Multichannel mapping of in vivo rat uterine myometrium exhibits both high and low frequency electrical activity in non-pregnancy.

The uterus exhibits intermittent electrophysiological activity in vivo. Although most active during labor, the non-pregnant uterus can exhibit activity of comparable magnitude to the early stages of labor. In this study, two types of flexible electrodes were utilized to measure the electrical activity of uterine smooth muscle in vivo in anesthetized, non-pregnant rats. Flexible printed circuit electrodes were placed on the serosal surface of the uterine horn of six anesthetized rats. Electrical activity was recorded for a duration of 20-30 min. Activity contained two components: high frequency activity (bursts) and an underlying low frequency 'slow wave' which occurred concurrently. These components had dominant frequencies of 6.82 ± 0.63 Hz for the burst frequency and 0.032 ± 0.0055 Hz for the slow wave frequency. There was a mean burst occurrence rate of 0.76 ± 0.23 bursts per minute and mean burst duration of 20.1 ± 6.5 s. The use of multiple high-resolution electrodes enabled 2D mapping of the initiation and propagation of activity along the uterine horn. This in vivo approach has the potential to provide the organ level detail to help interpret non-invasive body surface recordings.

Steady-state approximations for Hodgkin-Huxley cell models: Reduction of order for uterine smooth muscle cell model.

Multi-scale mathematical bioelectrical models of organs such as the uterus, stomach or heart present challenges both for accuracy and computational tractability. These multi-scale models are typically founded on models of biological cells derived from the classic Hodkgin-Huxley (HH) formalism. Ion channel behaviour is tracked with dynamical variables representing activation or inactivation of currents that relax to steady-state dependencies on cellular membrane voltage. Timescales for relaxation may be orders of magnitude faster than companion ion channel variables or phenomena of physiological interest for the entire cell (such as bursting sequences of action potentials) or the entire organ (such as electromechanical coordination). Exploiting these time scales with steady-state approximations for relatively fast-acting systems is a well-known but often overlooked approach as evidenced by recent published models. We thus investigate feasibility of an extensive reduction of order for an HH-type cell model with steady-state approximations to the full dynamical activation and inactivation ion channel variables. Our effort utilises a published comprehensive uterine smooth muscle cell model that encompasses 19 ordinary differential equations and 105 formulations overall. The numerous ion channel submodels in the published model exhibit relaxation times ranging from order 10-1 to 105 milliseconds. Substitution of the faster dynamic variables with steady-state formulations demonstrates both an accurate reproduction of the full model and substantial improvements in time-to-solve, for test cases performed. Our demonstration here of an effective and relatively straightforward reduction method underlines the particular importance of considering time scales for model simplification before embarking on large-scale computations or parameter sweeps. As a preliminary complement to more intensive reduction of order methods such as parameter sensitivity and bifurcation analysis, this approach can rapidly and accurately improve computational tractability for challenging multi-scale organ modelling efforts.

In silico modeling for the hepatic circulation and transport: From the liver organ to lobules.

The function of the liver depends critically on its blood supply. Numerous in silico models have been developed to study various aspects of the hepatic circulation, including not only the macro-hemodynamics at the organ level, but also the microcirculation at the lobular level. In addition, computational models of blood flow and bile flow have been used to study the transport, metabolism, and clearance of drugs in pharmacokinetic studies. These in silico models aim to provide insights into the liver organ function under both healthy and diseased states, and to assist quantitative analysis for surgical planning and postsurgery treatment. The purpose of this review is to provide an update on state-of-the-art in silico models of the hepatic circulation and transport processes. We introduce the numerical methods and the physiological background of these models. We also discuss multiscale frameworks that have been proposed for the liver, and their linkage with the large context of systems biology, systems pharmacology, and the Physiome project. This article is categorized under: Metabolic Diseases > Computational Models Metabolic Diseases > Biomedical Engineering Cardiovascular Diseases > Computational Models.

Modeling and experimental approaches for elucidating multi-scale uterine smooth muscle electro- and mechano-physiology: A review.

