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Themed Issue Papers |
1 Bioengineering Institute2 Department of Engineering Science3 Department of Physiology, The University of Auckland, New Zealand
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Abstract |
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 Top Abstract IntroductionReferences |
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(Received 10 October 2005;
accepted after revision 16 January 2006; first published online 23 January 2006)
Corresponding author M. Trew: Bioengineering Institute, Private Bag 92019, Auckland, New Zealand. Email: m.trew{at}auckland.ac.nz
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Introduction |
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TopAbstractIntroductionReferences |
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In contrast, our understanding of the factors that determine electrical activity in the intact heart is more qualitative. This is true, in particular, for the mechanisms that give rise to re-entrant arrhythmia and fibrillation. Normal and re-entrant activation are three-dimensional events that involve relatively large tissue volumes and are influenced by regional variation of the electrical properties of cardiac tissue and by the complex anatomy of the heart. Although electric potentials can be recorded with moderately high spatial and temporal resolution on the heart surface (Gray et al. 1997; Knisley, 1998) and on the transmural surfaces of wedge preparations (Sharifov & Fast, 2003), and with high spatial resolution in cultured myocyte preparations (Cheek et al. 2005), it is often difficult to relate these data to intramural electrical activity. It is possible to make minimally invasive intramural measurements of extracellular potential (Frazier et al. 1988; Rogers et al. 2002; Caldwell et al. 2005) and membrane potential (Hooks et al. 2001; Caldwell et al. 2005). It is also possible to infer underlying electrical activity from non-invasive measurements of magnetic density, from which potentials can be computed (Holzer et al. 2004). These techniques, however, lack the spatial resolution to reconstruct fully all aspects of the three-dimensional spread of electrical activation. Within this context, mathematical modelling provides a powerful tool with which to ‘bridge the scale gap’; a framework which can be used to interpret and interpolate experimental observations and to suggest questions for new experimental studies.
The role of computer modelling as an integrative tool in cardiac electrophysiology has been widely accepted. It is acknowledged that mathematical models which incorporate representations of observed structure have the potential to offer new insights into electrical effects and will become increasingly important for understanding critical aspects of cardiac electrophysiology, such as the generation and maintenance of re-entrant arrhythmias and ultimately their prevention (Spach, 2001).
In several recent studies, electrophysiological experiments have been interpreted using computer models that incorporate specific structural measurements. Rodríguez et al. (2005) linked optical potentials on the epicardial surface of rabbit hearts to the potentials predicted with a bidomain simulation model of the rabbit ventricles employing comparable stimulus protocols. They were able to demonstrate complex intramural transmembrane potential distributions following shock termination that would not be predicted on the basis of the epicardial maps. Simple models have been successfully used to elucidate relatively complex activation processes (e.g. White et al. 1998; Roth, 2000; Sambelashvili et al. 2004; Poelzing et al. 2005). In contrast, both Muzikant et al. (2002) and Vetter et al. (2005) have used geometrically simple but structurally realistic computer models to understand and match recorded potentials. A key feature of these models is that they are all based on specific measurements of myofibre orientation.
These examples illustrate that bidomain models of cardiac electrical activity, in which intracellular and extracellular potentials and transmembrane current flows are explicitly represented, can be used to qualitatively match and interpret experimental observations. However, these studies at best incorporate tissue-specific geometry and structure in a limited fashion. There remains a need to ‘develop more comprehensive models ... incorporating features of actual cardiac architecture at the microscopic level’ (Spach & Barr, 2000).
In this paper we describe tools and techniques that have been developed for linking experimental measurements of electrical activation to detailed tissue-specific computer models. On the basis of this approach, we argue that the standard view of ventricular myocardium as a uniformly coupled electrical continuum, transversely isotropic with respect to fibre direction, is likely to be incomplete. We have demonstrated that clefts between adjacent muscle layers in the heart: (i) may give rise to non-uniform electrical propagation from an intramural ectopic activation that is slow in the direction perpendicular to the clefts; and (ii) could play a significant role in the termination of fibrillation by an externally applied shock.
