Daniel L. Weiss*, David U.J. Keller, Gunnar Seemann, and Olaf Dössel. Introduction

Transcription

1 Europace (2007) 9, vi96 vi104 doi: /europace/eum213 The influence of fibre orientation, extracted from different segments of the human left ventricle, on the activation and repolarization sequence: a simulation study Daniel L. Weiss*, David U.J. Keller, Gunnar Seemann, and Olaf Dössel Institute of Biomedical Engineering, Karlsruhe Institute of Technology (KIT), Kaiserstrasse 12, Karlsruhe, Germany KEYWORDS Computational model; Human left ventricle; Heterogeneity; Anisotropy; DT-MRI measurement Aims This computational study examined the influence of fibre orientation on the electrical processes in the heart. In contrast to similar previous studies, human diffusion tensor magnetic resonance imaging measurements were used. Methods The fibre orientation was extracted from distinctive regions of the left ventricle. It was incorporated in a single tissue segment having a fixed geometry. The electrophysiological model applied in the computational units considered transmural heterogeneities. Excitation was computed by means of the monodomain model; the accompanying pseudo-electrocardiograms (ECGs) were calculated. Results The distribution of fibre orientation extracted from the same transversal section showed only small variations. The fibre information extracted from the equal circumferential but different longitudinal positions showed larger differences, mainly in the imbrication angle. Differences of the endocardial myocyte orientation mainly affected the beginning of the activation sequence. The transmural propagation was faster in areas with larger imbrication angles leading to a narrower QRS complex in pseudo-ecgs. Conclusion The model can be expanded to simulate electrophysiology and contraction in the whole heart geometry. Embedded in a torso model, the impact of fibre orientation on body surface ECGs and their relation to local pseudo-ecgs can be identified. Introduction Modelling of the electrical and mechanical procedures in the human ventricles and simulations of excitation conduction and deformation can enhance the understanding of fundamental cardiac processes. It is well known that the structure of the cardiac muscle influences the electrical propagation and force development throughout the myocardium. This is due to the anisotropy of the electrical and mechanical properties. The conductivity of ventricular tissue is determined by the direction of the fibres and is highest in the direction of their longitudinal axis. Realistic consideration of the cardiac fibre architecture 1 3 is an important factor in clinical applications, e.g. in the planning of patientspecific treatments and cardiac therapies. 4 The orientation of the fibres and their arrangement in laminar sheets has been studied in histological investigations for many years. 5 7 Diffusion tensor magnetic resonance imaging (DT-MRI) is a comparatively new imaging technique. Based on the attenuation of the MRI signal caused by the * Corresponding author. Tel: þ ; fax: þ address: daniel.weiss@kit.edu diffusion of water, this specialized sequence has made it feasible to measure the anisotropic properties of biological tissue as a function of the spatial position. It has previously been used to detect white matter in the human brain, a technique often referred to as fibre tractography. 8,9 Moreover, the feasibility of characterizing myocardial fibre orientation and the non-invasive measurement of the alignment of cardiac cells with the help of DT-MRI has been suggested Both qualitative and quantitative agreements were shown between the primary eigenvector of the diffusion tensor and the orientation of the long axis of myofibers in non-perfused as well as in perfused specimens of the heart. 10,15 22 Experimental studies in larger mammalian hearts have shown that the direction of the muscle fibres is reflected in the epicardial potential field. 23,24 Based on the details shown in such fields, information about the local fibre orientation, the amount of transmural fibre rotation, and the transmural direction of the travelling wave front have been estimated. 25 Transmural electrocardiograms (ECGs), also called pseudo-ecgs being the difference between the potentials at locations close to the endo- and epicardium, reflect the electrical processes & The European Society of Cardiology All rights reserved. For permissions, please journals.permissions@oxfordjournals.org