The uterus provides protection and nourishment (via its blood supply) to a developing fetus, and contracts to deliver the baby at an appropriate time, thereby having a critical contribution to the life of every human. However, despite this vital role, it is an under-investigated organ, and gaps remain in our understanding of how contractions are initiated or coordinated. The uterus is a smooth muscle organ that undergoes variations in its contractile function in response to hormonal fluctuations, the extreme instance of this being during pregnancy and labor. Researchers typically use various approaches to studying this organ, such as experiments on uterine muscle cells, tissue samples, or the intact organ, or the employment of mathematical models to simulate the electrical, mechanical and ionic activity. The complexity exhibited in the coordinated contractions of the uterus remains a challenge to understand, requiring coordinated solutions from different research fields. This review investigates differences in the underlying physiology between human and common animal models utilized in experiments, and the experimental interventions and computational models used to assess uterine function. We look to a future of hybrid experimental interventions and modeling techniques that could be employed to improve the understanding of the mechanisms enabling the healthy function of the uterus.

Dynamics of Structured Networks of Winfree Oscillators.

Winfree oscillators are phase oscillator models of neurons, characterized by their phase response curve and pulsatile interaction function. We use the Ott/Antonsen ansatz to study large heterogeneous networks of Winfree oscillators, deriving low-dimensional differential equations which describe the evolution of the expected state of networks of oscillators. We consider the effects of correlations between an oscillator's in-degree and out-degree, and between the in- and out-degrees of an "upstream" and a "downstream" oscillator (degree assortativity). We also consider correlated heterogeneity, where some property of an oscillator is correlated with a structural property such as degree. We finally consider networks with parameter assortativity, coupling oscillators according to their intrinsic frequencies. The results show how different types of network structure influence its overall dynamics.

A permutation method for network assembly.

We present a method for assembling directed networks given a prescribed bi-degree (in- and out-degree) sequence. This method utilises permutations of initial adjacency matrix assemblies that conform to the prescribed in-degree sequence, yet violate the given out-degree sequence. It combines directed edge-swapping and constrained Monte-Carlo edge-mixing for improving approximations to the given out-degree sequence until it is exactly matched. Our method permits inclusion or exclusion of 'multi-edges', allowing assembly of weighted or binary networks. It further allows prescribing the overall percentage of such multiple connections-permitting exploration of a weighted synthetic network space unlike any other method currently available for comparison of real-world networks with controlled multi-edge proportion null spaces. The graph space is sampled by the method non-uniformly, yet the algorithm provides weightings for the sample space across all possible realisations allowing computation of statistical averages of network metrics as if they were sampled uniformly. Given a sequence of in- and out- degrees, the method can also produce simple graphs for sequences that satisfy conditions of graphicality. Our method successfully builds networks with order O(107) edges on the scale of minutes with a laptop running Matlab. We provide our implementation of the method on the GitHub repository for immediate use by the research community, and demonstrate its application to three real-world networks for null-space comparisons as well as the study of dynamics of neuronal networks.

A Pipeline for the Registration of Calcium Transient Data to Structural Networks of the Interstitial Cells of Cajal.

Interstitial cells of Cajal (ICC) generate electrical pacemaker activity in the gastrointestinal (GI) tract known as slow waves, which regulate GI motility. ICC express both the Kit receptor tyrosine kinase protein and a Ca2+-activated Cl--channel, encoded by the anoctamin1 (Ano1) protein, which is an essential contributor to the Ca2+ cycling of ICC and slow wave pacemaking. Recent dye-loading imaging studies have demonstrated Ca2+ transients in ICC in isolated tissue preparations. The main aim of this study was to develop a method that allows Ca2+ transients to be registered to structural ICC network data. Confocal image stacks of ICC labeled for Kit or Ano1 and Ca2+ recording data were processed using a thresholding protocol. The Ca2+ transients were then registered to the ICC structural network. First, a general idea of the placement was found by mapping the field-of-view of the Ca2+ transient data to the distorted tissue that contained the ICC network image. The errors in the registration were then corrected for by warping the internal Ca2+ transient field according to the structural network. In data sets from tissues with induced, targeted knockdown of Ano1 expression in a subset of ICC, agreement between the Ca2+ transient data and structural network was 68 ± 10%. This level of agreement allowed selective extraction of Ca2+ data from Ano1-positive (Ano1+) and Ano1-negative (Ano1-) ICC. In the future, this technique will allow investigation into the functional properties of ICC in relation to the level of knockdown of specific ICC associated proteins.

