Spatiotemporal patterning of uterine excitation patterns in human labour

BioSystems 112 (2013) 63–72

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Spatiotemporal patterning of uterine excitation patterns in human labour Eleftheria Pervolaraki ∗ , Arun V. Holden School of Biomedical Sciences, University of Leeds, Leeds LS2 9JT, UK

a r t i c l e

i n f o

Keywords: Computational biology Uterus Pregnancy Birth

a b s t r a c t The mechanisms leading to the initiation of normal, premature or dysfunctional human labour are poorly understood, as animal models are inappropriate, and experimental studies are limited. Computational modelling provides a means of linking non-invasive clinical data with the results of in vitro cell and tissue physiology. Nonlinear wave processes – propagation in an excitable medium – provides a quantitatively testable description of mechanisms of premature and full term labour, and a view of changes in uterine electrophysiology during gestation as a trajectory in excitation and intercellular coupling parameter space. Propagation phenomena can account for both premature and full term labour. © 2013 Elsevier Ireland Ltd. All rights reserved.

1. Introduction There are about 135 million live births/year in the world, making human birth a common and normal process. In the developed world more than 90% of women aged 40–45 have given birth at least once, and although over 98% of births happen in a hospital environment, in the presence of professionals and monitoring instrumentation, the mechanisms of human parturition still remains poorly understood. This is partly because of the complexities of human birth, partly because the processes of human birth differ from those in intensively studied common laboratory and farm animals, and partly because clinical research information is limited and obtained at different sites and so is not effectively integrated. The geometry of the adult human pelvis is adapted for our upright, bipedal gait, being wider than deep at the pelvic inlet to the birth canal. At the outlet of the birth canal the depth is greater than the width. The development of a large brain leads to a neonatal skull size that can just squeeze through the birth canal by rotation of foetal head; the length is aligned first to the width, and then to the depth of the birth canal. This is followed by a further rotation of the shoulders (Weaver and Hublin, 2009). This complex process has been directly imaged using an open magnetic resonance imager (Phillips Panorama HFO) at Charité Hospital, Berlin (Bamberg et al., 2012) and underlies the need for birth assistance that is unique to humans. Pregnancy-related genes that show accelerated protein evolution have been identified in the human lineage, and may be associated with rapid evolution over the past 10–20 thousand years of the processes controlling the timing of birth: one of these

∗ Corresponding author. Tel.: +44 113 343 4251. E-mail addresses: [email protected], [email protected] (E. Pervolaraki). 0303-2647/$ – see front matter © 2013 Elsevier Ireland Ltd. All rights reserved. http://dx.doi.org/10.1016/j.biosystems.2013.03.012

human accelerated genes has been associated with familial preterm birth (Plunkett et al., 2011). Normal, full term human birth is peculiar, and in terms of allometric scaling relative to other primates, and comparative physiology relative to other mammals has some attributes of premature mammalian birth. Understanding the organ level, bio-mechanical 10 cm-scale processes requires quantitative information on the passive constitutive mechanical properties of the component tissues, and how they change during gestation and into labour, as well as the pattern of active contractions during the development of labour, and the mechanical interactions between the uterus, foetus and cervix. Ultrasound is widely used in clinical obstetrics to monitor foetal development, and can provide uterine geometric (Degani et al., 1998) and computed tension data during gestation. However, quantitative data is largely lacking and finite element modelling of the uterus and cervix is limited (House and Socrate, 2006). The cervix is cylindrical, with collagen fibres aligned along longitudinal, radial and circumferential directions giving mechanical anisotropy (Myers et al., 2010). The geometry of the clinical softening and ripening of the cervix as pregnancy progresses into labour can be monitored noninvasively by 3-D sonography (Lang et al., 2010), and the cervical strength estimated via its collagen content, estimated by light-induced fluorescence (Garfield et al., 2002). In many mammals (rat, mouse, rabbit, sheep, pig) birth is triggered by a fall in plasma progesterone level, which does not occur in humans, in whom the trigger for the initiation of labour is unknown. Multiple interactions between foetal, foetal membrane and placental, and maternal signalling systems lead to the release of maternal and foetal oxytocin, and placental estriol and prostaglandlins, that all act on receptors on uterine smooth muscle (myometrial) cells (Norwitz et al., 1999), altering protein expression and facilitating synchronised uterine contractions. During both

