Endogenous driving and synchronization in cardiac and uterine virtual tissues: bifurcations and local coupling

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.

Cell-to-cell modeling of the interface between atrial and sinoatrial anisotropic heterogeneous nets

The transition between sinoatrial cells and atrial cells in the heart is not fully understood. Here we focus on cell-to-cell mathematical models involving typical sinoatrial cells and atrial cells connected with experimentally observed conductance values. We are interested mainly in the geometry of the microstructure of the conduction paths in the sinoatrial node. We show with some models that appropriate source-sink relationships between atrial and sinoatrial cells may occur according to certain geometric arrangements.

Alterations in gap junction connexin43/connexin45 ratio mediate a transition from quiescence to excitation in a mathematical model of the myometrium

The smooth muscle cells of the uterus contract in unison during delivery. These cells achieve coordinated activity via electrical connections called gap junctions which consist of aggregated connexin proteins such as connexin43 and connexin45. The density of gap junctions governs the excitability of the myometrium (among other factors). An increase in gap junction density occurs immediately prior to parturition. We extend a mathematical model of the myometrium by incorporating the voltage-dependence of gap junctions that has been demonstrated in the experimental literature. Two functional subtypes exist, corresponding to systems with predominantly connexin43 and predominantly connexin45, respectively. Our simulation results indicate that the gap junction protein connexin45 acts as a negative modulator of uterine excitability, and hence, activity. A network with a higher proportion of connexin45 relative to connexin43 is unable to excite every cell. Connexin45 has much more rapid gating kinetics than connexin43 which we show limits the maximum duration of a local burst of activity. We propose that this effect regulates the degree of synchronous excitation attained during a contraction. Our results support the hypothesis that as labour approaches, connexin45 is downregulated to allow action potentials to spread more readily through the myometrium.

The location of pacemakers in the uteri of pregnant guinea pigs and rats

The pregnant uterus is a smooth muscle organ whose pattern of contraction is dictated by the propagation of electrical impulses. Such electrical activity may originate from one or more pacemakers, but the location of these sites has not yet been determined. To detect the location of the pacemaker in the gravid uterus, two approaches were used: 1) determine the site from where the contraction started using isolated uteri from the pregnant guinea pig, and videotape their contractions; and 2) record, in isolated uteri from pregnant term rats, with 240 extracellular electrodes simultaneously, and determine where the electrical bursts started. In both the contractile and electrophysiological experiments, there was not a single, specific pacemaker area. However, most contractions (guinea pig 87%) and bursts (rat 76%) started close to the mesometrial border (mean 2.7 ± 4.0 mm SD in guinea pigs and 1.3 ± 1.4 mm in rats). In addition, in the rat, most sites of initiations were located closer to the ovarial end of the horn (mean distance from the ovarial end 6.0 ± 6.2 mm SD), whereas such an orientation was not seen in the guinea pig. In both guinea pig and rat uteri at term, there is not one specific pacemaker area. Rather, contractile and electrical activity may arise from any site, with the majority starting close to the mesometrial border. Furthermore, in the rat, most activities started at the ovarial end of the horn. This may suggest a slightly different pattern of contraction in both species.

Simulating uterine contraction by using an electro-chemo-mechanical model

Contractions of uterine smooth muscle cells consist of a chain of physiological processes. These contractions provide the required force to expel the fetus from the uterus. The inclusion of these physiological processes is, therefore, imperative when studying uterine contractions. In this study, an electro-chemo-mechanical model to replicate the excitation, activation, and contraction of uterine smooth muscle cells is developed. The presented modeling strategy enables efficient integration of knowledge about physiological processes at the cellular level to the organ level. The model is implemented in a three-dimensional finite element setting to simulate uterus contraction during labor in response to electrical discharges generated by pacemaker cells and propagated within the myometrium via gap junctions. Important clinical factors, such as uterine electrical activity and intrauterine pressure, are predicted using this simulation. The predictions are in agreement with clinically measured data reported in the literature. A parameter study is also carried out to investigate the impact of physiologically related parameters on the uterine contractility.

