapping and in silico modeling

BACKGROUND

Gastric contractions are coordinated by slow waves, generated by interstitial cells of Cajal (ICC). Gastric surgery affects slow wave conduction, potentially contributing to postoperative gastric dysfunction. However, the impact of gastric cuts on slow waves has not been comprehensively evaluated. This study aimed to define consequences of surgical excisions on gastric slow waves by applying high-resolution (HR) electrical mapping and in silico modeling.

METHODS

Patients undergoing gastric stimulator implantation (n = 10) underwent full-thickness stapled excisions (25 × 15 mm, distal corpus) for histological evaluation, enabling HR mapping (256 electrodes; 36 cm(2) ) over and adjacent to excisions. A biophysically based in silico model of bidirectionally coupled ICC networks was developed and applied to investigate the underlying conduction mechanisms and importance of excision orientation.

KEY RESULTS

Normal gastric slow waves propagated aborally (3.0 ± 0.2 cpm). Excisions induced complete conduction block and wavelets that rotated around blocks, then propagated rapidly circumferentially distal to the blocks (8.5 ± 1.2 vs normal 3.6 ± 0.4 mm/s; p < 0.01). This 'conduction anisotropy' homeostatically restored antegrade propagating gastric wavefronts distal to excisions. Excisions were associated with complex dysrhythmias in five patients: retrograde propagation (3/10), ectopics (3/10), functional blocks (2/10), and collisions (1/10). Simulations demonstrated conduction anisotropy emerged from bidirectional coupling within ICC layers and showed transverse incision length and orientation correlated with the degree of conduction distortion.

CONCLUSIONS & INFERENCES

Orienting incisions in the longitudinal gastric axis causes least disruption to electrical conduction and motility. However, if transverse incisions are made, a homeostatic mechanism of gastric conduction anisotropy compensates by restoring aborally propagating wavefronts. Complex dysrhythmias accompanying excisions could modify postoperative recovery in susceptible patients.

A simplified biophysical cell model for gastric slow wave entrainment simulation

Gastric electrical activity, also termed slow wave activity, is generated by a class of pacemaker cells called the interstitial cells of Cajal (ICC), which are organized with decreasing intrinsic frequencies along the stomach. In the healthy stomach, slow waves of different intrinsic frequencies converge to a single frequency with a constant phase-lag, in a process called entrainment. The main aim of this study was to develop a simplified biophysical ICC model that is capable of modeling the self-excitatory behavior and standard morphology of gastric slow waves. Entrainment of gastric slow waves was simulated in a one-dimensional (1D) model, with a linear gradient of intrinsic slow wave frequencies. In a coupled 1D model, the simulated slow waves were entrained to a single frequency; whereas in an uncoupled 1D model, the simulated slow waves occurred at different frequencies, resulting in loss of entrainment. The new cell model presents an option for future large multi-scale simulations of gastric slow waves in intact ICC network and diseased conditions where the loss of entrainment may lead to slow wave dysrhythmias and diminished gastric motility.

A theoretical study of the initiation, maintenance and termination of gastric slow wave re-entry

Gastric slow wave dysrhythmias are associated with motility disorders. Periods of tachygastria associated with slow wave re-entry were recently recognized as one important dysrhythmia mechanism, but factors promoting and sustaining gastric re-entry are currently unknown. This study reports two experimental forms of gastric re-entry and presents a series of multi-scale models that define criteria for slow wave re-entry initiation, maintenance and termination. High-resolution electrical mapping was conducted in porcine and canine models and two spatiotemporal patterns of re-entrant activities were captured: single-loop rotor and double-loop figure-of-eight. Two separate multi-scale mathematical models were developed to reproduce the velocity and entrainment frequency of these experimental recordings. A single-pulse stimulus was used to invoke a rotor re-entry in the porcine model and a figure-of-eight re-entry in the canine model. In both cases, the simulated re-entrant activities were found to be perpetuated by tachygastria that was accompanied by a reduction in the propagation velocity in the re-entrant pathways. The simulated re-entrant activities were terminated by a single-pulse stimulus targeted at the tip of re-entrant wave, after which normal antegrade propagation was restored by the underlying intrinsic frequency gradient.