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Fig. 1. Schematic of interactions that modulate myometrial excitability during gestation and labour, and data sources for virtual tissue engineering of the uterus. As full term approaches maternal, foetal and placental hormonal and inflammatory response signalling all act on uterine smooth muscle cells, and their connections via gap junctions, and increase excitability and synchronisation of activity. They also act on the cervix, causing dilation and weakening. During pregnancy maternal blood samples are taken, allowing genetic information (e.g. single nucleotide polymorphisms SNPs) to be acquired, as well as noninvasive monitoring (ultrasound). There can be magnetic resonance imaging, or electromyography (EMG). During labour there can be noninvasive monitoring of uterine contractions (tocodynamometry), or invasive measurements of intrauterine pressure, or cervical dilation. At delivery placental and foetal membrane tissue, and (in Caesarean deliveries) myometrial tissue biopsies can be available for histological analysis, gene expression, molecular mapping, tissue culture and in vitro cell and tissue electrophysiological studies. These inform an integrated and distributed hierarchy of molecular, cell, tissue, organ, and pregnancy models. CNS: central nervous system; CRH corticotrophin releasing hormone; DHEAS: dhydroepiandrosterone sulphate (an androgen hormone produced by the ovaries and adrenal gland).

preterm (before 37 weeks gestation, full term is 40 weeks) and fullterm labour there is an increased inflammatory response, with an invasion of both the myometrium and cervix by inflammatory cells (Challis et al., 2009), and upregulation of components that mediate inflammation-cytokine production, interleukins and COX2. A genomic analysis of mRNA expression in full term myometrial and cervical biopsies identified genes that were upregulated in both the labouring myometrium and cervix, and identification of functional groups and specific pathways showed these genes were mostly concerned with inflammation and chemotaxis (Bollopragada et al., 2009). A quantitative predictive network model of the interacting processes (Fig. 1) that lead to myometrial activity would allow the interpretation of changes in maternal plasma biomarkers that are available from routine clinical blood samples in terms of the different triggers for parturition inferred from mRNA analysis from biopsies obtained at Caesarean section. Our poor understanding of the mechanisms of normal birth does have consequences: a commercial consequence is the paucity of new drugs, or their development, for maternal health (Fisk and Atun, 2008), even though disorders of human parturition (both preterm and dysfunctional parturition) are major contributors to human disease, and have major resource implications, for health services and society. Normal parturition requires appropriate positioning of the foetal head (engagement) within the pelvis, cervical ripening in addition to coordinated uterine contractions. However, uterine contractions can be considered a final common pathway for the mechanisms controlling parturition. These contractions not only lead to the delivery of the baby and placenta, but after delivery the uterus contracts down from an organ length of ∼30 cm to ∼7 cm, minimising blood loss through haemorrhage. Although most births proceed normally, disorders in the timing, pattern and effectiveness of uterine contraction occur.

1.1. Preterm parturition The global prevalence of preterm birth is 9.6%; it affects 12.9 million babies per year and is a direct cause of death of over one million babies worldwide annually. In developed countries, preterm birth is the single greatest contributor to neonatal mortality and morbidity, accounts for 50% of neurological disorders in infants and can lead to life-long morbidity. In addition to the devastating health, social and emotional effects that preterm birth has on babies, parents and the extended family, the financial costs to families and society are also huge, with estimated public sector costs alone for preterm birth in the UK in 2005 of ∼£3 billion. Spontaneous preterm labour (i.e. inappropriate early activation of uterine contractions) is the single biggest cause of preterm birth with data showing rising rates (Norman et al., 2009a,b). 1.2. Dysfunctional parturition Dysfunctional parturition manifest (ineffective myometrial contractions before delivery) is the second leading cause of Caesarean section rates, which in developed countries are around double the rate recommended by the World Health Organisation, and are associated with significant increased maternal mortality and morbidity in addition to extra health care costs. Obstetric haemorrhage is largely due to failure of the myometrium to contract effectively after delivery, again implying dysfunctional parturition, and is the leading cause of maternal death world wide, and accounts for over 132,000 maternal deaths annually. Increased understanding of uterine and cervical behaviour during gestation and labour, and the triggers for birth should help to alleviate these consequences, but animal studies are of little use because of the peculiarities of human birth. The pregnant woman, or a woman in labour, are not suitable subjects for