2 Influence of fibre on activation and repolarization sequence vi97 within the myocardium. It has been reported that T wave morphology of these pseudo-ecgs is greatly be influenced by the heterogeneous transmural distribution of the myocytes biophysical properties The aim of this study was to examine the influence of fibre orientation inside the ventricular wall as derived from a DT-MRI dataset, which provided the diffusion properties in a human heart. Previously, similar studies have been mainly based on non-human data. Our study was performed by means of computational methods, applied to a large segment of ventricular tissue. This enabled the study of the effect of fibre orientation in combination with their interaction with transmural inhomogeneities of electrophysiological tissue properties. The effects on the spread of activation are documented, as well as those on the waveforms of the pseudo-ecgs. Methods Extraction of fibre orientation from the diffusion tensor magnetic resonance imaging dataset This study was based on the fibre orientation measured in a human heart. The recently published anatomical data were provided by the Center for Cardiovascular Bioinformatics and Modelling (CCBM) of the Johns Hopkins University, Baltimore (Maryland, USA) based on the DT-MRI principle. 33 This specialized imaging sequence permits the estimation of the fibre orientation within the tissue analysed. Diffusion tensor magnetic resonance imaging measures the anisotropic diffusion properties of water inside biological tissues as a function of the spatial position. In muscular tissue, the diffusion properties are determined by the microscopic cell structure of the myocytes. The dominant direction of the diffusion is strongly correlated with the geometrical shape of the cells: diffusion is largest in the direction of the long axis of the myocytes. The diffusion process can be specified by diffusion coefficients, constituting a second order tensor D (a 33 matrix). D can be expressed by its eigenvalues l 1 to l 3 and the corresponding eigenvectors e 1 to e 3. In the case of isotropic diffusion, the order of magnitude of all eigenvalues is equal. For anisotropic diffusion with one preferential direction, one eigenvalue is much larger than the two others and diffusion is directed mainly along the eigenvector corresponding to this largest eigenvalue. For a more comprehensive discussion on DT-MRI, see Basser et al. 10,12 and Bammer. 11 The anatomical data included both ventricles and parts of the atria. To avoid changes in the ventricular geometry after the isolation, it was fixed at zero transmural pressure using an isotonic 3% formaldehyde solution. Several days after fixation, the heart was placed inside a 1.5 T GE CV/I MRI Scanner (GE, Medical System, Wausheka, WI, USA) and imaged using a 3D Fast Spin Echo DT-MRI sequence (40 mt/ m maximum gradient amplitude and 150 T/m/s slew rate). 16,33,34 The resulting dataset consists of 13 components defining scalar values for each volume element (voxel) analysed in 3D space: intensity, the three eigenvalues (l 1 to l 3 ) and nine specifying the Cartesian components of the corresponding eigenvectors (e 1(x, y, z) to e 3(x, y, z) ). The overall geometry of the heart was extracted from the intensity data. The resulting structure was segmented using digital image processing techniques such as thresholding, region growing, and interactive deformable meshes. 35,36 The tissue was segmented into left and right atria, left and right ventricular tissue, septum, left and right papillary muscles, the valves, and blood. Next, the left ventricular as well as septal tissue was extracted, describing the working myocardium of the entire left ventricle. A subsequent zoning was used, dividing the ventricle in n parts of similar size. In each part, the specific fibre orientation was extracted, thus yielding n distinguishable fibre orientations in different parts of the left ventricle. These could be used to study the influence of different fibre orientations on the excitation and repolarization sequence. The zoning applied was based on the recommendation of the American Heart Association for the segmentation of the left ventricle into 17 segments. 37 This segmentation was expanded such that the left ventricle was divided into n¼100 segments. It included ten slices, evenly cut along the long-axis of the ventricles. Ten pie-shaped sectors were defined on every slice. Each sector was anchored on the balance point of its slice and described an angle of 368 (Figure 1). The direction of the long axis of the cardiac myocytes was extracted from the datasets containing the Cartesian components of the eigenvectors. Justified by the highly anisotropic diffusion, fibre orientation was described by the direction of the eigenvector e 1 corresponding to the largest eigenvalue l 1. In each of the n zones, a line was constructed, passing through the centre of the zone, and oriented normal to the local epicardial surface. The line segment s between the intersection of this line with the endo- and epicardial surfaces specified a transmural path, its length representing local wall thickness. Along each line segment s, fibre orientation was extracted from the Cartesian components e 1x to e 1z of the eigenvector e 1 in any voxel crossed by s. In this way, a localized orthogonal coordinate system lc was determined in each voxel, according to the definition by Streeter. 5 This system is oriented perpendicular to the epicardium and on the longitudinal axis of the left ventricle: axis u was arranged normal to the local epicardial surface and points along the radial direction. Figure 1 Intensity data of the Diffusion tensor magnetic resonance imaging, depicting the zoning of the human left ventricle. Left: schematic representation. Anatomical structure segmented into 10 longitudinal evenly spaced slices, each circumferentially divided into 10 sectors. Right: visualization of the resulting n = 100 zones.

