Rabbit-specific ventricular model of cardiac electrophysiological function including specialized conduction system

Progress in Biophysics and Molecular Biology 107 (2011) 90e100

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Review

Rabbit-specific ventricular model of cardiac electrophysiological function including specialized conduction system R. Bordas a, K. Gillow c, Q. Lou e, I.R. Efimov e, D. Gavaghan a, P. Kohl a, d, V. Grau b, B. Rodriguez a, * a

Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK Oxford e-Research Centre, 7 Keble Road, Oxford OX2 3QF, UK c Oxford University Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK d National Heart & Lung Institute (Imperial College London), Heart Science Centre, Harefield Hospital, Harefield, Middlesex UB9 6JH, UK e Washington University in St Louis, Department of Biomedical Engineering, St Louis, MO 62130-4899, USA b

a r t i c l e i n f o

a b s t r a c t

Article history: Available online 13 June 2011

The function of the ventricular specialized conduction system in the heart is to ensure the coordinated electrical activation of the ventricles. It is therefore critical to the overall function of the heart, and has also been implicated as an important player in various diseases, including lethal ventricular arrhythmias such as ventricular fibrillation and drug-induced torsades de pointes. However, current ventricular models of electrophysiology usually ignore, or include highly simplified representations of the specialized conduction system. Here, we describe the development of an image-based, species-consistent, anatomicallydetailed model of rabbit ventricular electrophysiology that incorporates a detailed description of the free-running part of the specialized conduction system. Techniques used for the construction of the geometrical model of the specialized conduction system from a magnetic resonance dataset and integration of the system model into a ventricular anatomical model, developed from the same dataset, are described. Computer simulations of rabbit ventricular electrophysiology are conducted using the novel anatomical model and rabbit-specific membrane kinetics to investigate the importance of the components and properties of the conduction system in determining ventricular function under physiological conditions. Simulation results are compared to panoramic optical mapping experiments for model validation and results interpretation. Full access is provided to the anatomical models developed in this study. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: His-Purkinje system Activation sequence Electrophysiology simulation

Contents 1. 2.

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4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 Methods for anatomical model construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 2.1. Construction of the rabbit ventricular model from high resolution MR dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.2. Construction of the specialized conduction system model from high resolution MR dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.2.1. Segmentation of the free-running Purkinje system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.2.2. Skeletonization of the free-running Purkinje system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.2.3. Bundle of His and bundle branch generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.2.4. Endocardial bound Purkinje generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Functional model of rabbit ventricular and Purkinje electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.1. Rabbit-specific electrophysiological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.2. Numerical solution techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3. Stimulation protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Experimental panoramic optical mapping recording methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Rabbit ventricular activation sequence in simulations and experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.1. Rabbit-specific specialized conduction system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

* Corresponding author. Tel.: þ44(0)1865 610806. E-mail address: [email protected] (B. Rodriguez). 0079-6107/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.pbiomolbio.2011.05.002

R. Bordas et al. / Progress in Biophysics and Molecular Biology 107 (2011) 90e100

6.2. 6.3.

Impact of specialized conduction system in ventricular activation sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editors’ note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction The function of the ventricular specialized conduction system (SCS) is to ensure coordinated electrical activation and contraction of the ventricles, by providing a high-speed path between atrioventricular node and ventricular muscle. It is therefore critical to the overall function of the heart. Importantly, the SCS has also been implicated as a player in lethal ventricular arrhythmias such as ventricular fibrillation and drug-induced torsades de pointes (Scheinman, 2009; Silverman et al., 2006). Histological investigations have shown that the SCS consists of a network of thin fibres, electrically isolated from the myocardium for a significant part of their path, with three main components: the bundle of His, the bundle branches, and the Purkinje fibre network (Tawara, 1906, 2000). The Purkinje fibre network is electrically connected to the myocardium at discrete sites situated in the ventricular myocardium and termed Purkinje-Ventricular (PV) junctions. The location and spatial extent of PV junctions varies between species, with sheep and pig having almost fully transmural presence, while the ventricles of rabbit, dog or human contain PV junctions only in the sub-endocardial layers (Ansari et al., 1999; Holland and Brooks, 1976; Ryu et al., 2009; Tranum-Jensen et al., 1991; Tribulova et al., 2002). Whilst essential to ventricular function, the SCS is usually neglected in cardiac electrophysiology models (Bishop et al., 2010; Corrias et al., 2010; Rodriguez et al., 2005) or represented incompletely (Berenfeld and Abboud, 1996; Berenfeld and Jalife, 1998; Boyle et al., 2007; Romero et al., 2010a; Ten Tusscher and Panfilov, 2008; Vigmond and Clements, 2007). In some cases, algorithmically generated systems are used (Ijiri et al., 2008; Pollard and Barr, 1990; Romero et al., 2010b; Washio et al., 2007; Zimmerman et al., 2009). Only one previous study has modelled the free-running Purkinje system (FRPS) that traverses the ventricular cavities (Vadakkumpadan et al., 2009). However, the remainder of SCS was not included in that model. In spite of these limitations, the ventricular models mentioned above have highlighted the importance of the SCS in mechanisms underlying cardiac arrhythmias, defibrillation and cardiac resynchronization therapies. Therefore, further improvements in the geometrical description of the SCS in ventricular models of electrophysiology, and a better understanding of the role of the system in modulating function in health and disease, are needed. The main goal of this study is to describe techniques and methodologies required for the incorporation of a detailed description of the SCS into rabbit ventricular electrophysiology models, and the investigation of the role of the SCS in modulating ventricular activation sequence. Full access is provided to models developed in this study. In brief, model generation first involved the construction of an anatomically faithful representation of the FRPS, obtained by segmentation of the Oxford rabbit heart magnetic resonance (MR) dataset (Plank et al., 2009), followed by skeletonization to produce a computationally suitable cable model. Secondly, a custom tool was created to allow the addition of the bundle of His and bundle branches, based on descriptions in the literature. Finally, the distal endocardial bound Purkinje system was generated using an Lindenmeyer system (L-system), a rule-based