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experimental investigation, and information needs to be obtained as part of normal clinical observations and care, or in non- or minimally invasive clinically motivated investigations. Uterine tissue biopsies obtained at Caesarean section can be physiologically viable (in terms of electrical and mechanical activity) for more than 24 h, and so allow extensive in vitro experiments on full term tissue. Quantitative models of the dynamics of cells in these tissues can be coupled into spatially extensive tissue models – virtual tissue engineering (Holden, 2010), – to reconstruct spatio-temporal activity in the uterine tissue, or full term uterus. Parameters in these cell models may be modified to values estimated or inferred to occur earlier in gestation, or to reproduce behaviours characteristic of the cells earlier in gestation, and then to quantitatively predict the changes in the pattern of spatio-temporal behaviour of the uterine myometrium during gestation and leading into birth. Some of these predictions may be validated by non-invasive measures, and so the model can provide a tool for quantitatively testing hypotheses about uterine mechanisms. We aim to improve our understanding on the control of uterine organ level activity during human pregnancy and labour using the methods of virtual uterine tissue engineering – the construction and solution of biophysically and anatomically detailed computational models of uterine cell, tissue and organ electromechanics, combined with time series of uterine activity recordings during pregnancies (Aslanidi et al., 2011). This considers the behaviour of the uterus as the behaviour of an excitable medium, where synchronised activity – the contractions of labour – could be driven by a specialised pacemaker site (as in the heart), or emerge via synchronisation of weakly coupled oscillators, or by the propagation of endogenous nonlinear waves. These possible alternative control strategies depend on the excitation and coupling parameters of the medium, that change during gestation, and understanding the control strategy would help to target interventions aimed at dysfunctional contractions. If these interventions are pharmacological then detailed quantitative models of USMC membrane excitation are necessary. 2. Uterine smooth muscle cells electrophysiology and models At full term, the excitation and contraction of the human myometrium is almost entirely determined by the influx of extra-cellular calcium voltage gated calcium channels; there is little sarcoplasmic reticulum Ca2+ -induced Ca2+ -release in uterine smooth muscle cells (UMSC). 2.1. Electrophysiology of human uterine smooth muscle cells In vitro electrophysiological recordings can be performed on human myometrial cells and tissues from the lower segment of the uterus obtained at Caesarean sections (i.e. from 28 weeks to 40 weeks gestational age), before the uterus is contracted down by syntocin administration. These preparations are electrophysiologically viable for over 24 h and proteolytic enzyme digestion can provide single cells that are viable for recording membrane potentials. The cells may be resting or autorhythmic. The resting membrane potential changes during gestation from ∼75 mV at week 28 to about ∼50 mV at full term (week 40) (Parkington et al., 1999). These changes can be explained by regulatory changes in channel expression and the gradual depolarization of the membrane potential during gestation can be explained by down regulation of K+ selective conductance. Spontaneous or evoked (by an injected depolarising current or by oxytocin) spike and plateau action potentials lasting a few to hundreds of seconds (at 37 ◦ C) have been described. In tissue

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Fig. 2. (a) Isolated human uterine smooth muscle cell (USMC), obtained by proteolytic digestion of tissue biopsy at a full term Caesarean section. Confocal microscopy, with Ca2+ channels (red) localised to the sarcoplasmic reticulum surrounded by the F actin cytoskeleton (green), from Blanks et al. (2007). (b) Solitary action potential evoked by maintained depolarising current pulse in an isolated USMC, showing simple spike and plateau action potential. (c) Schematic representation of (a) the ionic channels, pumps and exchangers in the model for uterine smooth muscle cell electrophysics. Changes in [Ca2+ ]i follow membrane Ca2+ fluxes with separate uptake (Iup ) and release (Irel ) compartments in the sarcoplasmic reticulum. Ca2+ is transferred to the SR through the SERCA pump; Ca2+ is then diffused across (Itr ) to the release compartment before releasing back to the cytosol through the ryanodine receptor flux.

preparations, complex bursting action potentials are observed (Nakao et al., 1997) but human isolated USMCs only exhibit simple action potentials (spike, or spike with plateau – see Fig. 2b) (Bru-Mercier et al., 2007). Whole cell voltage clamp experiments have provided a detailed characterisation of the T type Ca2+ channel (Blanks et al., 2007; Shmigol et al., 2007) but an extensive voltage or patch clamp description of the magnitudes and kinetics survey of all the currents in human USMCs during gestation has yet to be achieved. However, the presence and approximate relative magnitudes of different conductance systems can be estimated from immunohistochemistry and mRNA mapping of tissue samples obtained. The kinetics of the different ionic channel types can be obtained from single channel patch clamp studies on human channels expressed in Xenopus oocytes or mammalian expression lines, and so human USMC models, based on data from human cells and tissues, are being constructed. Molecular biology provides a high throughput, and partially automated approach to obtaining the identity and kinetics of the ionic channels for construction of the human myometrial cell. The ionic channels, exchangers and pumps in that contribute to Iion correspond to the activities of membrane proteins, each composed of subunits that may have different isoforms. These proteins result from transcription from their genes, via mRNA, followed by protein synthesis in ribosomes, trafficking, insertion in the membrane where their activity and membrane turnover may be regulated. The presence of these proteins can be determined by Western blot and their localisation by immunofluorescence. However, uterine tissue also contains vascular smooth muscle, but laser capture microdissection techniques can isolate blocks of pure myometrial cells, and so the channel components of UMSC can be identified. For the each ionic channel system identified as being present in the uterine smooth muscle cell membrane, the kinetics can be