A spatial-temporal model for zonal hepatotoxicity of acetaminophen.

The metabolism zonation in liver lobules is well known yet its incorporation into the mathematical models of acetaminophen (APAP) metabolism is still primitive - only the oxidation pathway via reaction with the cytochrome P450 (CYP450) has been considered, yet the zonal heterogeneity exhibits in all three pathways including sulphation, glucuronidation and oxidation. In this paper we present a novel computational method where an intracellular APAP metabolism model is integrated into a Finite Element Model (FEM) of sinusoids, and the zonal heterogeneity in three metabolism pathways are all incorporated. We demonstrate that the degradation of APAP, detoxification via glutathione (GSH) and the formation of hepatotoxicity, are all affected profoundly by the zonal difference. Specifically, glucuronidation plays a major role in the degradation of APAP. Generation of GSH, its conjugation with the toxic NAPQI and the spatial distribution of CYP450 combined together determine the toxicity of APAP. We suggest that the current platform be used for further hepatotoxicity study of APAP by incorporating other heterogeneity factors.

A multi-scale spatial model of hepatitis-B viral dynamics.

Chronic hepatitis B viral infection (HBV) afflicts around 250 million individuals globally and few options for treatment exist. Once infected, the virus entrenches itself in the liver with a notoriously resilient colonisation of viral DNA (covalently-closed circular DNA, cccDNA). The majority of infections are cleared, yet we do not understand why 5% of adult immune responses fail leading to the chronic state with its collateral morbid effects such as cirrhosis and eventual hepatic carcinoma. The liver environment exhibits particularly complex spatial structures for metabolic processing and corresponding distributions of nutrients and transporters that may influence successful HBV entrenchment. We assembled a multi-scaled mathematical model of the fundamental hepatic processing unit, the sinusoid, into a whole-liver representation to investigate the impact of this intrinsic spatial heterogeneity on the HBV dynamic. Our results suggest HBV may be exploiting spatial aspects of the liver environment. We distributed increased HBV replication rates coincident with elevated levels of nutrients in the sinusoid entry point (the periportal region) in tandem with similar distributions of hepatocyte transporters key to HBV invasion (e.g., the sodium-taurocholate cotransporting polypeptide or NTCP), or immune system activity. According to our results, such co-alignment of spatial distributions may contribute to persistence of HBV infections, depending on spatial distributions and intensity of immune response as well. Moreover, inspired by previous HBV models and experimentalist suggestions of extra-hepatic HBV replication, we tested in our model influence of HBV blood replication and observe an overall nominal effect on persistent liver infection. Regardless, we confirm prior results showing a solo cccDNA is sufficient to re-infect an entire liver, with corresponding concerns for transplantation and treatment.

A mathematical model of the effects of anoctamin-1 loss on intestinal slow wave entrainment.

The interstitial cells of Cajal (ICC) generate electrophysiological events called slow waves that regulate the motility of the gastrointestinal (GI) tract. Recent studies have demonstrated that the Ca2+-activated Cl- -channel, encoded by the anoctamin1 (Ano1) protein, has a major role in regulating intestinal slow waves and motility. The main aim of this study was to develop a multi-scale mathematical model capable of simulating both normal slow wave entrainment and the effects of Ano1 knockout (KO) on the normal activity. A biophysically-based cell model was adapted to simulate the effects of Ano1 KO at the cellular level. A 10mm one-dimensional (1D) model was then developed to simulate entrained intestinal slow wave propagation. Cellular KO at levels of 100% and 20% were applied to a varying-sized middle region of the 1D model. The main finding was that the level of loss of entrainment increased as both cellular and spatial Ano1 KO levels increased, mostly manifesting as ectopic activation. In the future, this model will be extended and used in combination with Ca2+ -imaging data to quantitatively investigate the effects of Ano1 loss in ICC.