3 vi98 D.L. Weiss et al. The axis v was computed by the Cartesian product of the longitudinal axis Z of the ventricle and u and was used for describing local latitude. The third component w is the Cartesian product of u and v, lies in the plane defined by u and Z and thus points along the line of local longitude. Whereas the longitudinal axis Z was fixed and thereby identical for every voxel, the local epicardial surface, and hence the local coordinate system, can vary in each voxel. For a schematic representation of lc, see Fig. 26 in Streeter. 5 The primary eigenvector e 1 was extracted in each voxel from the CCBM dataset and interpreted in the corresponding local coordinate system lc. The helix angle a 1 of e 1 (inclination) in this system is defined in the v/w plane and characterizes the rotation of the fibre orientation through the ventricular wall. The transverse angle a 3 of e 1 (imbrication) within the u/v plane specifies to what extent the myocytes turn away from the epicardial surface. Both angles range from -908 to 908 and are suitable for describing the three-dimensional orientation of e 1 and thus of the myocyte. They were calculated by transforming the coordinates from the global CCBM to the local lc system. Computation of a 1 and a 3 was performed for every voxel lying on the line segment s. The transmural position of the voxel was calculated by normalizing its minimal distance d min to the endocardial surface to the length of s. Thus, the resulting distance d norm took values between 0 for endocardial and 1 for epicardial cells. The extracted values describe the transmural variation of the fibre orientation through the centre of the corresponding zone. The advantage of using the localized system lc is that the values of a 1 and a 3 are independent from the orientation of the whole ventricle. In this way, fibre orientation was specified in a general fashion. Combined with the transmural position normalized to wall thickness, this yielded a uniform method for specifying fibre orientation in each of the ventricular segments studied. Single ventricular segment including realistic fibre orientation The effect of different types of varying fibre orientation throughout the myocardium was studied by implementing these variations in a single tissue segment having a fixed geometry. The geometry represents a part of the free wall with a constant thickness (Figure 2). The height measured along the longitudinal axis of the ventricle was 12 mm and the width in circumferential direction was 24 mm. Inside this segment, cubic volume elements were distributed, having side lengths of 0.2 mm. These were arranged as layers forming ellipsoidal segments that followed the natural curvature of the ventricular wall. The number of layers determined the thickness of the resulting heart wall. The averaged thickness of the left ventricular wall extracted from the CCBM dataset was 14 mm. This permitted the placing of 70 layers parallel to the epicardium. The tissue segment comprised units, each representing a region to which the electrophysiological properties of active cardiomyocytes were assigned. For each of these units, the local diffusion properties related to fibre orientation were assigned by means of the method described above. The values of the a 1 and a 3 only depended on the transmural position. Thus, circumferential or longitudinal variations of the fibre orientation were not included. Figure 2 Tissue segment extracted from the human left ventricular wall. Thickness 14 mm; colour on the boundary codes the underlying 70 layers. The small white dots are situated on the tissue segment. The fibre orientations extracted from the distinctive zones in Figure 1 were inserted in the tissue. An example is visualized along the central line. Extracellular potentials F e were simulated at the four electrode locations indicated. The stimulation site was the endocardium at the point nearest to electrode location no. 1 (marked in red). Cellular model and electrophysiological heterogeneity The ionic model used for describing the dynamic electrophysiological properties of the units was the one formulated for the human ventricular cardiomyocyte by ten Tusscher et al. 38 It consists of a set of coupled non-linear ordinary differential equations describing state variables by Hodgkin-Huxley-like formalisms. 