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obtained via expression of its associated gene(s) in Xenopus eggs, or in a mammalian expression system. The kinetics of the channel depend on its microenvironment, e.g. changes in membrane surface charge shift the voltage dependence of the activation and inactivation gating processes, and so may not correspond to the in vivo kinetics. The quantitative polymerase chain reaction, in situ hybridisation, and immune-fluorescence can estimate the ratio of specific channel protein expression in different parts of the same tissue. This approach could be used to introduce spatial heterogeneities into models of uterine tissue electrophysiology (as only lower segment samples are available from Caesarean sections). However, although this approach has been successful in cardiac modelling (Chandler et al., 2009) it is semi-quantitative, and does not provide the magnitude of the channel density, from which the maximal conductances can be calculated. 2.2. Uterine smooth muscle cell models There are a number of detailed biophysical models for the electrophysiology of uterine smooth muscle cells (USMCs) – these are “mammalian” i.e. chimaeric, based on voltage clamp data from different species, predominantly the laboratory rat (Bursztyn et al., 2007; Rihana et al., 2009; Tong et al., 2011). These all relate the electrical activity of an isolated cell V(t) to the sum of membrane ionic current Iion flows through pumps, exchangers and channels; some of which are voltage and/or time dependent, and have a structure as in Fig. 2:

 Cm dV = Iion . dt

(1)

A recent model of a rat USMC has 15 ionic currents and simple calcium dynamics and is described by 25 dynamical variables and >125 parameters. The currents include: Inward currents: L-type Ca2+ current (ICaL ), attributed to be the major inward current, fast Na+ current (INa ), T-type Ca2+ current

(ICaT ) and a hyperpolarisation-activated current (Ih ). A nonspecific cation current (INSCC ) is also included, with a reversal potential within the reported AP amplitude range. Outward currents: Fast A-type transient K+ current (IKa ), two voltage-gated K+ currents (IK1 and IK2 ), Ca2+ -activated K+ current (IK(Ca) ) and a leak background current (IKleak, I b ). Electrogenic exchangers: Na+ –Ca2+ exchanger current (INaCa ) and Na+ –K+ pump current (INaK ) and Ca2+ –Cl− exchanger (IClCa ). Calcium fluxes: The [Ca2+ ]i dynamics, the Na+ –Ca2+ exchanger (JNaCa ) and the plasma membrane Ca2+ -ATPase (JPMCA ) is modified from a simple uterine excitation–contraction model to include the kinetics from membrane calcium channels and electrogenic INaCa . Numerical solutions of the Tong et al. (2011) USMC model show spiking and plateau action potentials (see Fig. 3), and can be modified to reproduce the effects of hormones via its action on the magnitude and kinetics of Ca2+ and K+ currents, and reproduce V(t), [Ca2+ ]i and force under a variety of conditions. The excitability and action potentials of mammalian USMCs from the pregnant uterus change during gestation and with changes to the foetal location within the uterus. Parameters in the cell model can be modified to simulate these changes in uterine electrophysiology. During gestation the state of the USMC, defined by its set of parameter values, may be considered to be moving in parameter space. Trafficking, channel turnover and sequestration within caveolae also influence the density of the ion channels on the cell surface, and so the membrane parameters of the USMC model may fluctuate. Biophysically detailed models are needed for V(t) of human uterine smooth muscle cells, from different parts of the uterus (in case of spatial heterogeneities) and at different times during gestation. Isolated mammalian USMCs show a wide range of electrophysiological activity, from spike like action potentials, repetitive spiking, action potentials with long plateaux and bursts of spikes on the plateau. USMC models need to reproduce these different behaviours, in different regions of parameter space. Bifurcation analysis of USMC models, via the use of continuation algorithms, identifies

Fig. 3. Numerical solutions of UMSC model: (a) plateau action potential evoked by 5 s current pulse; membrane potential (solid line), normalised Ca2+ transient (dotted line) and contraction (dashed line); (b–e) transition from simple spiking (b), through burst of spikes (c) to burst (d), to spike and plateau (e) (Tong et al., 2011). (f) Bifurcation diagram for the uterine smooth muscle cell model as a constant external current (hold) as the bifurcation parameter is varied. The bifurcation diagram shows different system behaviours, changing from a low resting stable at 50 mV, via two bistable regions to a more depolarised steady state at 20 mV as the bifurcation parameter changes; the stability of the equilibrium solution (line) is lost at a Hopf bifurcation HB and a limit point LP, and numerical tracking of stable (solid line) and unstable steady (dashed line) and stable (solid circles) and unstable (open circles) oscillatory solutions. The bifurcation parameter “hold” represents an applied current, for example a stretch activated current.

