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.

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.

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 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.

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.

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.