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iterative growth algorithm, parameterized to give an anatomically realistic endocardial network (Ijiri et al., 2008). The structural conduction system model was integrated into the recently published anatomical finite element mesh of the rabbit ventricles, developed based on the same MR dataset (Bishop et al., 2010). Discrete PV junctions coupling the SCS to the myocardium were stochastically distributed with varying density. Simulation of electrical activity through the SCS and the ventricular models is performed using a monodomain model of electrical propagation with rabbit-specific membrane kinetics in both Purkinje and myocardial tissue (Corrias et al., 2011; Mahajan et al., 2008). Simulations of normal sinus rhythm are conducted using the full specialized conduction system model and a simplified model, which omits the FRPS. The resulting findings allow investigation into how the degree and granularity of the SCS integrated into the ventricular model affects ventricular function. Simulation results are compared to panoramic optical mapping of epicardial activation in different rabbit hearts during sinus rhythm. By providing access to the SCS model developed in this study we hope to facilitate more widespread investigation into the use of an anatomically representative model of the SCS in the rabbit heart for cardiac electrophysiology research. 2. Methods for anatomical model construction Development of both the ventricular and the SCS models presented here is based upon a previously published high-resolution MR dataset of a rabbit heart (see (Bishop et al., 2010), supplemental data). Details of sample preparation of the heart and MR data reconstruction are described in (Burton et al., 2006; Schneider et al., 2004). The use of a single high resolution dataset as the basis for both models is essential, as many FRPS fibres attach to endocardial structures, such as papillary muscles, that may vary in size and location between hearts. 2.1. Construction of the rabbit ventricular model from high resolution MR dataset Construction of a tetrahedral model of the ventricles from the MR dataset has been described previously (Bishop et al., 2010). Briefly, the myocardium was segmented from the MR dataset using a pipeline of level-set segmentation filters. The segmentation pipeline consisted of an initial approximate segmentation using an intensity based level-set filter, initialized with automatically selected seed points. The second stage of the pipeline applied a geodesic active contour level-set filter using the output from the threshold filter as an initial level set to remove artifacts due to the MR bias field. In the final stage of the pipeline a Laplacian level-set filter was used to fine-tune the segmentation. A series of post-processing operations were performed on the segmentation. A set of morphological operations were applied to improve the segmentation. Isolated regions of the segmentation were removed using a connected components algorithm. Manual corrections were made to thin-walled blood vessels on the outersurface of the heart that were not defined during segmentation. A smooth surface was fitted to a band of connective tissue along the

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atrio-ventricular border. All tissue above the surface was removed to leave only the ventricles. Finite element approximations of the equations modelling electrical activity in the heart require the geometry to be discretized into a mesh. The segmentation was processed using Tarantula (www.meshing.at), an octree-based mesh generator, to generate an unstructured, conforming tetrahedral mesh. Tarantula was used to produce a mesh with a node discretization of w6 voxels. Both the myocardium and the surrounding bath were meshed, although only the myocardial mesh was required for this study. The myocardium was meshed with a mean tetrahedral edge length of w250 mm. This produced a mesh that consisted of 3,575,122 tetrahedral elements and 644,167 node points. The resolution of the mesh was only half the resolution used in (Bishop et al., 2010). However, the resolution is sufficient to ensure numerical convergence and to model the endocardial surface detail required for integration with the SCS model. The directional arrangement of cells in the myocardium (often termed “fibre orientation”) was projected from the mesh described by (Bishop et al., 2010) onto the lower resolution mesh used here. The original fibre orientation was determined using a mathematical model developed by (Potse et al., 2006), itself based on experimental data collected by (Streeter et al., 1969). Fibre orientations were axisymmetric, with no sheet direction defined. 2.2. Construction of the specialized conduction system model from high resolution MR dataset The SCS model is built in three distinct phases, with both the MR dataset and the ventricular mesh being used as a starting point. At each stage the model is integrated into the ventricular mesh. The initial model is built by segmenting and skeletonizing the FRPS as seen in MR data. The bundle of His and bundle branches are added to the model based on the intersection locations of the FRPS with the endocardial surface of the septum in the ventricular tetrahedral model. Finally, the distal endocardial bound Purkinje system is algorithmically generated using an L-system based algorithm parameterized to produce a realistic endocardial network. The intersection locations of the FPRS with the endocardial surface are used as initial points for the generation algorithm. 2.2.1. Segmentation of the free-running Purkinje system Direct segmentation of the FRPS from MR data is complicated by the small size of Purkinje fibres relative to the myocardium. Full segmentations of the FRPS have previously been created by manual extraction (Vadakkumpadan et al., 2009) and using the method described here (Bordas et al., 2010). Isolated segments of the FRPS have also been segmented using a non-linear orientation filter (Cetingul et al., 2009). However, a complete segmentation of the system was not obtained using this method. The segmentation method presented here proceeds indirectly: Our method first segments the myocardium, which is then subtracted from the original images, significantly simplifying the subsequent segmentation of the FPRS. The segmentation process is performed in several stages. In the first step, a grey-level threshold is applied to the image stack. Otsu’s method (Otsu, 1979) is used to select the threshold parameter. The FPRS is located within the ventricular cavities, which are towards the centre of the MR dataset. The effect of the MR bias field is minimized towards the centre of the image, removing the need for level-set type segmentation. A morphological closing operation using a disk shaped structuring element, with a radius of 6 pixels (158.4 mm), is performed on each transverse slice of the image stack. The closing operation removes small holes in the segmented myocardium. A morphological opening operation is subsequently