Spatial heterogeneity enhances and modulates excitability in a mathematical model of the myometrium

The muscular layer of the uterus (myometrium) undergoes profound changes in global excitability prior to parturition. Here, a mathematical model of the myocyte network is developed to investigate the hypothesis that spatial heterogeneity is essential to the transition from local to global excitation which the myometrium undergoes just prior to birth. Each myometrial smooth muscle cell is represented by an element with FitzHugh-Nagumo dynamics. The cells are coupled through resistors that represent gap junctions. Spatial heterogeneity is introduced by means of stochastic variation in coupling strengths, with parameters derived from physiological data. Numerical simulations indicate that even modest increases in the heterogeneity of the system can amplify the ability of locally applied stimuli to elicit global excitation. Moreover, in networks driven by a pacemaker cell, global oscillations of excitation are impeded in fully connected and strongly coupled networks. The ability of a locally stimulated cell or pacemaker cell to excite the network is shown to be strongly dependent on the local spatial correlation structure of the couplings. In summary, spatial heterogeneity is a key factor in enhancing and modulating global excitability.

The canine virtual ventricular wall: A platform for dissecting pharmacological effects on propagation and arrhythmogenesis

We have constructed computational models of canine ventricular cells and tissues, ultimately combining detailed tissue architecture and heterogeneous transmural electrophysiology. The heterogeneity is introduced by modifying the Hund-Rudy canine cell model in order to reproduce experimentally reported electrophysiological properties of endocardial, midmyocardial (M) and epicardial cells. These models are validated against experimental data for individual ionic current and action potential characteristics, and their rate dependencies. 1D and 3D heterogeneous virtual tissues are constructed, with detailed tissue architecture (anisotropy and orthotropy, due to fibre orientation and sheet structure) of the left ventricular wall wedge extracted from a diffusion tensor imaging data set. The models are used to study the effects of tissue heterogeneity and class III drugs on transmural propagation and tissue vulnerability to re-entry. We have determined relationships between the transmural dispersion of action potential duration (APD) and the vulnerable window in the 1D virtual ventricular wall, and demonstrated how changes in the transmural heterogeneity, and hence tissue vulnerability, can lead to generation of re-entry in the 3D ventricular wedge. Two class III drugs with opposite qualitative effects on transmural APD heterogeneity are considered: d-sotalol that increases transmural APD dispersion, and amiodarone that decreases it. Simulations with the 1D virtual ventricular wall show that under d-sotalol conditions the vulnerable window is substantially wider compared to amiodarone conditions, primarily in the epicardial region where unidirectional conduction block persists until the adjacent M cells are fully repolarised. Further simulations with the 3D ventricular wedge have shown that ectopic stimulation of the epicardial region results in generation of sustained re-entry under d-sotalol conditions, but not under amiodarone conditions or in control. Again, APD increase in M cells was identified as the major contributor to tissue vulnerability--re-entry was initiated primarily due to ectopic excitation propagating around the unidirectional conduction block in the M cell region. This suggests an electrophysiological mechanism for the anti- and proarrhythmic effects of the class III drugs: the relative safety of amiodarone in comparison to d-sotalol can be explained by relatively low transmural APD dispersion, and hence, a narrow vulnerable window and low probability of re-entry in the tissue.

Physiology and electrical activity of uterine contractions

Presently, there is no effective treatment for preterm labor. The most obvious reason for this anomaly is that there is no objective manner to evaluate the progression of pregnancy through steps leading to labor, either at term or preterm. Several techniques have been adopted to monitor labor, and/or to diagnose labor, but they are either subjective or indirect, and they do not provide an accurate prediction of when labor will occur. With no method to determine preterm labor, treatment might never improve. Uterine electromyography (EMG) methods may provide such needed diagnostics.

