Part I: tissue specific model generation
The model construction methods are based on those used by Costa et al. [15], and LeGrice et al. [10, 11], with several modifications. All hearts used in model development were acquired following in-vivo electrophysiological measurement made in open chest pigs of weight 42–60 kg. The models were specifically developed to aid comparison of patterns of electrical propagation measured by an array of plunge electrodes in the in-vivo heart, and those predicted by a bidomain computational model of electrical activation [19]. All animal procedures were carried out under approval of the University of Auckland Animal Ethics Committee. In each animal, the heart was arrested by injection of cardioplegic solution into the LV, and then rapidly excised from the animal. The coronary arteries were flushed of blood using chilled cardioplegic solution, and electrodes replaced with styrene rod markers. The heart was then fixed by slow injection of 3% formalin in phosphate buffer solution simultaneously into all three coronary arteries.
A cardiac coordinate system (X1, X2, X3) was defined as aligned with the local circumferential (X1), longitudinal (X2), and radial (X3) axes of the LV (Figure 1). The block of tissue to be modelled is excised from the mid-anterior LV freewall, by cutting in the X2-X3 and X1-X3 planes (Figure 1). The block centres were located from the apex between one-third and one-half the distance from the apex to the atrioventricular sulcus, and in the area of watershed between the left anterior descending and circumflex artery supply territories. Typical block sizes were ~15 × 15 × 15 mm. The block is then bisected longitudinally to yield two blocks (a and b, see Figure 1). These blocks are frozen rapidly in a cryostat (Leica; CM1510) cooled to -35°C. At this stage, distance measurements between adjacent styrene rods, marking the in-vivo electrode locations along the longitudinal (X2) axis, could be used to assess the degree of tissue shrinkage sustained during the fixation and freezing processes. Geometrical measurements made in the frozen tissue were later scaled in an isotropic fashion according to these measurements.
Using the cryostat, frozen sections of tissue (20 μm thickness) are then taken from both tissue blocks, transferred onto glass slides, and photographed. Serial sections in the plane of the epicardium (X1-X2 plane) are collected every 500 μm from block a, and used to characterise the variation in myofibre angle throughout the ventricular wall. From block b, a single slice is taken in the base-apex (X2-X3) plane, followed by serial sections (collected every 200 μm) in the circumferential (X1-X3) plane (Figure 1). The base-apex section, and the series of circumferential plane sections, all intersect myolaminae, the angles of which can be measured relative to the cardiac coordinate system. Registration of circumferential plane sections to the base-apex section was aided by the placement of two fiducial rods through block b in the X1 direction (Figure 1). It was found that much finer resolution of myolaminal organisation was possible with 20 μm thick sections compared to thicker (0.1–1 mm) sections used in previous studies [11, 15].
Tissue sections from one processed myocardial tissue block (Ex07) are shown in Figure 2. The base-apex (X2-X3) plane slice is shown at upper-left. To the right of this section are three representative circumferential (X1-X3) plane slices, registered to the base-apex slice at three X2 locations. Beneath are five epicardial (X1-X2) plane sections showing fibre angle orientations at different transmural depths in the tissue. Microstructural angles can be measured from sections in each of the three planes. Under the assumption that myofibres run in-plane to the epicardium (imbrication angle is zero), angles measured in the epicardial plane slices represent myofibre orientation. Angles measured on the base-apex and circumferential plane slices represent the local angle of intersection of myolaminae with the slice plane. Microstructural angles are determined in each plane automatically from tissue section images using a gradient intensity algorithm [20, 21]. Applied to the base-apex plane section, the algorithm was used to determine angles on a 30 × 30 grid (Figure 2; lower-right inset). Each circumferential plane section was similarly processed to obtain a set of 30 angles along the edge of the tissue that abutted the base-apex plane section. A single microstructural angle was computed for all epicardial plane sections. Following notation used previously [15], the microstructural angles measured on epicardial plane sections are termed α, those measured on the base-apex slice are termed β', and those on the circumferential plane slices β". Fibre angles (α) were measured relative to the cardiac X1 axis, whilst β' and β" angles were both measured relative to the X3 axis. Angles were signed positive or negative in accordance with previously developed convention [15].
The model allowed variation in α along the X3 direction, whilst β' and β" both varied in X2 and X3 directions.
Myolamina orientation was usually difficult to discern from base-apex and circumferential sections immediately adjacent to the epi- and endo-cardium, where coupling of adjacent laminae is known to be tightest [14, 22]. In these areas, and where blood vessels obscured the laminar structure, the microstructural angle β' or β", computed by the gradient intensity algorithm, was manually rejected.
Complete sets of β' and β" angles are shown for the same tissue block (Ex07) in Figure 3. Areas where microstructural angles could not be determined confidently are shown in grey. The set (30 × 30 elements) of β' angles (Figure 3; left panel) are derived directly from the base-apex plane image shown in Figure 2 (upper-left panel). The corresponding set of β" angles (Figure 3; middle panel) consists of 30 rows each of which is derived from a single circumferential plane tissue section, along its edge that abuts the base-apex plane section.
A continuous description of fibre angle variation through the wall was generated by fitting 10 linear finite elements to the fibre angle data (Figure 3; right panel). Fitted fibre angles at a given transmural depth (X3 location) could then be combined with measured β' or β" angles at the same transmural depth to derive sheet angles (β) at that depth, using the formulae [15]:
tan β = tanβ'cosα (|α| ≤ 45°)
tan β = -tanβ"sinα (|α| > 45°)
Equation 1 was applied where the absolute fibre angle was ≤ 45°, as in this range β' provided the most accurate projection of sheet angle, whilst equation 2 was applied outside this range, where β" provided the most accurate projection [15]. The final tissue model (Figure 3; right panel) consisted of the continuous fibre angle description across the wall, and up to 900 discrete sheet angles (β) mapped to locations in the base-apex plane. The discrete description of sheet angles allowed accurate representation of discontinuities in sheet angle across the wall.
Part II: application to electrical simulation
The electrical action of the heart can be best represented by a bidomain model in which the intra- and extra- cellular spaces of the heart are conceptualised as interpenetrating, and extending throughout the entire model volume. The governing equations of the model [3, 19] are solved within the CMISS [23] computational framework using a number of different techniques, however, for this study, a grid-based finite element solution technique is used as previously described [3, 24]. This solution technique allows the piecewise discontinuous incorporation of myofibre and myosheet angles on an element-by-element basis over the solution domain.
Current of 0.03 mA is withdrawn from the extracellular space of two example model tissues, at a point located centrally in the model volume (~8.5 mm below the epicardium). Equal magnitude (but opposite sign) current is uniformly distributed amongst all the tissue boundaries except for the epicardium, to match the experimental case of recordings taken from an open chest pig. The steady-state extracellular potential (Φ
e
) field generated by the current sink (cathode) is simulated first in the absence of any cellular response to the current. Subsequently, the actively propagated wavefront generated by the cathodal current is examined by simulation of the activation time (AT) field over the model volumes. A simple cubic model of the cardiac action potential is utilised for the active models [3, 25]. The conductivities used in the models (chosen in approximate accordance with previous work) [3] are (in units of S/m) g
il
= 0.263, g
it
= 0.0263, g
in
= 0.008, g
el
= 0.263, g
et
= 0.245, and g
en
= 0.1087. Membrane capacitance of the models was set to 0.01 μF/mm2, and membrane conductivity to 0.004 mS/mm2. The cubic action potential model had a resting potential of -85 mV, a threshold potential of -80 mV, and a plateau potential of 15 mV.