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equilibrium solutions, their stability, and perhaps bistability; periodic solutions and their stability, and how these interact to produce bursting activity (Fig. 3). UMSC models, as in Eq. (1), are dynamical systems, with a (large) number of variables and parameters. They have equilibrium solutions, at which dV/dt = 0, where V is a vector of all the variables (potential, ionic concentrations, channel gating variables), and the equilibrium solution changes as a parameter is varied. If the stability of the equilibrium solution changes between stable and unstable at a particular value of a parameter, there is a bifurcation, and a qualitative change in behaviour. The stability can be tracked using continuation algorithms that compute and follow, in parameter space, eigenvalues (for the stability of equilibria) and Floquet multipliers (for the stability of periodic solutions) e.g. XPPAUT (Ementrout, 2000) and bifurcation points identified. At a Hopf bifurcation, a stable equilibrium loses its stability as a single isolated complex conjugate pair of eigenvalues crosses the imaginary axis, and periodic solutions emerge. Bifurcation analysis allows the characterisation of complex bursting patterns in terms of how they arise dynamically, rather than by what individual ionic mechanisms contribute to their generation (Bertram et al., 1995). A qualitative understanding of the bifurcational structure of the USMC model in parameter space provides a view of how the excitation dynamics can change during gestation, and a detailed numerical bifurcation analysis of specific USMC models can quantitatively account for specific quantitative experimental results e.g. the effects of oestradiol. The excitability and action potentials of mammalian USMCs from the pregnant uterus change during gestation and with changes to the foetal location within the uterus. Parameters in the cell model can be modified to reproduce these changes in uterine electrophysiology. In vitro electrophysiological recordings can be performed on human myometrial cells and tissues from the lower segment of the uterus obtained at Caesarean sections (i.e. close to full term). These changes in electrophysiology are due to up/down regulatory changes in channel expression – e.g. the slow drift in human USMC membrane potential from ∼−75 mV at 28 weeks to ∼−55 mV at full term (Partington et al., 1999) can be accounted for by a slow upregulation of K+ selective conductances. During gestation the state of the USMC, defined by its set of parameter values, may be considered to be moving in parameter space. An extensive survey of the electrophysiology of human USMCs during gestation is not possible, but the presence and approximate magnitudes of different conductance systems can be estimated from immunohistochemistry and mRNA mapping of tissue samples obtained. The kinetics of the different ionic channels types can be obtained from single channel patch clamp studies on human channels expressed in Xenopus oocytes or mammalian expression lines, and so human USMC models, based on data from human cells and tissues, are being constructed. Isolated mammalian USMCs show a wide range of electrophysiological activity, from spike like action potentials, repetitive spiking, action potentials with long plateaux and bursts of spikes on the plateau. USMC models need to reproduce these different behaviours, in different regions of parameter space. Bifurcation analysis of USMC models, via the use of continuation algorithms, identifies equilibrium solutions, their stability, and perhaps bistability; periodic solutions and their stability, and how these interact to produce bursting activity (Fig. 3). Trafficking, channel turnover and sequestration within caveolae also influence the density of the ion channels on the cell surface, and so the membrane parameters of the USMC model may fluctuate. Bifurcation analysis allows the characterisation of complex bursting patterns in terms of how they arise dynamically, rather than by what individual ionic mechanisms contribute to their generation (Bertram et al., 1995). A qualitative understanding of the

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Fig. 4. Visualisation of bursting excitation in myometrial tissue by confocal microscopy of a stretched (1.5 times resting length) human myometial tissue sample obtained at a Caesarean section. (a) A single frame from a movie showing Fluo-4 signal from a thin (∼200 ␮m) slice; (b) time series of calcium bursts as the total sum of fluorescent signal within a small area of the frame. The Ca2+ bursts provide an index of the electrical activity.

bifurcational structure of the USMC model in parameter space provides a view of how the excitation dynamics can change during gestation, and a detailed numerical bifurcation analysis of specific USMC models can quantitatively account for specific experimental results e.g. the effects of oestradiol (Fig. 4). 3. Emergence of tissue electrophysiology Excitation in smooth muscle is myogenic, driven by endogeneous activity within the tissue, maybe from specialised pacemaker cells or tissue, or in response to stretch, and propagates through the muscle tissue. Propagation through in a tissue requires intercellular coupling: in the myometrium this is via gap junctions, composed of connexins that provide low resistance pathways between cells. During most of gestation the intercellular coupling is weak, and there is an increase in gap junctional expression that precedes normal birth (Fig. 5). 4. Connexins, coupling and conduction velocity The control of muscle activity can be neurogenic, as in skeletal muscles, where motor-neuronal activity initiates, via neuromuscular transmission, muscle excitation, or myogenic, where excitation can be driven by a pacemaker region, or in response to stretch, or myogenic activity modulated by neuronal efferent activity, as in the heart. Coordinated myogenic activity in a tissue requires intercellular coupling: in cardiac and smooth muscle tissue this is via gap junctions, composed of connexins, that provide low resistance pathways between cells. During most of gestation the intercellular coupling is weak, and there is an increase in gap junctional expression that precedes normal birth. In laboratory animals this increased connexion expression is regulated by the rising oestrogen and falling progesterone levels. If there was no gap junctional coupling between the USMCs, emphatic coupling, where a

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Fig. 5. Non-invasive abdominal surface recordings of uterine electrical activity. (a) Recordings of electrohysterogram during pregnancy at 28, 32, and 40 weeks of gestation, the ECG components have been filtered out. (b) Raw recordings from an abdominal electrode: as well as uterine activity the maternal and foetal (R wave marked by ellipses) electrocardiograms are apparent.