MAIN FINDINGS

(i) the stability of re-entry is regulated by stimulus timing, intrinsic frequency gradient and conductivity; (ii) tachygastria due to re-entry increases the frequency gradient while showing decreased propagation velocity; (iii) re-entry may be effectively terminated by a targeted stimulus at the core, allowing the intrinsic slow wave conduction system to re-establish itself.

The down-regulation of neuroligin-2 and the correlative clinical significance of serum GABA over-expression in Hirschsprung's disease

The goal of this study was to investigate the expression level of neuroligin-2 in different colon tissue segments of children with Hirschsprung's disease (HSCR) and the correlative clinical significance of serum Gamma-Aminobutyric Acid (serum GABA) in HSCR. Neuroligin-2 was assessed by Immunohistochemistry staining method on routine paraffin section from different colon tissue segments of HSCR (ganglionic colonic segment, transitional colonic segment and aganglionic colonic segment). Western-blot analysis and real-time fluorescence quantitative PCR(qRT-PCR) were applied to compare and evaluate the expression levels of neuroligin-2 from three segments of HSCR, and we used Enzyme-linked Immunosorbent Assay (ELISA) method to detect and compare the serum GABA between HSCR and non-HSCR. Immunohistochemistry staining demonstrated that intensive neuroligin-2 staining was detected in the ganglion cells in the ganglionic colonic and transitional colonic segments from the HSCR children; however, neuroligin-2 staining was down-regulated significantly in the aganglionic colonic segments. The expression levels of neuroligin-2 mRNA and protein in the aganglionic colonic segment were decreased compared to the ganglionic colonic segment and transitional colonic segment (P < 0.05). And the level of serum GABA was significantly higher in HSCR than that in non-HSCR. The expression of neuroligin-2 varies from different segments of HSCR. The down-regulation of neuroligin-2 in aganglionic colonic segments may be correlated with the excessive intestine contraction and further result in HSCR. The over-expression of serum GABA may be considered as a new diagnostic method of HSCR.

A continuum neuronal tissue model based on a two-compartmental representation of cells

Although significant advances have been made in continuum modeling of cardiac and smooth muscle tissue, the progress in neuronal continuum modeling has been slower. In this paper, a continuum neuronal tissue model based on a two-compartmental representation of cells is presented. Each neuron is described using both a somatic compartment modeled by the classical Hodgkin-Huxley current kinetics and a dendritic compartment based on a passive RC formulation. In addition, a synaptic current is fed into the dendritic compartment to account for the presynaptic influence of cells located within the dendritic field of each soma. A number of cases are simulated, including intracellular current injection into either the dendritic or somatic compartments, as well as extracellular current stimulation with and without synaptic input into neurons. The model incorporates a number of parameters controlling neuronal excitability which can be adjusted to validate each neuron's responses against experimental data, allowing for the modeling of different neuronal cell types and behaviors.

Expression and significance of neuroligins in myenteric cells of Cajal in Hirschsprung's disease

PURPOSE

The aim of this study was to investigate the expression and significance of neuroligins in myenteric cells of Cajal (ICC-MY) in Hirschsprung's disease (HSCR).

METHODS

Longitudinal muscle with adherent myenteric plexus (LMMP) from surgical excision waste colon of HSCR children were prepared by peeling off the mucous layer, sub-mucosal layer and circular muscle. Neuroligins, c-Kit (c-Kit-immunoreactivity representing ICC) and their relationship were assessed by double labeling immunofluorescence staining. ICC-MY were dissociated and cultured from LMMP by enzymolysis method, and were purified and analyzed using a combination of magnetic-activated cell sorting (MACS) and flow cytometry (FCM). Western-blot analysis was applied to compare and evaluate the expression levels of neuroligins in ICC-MY which were dissociated from different segments of HSCR (ganglionic colonic segment, transitional colonic segment and aganglionic colonic segment).