Cardiac ventricular structure and tissue-specific structural models
The mechanical function of the heart is heavily dependent on its muscular architecture, and this structure also contributes to the anisotropic electrical properties of cardiac tissue. Streeter & Bassett (1966) described the ventricular myocardium as a structure in which the myofibre orientation varied smoothly across the ventricular wall. They measured the transmural variation of myocyte orientation at a number of ventricular sites in different species and showed that the fibre angle can vary transmurally up to 180 deg. Figure 1 highlights this variation schematically. Detailed measurements of transmural myofibre orientation from both dogs (Nielsen et al. 1991; LeGrice et al. 1995a, 1997) and pigs (Stevens et al. 2003; Vetter et al. 2005) have shown that there can be significant local variation of fibre orientation. This is particularly marked at the junctions of the free walls of the right ventricle (RV) and the left ventricle (LV), in the interventricular septum and in the subepicardial region of the RV.
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There is now a significant body of work in the cardiac mechanics literature which addresses the role of laminar structure in ventricular myocardium. Magnetic resonance imaging of intact, unfixed hearts has demonstrated preferred diffusion directions within the ventricular wall, consistent with the muscle layer arrangement outlined above (Dou et al. 2003; Harrington et al. 2005; Helm et al. 2005). Moreover, experimental studies employing ciné X-ray with embedded radio-opaque markers indicate that intramural LV shear deformations coincide with three-dimensional layer orientations (Costa et al. 1999; LeGrice et al. 1995b). These results have been interpreted as showing that the laminar arrangement of ventricular myoardium contributes to the marked changes in LV wall thickness that occur throughout the cardiac cycle by facilitating slippage and rearrangement of adjacent groups of myocytes. Studies of the material properties of isolated non-contracting ventricular myocardium (Dokos et al. 2002) and kinematic analyses of LV deformation (Arts et al. 2001) support this view.
In contrast, the extent to which the laminar architecture of ventricular myocardium affects the spread of electrical activation in the heart is less clear. Intercellular electrical coupling is determined by the numbers of gap junctions spanning the membranes of adjacent myocytes. In adult ventricular myocardium, the principal gap junction protein is connexin43 (Cx43) and it is concentrated at the intercalated discs (Smith et al. 1991) with little other lateral coupling between contiguous myocytes. On this structural basis, it may be argued that electrical activation spreads between adjacent muscle layers via muscle branches and that interlaminar clefts present a barrier to electrical propagation (LeGrice et al. 1995a; Kléber et al. 2000; Hooks et al. 2002). The corollary to this is that the spread of electrical activation from a point stimulus will be most rapid in the direction of the myocyte axis and will spread more slowly within layers, transverse to the myocyte axis, via the coupling between adjacent cells. Propagation will be least rapid between neighbouring layers normal to the cleavage planes that separate them. This view is at variance with the widely held assumption that ventricular myocardium is a uniformly coupled syncytium in which electrical properties are transversely isotropic with respect to the fibre direction.
Computer models that incorporate detailed structural information provide a means of investigating the possible effects of myocardial architecture on electrical activation (Street & Plonsey, 1999; Hooks et al. 2002, 2006; Sands et al. 2004; Trew et al. 2005a,b; Trew & Sands, 2005). However, acquisition and reconstruction of the necessary structural data can be a labourious process. For example, assembly of the high-resolution, extended volume images of the rat LV wall presented by Young et al. (1998) required weeks of imaging and image registration. Similarly, an anatomically detailed three-dimensional reconstruction of the rabbit sino-atrial node (Dobrzynski et al. 2005) took 18 months of painstaking work.
An automated system has recently been developed to enable extended volume images to be acquired more efficiently (Sands et al. 2005). The system consists of a confocal microscope (or digital camera), a high-precision three-axis translation stage and an ultramill, all of which are controlled by a computer using custom-written software. The design makes it possible to obtain serial images over large areas of the upper surface of embedded tissue specimens. We use the confocal microscope for high-resolution volume images (0.5–4.0 µm). Typically, a series of extended optical sections is acquired to a depth of 60–80 µm, the ultramill is then used to plane around 50 µm from the upper surface of the specimen, and the sequence is repeated until the volume of interest has been completely imaged. Lower resolution images (4–50 µm) are acquired using the digital camera after etching and topically staining the upper surface of the tissue specimen. An important feature of the approach is that spatial registration between individual images is intrinsically preserved, and it is therefore possible to reconstruct three-dimensional morphology with high accuracy throughout a tissue sample. Digital reslicing, segmentation and volume rendering methods can be applied to the reconstructed volumes to provide quantitative information about the three-dimensional organization of myocytes, extracellular collagen matrix and the blood vessel network in the heart that has previously been difficult to attain. In Fig. 2, data obtained from a three-dimensional reconstruction of a transmural segment of LV myocardium from the free wall of the rat heart are shown. Systematic reconstructions are also being carried out for much larger transmural specimens from the pig LV free wall (see Fig. 4).