performed on each slice of the image stack, using the same structuring element. The opening operation removes small objects in the ventricular cavity, such as Purkinje fibres. The structuring element size is chosen to remove objects the size of Purkinje fibres, whilst retaining larger features such as papillary muscles, trabeculae and the myocardium. A three-dimensional connected components algorithm is applied to the resulting image stack and the largest connected component, a segmentation of the myocardium, papillary muscles and trabeculae, is extracted. This segmentation is applied as a mask on the initial set of thresholded images. The mask removes the myocardium and associated features, leaving only small features, such as the Purkinje network, chordae tendineae (CT) and parts of the valves. The remaining features are overlaid on top of the original image stack to allow visual inspection. Seed points are chosen that identify features corresponding to Purkinje fibres and avoid regions such as the CT. The seed point selection process involves a user scanning through the image stack in each of the three principal directions (transverse, frontal and sagittal) to select the points. Connected components are individually labelled to extract the Purkinje fibres containing these seeds points. The result of the segmentation is visually validated and the seed point selection performed iteratively until all of the Purkinje system is identified. Due to the highly interconnected nature of the FRPS only a small number of seed points (w25) are required for each ventricle. 2.2.2. Skeletonization of the free-running Purkinje system The segmentation was skeletonized using a homotropic thinning algorithm (Pudney, 1998), followed by centre line correction (Zhou and Toga, 1999), to produce a one-dimensional branched cable model. To simplify implementation of the numerical simulation method, the cable model is mapped onto the nearest edges of the full tetrahedral mesh (incorporating the bath mesh). First, each pair of nodes that make up a segment in the cable model are mapped onto the nearest nodes in the tetrahedral mesh. The shortest path along the edges of tetrahedra between each pair of mapped nodes is calculated using the A* algorithm (Hart et al., 1968) and the mapped nodes connected using this path. In some cases, due to the use of separate segmentation techniques in construction of the ventricular and FRPS models, the end-points of the FRPS terminate just short of the endocardial wall. To correct this artifact, FRPS end-points within 250 mm of the endocardial wall are identified and extra connections are added between these points and the wall. The location of PV junctions cannot be obtained from MR data. In the absence of this data, PV junctions are defined at each point where the FRPS model meets the endocardial surface. The final FRPS cable model with PV junctions is shown in Fig. 1a. 2.2.3. Bundle of His and bundle branch generation The tetrahedral mesh of the ventricles and the skeletonized FRPS mesh are used as a starting point to construct the bundle of His and the bundle branches. The bundle of His runs from the atrioventricular node into the septum. Two nodes are manually specified in the ventricular mesh to represent the start and end of the bundle of His. The A* algorithm is used to join the two nodes via the shortest path along the edges of the tetrahedral mesh. The bundle of His subdivides into the left and right bundle branches, which run along the surface of the septal wall. A further two nodes are manually specified on the left and right endocardial surfaces to represent the start of each bundle branch. The nodes are joined via the shortest path to the distal end of the bundle of His. The left and right endocardial surfaces of the septum are extracted from the tetrahedral ventricular mesh using Paraview (www.paraview.org). Nodes at the intersection of the FRPS and the

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Fig. 1. 1a shows a transerve slice through the high resolution rabbit MRI dataset, the left ventricle (LV) and right ventricle (RV) are labelled. Arrows highlight example free-running Purkinje fibres (labelled PF). 1b and 1c show segmentations of the free-running Purkinje system (FRPS), overlaid on slices through the MR dataset, in the left and right ventricles, respectively.

endocardial surface of the septum are identified. These intersection points are joined via shortest paths along the edges of the endocardial surface triangulation to the start node of the appropriate bundle branch. The final bundle of His and bundle branch model is shown in Fig. 2b. The resulting network is a good approximation of the bundle of His and the bundle branches, as observed in photographs of the SCS (Ansari et al., 1999; Tawara, 1906, 2000). 2.2.4. Endocardial bound Purkinje generation The distal endocardial bound Purkinje network is generated using a parameterized rule-based method based on a L-system described in (Ijiri et al., 2008) and briefly described here. The characteristics of the network generated by the algorithm are determined by a set of input parameters. The parameters describe the statistical distribution of branch lengths, the angle at Purkinje branch points, the degree of curvature in branches and the probability that a point in the Purkinje system is assigned as a PV junction. In order to produce a realistic endocardial network,

consistent with descriptions in the literature, parameters have to be carefully specified (see Table 1). A parameter set validated against photographs of endocardial Purkinje network in sheep (Ijiri et al., 2008) was used as the basis for the parameters developed here. The parameters were adjusted for the smaller dimensions and larger surface detail of the rabbit heart mesh using parameter sweeping and visual evaluation of the resulting networks. The end points of the FRPS are used as starting points for the endocardial generation algorithm. To initialize the algorithm, four growth apices are placed emanating from each start point. The apices are given a initial directions pointing along the endocardial surface at right angles to each other. Each apex represents the start of a branch with an initial direction and a target branch length. The target branch length is assigned by sampling from a normal distribution with mean lbra and variance s2. The initial apices are inserted into a queue at random. The growth algorithm proceeds iteratively: the first apex in the queue is taken and a branch is grown, after growth the apex is discarded and the next apex in the