small fraction (membrane resistance ≫ extracellular resistance) of the extracellular current produced by activity in one cell could produce very weak coupling between neighbouring USMCs, allowing synchronisation of oscillatory activity but not propagation of waves. Gap junctional coupling provides low resistance pathways that can allow propagation of activity. Young (1997) has suggested two possible mechanisms for intercellular propagation, the global electrical propagation at ∼10 cm/s of action potentials via the diffusive spread of current, and the local spread of Ca2+ waves at ␮m/s. The propagation of action potentials can be modelled by a reaction diffusion system that provides a simple continuum model for a tissue composed of coupled excitable cells. ∂V = ∇ (D∇ V ) − Iion . ∂t

(2)

Here V (mV) is the membrane potential,  is a spatial gradient operator; t is time (ms). D is the diffusion coefficient tensor (mm2 ms−1 ) that characterises electrotonic spread of voltage. Iion is the total membrane ionic current density (␮A ␮F−1 ) described as in Tong et al. (2011). D is a measure of the electronic effects of the gap junctional density. For a one-dimensional excitable model, for a particular Iion , if D is less than some critical value Dc i.e. weak coupling, an action potential cannot propagate, for D > Dc the velocity v0 of a solitary wave increases as the square root of D. The velocity of a solitary wave solution increases linearly with the magnitude of the fast inward current of (1) and the maximum dV/dt of the upswing of the action potential. For periodic wave trains the conduction velocity decreases at higher rates, as one action potential propagates into the refractory wake of the preceding action potential: this is conduction velocity restitution. In two dimensions the propagation velocity can be anisotropic, faster in one direction than another, and in a homogeneously anisotropic two-dimensional medium the response to a localised suprathreshold excitation is an ellipsoidal, not a circular, travelling wave, with the long axis of the ellipsoid orientated along the predominant fibre direction. The velocity v of the wavefront also

depends on its curvature k = 1/r, where r is the local radius of curvature, by the eikonal relation

v(k) = v0 − kD,

(3)

where v0 is the velocity of a solitary plane wave. In three dimensions there are two principal radii of curvature. The spatio-temporal pattern of propagation of an electrical excitation wave into a resting excitable medium is determined by geometry, rather than details of excitability, and so can be modelled as in Fig. 7 using a simple caricature of excitation. If the wave is propagating into the aftereffects of previous activity i.e. into partially recovered, refractory tissue, the spatio-temporal details of the excitation dynamics will influence the pattern of propagation, and so a numerical solution of (2) with a biophysically detailed description of Iion is necessary. The possible spatio-temporal patterns of activity in the uterus depends on its excitability, anisotropic coupling, and geometry, all of which change during gestation. Conduction velocities can be estimated using invasive, intrauterine electrodes (Wolfs and van Leeuwen, 1979) and from non-invasive multi-channel recordings of uterine electrical activity from the abdominal surface. These propagation velocities are estimated from correlation lags or delay times between pairs of electrodes, and so are estimates of propagation velocity only if the wavefront is orthogonal to the lines between the electrodes (Lucovnik et al., 2011). At full term the velocity ranges from 2.5 to 5 cm/s. For a complex action potential lasting 50s with a conduction velocity of 5 cm/s, the “wavelength” (product of velocity and duration) would be 250 cm i.e. larger than the size of the uterus, so a single travelling wave propagates as its wavefront, and would produce a spatially uniform contraction. If endogeneous uterine excitation was initiated at a consistent pacemaker site (analogous to the sino-atrial node of the heart), or at a specific site (say the entry of the fallopian tube, at the upper “corner” of the uterus) due to boundary conditions then the direction of propagation would be consistent. Propagation from the upper towards the lower segment might be expected. Non-invasive multi-channel electrophysiological recordings do not show any such preferred direction of propagation. The conduction velocity

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Fig. 6. Full term uterus as reconstructed from in vivo MRI images of a pregnant abdomen: (a) transverse, (b) coronal, (c) sagittal images; (d and e) surface visualisation of reconstructed abdomen and geometry of pregnant uterus (f) Diffusion Tensor MRI (DTMRI) visualisation of the architecture of a segment of the wall of a human uterus removed after delivery by hysterectomy; illustration of the orientation streamlines of the primary eigenvector field in a tissue segment. These streamlines track the fibre and bundle orientation.