Modeling calcium waves in an anatomically accurate three-dimensional parotid acinar cell.

We construct a model of calcium waves in a three-dimensional anatomically accurate parotid acinar cell, constructed from experimental data. Gradients of inositol trisphosphate receptor (IPR) density are imposed, with the IPR density being greater closer to the lumen, which has a branched structure, and inositol trisphosphate (IP3) is produced only at the basal membrane. We show (1) that IP3 equilibrates so quickly across the cell that it can be assumed to be spatially homogeneous; (2) spatial separation of the sites of IP3 action and IP3 production does not preclude the formation of stable oscillatory Ca2+ waves. However, these waves are not waves in the mathematical sense of a traveling wave with fixed profile. They result instead from a time delay between the Ca2+ rise in the apical and basal regions; (3) the ryanodine receptors serve to reinforce the Ca2+ wave, but are not necessary for the wave to exist; (4) a spatially independent model is not sufficient to study saliva secretion, although a one-dimensional model might be sufficient. Our results here form the first stages of the construction of a multiscale and multicellular model of saliva secretion in an entire acinus.

The impact of human and vector distributions on the spatial prevalence of malaria in sub-Saharan Africa.

Eradication of malaria from the world in the latter part of the twentieth century proved an elusive, albeit desirable, objective. Unfortunately, resurgence of malarial incidence is currently underway. Key to understanding effective control schemes such as indoor residual spraying (spraying insecticide inside houses to kill the malarial vector mosquitoes) is the impact of spatial distributions for communities exposed to the malarial vector mosquito populations. Densities of human dwellings in small communities vary considerably in regions exposed to larval breeding sites. We extend prior modelling work to explore the spatial impact and diffusive transport of mosquito population densities on various distributions of human populations on relatively small landscape representations. Bistable dynamics of our reaction-diffusion model, which excludes advective transport, suggest two temporal phases for infection. An initial rapid phase occurs during transitions from initial homogeneous or spatially confined infections to peak levels over the course of days, and a relaxation phase develops to a steady state over weeks or months, suggesting successful intervention methods likely require recognising the phase of infection. We further observe a strong dependence of human infection and recovery on distributions of susceptible human populations with some degree of independence from mosquito distributions given an adequate supply of mosquito vectors to sustain infections. A subtle and complex interplay between human dwelling densities, mosquito diffusion and infection rates also emerges. With a sufficiently fast diffusive transport of mosquitoes, our model indicates that relative timescales for infection rates are slower, leading to lower rates of infection. This suggests that, although we here only include diffusive transport, if mosquitoes are subject to rapid enough movement (e.g., wind), communities situated in windy areas are exposed to less infectious risk than those in non-windy areas. This should help to guide intervention strategies with geographical considerations in mind. Our implementation of a reaction-diffusion model here further reveals some issues regarding continuum methods for population and infectious disease models that suggest consideration of discrete spatial methods (e.g., agent-based) for future work.

Mitochondrial calcium handling within the interstitial cells of Cajal.

The interstitial cells of Cajal (ICC) drive rhythmic pacemaking contractions in the gastrointestinal system. The ICC generate pacemaking signals by membrane depolarizations associated with the release of intracellular calcium (Ca(2+)) in the endoplasmic reticulum (ER) through inositol-trisphosphate (IP3) receptors (IP3R) and uptake by mitochondria (MT). This Ca(2+) dynamic is hypothesized to generate pacemaking signals by calibrating ER Ca(2+) store depletions and membrane depolarization with ER store-operated Ca(2+) entry mechanisms. Using a biophysically based spatio-temporal model of integrated Ca(2+) transport in the ICC, we determined the feasibility of ER depletion timescale correspondence with experimentally observed pacemaking frequencies while considering the impact of IP3R Ca(2+) release and MT uptake on bulk cytosolic Ca(2+) levels because persistent elevations of free intracellular Ca(2+) are toxic to the cell. MT densities and distributions are varied in the model geometry to observe MT influence on free cytosolic Ca(2+) and the resulting frequencies of ER Ca(2+) store depletions, as well as the sarco-endoplasmic reticulum Ca(2+) ATP-ase (SERCA) and IP3 agonist concentrations. Our simulations show that high MT densities observed in the ICC are more relevant to ER establishing Ca(2+) depletion frequencies than protection of the cytosol from elevated free Ca(2+), whereas the SERCA pump is more relevant to containing cytosolic Ca(2+) elevations. Our results further suggest that the level of IP3 agonist stimulating ER Ca(2+) release, subsequent MT uptake, and eventual activation of ER store-operated Ca(2+) entry may determine frequencies of rhythmic pacemaking exhibited by the ICC across species and tissue types.