Fig. 2. Three phases of specialized conduction system (SCS) model construction overlaid on the endocardial surfaces. Purkinje-Ventricular (PV) junction points are shown as green spheres. The LV is on the right hand side in each case. 2a shows the skeletonized cable model of the FRPS developed based on segmentation of the MR dataset. 2b shows the bundle of His and bundle branches developed based on descriptions in the literature and the intersection of the FRPS with the septal wall. 2c shows the distal endocardial bound Purkinje system generated using the L-system based algorithm Ijiri et al. (2008).

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Table 1 Parameters used to generate the endocardial bound Purkinje system. Param

Description

Value

Units

lbra Ns

Mean branch length Number of segments per branch Branch length variance Child branch angle Gradient weight Surface tolerance Collision tolerance PV junction probability

4 40 0.04  lbra 45 2/Ns 0.002  lbra 0.1  lbra 2

mm dimensionless mm2 degrees dimensionless mm mm %

s2 q

wl stol ktol Ppvj

queue is processed. The algorithm continues until the queue is empty. A branch is approximated by Ns segments of length lbra/Ns. To generate curved branches and minimize collisions the branch segments are curved away from the nearest existing branch. The direction of each branch segment is given by

dorig þ wl *dnearest ; d ¼ dorig þ wl *dnearest

(1)

where dorig is the unit vector in the direction of the preceding branch segment, dnearest is the unit vector from the nearest existing branch to the branch segment, wl is a parameter governing the degree of curvature and jj$jj is the Euclidean distance. This results in branches that curve gradually away from existing branches. To constrain the growing Purkinje network to lie on the endocardial surface, the direction d is projected on to the endocardial surface at each step:

d0 ¼ d  ðd$nÞn;

(2)

where n is the normal vector of the nearest point on the endocardial surface. Despite projecting the growth direction onto the endocardial surface, the irregular nature of the endocardial wall means that branches can still grow away from the surface. When a branch grows more than a specified tolerance, stol, away from the surface growth is stopped. Due to the high level of endocardial surface detail present in the rabbit ventricular mesh described in Section 2.1, the branches grown by the method have a high propensity to move away from the endocardial surface. To address this issue, a higher number of branch segments are used than previous work (Ijiri et al., 2008; Romero et al., 2010b). Growth is stopped if a growing branch collides with an existing branch, with collision determined based on a specified tolerance, ktol. If a collision occurs the growing branch is connected to the existing branch and growth is halted. It is important to note that, due to branches leaving the endocardial surface and collisions between fibres, the mean branch length produced by the algorithm is considerably less than that specified by lbra and the distribution of branch lengths is non-Gaussian. After all segments of the branch have been grown the initial apex is removed from the queue and two child apices are added to the end queue. The initial direction of the child apices is governed by the branch angle parameter, q. The apices are assigned an initial direction at q degrees to the previous direction on the plane of the endocardial surface, such that


dchild ¼ d0 cosðqÞ  d0  n sinðqÞ

(3)

where d0 is the previous direction and dchild is the initial direction of the child apex. To allow use in electrophysiology simulation, the generated endocardial bound Purkinje system is mapped onto the edges of the tetrahedral ventricular mesh using the method described for the FRPS. To integrate the endocardial bound model with the ventricular