RESULTS

Neuroligins and c-Kit were expressed on the same cells (ICC-MY); ICC-MY were dissociated, cultured and purified. For HSCR, neuroligins were expressed significantly in ICC-MY from ganglionic colonic segments, moderately in those from transitional colonic segments and down-regulated significantly in those from aganglionic colonic segments.

CONCLUSIONS

Neuroligins were expressed in ICC-MY of human beings, and the expression varies from different segments of HSCR. This abnormal expression might play an important role in the pathogenesis of this disease through affecting the synaptic function of ICC-MY.

Multiscale computational models of complex biological systems

Integration of data across spatial, temporal, and functional scales is a primary focus of biomedical engineering efforts. The advent of powerful computing platforms, coupled with quantitative data from high-throughput experimental methodologies, has allowed multiscale modeling to expand as a means to more comprehensively investigate biological phenomena in experimentally relevant ways. This review aims to highlight recently published multiscale models of biological systems, using their successes to propose the best practices for future model development. We demonstrate that coupling continuous and discrete systems best captures biological information across spatial scales by selecting modeling techniques that are suited to the task. Further, we suggest how to leverage these multiscale models to gain insight into biological systems using quantitative biomedical engineering methods to analyze data in nonintuitive ways. These topics are discussed with a focus on the future of the field, current challenges encountered, and opportunities yet to be realized.

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.

Convolution based method for calculating inputs from dendritic fields in a continuum model of the retina

Computational models are valuable tools that can be used to aid the design and test the efficacy of electrical stimulation strategies in prosthetic vision devices. In continuum models of retinal electrophysiology, the effective extracellular potential can be considered as an approximate measure of the electrotonic loading a neuron's dendritic tree exerts on the soma. A convolution based method is presented to calculate the local spatial average of the effective extracellular loading in retinal ganglion cells (RGCs) in a continuum model of the retina which includes an active RGC tissue layer. The method can be used to study the effect of the dendritic tree size on the activation of RGCs by electrical stimulation using a hexagonal arrangement of electrodes (hexpolar) placed in the suprachoroidal space.

Rapid high-amplitude circumferential slow wave propagation during normal gastric pacemaking and dysrhythmias

BACKGROUND

Gastric slow waves propagate aborally as rings of excitation. Circumferential propagation does not normally occur, except at the pacemaker region. We hypothesized that (i) the unexplained high-velocity, high-amplitude activity associated with the pacemaker region is a consequence of circumferential propagation; (ii) rapid, high-amplitude circumferential propagation emerges during gastric dysrhythmias; (iii) the driving network conductance might switch between interstitial cells of Cajal myenteric plexus (ICC-MP) and circular interstitial cells of Cajal intramuscular (ICC-IM) during circumferential propagation; and (iv) extracellular amplitudes and velocities are correlated.

METHODS

An experimental-theoretical study was performed. High-resolution gastric mapping was performed in pigs during normal activation, pacing, and dysrhythmia. Activation profiles, velocities, and amplitudes were quantified. ICC pathways were theoretically evaluated in a bidomain model. Extracellular potentials were modeled as a function of membrane potentials.

KEY RESULTS

High-velocity, high-amplitude activation was only recorded in the pacemaker region when circumferential conduction occurred. Circumferential propagation accompanied dysrhythmia in 8/8 experiments was faster than longitudinal propagation (8.9 vs 6.9 mm s(-1) ; P = 0.004) and of higher amplitude (739 vs 528 μV; P = 0.007). Simulations predicted that ICC-MP could be the driving network during longitudinal propagation, whereas during ectopic pacemaking, ICC-IM could outpace and activate ICC-MP in the circumferential axis. Experimental and modeling data demonstrated a linear relationship between velocities and amplitudes (P < 0.001).

CONCLUSIONS & INFERENCES

The high-velocity and high-amplitude profile of the normal pacemaker region is due to localized circumferential propagation. Rapid circumferential propagation also emerges during a range of gastric dysrhythmias, elevating extracellular amplitudes and organizing transverse wavefronts. One possible explanation for these findings is bidirectional coupling between ICC-MP and circular ICC-IM networks.