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We are currently using two methods for segmenting the tissue structures: one manual (Hooks et al. 2002; Trew et al. 2005a) and one automatic (Sands et al. 2004). The manual segmentation represents the most significant cleavage planes as two-dimensional surfaces in the three-dimensional space. The automatic segmentation is performed at a predetermined resolution, and non-myocardial features such as cleavage planes or blood vessels are identified and form a template to be used for modelling purposes. The size of the porcine sample (see Fig. 4) precludes high-resolution scanning, and we are developing techniques to segment structure reliably and efficiently from these lower resolution scans. Figure 3 shows two segmented representations of the interlaminar clefts (or cleavage planes) and a representation of the transmural fibre orientation.
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In order to characterize the effects of muscular architecture on electrical function, it is necessary to relate intramural electrical activity to the local three-dimensional structure. Intramural extracellular potentials have been measured in the LV free wall using arrays of plunge needles containing multiple electrodes (Frazier et al. 1988), and this study confirmed the importance of myofibre orientation in determining the spread of electrical activation. With the development of improved probes (Rogers et al. 2002), it is now possible to record intramural extracellular recordings at a higher level of spatial detail than has previously been possible (Huang et al. 2005). However, spatial resolution remains relatively coarse and the data obtained are generally not directly related to local tissue structures. Moreover, it has not previously been possible to measure transmembrane potentials across the wall of the intact heart. An alternative approach has been to record electrical activity on the transmural surface of an LV wedge preparation (Poelzing et al. 2005). Using this preparation, intramural transmembrane potentials can be mapped optically at high resolution. However, interpretation of these data is complicated by the fact that they are two-dimensional rather than three-dimensional, and that electrical loading at the cut transmural surface of the wedge preparation differs from the intact heart. Recently we have developed techniques that enable both extracellular and optical action potentials to be measured at multiple intramural sites in the LV, and we are extending these capabilities through the use of high-density transmural extracellular recording arrays.
Experimental methods and protocols. Intramural optical action potentials are recorded using a novel optical probe or optrode (Hooks et al. 2001) that consists of seven hexagonally packed optical fibres inserted into a tapered glass micropipette (400 µm o.d.; Fig. 4A). Fibres terminate with 1.4 mm spacing and address a tissue volume radial to the optrode, each staggered by 60 deg. They are coupled to an optical mapping system, described in detail by Hooks et al. (2001). Excitation light (488 nm) is delivered from a water-cooled argon ion laser to the optrode and excites the membrane potential-sensitive dye di-4-ANEPPS (Molecular Probes, Eugene, OR, USA) adjacent to the fibre ends. Fluorescent light returns via the same path and is split into long- and short-wavelength bands that are routed to separate photodetectors. Either dual wavelength ratiometry or a modified subtraction technique (Tai et al. 2004) is used to minimize light source noise and remove artefact due to motion or fluorescence bleaching. This is possible because di-4-ANEPPS is a ratiometric dye. Modulation from a change in the membrane potential is opposite in sense in the short- and long-wavelength bands. However, photobleaching and motion artefact produce comparable changes in both bands.