model the location of PV junctions must be specified. The distribution of PV junctions is unknown and cannot be observed in the available imaging data. In our model, junctions are placed at nodes in the endocardial bound system at random, with each node having a user specified probability (Ppvj) of being assigned as a junction. The complete endocardial bound Purkinje system model, with PV junctions, is shown in Fig. 2c. 3. Functional model of rabbit ventricular and Purkinje electrophysiology To allow investigation of the role of anatomical structure of the SCS on ventricular function, the subject-specific anatomical model described in the previous section was used in conjunction with a rabbit-specific model of ventricular and Purkinje electrophysiological function. 3.1. Rabbit-specific electrophysiological model The bidomain model is considered to offer a sufficiently complete description of electrical activity in cardiac tissue (Keener and Sneyd, 1998). Bidomain equations explicitly model the potential in both the intracellular and the extracellular space, allowing investigation into mechanisms of cardiac electrical propagation as well as effects of externally applied electric currents on cardiac tissue, such as those used in defibrillation studies. The bidomain equations may be simplified to monodomain equations, which only explicitly model the transmembrane potential. Previous studies have demonstrated that monodomain equations can be used to simulate normal electrical propagation through cardiac tissue with minimal error compared to the bidomain equations (Potse et al., 2006). In this study we focus on investigating the role of the SCS in propagation of electrical excitation through the ventricles, and so we use the monodomain equations to model electrical activity in both ventricular tissue and SCS. Membrane dynamics within the myocardium were modelled using the MahajaneShiferaw rabbit ventricular action potential model (Mahajan et al., 2008). Conductivity was taken to be 0.174 S/ m parallel to the myocyte fibre direction and 0.019 S/m perpendicular to it. Conductivity values were calculated from the intracellular and extracellular conductivities presented in (Clerc, 1976) using the harmonic mean. The SCS is modelled by a one-dimensional version of the monodomain equations. We note that individual fibres are modelled as multicellular bundles, rather than unicellular fibres, such that the cell surface-to-volume ratio is not dependent on the fibre radius (Plonsey, 1988). Conductivity in the SCS was set to 0.4 S/m (Vigmond and Clements, 2007). In the absence of radii data, all SCS fibres were assigned a radius of 50 mm. To provide a fully integrated, species consistent, functional model of rabbit ventricular electrophysiology, the recent Corrias model of rabbit Purkinje action potential was used to govern membrane dynamics in the SCS (Corrias et al., 2011). The Corrias rabbit Purkinje action potential model is biophysically-detailed and includes a mathematical description for transmembrane currents and calcium dynamics kinetics based on experimental data from rabbit, when available. The majority of the SCS is electrically isolated from the myocardium, with the exception of discrete PV junctions. We model PV junctions as a resistor coupling discrete points on Purkinje fibres to a volumetric region placed on the endocardial wall, as described by (Azzouzi et al., 2010; Vigmond and Clements, 2007). In previous work, the influence of the PV junctions on Purkinje fibres has been treated as a Robin boundary condition at the end of the fibre (Azzouzi et al., 2010; Vigmond et al., 2009). However, this is not in keeping with the micrograph studies of PV junctions

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(Tranum-Jensen et al., 1991), which show that Purkinje fibres are coupled by thin strands to transitional cells, with the transitional cell coupled in turn to the myocardium. The coupling strands can occur along the length of Purkinje fibres and not only at their end points. Thus, the influence of PV junctions on the Purkinje fibres is treated as a source term in the Purkinje fibre equations. The Purkinje source term and the volumetric source term balance total current flow to ensure conservation of current. PV junctions were coupled to a hemispherical region of 1 mm radius on the endocardium. PV junction resistance was set uniformly to 20 MU (Romero et al., 2010b; Vigmond and Clements, 2007). 3.2. Numerical solution techniques Numerical solutions to both the myocardial tissue and SCS fibre equations are calculated using a semi-implicit discretization in time (Keener and Bogar, 1998; Whiteley, 2006). The semi-implicit discretization results in a single linear system being updated at each time-step to solve for the transmembrane potential. The remainder of the equations to be solved, consisting of ionic current cell models, uncouple into small systems of ordinary differential equations. Under this time discretization the PV junction equations were treated explicitly, in a similar manner to the ionic cell models. The time discrete system is solved using the Galerkin finite element method with first order Lagrange basis functions. The numerical method was implemented using the Libmesh finite element library (Kirk et al., 2006). A time-step of 0.1 ms was used for all simulations. 3.3. Stimulation protocol The stimulation protocol used to elicit wavefront propagation throughout the SCS model consisted of stimulation of the proximal end of the bundle of His. The stimulation was designed to replicate activation of the ventricles under normal sinus rhythm. A transmembrane current pulse of was delivered to the bundle of His over a 1 ms duration. The time at which the transmembrane potential crossed a level of 20 mV at each node was recorded and used to make activation plots. The activation time at each node was interpolated onto each element and weighted by element volume to allow the distribution of activation time over the tissue to be calculated.

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4. Experimental panoramic optical mapping recording methods The panoramic optical mapping system and the data analysis methods have been previously described in detail (Lou et al., 2008; Qu et al., 2007). Briefly, this system is capable of optically mapping action potentials from almost the entire ventricular epicardium of a rabbit heart by three photo-diode arrays (PDA), which centre around the heart and spaced 120 apart. A three-dimensional epicardial surface of every experimented rabbit heart is first reconstructed; and the optical signals recorded by the PDAs are then registered and texture mapped onto the reconstructed surface. Finally, various parameters, such as activation pattern, can be quantified and visualized on the realistic heart surface. 5. Rabbit ventricular activation sequence in simulations and experiments The structural and functional models of rabbit ventricular electrophysiology described in the previous section were used to simulate the activation sequence of the rabbit ventricles. In order to asses the importance of the different components of the SCS in determining activation sequence in sinus rhythm, simulations were conducted using the full SCS model and a simplified model in which the FRPS was removed and replaced by endocardial bound connections to the bundle branches. Simulations were also performed in the absence of the endocardial bound system but results were almost identical to those obtain with the full SCS model. Fig. 3 shows the epicardial activation sequence obtained in the simulations using the rabbit ventricular models with and without FRPS. The initial wavefront breakthrough point is found towards the posterior of the RV free wall in both cases, corresponding to the location of minimum wall thickness in the model. The full model, incorporating FRPS, displays earlier and more uniform activation of the right ventricle free wall than the simplified model. Similarly, activation of the LV free wall is more uniform in the full model, though the effect is less pronounced than in the RV. Fig. 4 shows an example of an epicardial activation sequence recorded using optical mapping of a rabbit heart during sinus rhythm. The experimentally recorded activation pattern qualitatively matches that of both the full and simplified models. However, the coordinated activation of

Fig. 3. Time of first activation of the epicardium after stimulation of the bundle of His. 3a and 3b show the epicardial activation patterns on the left and right ventricle, respectively, for the simplified model without the free-running Purkinje system (FRPS). 3c and 3d show epicardial activation for the full model incorporating FRPS. Activation times are given relative to the time of first epicardial breakthrough, which occurred 16 ms after stimulation of the bundle of His in both models. First breakthrough also occurred at the same location in each model: towards the posterior of the right ventricle free wall.