Spatio-temporal calcium dynamics in pacemaking units of the interstitial cells of Cajal.

The interstitial cells of Cajal (ICC) are responsible for producing pacemaking signals that stimulate rhythmic contractions in the gastro-intestinal system. The pacemaking signals are generated by membrane depolarizations, which are in turn linked to the integrated transport of calcium between the endoplasmic reticulum (ER), through inositol-trisphosphate receptor (IP(3)R) release, and mitochondria, through the uniporter. A non-specific cation channel (NSCC) is associated with the membrane depolarizations, and is inhibited by intracellular calcium. One theory proposes that the integrated calcium transport occurs within specific regions of the ICC called "pacemaker units," and results in localized calcium concentration reductions within these units, which in turn activate the NSCC and depolarize the membrane. We have constructed a model of the spatio-temporal calcium dynamics within an ICC pacemaker unit to determine under what conditions the local calcium concentrations may reduce below baseline. We obtain reductions of calcium concentrations below baseline but only under certain conditions. Without strong and persistent stimulation of the IP(3)R, reductions of calcium below baseline occur only with a non-physiological, time-dependent uniporter. Alternatively, sufficient IP(3)R release leads to reductions of calcium below baseline, due to depletion of the ER calcium store over the time scale of seconds, although these reductions require strong mitochondrial and ER calcium uptake.

Reaction diffusion modeling of calcium dynamics with realistic ER geometry.

We describe a finite-element model of mast cell calcium dynamics that incorporates the endoplasmic reticulum's complex geometry. The model is built upon a three-dimensional reconstruction of the endoplasmic reticulum (ER) from an electron tomographic tilt series. Tetrahedral meshes provide volumetric representations of the ER lumen, ER membrane, cytoplasm, and plasma membrane. The reaction-diffusion model simultaneously tracks changes in cytoplasmic and ER intraluminal calcium concentrations and includes luminal and cytoplasmic protein buffers. Transport fluxes via PMCA, SERCA, ER leakage, and Type II IP3 receptors are also represented. Unique features of the model include stochastic behavior of IP3 receptor calcium channels and comparisons of channel open times when diffusely distributed or aggregated in clusters on the ER surface. Simulations show that IP3R channels in close proximity modulate activity of their neighbors through local Ca2+ feedback effects. Cytoplasmic calcium levels rise higher, and ER luminal calcium concentrations drop lower, after IP3-mediated release from receptors in the diffuse configuration. Simulation results also suggest that the buffering capacity of the ER, and not restricted diffusion, is the predominant factor influencing average luminal calcium concentrations.

Interplay of ryanodine receptor distribution and calcium dynamics.

Spontaneously generated calcium (Ca2+) waves can trigger arrhythmias in ventricular and atrial myocytes. Yet, Ca2+ waves also serve the physiological function of mediating global Ca2+ increase and muscle contraction in atrial myocytes. We examine the factors that influence Ca2+ wave initiation by mathematical modeling and large-scale computational (supercomputer) simulations. An important finding is the existence of a strong coupling between the ryanodine receptor distribution and Ca2+ dynamics. Even modest changes in the ryanodine receptor spacing profoundly affect the probability of Ca2+ wave initiation. As a consequence of this finding, we suggest that there is information flow from the contractile system to the Ca2+ control system and this dynamical interplay could contribute to the increased incidence of arrhythmias during heart failure.