This technique has been validated by comparing adjacent intramural optical and extracellular potentials recorded in an isolated Langendorff-perfused heart preparation (Caldwell et al. 2005). Intramural extracellular potentials were recorded using epoxy-coated plunge needles, each containing up to 13 unipolar silver wire (70 µm) electrodes at 1 mm spacing (Rogers et al. 2002). To provide an in vivo control, intramural extracellular potentials were first recorded in anaesthetized pigs using identical experimental protocols. Young pigs (20–35 kg) were anaesthetized initially with tiletamine–zolazepam (Zoletil; Virbac Animal Laboratories (NZ), Auckland, New Zealand; 10 mg kg–1, I.M.) and maintained with halothane (2–5%) in oxygen. The heart was exposed via a thoracotomy, and three needle probes were introduced into the anterior free wall of the LV. An intramural bipolar pacing probe was placed adjacent to the extracellular recording probes. Extracellular potentials were monitored at all 36 intramural sites until ST segment elevation returned to baseline, and then in sinus rhythm and during ventricular pacing (1–3 Hz) using a constant current stimulator (duration 2 ms; current 1.5 x capture threshold). The heart was then excised with needle probes in place and mounted in a modified Langendorff perfusion apparatus. The heart was perfused with oxygenated Tyrode solution (37°C, 95% O2 and 5% CO2). The electromechanical uncoupler 2,3-butanedione monoxime (BDM; 7.5–12.5 mmol l–1) was added to the perfusate to suppress motion artefact, and di-4-ANEPPS (15 ml, 75 µmol l–1) was infused into the left anterior descending coronary artery. An optrode was positioned at the centre of the dyed region adjacent to the pacing probe, and the pacing protocols performed in vivo were repeated in the isolated heart. Extracellular potentials and optical signals were acquired at 1 kHz and stored. Data were averaged over 8–12 successive heart beats. In three hearts intracellular action potentials were recorded at epicardial sites adjacent to the optrode using glass micropipettes. More details can be found in Caldwell et al. (2005).
Experimental results and discussion.
Typical intramural optical action potentials in sinus rhythm are shown in Fig. 4B, overlaid on adjacent extracellular potentials at equivalent depths through the LV wall. Intramural optical action potentials were very reproducible, with little variation in activation times for repeated recordings at the same site. In sinus rhythm and during atrial pacing, activation across the wall was rapid (mean time to activate all 6 intramural sites 
8 ± 4 ms), but the transmural spread of activation was substantially slower with subepicardial pacing. Two further points are noteworthy. Firstly, optical action potential morphology was similar in all hearts studied, with no systematic transmural differences observed. Secondly, the spread of activation along the optrode in sinus rhythm and atrial pacing was less uniform than expected.
In vivo extracellular potentials recorded during sinus rhythm and atrial pacing displayed a smooth negative deflection of short duration, and there was a rapid spread of activation from subendocardium to subepicardium. During intramural pacing, extracellular potentials recorded from plunge needles close to the stimulus probe were substantially broadened and, unlike sinus rhythm and atrial pacing, multiple deflections were consistently observed in the downstroke of the action complex. Such polyphasic, or fractionated, electrograms are commonly associated with myocardial infarctions and ischaemia (Ino et al. 1995; Khoury et al. 1998) and are attributed to slow, non-uniform or discontinuous activation. That electrograms with fractionated downstrokes can also be recorded in externally stimulated normal myocardium is an interesting observation and is consistent with results presented elsewhere (Khoury et al. 1998; Punske et al. 2003). Typical intramural extracellular potential traces from the in vivo porcine data sets of Caldwell et al. (2005) (not shown in that reference) are presented in Fig. 4C. These traces show the transmural spread of excitation from a bipolar stimulus located between the epicardium and the mid-wall. Distinct activation events in the same trace (points of maximum negative slope) separated by up to 5–6 ms can be clearly seen. Similar features can be observed in other published results (Punske et al. 2003). The extracellular potentials recorded in an isolated heart preparation were generally very similar to those seen for comparable experimental protocols in vivo. However, propagation was slower and there was greater variability of activation times, and consequently the extent of fractionation was significantly greater.
A previous study by Colli Franzone et al. (2000) has convincingly related certain aspects of electrogram morphology to tissue structure. There the authors showed that polyphasic deflections during the post-stimulus and preactivation phase of experimentally recorded electrograms in canine preparations could be reliably reproduced by computer models that included generic continuous tissue structural features such as anisotropic conductivities and transmural fibre rotations. However, unlike the results presented in Fig. 4C, and in Punske et al. (2003), there was no discernable evidence of fractionation in the activation downstrokes. This suggests that continuous structural features, while contributing to the electrogram morphology of the preactivation phase, are not the principal source of fractionation in the activation downstroke phase.