is determined by the diffusion coefficient in Eq. (2), which includes the effects of the gap junctional coupling between USMCs. If the intercellular coupling is very low, there is no propagation; as the gap junctional coupling increases propagation becomes possible, and the propagation velocity increases as the square root of the gap junctional coupling conductance. The very low gap junctional expression during pregnancy would prevent propagation; a small increase in gap junctional expression would allow propagation of slow velocity short wavelength waves, and perhaps even re-entrant waves. 5. Time series recordings Electrohysterography (EHG) is the non-invasive monitoring of uterine electrical activity (which are weaker than a few mV and distributed over 0.01–3 Hz) via electrodes placed on the abdomen. Since it is non-invasive, the EHG can be used to monitor uterine activity throughout pregnancy and into labour, as a research tool, and to provide a quantitative, objective measure of uterine activity for the management of labour (Buhimshi et al., 1997). Changes in the relative power in two frequency bands (0.2–0.45 Hz and 0.8–3 Hz) have been described with the progression of pregnancy. Bursts lasting several minutes can be correlated with the timing of contractions. The use of multiple electrodes for EHG recordings allows conduction velocities to be estimated. These recordings of electrical activity during pregnancy also provide maternal and foetal heart ECGs, and the bursts of activity correlate with recorded and perceived contractions. 6. Uterine tissue geometry and architecture Current clinical imaging of the foetus inside the uterus is via ultrasound, to assess the development of pregnancy. This occasionally leads to a clinical need for magnetic resonance imaging

that provides high resolution, three-dimensional geometry, from which the geometry of the uterus can be extracted. Clinical imaging of a full term pregnancy is illustrated in Fig. 6, together with an extracted uterine geometry. Clinical MRI of a full term in vivo uterus on a 3 T system does not provide sufficient resolution for reconstructing the architecture of the myometrium. This is necessary for understanding how complex action potentials emerge from the myometrial network, and lead to effective contractions. The detailed architecture has been obtained from ex vivo MRI of tissue blocks of a uterus that has been removed after delivery. Diffusion Tensor MRI (DTMRI) provides an index and measures of the myometrial fibre or fibre bundle organisation. The uterus visualised in Fig. 6a–c was removed by hysterectomy after delivery. Immediately after delivery by a Caesarean section the uterus was flaccid, and had the same size as before delivery; injection of syntocin caused the uterus to contract down to ∼9 cm. After a normal vaginal delivery oxytocin (released in response to the infant suckling), or syntocin injection by a midwife triggers such a contraction, which prevents extensive post-partum haemorrhage. Transmural blocks of uterine obtained from a post-partum hysterectomy, after syntocin administration, were scanned using diffusion tensor imaging on a 3 cm bore 9.4 T Bruker (Ettlingen, Germany) spectrometer, and the orientation of the primary eigenvectors estimated. The fibre orientation was tracked by following the primary eigenvectors from an initial seeding point. If the myometrial cells are organised into bundles the lines would track these “fibre bundles”, and if the fibre bundles were organised into separate circular and longitudinal sheets they would be apparent. In the postpartum human uterine wall Fig. 6f shows a feltwork of interweaving fibres, with some bundles crossing the wall, from the serosal border to the decidua. The myometrium appears organised into bundles with the fibre bundles interweaving and coalescing into higher order, thicker fasciculae, and they often branch away from the main fibre stream and into neighbouring bundles. The fibre tracking Fig. 6f is consistent with the fibres bundles seen

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Fig. 7. Surface views of computed excitation pattern, with white/colour excited, blue recovered tissue. The blue and white figures have the wave at different times superimposed. (a) Solitary wave initiated at the fundus, with excitation wave front shown at three successive times T. (b) Spiral pair initiated in the fundus appears as persistent periodic activity, as a site emitting annular waves: the diffusion coefficient is reduced to give a shorter wavelength.

histologically, and provides 3D, anisotropic tissue geometry for computational simulation of tissue electrophysiology. Propagation spread is along and through this feltwork of bundles, but the width of the fast upswing of the myometrial action potential is ∼1 cm and so a continuum PDE description of propagation is justified. 7. Integrated model During most of gestation and early labour the uterus is quiescent, and so any propagation is into resting tissue, and so is determined primarily by the geometry, curvature and eikonal relation (Eq. (3)). Simplified models that caricature excitation may be used to explore the possible emergent spatiotemporal patterns of excitation in the uterine geometry; it is only necessary to have appropriate space and time scales i.e. wavelength of excitation, uterine size, and propagation velocity. Fig. 7 illustrates idealised patterns of activity on the uterine surface computed for the isotropic geometry of the in vivo uterus of Fig. 6 using a FitzHugh–Nagumo two-variable caricature for excitation, with standard parameter values (Winfree, 1991; for details of the excitation model). Although the uterus is three-dimensional, it is thin walled, which motivates the assumption that propagation is dominated by the one and two-dimensional effects of the rate and curvature dependence of propagation velocity. The parameters are for an excitable medium, not an autorhythmic medium, and so activity has been initiated by stimulation. In Fig. 7(a) excitation is initiated in the fundus, producing an annular wave front propagating in the homogenous, isotropic medium towards the cervix: this represents the trigger for a single contraction. Even though the model is isotropic and homogenous, there are irregularities in propagation velocity due to the uterine geometry and curvature effects. These irregularities are enhanced by static changes in the diffusion coefficient, that simulate