39 The differential equations were solved by numerically integration using the forward Euler method with time increments of 10 ms. Transmural heterogeneity of electrophysiological parameters was implemented by assigning different values to the units across the myocardial wall. The transmural profiles specified below are characterized by parameter values at the endocardium, mid-myocardium, and epicardium. The mid-myocardial region was taken in closer proximity to the endocardium than to the epicardium. A cubic spline interpolation was used to assign parameter values throughout the myocardial wall based on the values at endo-, mid-, and epicardial locations, resulting in a smooth transition of the heterogeneous parameters across the ventricular wall. The heterogeneous parameters included the maximum conductance of the slow component of the delayed rectifier current I Ks and the transient outward current I to. Slower recovery from inactivation for endocardial than for epicardial I to was incorporated by the implementation of heterogeneous time constants. In addition to the proposed inhomogeneities and according to experimental results, 40 the maximum conductance g Ks of I Ks was reduced to 92% in the endocardium and to 79% in the mid-myocardial units. Heterogeneity in the sodium calcium exchanger current I NaCa was introduced by assigning its largest value in mid-myocardial and smallest value in endocardial units (k NaCa,Endo = 1 pa/pf, k NaCa,mid = 1.5 pa/pf, k NaCa,Epi = 1.39 pa/pf). 41

4 Influence of fibre on activation and repolarization sequence vi99 The effect of assigned, distinctive parameterizations of the cell model were pre-calculated in a single cell environment, activated for 25 s with a basic cycle length of 1 s, in order to achieve steady-state initial conditions for the subsequent evaluations. The intrinsic action potential duration (APD) to 90% repolarization (APD 90 ) in this non-coupled environment varied from 268 ms at epicardial units, increasing to 317 ms mid-myocardially, and lowering to 278 ms at endocardial units. Comparable to the handling of the fibre orientation, the parameter setting for each voxel was only determined by its transmural position. This leads to a layered parameterization of the electrophysiology within the geometry that disallowed the provision for apico-basal or circumferential heterogeneities. Propagation of excitation and recovery The propagation of excitation and recovery results as the consequence of the current flowing through gap junctions between myocytes, passing membranes, and extracellular space. The numerical evaluation of this process can be carried out by using the bidomain theory. A complete bidomain diffusion model can handle the effect of gap junctions as well as the anisotropy of intra- and extracellular conductivity. 42,43 When assuming intra- (i) and extracellular (e) domains as having equal longitudinal (l) to transverse (t) conductivity ratios, the bidomain model can be reduced to the so-called monodomain model. The advantage of this approach is the reduced complexity of the equations: they contain only one unknown: the transmembrane voltage V m (x, y, z). This is the method used in this paper. Sealed-end or mathematically called von Neumann boundary conditions were imposed on all edges or faces of the domain assuming that no current is leaking out of the tissue. The values for the anisotropic conductivity tensors for the monodomain model were derived from the bidomain values proposed by Colli Franzone et al. 44 The transverse conductivity s t was V 21 cm 21, the longitudinal conductivity s l was V 21 cm 21. The resulting propagation velocity transverse to the fibres, and thus in the transmural direction, was in the order of 0.4 m/s. The parabolic equation was implemented using the Finite Differences Method and was solved with a time step of 20 ms. Conductivity between the computational cells via the gap junctions was assumed to be constant within the geometry. The tissue was excited by applying a spherical suprathreshold stimulation. To this end, a transmembrane current lasting 2 ms was impressed spatially over a region specified by a three-dimensional normal curve of distribution (amplitude 25 pa/pf, s = 0.8 mm) in the endocardium close to electrode no. 1 (Figure 2). This position of the stimulus yielded the longest possible activation pathway within the tissue. The pseudo-electrocardiogram The ECG reflects the complex process of activation and repolarization of cardiac tissue as expressed by the time course of the potential difference observed between any two tracking electrodes that were influenced by the electric current generated by the heart. When the electrodes are situated close to the endocardium and/or the epicardium, the resulting signals are commonly referred to as electrograms. The signals that can be observed if one electrode is in the direct vicinity of the epicardium and