Virtual cell and tissue dynamics of ectopic activation of the ventricles

Cardiac ventricular cells and tissues are normally excitable, and are activated by propagating waves of excitation that are initiated in the specialized pacemaking region of the heart. However, isolated or repetitive activity can be initiated at abnormal (ectopic) sites in the ventricles. To trigger an endogenous ectopic beat, there must be a compact focus of cells with changed membrane excitation parameters and kinetics, which initiate activity by after-depolarizations triggered by propagating activity, or that have bifurcated into autorhythmicity. This ectopic focus needs to be large enough, and adequately coupled, to drive the surrounding tissue. We investigate the initiation of ectopic excitation in computational models of human ventricular cells triggered by after-depolarizations and by up/down regulation of specific membrane conductance systems, and the propagation and evolution of ectopic activity in homogeneous or heterogeneous and isotropic, anisotropic, or orthotropic tissues.

Modelling transcriptional feedback loops: the role of Gro/TLE1 in Hes1 oscillations

The transcriptional repressor Hes1, a basic helix-loop-helix family protein, periodically changes its expression in the presomitic mesoderm. Its periodic pattern of expression is retained in a number of cultured murine cell lines. In this paper, we introduce an extended mathematical model for Hes1 oscillatory expression that includes regulation of Hes1 transcription by Drosophila Groucho (Gro) or its vertebrate counterpart, the transducine-like enhancer of split/Groucho-related gene product 1 (TLE1). Gro/TLE1 is a necessary corepressor required by a number of DNA-binding transcriptional repressors, including Hes1. Models of direct repression via Hes1 typically display an expression overshoot after transcription initiation which is not seen in the experimental data. However, numerical simulation and theoretical predictions of our model show that the cofactor Gro/TLE1 reduces the overshoot and is thus necessary for a rapid and finely tuned response of Hes1 to activation signals. Further, from detailed linear stability and numerical bifurcation analysis and simulations, we conclude that the cooperativity coefficient (h) for Hes1 self-repression should be large (i.e. h>or=4). Finally, we introduce the characteristic turnaround duration, and show that for our model the duration of the repression loop is between 40 and 60 min.

Asymptotic properties of mathematical models of excitability

We analyse small parameters in selected models of biological excitability, including Hodgkin-Huxley (Hodgkin & Huxley 1952 J. Physiol.117, 500-544) model of nerve axon, Noble (Noble 1962 J. Physiol.160, 317-352) model of heart Purkinje fibres and Courtemanche et al. (Courtemanche et al. 1998 Am. J. Physiol.275, H301-H321) model of human atrial cells. Some of the small parameters are responsible for differences in the characteristic time-scales of dynamic variables, as in the traditional singular perturbation approaches. Others appear in a way which makes the standard approaches inapplicable. We apply this analysis to study the behaviour of fronts of excitation waves in spatially extended cardiac models. Suppressing the excitability of the tissue leads to a decrease in the propagation speed, but only to a certain limit; further suppression blocks active propagation and leads to a passive diffusive spread of voltage. Such a dissipation may happen if a front propagates into a tissue recovering after a previous wave, e.g. re-entry. A dissipated front does not recover even when the excitability restores. This has no analogy in FitzHugh-Nagumo model and its variants, where fronts can stop and then start again. In two spatial dimensions, dissipation accounts for breakups and self-termination of re-entrant waves in excitable media with Courtemanche et al. kinetics.

Mathematical models in physiology

Computational modelling of biological processes and systems has witnessed a remarkable development in recent years. The search-term (modelling OR modeling) yields over 58000 entries in PubMed, with more than 34000 since the year 2000: thus, almost two-thirds of papers appeared in the last 5-6 years, compared to only about one-third in the preceding 5-6 decades. The development is fuelled both by the continuously improving tools and techniques available for bio-mathematical modelling and by the increasing demand in quantitative assessment of element inter-relations in complex biological systems. This has given rise to a worldwide public domain effort to build a computational framework that provides a comprehensive theoretical representation of integrated biological function-the Physiome. The current and next issues of this journal are devoted to a small sub-set of this initiative and address biocomputation and modelling in physiology, illustrating the breadth and depth of experimental data-based model development in biological research from sub-cellular events to whole organ simulations.