spatial granularity in intercellular coupling, or by use of a diffusion tensor obtained from diffusion tensor imaging and that represents anisotropy in intercellular coupling and tissue conductivity producing anisotropy in propagation velocity. In Fig. 7(b) a spiral wave pair is initiated at the fundus, and appears as a periodic source. Such re-entrant waves can be produced by a localised excitation applied at an appropriate time in the wake of a travelling wave and so provide a possible mechanism for premature contractions in a physiologically normal pre-term uterus. Re-entrant waves in two-dimensional media are born in pairs of opposite chirality – and can be characterised by the location of their tips, identified as a phase singularity in physiological recordings, or by the intersection of isolines in computational simulations, and by their frequency. Spiral waves have been induced and recorded in vitro using multiple electrodes in tissue experiments on the internal surface of the guinea pig uterus (Lammers et al., 2008), and may be anticipated to occur in the human pregnant uterus. Their demonstration would require high resolution spatial mapping of uterine activity, and a multiple channel multi-channel SQUID array has been developed to record the spatiotemporal activity in the human uterus (Eswaran et al., 2002, 2004, 2009) and characterise synchronisation (Ramon et al., 2005). Multi-channel monitoring of spatiotemporal activity of the uterus is beginning to be used in research and to provide spatiotemporal data of uterine activation patterns before and during early labour. Reentrant waves are an expected behaviour of an excitable tissue and a computationally predicted behaviour for the uterus. They are a peculiar (pathological) behaviour of a normal (physiological) system, and so are an example of a dynamical disease (May, 1978), and are produced by unusual initial conditions in a normal system. During gestation the uterine myometrium is resting, but can be excitable; localised excitation in the wake of an excitation waves could initiate a pair of spiral waves. A possible mechanism

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would be an appropriately timed pair of kicks by the foetus, each initiating a localised excitation. This could provide a mechanism for the onset of premature contractions in physiologically normal myometrium. Topologically, the pregnant myometrium is an excitable, thin walled sphere, with an inexcitable hole (the cervix). A pair of spiral waves in such a system would be expected to meander (the tips move in a cycloid fashion) and drift (a much slower directed motion of the tips); both these are at velocities less than the propagation velocity. These velocities are all increased by any spatial gradients in parameters. Given the geometry, one tip would reach the inexcitable boundary with the cervix, where it would continue rotating around the cervix, leaving one free to in the uterus. This free tip could persist, giving maintained repetitive excitation, or could be extinguished by propagation failure (an accumulation of refractoriness, or a dissipation of excitation), or also drift to the cervix, and extinguish the re-entry. Thus re-entrant excitation could be persistent and pharmacologically intractable (as the cell and tissue parameters are all normal), or self-terminating. 8. Discussion and conclusions The mechanisms of human birth are difficult to study, as patient care has a higher priority than research. Computational modelling provides a route for quantitatively linking non-invasive clinical measurements with the results of in vitro cell and tissue studies on biopsies obtained during Caesarean sections. These lead to a view of the control of uterine activity by nonlinear wave processes in excitable media, and to changes in uterine electrophysiology during pregnancy as a trajectory in excitation and coupling parameter space. During most of pregnancy the uterus is quiescent, has a low excitability, and weak coupling prevents propagation and global contractions. As intercellular coupling increases during gestation propagation can occur, with a slow conduction velocity that gives a short wave length that could allow re-entrant propagation. Reentrant waves (an abnormal pattern of activity in a physiologically normal system) could be one mechanism of premature labour, but would be expected to self-terminate unless pinned to some structural heterogeneity. Further increases in coupling gives longer wavelength that could account for the synchronous, global contractions of normal labour. This quantitative approach is focused on the electrophysiology of the uterine smooth muscle, and views uterine contraction as the primary control target. This neglects the necessity for mechanical changes in cervix (dilation and “ripening”), and the extensive maternal, foetal and placental endocrine interactions that influence uterine myometrial excitability and intercellular coupling, and cervical mechanics. Acknowledgements This has developed from an eScience Institute, Edinburgh minitheme “Eulerian and Lagrangian approaches towards quantitative prediction of premature labour, normal and dysfunctional full term labour”, held January–July 2011. References Aslanidi, O., Atia, J., Benson, A.P., van den Berg, H.A., Blanks, A.M., Choi, C., Gilbert, S.H., Goryanain, I., Hayes-Gill, B.R., Holden, A.V., Li, P., Norman, J.E., Shymygol, A., Simp-son, N.A.B., Taggart, M.J., Tong, W.C., Zhang, H., 2011. Towards a computational reconstruction of the electrodymamics of premature and full term labour. Prog. Biophys. Mol. Biol. 107, 182–192. Bamberg, C., Rademacher, G., Güttler, F., Teichgräber, U., Cremer, M., Bührer, Spies, C., Hinkson, L., Henrich, W., Kalache, K., Dudenhausen, J., 2012. Human birth observed in real-time open magnetic resonance imaging. Am. J. Obstet. Gynecol. 206, 505.e1–505.e6. Bertram, R., Butte, M.J., Kiemel, T., Sherman, A., 1995. Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57, 413–439.

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