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Fig. 4. Epicardial activation of normal sinus rhythm in rabbit heart from the panoramic optical mapping of Langendorff-perfused rabbit heart. Activation maps of the right ventricle (RV, top) and the left ventricle (LV, bottom) epicardium are shown. Each colour in the activation map spans an interval of 1 ms. The corresponding views of the rabbit heart are shown on the left. Electrocardiogram (ECG) and action potentials (AP) at three different locations marked by numbers in the activation maps are shown on the right. Note that the upstroke of AP3 appears later compared with the upstroke of AP1. It can be seen that the first activation on the epicardium starts in the RV, which is qualitatively similar to what is shown in our simulation (Fig. 3).

the RV free wall and point of first breakthrough is closer to the optical mapping recording in the full model. Even though simulations and experiments yielded similar experimental epicardial activation patterns, differences exist that could be due to differences in ventricular and Purkinje anatomy and electrophysiology between the two hearts and also photon scattering effects in the experiments (Bishop et al., 2006). Fig. 5 shows a comparison of the endocardial activation maps following stimulation of the proximal end of the bundle of His using both the full and simplified SCS models. Global patterns of activation are similar in both models but the presence or absence of the FRPS results in a number of important differences in activation sequence in the endocardium. Following stimulation of the bundle of His electrical, excitation quickly propagates throughout the SCS before traversing the PV junctions towards the myocardium. In both models, the first site of activation in the myocardium is the left-ventricular (LV) septal wall. In the full model, the FRPS provides a shorter link between the endocardial bound Purkinje system on the free walls and papillary muscles and the bundle branches. As a result, activation times for the papillarly muscles and free walls are shorter in the presence of the FRPS in the full model as opposed to the simplified model without FRPS. Activation times for the two models are further quantified in Fig. 6, which shows the distribution of activation times in the myocardium after stimulation of the bundle of His, in both the full and simplified models. The distribution of activation times without the FRPS shows a greater spread compared to the case in which FRPS is present. Mean activation time for the full model is 24.5 ms, with a time to total activation of 46 ms whereas the absence of the FRPS results in an increase to 26.3 ms of mean activation time, and 49 ms total activation time respectively. The integration of the FRPS

into the SCS model primarily results in faster, more uniform activation of the myocardium. The change in activation time and sequence is due to the shorter propagation paths provided by the FRPS compared to the endocardial bound Purkinje system in the simplified model. The presence of the latter explains why the difference in activation times is relatively small for this model. 6. Discussion In this study, we describe novel techniques and methodologies required for the construction of a high resolution MR-based anatomically-detailed rabbit-specific model of ventricular electrophysiology including a detailed description of the SCS. The anatomical model of rabbit ventricular and the free-running component of the Purkinje system are obtained from the same high-resolution MR data. A semi-automatic segmentation method is developed for the extraction of the FRPS and a parameterized generation algorithm is used to generate the endocardial bound Purkinje system. The bundle branches are also (manually) included. The highly detailed anatomical model of the integrated ventricular and SCS rabbit model is then used in combination with a functional model of rabbit ventricular and Purkinje electrophysiology to simulate the impact of the presence of FRPS in determining the activation sequence under normal sinus rhythm. With this work, we address questions of interoperability and interfaces between models in cardiac electrophysiology, related to the integration of structural and functional models from the ionic to the whole ventricular level, the construction of anatomically-based models from high resolution MRI datasets, and comparison of electrophysiological activity in simulations and experimental optical mapping recordings.

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Fig. 5. Time of first activation in the endocardium after stimulation of the bundle of His. 5a and 5b show activation when the FRPS is removed from the SCS model. 5c and 5d show activation with the FRPS integrated into the SCS model. The FPRS results in faster and more uniform activation of the endocardial surface. Activation of the LV papillary muscles, the lower septum and both free walls occurs noticeably earlier when the FRPS is integrated.

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Fig. 6. 6a and 6b show distribution of first activation times in the ventricular model, with the simplified and full SCS model, respectively. The integration of FRPS results in faster and more uniform activation throughout the myocardium.