For subepicardial pacing in the isolated heart preparations, monophasic optical action potentials were recorded at adjacent spatial locations to electrograms with fractionated downstrokes. These parallel recordings suggest that subepicardial pacing generates non-uniform electrical activation in a volume greater than that addressed by the optrode: around 1003– 2003µm3 according to studies recently carried out in our laboratory. Extracellular potentials reflect the integration of current flows over a much larger volume, and include both near- and far-field effects (Hooks et al. 2001). Based on these experimental observations and our computer modelling (see the following section) we believe that fractionated downstrokes in electrograms result from functional discontinuities due to mesoscale features such as cleavage planes. Within this context, the optical action potential upstroke is due to activation of the collection volume of the optrode. Spach & Barr (2000) have discussed variations in the morphology of recorded action potentials and they related this to discontinuous activation, although at the microscopic levels of capillaries, the dimensions of interstitial space and the role of cellular scaling.
Pertsov (1997) elegantly argued that only structures with dimensions greater then the width of the depolarization wavefront could contribute directly to discontinuous propagation. Microscale features act only to modulate the propagation. In normal myocardial tissue the width of the depolarization wavefront is typically 1 mm, suggesting that structures of dimension > 1 mm are contributing most significantly to discontinuous activation. Electrograms with fractionated downstrokes have also been reproduced in two-dimensional models of coupled ‘cell’ segments, when electrical coupling was non-uniform at dimensions between 0.1 to 1 mm (Ellis et al. 1995). The experimental results that we discuss here and the modelling results of the following section add further to the scale arguments of Pertsov (1997) and the results of Ellis et al. (1995). Taken together, they suggest that microstructural features such as capillaries or local membrane properties are likely to be weaker contributors to the discontinuous activation, which is ultimately observed in fractionated downstrokes in electrograms.
The study of Caldwell et al. (2005) demonstrates that the optrode provides reliable intramural optical action potential recordings in the isolated Langendorff-perfused pig heart preparation. However, such preparations have limitations in cardiac electrophysiology research. Heart isolation and electromechanical uncoupling with BDM slow electrical propagation and depress restitution relationships, while oedema constrains the time for which the preparation can be used (Qin et al. 2003; Caldwell et al. 2005). We are currently using arrays of needle probes to map two-dimensional and three-dimensional intramural extracellular potentials in an in vivo pig preparation. These data are being related to detailed reconstructions of tissue structure in the region bounded by the probes (see Fig. 5). Preliminary results from these studies indicate that the spread of electrical activation from an intramural point stimulus is not transversly isotropic with respect to the local myofibre orientation, but spreads relatively slowly and non-uniformly in directions normal to that orientation.
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Tissue-specific mathematical models of ventricular electrical activation
It is relatively straightforward to specify the features of the computer models necessary to understand tissue-level processes at submeasurement but supracellular scales (or mesoscales). The models must include descriptions of the architecture of groups (or sheets) of myocytes and the cardiac boundary geometry at a scale appropriate to the problems addressed, an adequate representation of the main time-dependent processes that determine cellular electrical activity, and possibly a realistic description of the spatial variation of cellular electrical properties. However, such models also need to be computationally efficient to enable the simulation of electrical activity in adequate tissue volumes within a reasonable time frame. Finally, model validation is a critical element in this process. As discussed in the previous section, new and innovative techniques must be developed for mapping both transmembrane and extracellular potentials intramurally to provide comprehensive data for model validation. Although it has not previously been possible to realize all of the aforementioned model requirements simultaneously, they are now beginning to come within our grasp. In this section we describe the advanced computer modelling techniques that have been developed by our group, and some of the key results obtained to date.
The bidomain model.