the other one close to endocardium have been named transmural ECG, or even pseudo-ecg (p-ecg). Here, the notation p-ecg is used. It is computed to study the relationship between the relatively simple source configuration of the tissue segment and its expression in the form of ECG like signals. Based on the ionic model of cardiac myocytes and by using the monodomain model of diffusion, electric source ejecting current I m into the passive, resistive extracellular medium is I m ¼ I ion þ I c ; ð1þ with I ion the current resulting from the active membrane processes and I c = C m /@t the part of I m involved in charging up the membrane capacitance C m to the transmembrane voltage V m. The potential field F e (P, t) set up at position P in the extracellular medium can be approximated by F e ðp; tþ ¼ 1 ð I m ðtþ dv; ð2þ 4ps vol r with the distance r from the observation point P to the current source, and the integral taken over the active source region (the myocardium) The approximation assumes that the electric conductivity of the extracellular domain and the medium external to the tissue are both equal to s. The p-ecg was computed as the difference between the potentials at positions P epi close to epicardium and P endo close to endocardium p-ecgðtþ ¼F e ðp epi ; tþ F e ðp endo ; tþ: ð3þ The major wavelets observed in the p-ecg have been referred to in the literature as QRS complex and T wave, 27,28,31 similar to the convention for the surface ECG. Note however that a direct comparison between such signals is impossible because of the extreme differences in the source configuration, lead placement, and the volume conductor properties involved. The extracellular potential during computations in this study was simulated at two pairs of electrode locations. Two electrodes were placed at the endocardial, two at the epicardial tissue boundary. The distance between electrode tips and tissue was 4 mm. One electrode pair was placed close to the stimulation site, the other pair was set more distally (Figure 2). Computational environment Each simulation was conducted on an Apple XServe G5 2 GHz processor. The computation of a heart beat (500 ms) in the three-dimensional ventricular heart wall took 24 h. Subsequent calculation of the extracellular potential in the four electrodes was finished in 1 h. Results Fibre orientation The transverse angle a 3 between the epicardial surface and the longitudinal orientation of the myocyte was in the range of 2208 to 208 in the DT-MRI measurements. Examples of the transmural profiles of a 1 and a 3 are shown in row B of

5 vi100 D.L. Weiss et al. the Figures 3 and 4, which relate to segments taken from one slice or from different slices, respectively. The distribution of fibre orientation showed only small variations when the segments belonged to one slice (Figure 3B). The positive helix angle a 1 in endocardium decreased across the wall and was negative in the epicardium. Rearrangements of both value and slope of the distribution of the transverse angle a 3 were small through the heart wall. The fibre orientation in segments taken from the same circumferential but different longitudinal positions showed higher differences (Figure 4B). While the helix angle in the different configurations showed the twist also reported for animal data, 16,48 the slope of the transverse angle varied greatly. In the basal fraction (slice no. 3) it was negative, a 3 has positive values in the endocardium and negative values in the epicardium. In the apical fraction of the ventricle (slice no. 9), the slope of a 3 was positive and the values range from negative in the endocardium to positive in the epicardium (Figure 4B). Effect on propagation The computation of the activation and repolarization sequences was performed for all n = 100 fibre orientation distributions obtained from the different zones of the DT-MRI dataset, applied to the tissue segment as shown in Figure 2. The preferential direction of the wave front followed the orientation of the muscle fibres. 43 Generally, the fastest Figure 3 Simulation of the electrical activity in the tissue segment model. Fibre orientation distribution extracted from three different positions (sector no. 1 left column, sector no. 5 central column, sector no. 9 right column) within the basal slice no. 3 (Figure 1). Stimulus site as indicated. (A1 and A2) Transmembrane voltage at two different time steps. Colour coding: blue: resting state; red: activated tissue. (B) Transmural distribution of helix and transverse angle. (C) Corresponding p-ecgs derived from electrode nos. 4 and 1 (Lead 4.1) as well as nos. 3 and 2 (Lead 3.2).