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6.1. Rabbit-specific specialized conduction system model A major goal of this study was to create the methods for the generation of a full geometrical model of the SCS in rabbit and its integration in an anatomically-based ventricular model of rabbit electrophysiology. The model construction methods described in this study can be applied to the development of a SCS for any ventricular model. Our methods could therefore facilitate the construction of individualized computational models of cardiac function where structural MR data are used to extract information on anatomical features of the ventricles and the SCS. Ventricular and SCS anatomies in our model are based, where possible, upon the same high resolution MR rabbit dataset. To do so, a semi-automatic method for segmentation of the FRPS from MR data was developed, which can be applied to other datasets from rabbit or other species [for example, rat data (Bordas et al., 2010)]. Whilst performance of the segmentation method has been validated with respect to the MR data, it is difficult to determine whether all free-running Purkinje fibres are visible in the MR data without access to an additional imaging modalities. The quality of the segmentation and the density of the segmented system would indicate that at least a large proportion of it is captured in the data. We also cannot rule out the possibility that not all free-running fibres in the image are part of the Purkinje system. However, Purkinje cells have been consistently detected in the ‘false tendons’ in lu et al., 2003). This, coupled with the animal hearts (Kervanciog highly interconnected nature of the segmented network and the realistic activation sequences produced by the model even without the endocardial bound network, indicates that the segmentation of the MR data is an acceptable representation of the FRPS. Whilst the MR dataset provides a basis for extraction of anatomical features, the lack of data on the endocardial bound Purkinje system means that, by necessity, the endocardial bound Purkinje system was algorithmically generated. The iterative Lsystem method used here plays an important role in doing this. The endocardial networks generated by the method have been shown to agree well with photographs of the endocardial bound Purkinje system in sheep (Ijiri et al., 2008). In addition, the algorithm allows investigation of the role of anatomical variability in the endocardial bound Purkinje system. A set of parameter values, identified by parameter sweeping and user evaluation, suitable for use with the highly detailed rabbit ventricular model is given in Table 1. Whilst these parameters provide a first approximation, they could be further refined using measurements of the endocardial bound system anatomy in rabbit. The importance of each of the parameters defining the endocardial system in determining the activation sequence could also be investigated by performing a simulation study. The use of additional imaging modalities such as histology (Burton et al., 2006) could help in the full characterization of the SCS in specific hearts, and to optimize the model parameter set. Whole rabbit ventricle histology data provide information on different cell-type distribution and myocyte orientation. Specific staining for Purkinje system fibres and PV junctions in histological images would provide very valuable information for the refining of ventricular electrophysiological models such as the one described here. However, the registration of 3D histological tissue volume to remove distortion introduced during histological processing still represents a challenge. 6.2. Impact of specialized conduction system in ventricular activation sequence

ventricles in sinus rhythm. Simulation results were compared to panoramic optical mapping recordings from a rabbit heart for model validation and result interpretation. The role of the FRPS e often neglected in SCS models e in ventricular activation under normal sinus rhythm was investigated. Inclusion of the FRPS resulted in slightly faster and more coordinated activation of the ventricles compared to a simplified model neglecting this structure. The synchronization effect of the FRPS is due to it providing a shorter path between the septum and structures towards the free-walls than the endocardial bound Purkinje system. The results suggest that the FRPS plays a surprisingly small role in synchronizing activation in the ventricles beyond that played by the endocardial bound Purkinje system in the healthy heart. This is in keeping with experimental studies, where chemical ablation of the FRPS resulted in only minor changes in the normal heart, which were enhanced upon induction of additional pathologies, such as long-duration ventricular fibrillation Dosdall et al. (2008). Epicardial activation sequences in simulations and experiments shown here were very similar, particularly for the full model incorporating FRPS, indicating that the rabbit model of cardiac electrophysiology and anatomy developed here is realistic. It is important to note, however, that a high intersubject variability exists in the anatomical and functional features included in the model (Bordas et al., 2010; Rodriguez, 2010). Furthermore, even though the model described here includes a subject-specific description of the geometry of the ventricular surfaces and the FRPS based on MRI, it hasn’t been constructed to capture subjectspecific structural heterogeneities and fibre orientation. This could be extracted from histology and DT-MRI datasets. Moreover, the functional electrophysiological model used in this study is rabbitspecific in the sense that membrane kinetics in myocardium and SCS are described by biophysically-detailed ventricular and Purkinje action potential models mostly based on voltage-clamp and microelectrode recordings obtained from rabbit preparations. A systematic analysis of two rabbit action potential models and their ability to reproduce rabbit-specific electrophysiological properties is provided elsewhere (Romero et al., 2011). Cellular electrophysiological properties in our model are however not subject-specific as this would require the development of new model construction techniques using alternative modelling approaches such as the phenomenological ones described in (Walmsley et al., 2010; BuenoOrovio et al., 2008), and no gradients (baseeapex; lefteright, transmural) in cell electrophysiology properties have been introduced for the present study. Whilst the simulations shown here have been limited to normal physiological conditions, the tools developed in this study allow investigation into the interaction of anatomical structure of the SCS with ventricular function in other situations. Pathological conditions, such as bundle branch block, AV dissociation and drug induced torsades de pointes, in which the anatomical structure of the SCS impacts electrophysiological function can be investigated using the model tools developed in this study. 6.3. Model access Oxford rabbit heart MR data and the computational finite element ventricular mesh incorporating the SCS model are available via the “Download” link at http://web.comlab.ox.ac.uk/chaste/ index.html. Editors’ note

The integrated structural and functional models of rabbit electrophysiology were used in conjunction with numerical techniques to simulate the activation sequence throughout the rabbit

Please see also related communications in this issue by Romero et al. (2011) and Clayton et al. (2011).