The electrical behaviour of cardiac tissue can be represented by the bidomain model (Henriquez, 1993), which is solved in a highly complex geometry, as described by the tissue-specific structural models shown in Fig. 3. In the bidomain model, intracellular (myocardial) and extracellular domains communicate through the membrane ionic current, Iion. The solution parameters are the space and time varying transmembrane potential, Vm, and the extracellular potential, 
e, both experimentally measurable quantities. The bidomain conservation of current expressions are:
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i and e are the intra- and extracellular conductivity tensors and ie is a current injection per unit volume into the extracellular space. In the work described in this paper, the ionic current is determined using a cubic activation model (Hunter et al. 1975) or a defibrillation modified Beeler–Reuter model (Skouibine et al. 2000). Assuming isolated tissue, these equations are subject to no-flux current boundary conditions on far-field external boundaries and at a set of internal boundaries in the intracellular domain, i.e. cleavage planes. Further details can be found in Trew et al. (2005a). The bidomain equations have been numerically solved by researchers using classical techniques such as finite difference, finite element, finite volume and collocation methods (Buist et al. 2003; Trew et al. 2005a,c). For the computer modelling of a discontinuous myocardial structure with no-flux boundaries, methods based on the weak or integral forms of the bidomain equations are desirable. Initial modelling of this nature used a finite element method (Hooks et al. 2002); however, recent modelling has used a new, efficient finite volume method (Trew et al. 2005a) that takes advantage of the regular geometry of the tissue samples being considered. Simulating electrical activation on tissue-specific geometries is a computationally intensive process, since it requires spatial resolutions of 10–20 µm. These resolutions are an order of magnitude smaller than the 100 µm resolution (based on adequately resolving a tissue space-constant of 700–1200 µm) that is usually required for simulations in continuous tissue (e.g. Poelzing et al. 2005). The studies described in this paper have simultaneously solved between 8 and 16 million equations, describing intra- and extracellular current conservation, at a set of instances in time. In general, solutions of up to 20 ms simulated time have been computed in 12 h or less of elapsed real time (using 8–12 power 5, 1.9 GHz processors on a 210 Gb shared memory IBM Regatta). The scale of these problems places our modelling efforts among the largest reported in the field of cardiac electrical activation simulations. To maintain the viability of solving such large problems, we continue to develop our computer simulation tools (e.g. Austin et al. 2005a,b).
Model results and discussion. We present a selection of results that highlight hypotheses tested by our computer models, in particular that: (i) electrical discontinuities arising from the cleavage planes have both local and global effects; and (ii) certain features of electrical activation are difficult or impossible to understand or elucidate without the direct modelling of explicit and discontinuous tissue features such as cleavage planes. Computer models allow these hypotheses to be studied more directly than is currently possible given the scale limitations of the intramural experimental context.
Figure 6 shows simulation results from midwall stimuli applied to the detailed tissue-specific geometric models shown in Fig. 3A and B. These results illustrate that the initial propagation from a focal activation (bipolar in Fig. 6A–C or ectopic in Fig. 6D) is significantly impacted by the local laminar tissue structure. Similar results have been obtained for unipolar midwall stimulation (Trew et al. 2005b). As the activation wavefront becomes more developed, the impact of the laminar structure is reduced, although the anisotropic conductivity remains important. A comparison of simulation results with and without tissue-specific structural features (Fig. 6A and B) shows similar fully developed activation patterns. The model without the detailed geometry used a reduced sheet-normal conductivity suggested by Hooks et al. (2002). An interesting observation from Fig. 6C is the classic ‘dog-bone’ pattern of virtual sources along the midwall fibre orientation arising from the extracellular stimulation. The formation of virtual sources following unipolar extracellular current injection or extraction was predicted using the bidomain model before their existence could be verified experimentally by fluorescence imaging on the epicardium (Roth et al. 1998). The influence of the cleavage planes on the midwall formation of the virtual sources is such that, although details of early and late activation vary, the principal features remain consistent.
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The results of Figs 6 and 7 were derived from a near-threshold extracellular current injection or a direct membrane depolarization. When the tissue is stimulated by a strong electric field (e.g. a defibrillation-strength shock), myocardial electrical discontinuities produced by the laminar structure play a significant role in tissue activation. This is illustrated in Fig. 8, where a 10 V cm–1 transmural electric field has been placed across a detailed tissue model, with the cathode on the epicardium (left) and the anode on the endocardium (right). The figure shows the three-dimensional development of virtual sources on the anodal (endocardial) side of certain intracellular discontinuities. These results contribute to observations from previous modelling (Hooks et al. 2002) and experimental studies (White et al. 1998; Sharifov & Fast, 2003). The significance of such results is that they suggest a mechanism by which a strong electric shock can rapidly activate cardiac tissue and thereby extinguish re-entrant behaviour. In recent work, Hooks et al. (2006) have used computer modelling to support a link between tissue structure and the optimal shock strengths for rapid transmural activation. Thus, modelling supports the hypothesis that the application of a sufficiently strong extracellular potential field can result in rapid global activation and that to observe these effects requires the direct explicit modelling of electrically discontinuous tissue features.