6 Influence of fibre on activation and repolarization sequence vi101 Figure 4 The effect of variation in fibre orientation in sector no. 8 taken from different slices (basal slice no. 3 left column, equatorial slice no. 6 central column, apical slice no. 9 right column). The units distal to the stimulation were activated from the mid-myocardium in slice no. 9. propagation of the wave front was in the circumferential direction. Approximately 22 ms after the stimulation, the circumferential excitation was completed. Transmural propagation through the 14 mm thick wall took 25 ms (data given for simulation in slice no. 9, sector no. 8). The activating wave front ended always in the epicardial region closest to electrode no. 3. The most notable results were observed when comparing the influence of the fibre orientations taken from different sectors within one and the same slice (circumferential direction of the ventricle), and for sectors taken from different slices (longitudinal direction of the ventricle). These are the situations for which examples of fibre orientation are specified in rows B of Figures 3 and 4. Figure 3 indicates that the transmural differences of the transverse angle a 3 were small. Accordingly, transmural activation as well as repolarization sequences based on fibre orientations taken from one and the same slice were similar. For the situations illustrated in Figure 4, larger endo- to mid-myocardial absolute values of a 3 resulted in a faster transmural propagation. Isotonic interaction In tissue, the electrotonic coupling between the units only marginally reduced the intrinsic APD 90 values in the midmyocardial region, but clearly prolonged them in endocardial units. In the epicardial units, a slight increase of APD 90, up to 278 ms was observed. The action potential waveforms of epicardial cells were only slightly affected by the different parameter setting in the mid-myocardial units. With the assigned inhomogeneous parameter settings, the activation delay between endo- and epicardium was smaller than the difference of their APD 90 values. Correspondingly, repolarization was earlier at the epicardium, delayed at endocardium, and generally ended

7 vi102 D.L. Weiss et al. on a 3 was observable in regions distal to the stimulation: the larger its absolute value, the more mid-myocardially did the excitation in this area start (Figure 4A). However, any such changes in the T wave of the distal Lead 3.2 were not significant (Figure 4C). Figure 5 Transmembrane voltage after stimulation at three units within the tissue, along the line connecting electrodes 1 and 4 (Figure 2). Colour coding: red, green, and blue for endocardial, midmyocardial, and epicardial units, respectively. Data shown pertain to the fibre orientation of slice no.9, sector no. 8. in the mid-myocardial region. This is illustrated in Figure 5, which displays the action potentials of three myocardial units along the line from electrode positions 1 to 4. These results pertain to the fibre orientation shown in Figure 4 (slice no. 9), but the effect was observed quite generally for the other cases. The action potentials are those of an endocardial, a mid-myocardial, and an epicardial unit. Pseudo electrocardiograms The simulated pseudo-ecgs in Lead 4.1 and Lead 3.2 corresponding to the activation sequences illustrated in Figures 3A and 4A are shown in panels C of the same figures. In all cases, the p-ecgs showed narrow positive QRS complexes corresponding to a propagation of the depolarizing wave front from endocardium to epicardium. The subsequent J wave was found to be the impact of the heterogeneously distributed transient outward current. A long, nearly isopotential line segment followed, which reflects the nearly constant plateau of the transmembrane voltages. Since APD plus activation delay at the epicardium (mean values: 278 ms and 25 ms, respectively) was smaller than at the endocardium (mean value: 309 ms), the repolarization started in epicardium and finally ended in the mid-myocardial area. The positive T wave in the p-ecg was generated by this complex inverted repolarization sequence, as was also shown in measurements and simulations As shown in the Figures 3A1 and 3A2, the activation sequences within a horizontal section of the left ventricle were similar. As a consequence, the shapes of the QRS complexes and T waves in the p-ecgs were comparable between both leads (Figure 3C). The delayed onset of both waves in Lead 3.2 can be explained by the delayed activation of the tissue in the nearby region. For the cases illustrated in Figure 4, the QRS complexes were narrower for larger absolute angles a 3 than for angles in the range of zero degrees, as they are present in the basal and equatorial parts (Figure 4B). The T wave in Lead 4.1 appeared later and exhibited a reduced amplitude in slices with a higher index (Figure 1), which were closer to the apex: (299 ms/ 2.30 mv, 299 ms/2.41 mv, and 304 ms/1.90 mv for slices nos. 3, 6, and 9, respectively). Another effect depending Discussion Measurements showed that the extracellular potential on the epicardium is influenced by the fibre orientation of the underlying tissue. 24 This computational study determined the impact of structural differences in ventricular tissue related to fibre orientation on extracellular potentials. The utilized anatomical structure was a 3D segment of the human left ventricular free wall, which included the anisotropic diffusion properties based on fibre orientation. The integrated course of the myocardial fibre was gathered from multiple positions of a DT-MRI measurement of the human left ventricle. The impact of fibre orientation distributions extracted from a transverse section in the basal region as well as taken from segments having the same circumferential but different longitudinal positions were investigated. Differences in the direction of the endocardial myocytes mainly affected the beginning of the activation sequence. A helix angle a 1 of zero degree points in a circumferential direction and has no longitudinal contribution while a helix angle with an absolute value of 908 points along the longitudinal axis of the ventricle. 