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Acknowledgements This study was partially supported by the European Commission preDiCT project (grant reference DG-INFSO - 224381) and a BBSRC technology development grant (no. E003443). RB was supported through an EPSRC funded Life Sciences Dotoral Training Centre studentship (grant reference EP/E501605/1) and a industrial sponsorship award funded by Fujitsu Laboratories of Europe Ltd. QL and IE were supported by grants from NIH: R01-HL067322 and R01-HL085369. PK is a Senior Fellow of the British Heart Foundation. BR is supported by an UK Medical Research Council (MRC) Career Development Award (grant reference G0700278). The authors would like to thank Dr Martin Bishop (Computational Biology Group, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD) for providing the rabbit ventricles tetrahedral mesh and the Oxford 3D Heart project who provided the underlying data. The authors would like to thank Dr Bum-Rak Choi and Prof. Andrew Pollard for the helpful discussions. References Ansari, A., Ho, S., Anderson, R.H., 1999. Distribution of the Purkinje fibres in the sheep heart. Anatomical Rec. 254, 92e97. Azzouzi, A., Coudiere, Y., Turpault, R., 2010. A mathematical model of the ventricular conduction system. ICNAAM 2010. AIP Conf. Proc. 1281, 403e406. Berenfeld, O., Abboud, S., 1996. Simulation of cardiac activity and the ecg using a heart model with a reaction-diffusion action potential. Med. Eng. Phys. 18, 615e625. Berenfeld, O., Jalife, J., 1998. Purkinje-muscle reentry as a mechanism of polymorphic ventricular arrhythmias in a 3-dimensional model of the ventricles. Circ. Res. 82, 1063e1077. Bishop, M., Rodriguez, B., Eason, J., Whiteley, J.P., Trayanova, N.A., Gavaghan, D., 2006. Synthesis of voltage-sensitive optical signals: application to panoramic optical mapping. Biophys. J. 90, 2938e2945. Bishop, M.J., Plank, G., Burton, R.A.B., Schneider, J.E., Gavaghan, D.J., Grau, V., Kohl, P., 2010. Development of an anatomically detailed MRI-derived rabbit ventricular model and assessment of its impact on simulations of electrophysiological function. Am. J. Physiol-Heart 298, H699eH718. Bordas, R., Grau, V., Burton, R.A.B., Hales, P., Schneider, J.E., Gavaghan, D.J., Kohl, P., Rodriguez, B., 2010. Integrated approach for the study of anatomical variability in the cardiac Purkinje system: from high resolution MRI to electrophysiology simulation. Eng. Med. Biol. Soc. (EMBC) 2010 Annu. Int. Conf. IEEE, 6793e6796. Boyle, P.M., Deo, M., Vigmond, E.J., 2007. Behaviour of the Purkinje system during defibrillation-strength shocks. Eng. Med. Biol. Soc. (EMBC) 2007 Annu. Int. Conf. IEEE, 419e422. Bueno-Orovio, A., Cherry, E.M., Fenton, F.H., 2008. Minimal model for human ventricular action potentials in tissue. J. Theor. Biol. 253, 544e560. Burton, R.A.B., Plank, G., Schneider, J.E., Grau, V., Ahammer, H., Keeling, S.L., Lee, J., Smith, N.P., Gavaghan, D.J., Trayanova, N.A., Kohl, P., 2006. Three-dimensional models of individual cardiac histoanatomy: tools and challenges. Ann. N. Y. Acad. Sci. 1080, 301e319. Cetingul, H.E., Plank, G., Trayanova, N.A., Vidal, R., 2009. Estimation of multimodal orientation distribution functions from cardiac MRI for tracking Purkinje fibers through branchings. IEEE Int. Symp. Biomed. Imaging, 839e842. Clayton, R.H., Nash, M.P., Bradley, C.P., Panfilov, A.V., Paterson, D.J., Taggart, P., 2011. Experiment-model interaction for analysis of epicardial activation during human ventricular fibrillation with global myocardial ischaemia. Prog. Biophys. Mol. Biol. 107, 101e111. Clerc, L., 1976. Directional differences of impulse spread in trabecular muscle from mammalian heart. J. Physiol. 255, 335e346. Corrias, A., Giles, W., Rodriguez, B., 2011. Ionic mechanisms of electrophysiological properties and repolarization abnormalities in rabbit Purkinje fibers. Am. J. Physiol. Heart Circ. Physiol. 300, H1806eH1813. Corrias, A., Jie, X., Romero, L., Bishop, M.J., Bernabeu, M., Pueyo, E., Rodriguez, B., 2010. Arrhythmic risk biomarkers for the assessment of drug cardiotoxicity: from experiments to computer simulations. Phil Trans. Roy Soc. A 368, 3001e3025. Dosdall, D.J., Tabereaux, P.B., Kim, J.J., Walcott, G.P., Rogers, J.M., Killingsworth, C.R., Huang, J., Robertson, P.G., Smith, W.M., Ideker, R.E., 2008. Chemical ablation of the Purkinje system causes early termination and activation rate slowing of long-duration ventricular fibrillation in dogs. Am. J. Physiol. Heart Circ. Physiol. 295, H883eH889. Hart, P., Nilsson, N., Raphael, B., 1968. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybernetics 4, 100e107. Holland, R.P., Brooks, H., 1976. The qrs complex during myocardial ischemia. An experimental analysis in the porcine heart. J. Clin. Invest. 57, 541e550.

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Washio, T., ichi Okada, J., Hisada, T., 2007. A parallel multilevel technique for solving the bidomain equation on a human heart with Purkinje fibers and a torso model. SIAM J. Sci. Comput. 30, 2855e2881. Whiteley, J.P., 2006. An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE T. Bio-Med Eng. 53, 2139e2147. Zhou, Y., Toga, A., 1999. Efficient skeletonization of volumetric objects. IEEE T. Vis. Comput. Gr 5, 196e209. Zimmerman, V., Sebastian, R., Bijnens, B.H., Frangi, A.F., 2009. Modeling the Purkinje conduction system with a non deterministic rule based iterative method. Comput. Cardiol. 36, 451e464.