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We have already shown that our computer models directly include the details of specific geometric tissue structures. The modelling results indicate that geometric structure is an important contributor to many aspects of tissue electrophysiology, and we are able to represent these details in our models. As discussed previously, other studies have acknowledged the importance of anatomical fidelity in computer modelling and have demonstrated the importance of its accurate representation (Muzikant et al. 2002; Vetter et al. 2005; Rodriguez et al. 2005).
The bidomain equations represent the physics of current conservation between an extra- and an intracellular domain. The impact of the active membrane ionic current is also included, for which there are a range of possible models available (Beeler & Reuter, 1977; DiFrancesco & Noble, 1985; Luo & Rudy, 1994; Courtemanche et al. 1998; among others). The bidomain equations are based on rigorous mathematical derivations, with a set of free parameters whose values are estimated from experimental measurements (Henriquez, 1993; Neu & Krassowska, 1993). These free parameters include homogenized or ‘multiscaled’ values for the surface-to-volume ratio of cell membranes, the specific membrane capacitance and the myocardial and interstitial electrical conductivities. The specification of these free parameters, particularly the conductivities which are structurally dependent, can strongly influence the quantitative validation of the bidomain equations. In our modelling work we deliberately express tissue geometry to a high fidelity but do not attempt to include overly complex features of the active membrane in the ionic current models, since the hypotheses that we are testing are related primarily to geometry. The model validation process includes appreciating which aspects of electrical activation are being modelled and which are not.
There have been a number of studies which have validated the ability of a bidomain model to qualitatively predict the general physics of cardiac electrical activity. A classic example of this is the observed polarization pattern in the vicinity of a unipolar electrode, which shows a ‘dog-bone’ shape of virtual cathodes and anodes (Sepulveda et al. 1989), similar to those presented in Fig. 6. This observation, based on computer bidomain modelling, was later confirmed experimentally on the epicardium by Wikswo et al. (1991). Computer modelling using the bidomain model has also been able to qualitatively reproduce features of anodal and cathodal strength–interval curves observed experimentally (Roth et al. 1998).
The formation of virtual sources on the anodal side of myocardial discontinuities (cf. Fig. 8) was studied by White et al. (1998). They performed a series of experiments on canine hearts in which they introduced a transmural incision, closed and held by sutures to form a continuous extracellular space, and used a plaque of electrodes to map the epicardial response to current pulses. Their results clearly showed evidence that a myocardial discontinuity remote from a pacing electrode can give rise to an activation wavefront. To test the possible impact of injury potentials near the incision, they employed a one-dimensional bidomain model to examine the effects of elevated extracellular potassium, a factor associated with acute injury. Their computer simulation results clearly mimicked features of what they observed in vivo.
The recent work of Vetter et al. (2005) used detailed tissue-specific structural data of fibre orientation, particularly in the subepicardial region, to predict the features of experimentally recorded activation wavefronts on the epicardium using a computer model. The computer model was a simplified version of the bidomain equations. The work showed how the varying subepicardial fibre patterns in three different preparations strongly influenced the geometric shape of the activation wavefronts. These qualitative comparisons highlighted the importance of including accurate structural details in a computer model.
Muzikant et al. (2002) explored the predictive capability of the bidomain equations in three dimensions. They compared model and experimental epicardial extracellular potential recordings for both surface and intramural pacing. The experimental recordings were made using epicardial electrode plaques on dog hearts. Qualitative agreement between measurements and model results was obtained, and they discussed a number of possible reasons for quantitative discrepancies. The authors suggested that the details of the laminar structure needed to be more accurately represented and that other heterogeneities in the tissue material properties may also need to be included in their model.