Principles of model building: an experimentation-aided approach to development of models for signaling networks

Living cells continuously probe their environment and respond to a multitude of external cues. The information about the environment is carried by signaling cascades that act as "internal transducing and computing modules," coupled into complex and interconnected networks. A comprehensive understanding of how cells make decisions therefore necessitates a sound theoretical framework, which can be achieved through mathematical modeling of the signaling networks. In this chapter, we conceptually describe the typical workflow involved in building mathematical models that are motivated by and are developed in a tight integration with experimental analysis. In particular, we delineate the steps involved in a generic, iterative experimentation-driven model-building process, both through informal discussion and using a recently published study as an example. Experiments guide the initial development of mathematical models, including choice of appropriate template model and parameter revision. The model can then be used to generate and test hypotheses quickly and inexpensively, aiding in judicious design of future experiments. These experiments, in turn, are used to update the model. The model developed at the end of this exercise not only predicts functional behavior of the system under study but also provides insight into the biophysical underpinnings of signaling networks.

A preliminary model of gastrointestinal electromechanical coupling

Gastrointestinal (GI) motility is coordinated by several cooperating mechanisms, including electrical slow wave activity, the enteric nervous system (ENS), and other factors. Slow waves generated in interstitial cells of Cajal (ICC) depolarize smooth muscle cells (SMC), generating basic GI contractions. This unique electrical coupling presents an added layer of complexity to GI electromechanical models, and a current barrier to further progress is the lack of a framework for ICC-SMC-contraction coupling. In this study, an initial framework for the electromechanical coupling was developed in a 2-D model. At each solution step, the slow wave propagation was solved first and [Ca(2+)](i) in the SMC model was related to a Ca(2+)-tension-extension relationship to simulate active contraction. With identification of more GI-specific constitutive laws and material parameters, the ICC-SMC-contraction approach may underpin future GI electromechanical models of health and disease states.

A stochastic multi-scale model of electrical function in normal and depleted ICC networks

Multi-scale modeling has become a productive strategy for quantifying interstitial cells of Cajal (ICC) network structure-function relationships, but the lack of large-scale ICC network imaging data currently limits modeling progress. The single normal equation simulation (SNESIM) algorithm was utilized to generate realistic virtual images of small real wild-type (WT) and 5-HT(2B)-receptor knockout (Htr2b(-/-)) mice ICC networks. Two metrics were developed to validate the performance of the algorithm: 1) network density, which is the proportion of ICC in the tissue; and 2) connectivity, which reflects the degree of connectivity of the ICC network. Following validation, the SNESIM algorithm was modified to allow variation in the degree of ICC network depletion. ICC networks from a range of depletion severities were generated, and the electrical activity over these networks was simulated. The virtual ICC networks generated by the original SNESIM algorithm were similar to that of their real counterparts. The electrical activity simulations showed that the maximum current density magnitude increased as the network density increased. In conclusion, the SNESIM algorithm is an effective tool for generating realistic virtual ICC networks. The modified SNESIM algorithm can be used with simulation techniques to quantify the physiological consequences of ICC network depletion at various physical scales.

Biophysically based modeling of the interstitial cells of cajal: current status and future perspectives

Gastrointestinal motility research is progressing rapidly, leading to significant advances in the last 15 years in understanding the cellular mechanisms underlying motility, following the discovery of the central role played by the interstitial cells of Cajal (ICC). As experimental knowledge of ICC physiology has expanded, biophysically based modeling has become a valuable tool for integrating experimental data, for testing hypotheses on ICC pacemaker mechanisms, and for applications in in silico studies including in multiscale models. This review is focused on the cellular electrophysiology of ICC. Recent evidence from both experimental and modeling domains have called aspects of the existing pacemaker theories into question. Therefore, current experimental knowledge of ICC pacemaker mechanisms is examined in depth, and current theories of ICC pacemaking are evaluated and further developed. Existing biophysically based ICC models and their physiological foundations are then critiqued in light of the recent advances in experimental knowledge, and opportunities to improve these models are identified. The review concludes by examining several potential clinical applications of biophysically based ICC modeling from the subcellular through to the organ level, including ion channelopathies and ICC network degradation.