5 Therefore, larger values of a 1 increased the longitudinal portion of the preferential direction of the activation wave. The size of the transverse angle a 3 mainly affected the transmural activation sequence. This angle is zero when the myocyte is aligned tangentially to the epicardial surface. The maximum value of 908 is reached as soon as the main direction of the myocyte is normal to the epicardium. In this case, the preferential route of the conducting wave is in transmural direction. While no considerable variations of a 1 or a 3 were detected in the circumferential distribution, the longitudinal dispersion of the fibre orientation within the left ventricle showed a significant change of slope of a 3 (Figure 4B). The absolute value of a 3 in endocardial regions increased towards the apex slices nos. 3 to 9. Therefore, the transmural activation sequence close to the apex was faster than in the basal portion of the ventricle. Additionally, larger values of a 3 in endocardium led to a more midmyocardial beginning of the activation in areas distal to the stimulation. In the pseudo-ecgs, changes of the excitation and repolarization sequences were visible, showing differences in widths and amplitudes of the corresponding waves. The temporal differences associated with different fibre orientation were in the range of milli-seconds. The largest changes due to variations in a 3 were mainly visible in the T wave in Lead 4.1, simulated for the longitudinally varying configuration (Figure 4). The stimulation of the three-dimensional tissue block was set only in a small-sized area. With this setup, an impact of varying fibre orientation on the longitudinal and circumferential direction of the exciting wave was enabled. In the real heart, the Purkinje fibre network initiates the excitation. Due to its uniform distribution, large longitudinal or circumferential activation pathways are avoided and

8 Influence of fibre on activation and repolarization sequence vi103 the initiation of the excitation sequence in different regions of the heart is comparable. Limitations Due to the cardiac motion, the acquisition of spatially highresolution in vivo DT-MRI dataset is a very challenging task. 49 Generally, the reconstructed myofibre morphology has a resolution of 1 to 2 orders of magnitude lower than histological measurements. However, this resolution suffices for extracting the macroscopic expression of fibre structure in the form of the diffusion tensor. The monodomain model combines the intra- and extracellular space by assuming a constant factor between their longitudinal to transverse conductivity ratio. In contrast to the bidomain model, the extracellular potentials cannot be calculated explicitly and the effect of unequal anisotropy ratios between both spaces cannot be evaluated. However, Potse et al. 50 tested the accuracy of a monodomain model by comparing the results with a bidomain model and showed that differences between bidomain and monodomain results were extremely small, even for heterogeneous electrophysiology and anisotropic conductivities. In their computational study, only a small difference between the conduction velocities calculated by the two models was identified. All other properties of the membrane voltage were accurately reproduced by the monodomain model. 50 They have shown that the extracellular potentials computed from the membrane currents produced by the monodomain model are very close to those of a bidomain model. Their conclusion was that simulations of pacing or defibrillation necessitate the usage of the bidomain model, but propagating action potentials in tissue on the scale of the human heart can be studied with the monodomain model. We have performed similar experiments with propagating wave fronts set up in a small tissue segment, which were calculated with monodomain as well as the bidomain formulation and came to the same result (not shown here). Hence, the shapes of the wave fronts in our study can be considered as realistic. The assumption of equal anisotropy ratios prohibits any direct expression of prolonged APD 90 values of the mid-myocardial units in p-ecgs. 51 However, their strong, indirect effect on endo- and epicardial action potentials due to electrotonic coupling does become expressed. In a future study, we will calculate the electrical activity in the realistic whole heart anatomy that belongs to the DT-MRI dataset. In this case, the fibre orientation is specified in each volume element, based on the full data of the DT-MRI measurement. The complete heart model can be inserted into a torso model comprising different electrical properties of the several organs. Simulations with this environment can study the impact of fibre orientation variations on body surface potential maps and can identify the relation between local transmural and body surface ECGs. Additionally, the Purkinje fibres adjacent to the endocardium will be incorporated in the model. Their presence is likely to also affect the repolarization sequence. Furthermore, the heterogeneous nature of gap junction distribution across the heart wall 52 as well as apico-basal and interventricular gradients of the biophysical properties of the myocytes 32,53 55 remain to be included in prospective simulations. Future calculations of force development and deformation with such models can determine the influence of varying fibre orientations on mechanical activity and contraction of the heart. Conflict of interest: none declared. Acknowledgements We thank Patrick A. Helm and Raimond L. 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