A recent study by Hooks et al. (2006) showed that geometrically detailed bidomain computer models could qualitatively capture experimentally predicted patterns of latency to first activation and total transmural activation times for electric shocks of varying strengths (Sharifov & Fast, 2003). Hooks et al. (2006) suggested that the quantitative differences between measurement and model could in part be attributed to the resolution of the experimental optical mapping. Their basis for this argument was that the model under-predicted the activation latencies and over-predicted the transmural activation times, compared to the experimentally measured values. As tissue approaches total activation, the dominant tissue state is depolarization, and small regions of hyperpolarization are obscured by spatial averaging over the photodiode recording volume, thus leading to faster observed transmural activation. Similarly, when measuring activation latency, instances of small regions of depolarization are obscured by larger volumes of resting or hyperpolarized tissue. Spatial averaging then predicts an observed activation latency that is greater than the high-resolution model prediction.
A number of studies have validated the computer software implementations of our bidomain models (Buist et al. 2003; Trew et al. 2005a,c). These studies have compared various numerical methods for solving the bidomain equations, matched analytically derived conduction velocities, and showed that computer solutions converged to a fixed function as the spatial resolution in a discontinuous model was increased.
Consequently, we contend that the weight of proof is that our computer modelling-based predictions are valid indicators of the true tissue behaviour. A full quantification of uncertainty rests on the understanding of the features of tissue electrophysiology that are being represented by the models. It is likely that, as our knowledge and understanding of the impact of tissue structure on activation increases and our capacity increases to solve computer models that both directly and indirectly account for that structure, measurement and modelling results will begin to converge even further.
Future modelling directions. In this section we have used results from our computer simulations to argue that the direct modelling of tissue-specific geometry is important, particularly when interpreting and understanding extracellular potential traces and the response of cardiac tissue to strong electric fields. However, such modelling is only tractable in relatively small tissue volumes. Consequently we are also developing rigorous mathematical methods for expressing mesoscale discontinuities in such a way as to implicitly include their influence in models at the tissue and organ macroscale. The motivation for this multiscale approach is to reproduce and understand the features of three-dimensional transmural activation derived from the combined electrode array measurements and tissue structural information, shown in Fig. 5.
Concluding remarks
In this article we have presented evidence to support the argument that controlled experimental measurements and computer models should be used as complementary parts of an iterative process towards developing a fuller understanding of cardiac tissue electrophysiology. The sequence of this process is scale dependent. Experimental measurement of electrophysiology at both the cellular microscale and the tissue macroscale is accessible with current technology. Computer modelling of electrical activation enables hypotheses to be tested and insights to be gained at the ‘in-between’ mesoscale, so bridging the scale gap. The key factor in this modelling is the detailed and accurate representation of the geometric tissue structures.
The role of interlaminar clefts in the development and maintenance of re-entrant electrical activity has not yet been resolved. The results outlined here indicate that the spread of electrical activation from an ectopic stimulus is slow in the direction perpendicular to the cleavage planes. This could contribute to the formation of macroscopic re-entrant electrical circuits, particularly in the ischaemic heart. However, it is more difficult to establish whether discontinuities associated with muscle layers provide a potential substrate for micro re-entry. As already discussed, and shown by the preliminary data in Fig. 5, we are gathering detailed structural information in larger tissue samples. We are also currently developing heart-bypass experimental protocols which will allow the plunge electrode array to be used for gathering three-dimensional transmural recordings of re-entrant behaviour and fibrillation in the in vivo pig heart.
In conclusion, we are using a joint measurement and modelling paradigm to investigate the effects of discontinuities associated with muscle layers on the spread of electrical activation in ventricular myocardium. Detailed, tissue-specific structural models have been constructed, and novel experimental techniques for characterizing intramural electrical activity in the intact heart have been developed. On the basis of initial results obtained using these tools, we have argued that the standard view of the ventricular myocardium as a uniformly coupled electrical continuum, with transversely isotropic conductance normal to the fibre direction, is likely to be incorrect (Hooks et al. 2002; Trew et al. 2005a). We have also shown that interlaminar clefts could play a significant role in the termination of fibrillation by an applied shock (Hooks et al. 2002, 2006; Trew & Sands, 2005).
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