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Simulation of QRST Integral Maps With a Membrane-Based Computer Heart Model Employing Parallel Processing Marie-Claude Trudel, Bruno Dubé, Mark Potse*, Ramesh M. Gulrajani, Fellow, IEEE, and L. Joshua Leon, Member, IEEE
In memory of Dr. Ramesh M. Gulrajani (1944–2004). The authors would like to dedicate this paper to their late colleague. Abstract—The simulation of the propagation of electrical activity in a membrane-based realistic-geometry computer model of the ventricles of the human heart, using the governing monodomain reaction-diffusion equation, is described. Each model point is represented by the phase 1 Luo–Rudy membrane model, modified to represent human action potentials. A separate longer duration action potential was used for the M cells found in the ventricular midwall. Cardiac fiber rotation across the ventricular wall was implemented via an analytic equation, resulting in a spatially varying anisotropic conductivity tensor and, consequently, anisotropic propagation. Since the model comprises approximately 12.5 million points, parallel processing on a multiprocessor computer was used to cut down on simulation time. The simulation of normal activation as well as that of ectopic beats is described. The hypothesis that in situ electrotonic coupling in the myocardium can diminish the gradients of action-potential duration across the ventricular wall was also verified in the model simulations. Finally, the sensitivity of QRST integral maps to local alterations in action-potential duration was investigated. Index Terms—Action-potential duration, cardiac membrane model, computer heart model, parallel processing, QRST integral maps.
I. INTRODUCTION
C
OMPUTER models have long been used for the simulation of electrical activity in the heart and the subsequent computation of the electrocardiogram (ECG). Many of these models were of the “cellular automaton” type, with a rule-based Manuscript received July 8, 2002; revised November 11, 2003. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The work of M.-C. Trudel was supported by la Fondation J.A. de Sève and by FCAR, Québec. The work of M. Potse was supported by FCAR, Québec, and by the Netherlands Organization for Scientific Research (NWO). Asterisk indicates corresponding author. M.-C. Trudel was with the Institute of Biomedical Engineering, Université de Montréal, Montréal, QC H3C 3J7, Canada. She is now with EBS Inc., Montréal, QC H2X 3V8, Canada. B. Dubé is with the Institute of Biomedical Engineering, Université de Montréal, Montreal, QC H3C 3J7, Canada (e-mail:
[email protected]). *M. Potse is with the Institute of Biomedical Engineering, Université de Montréal, Montreal, QC H3C 3J7, Canada (e-mail:
[email protected]). R. M. Gulrajani, deceased, was with the Institute of Biomedical Engineering, Université de Montréal, Montreal, QC H3C 3J7, Canada. L. J. Leon was with the Institute of Biomedical Engineering, Université de Montréal, Montréal, QC H3C 3J7, Canada. He is now with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada. Digital Object Identifier 10.1109/TBME.2004.827934
excitation algorithm based on the known propagation velocities of the action potential along and across the cardiac fibers. An example is the model of Lorange and Gulrajani [1]. Once the excitation reached the individual model cells, a predetermined action-potential waveform was started at each cell. A notable exception to this cellular automaton approach was the model of Leon and Horáˇcek [2] that used subthreshold electrotonic conduction to bring the cells to a threshold membrane potential for excitation, with a predetermined action-potential waveform started at this excitation potential as before. More recently, reaction-diffusion equations have been used to characterize both subthreshold and suprathreshold activity, initially with a simple FitzHugh–Nagumo-type membrane equation characterizing the model cells [3]–[5] and later with a modified Beeler–Reuter membrane equation to represent the model cells by Huiskamp [6]. Here, we present a heart model, similar to Huiskamp’s, that also uses the reaction-diffusion equation but with the modified Beeler–Reuter membrane equation replaced by a modified version of the more recent Luo–Rudy phase 1 membrane equation [7]. The anatomy of our heart model is the same as that of the earlier one by Lorange and Gulrajani [1]. To enable the numerical integration of the reaction-diffusion equation, however, it was necessary to augment the spatial resolution of the Lorange–Gulrajani model from 1 to 0.25 mm by the addition of intermediate points. Like Huiskamp, we too only simulated excitation of the ventricles, using starting points and starting times at the endocardium taken from the Lorange–Gulrajani model. Since the spatial resolution is much finer than that of Huiskamp’s model (0.25 as opposed to 0.6 mm), simulations with the much larger number of points (approximately 12.5 million as opposed to 800 000 in Huiskamp’s model) necessitated parallel processing on a multiprocessor Silicon Graphics Origin 2000 computer. Also incorporated into the heart model was a region of “M-cells” in the ventricular midwall, as has been documented in both dogs [8] and in humans [9]. These M-cells have been measured to have an action-potential duration that is longer than those of endocardial or epicardial cells. This sets up a transmural gradient of membrane potential during repolarization that is largely responsible for the T-wave in the ECG being of the same polarity as the QRS complex. Consequently, the integrated area under each ECG lead during the QRS and T complexes is nonzero. A plot of this integrated area versus the two-dimensional (2-D) spatial location of the ECG lead on the
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body surface yields a so-called QRST integral map, and it has long been known that these QRST integral maps are relatively independent of the sequence of heart activation but sensitive to gradients of action-potential duration [10], [11]. In other words, the QRST integral maps are uniquely sensitive to the dispersion of action-potential durations. Since this dispersion can facilitate reentrant arrhythmic circuits, it is useful to test the sensitivity of QRST integral maps to changes in action-potential durations across the ventricles. In simulations of normal and ectopic activation, we verify the invariance of QRST integral maps to changes in activation sequence and then show that local changes in action-potential duration are mirrored by changes in the QRST integral maps. Preliminary versions of portions of this work have been published earlier as short conference proceedings [12]–[14].
TABLE I SIMULATION PARAMETERS
II. METHODS A. Reaction-Diffusion Equation The governing reaction-diffusion equation may be derived from the bidomain representation for the myocardium [15], in which the intracellular and interstitial domains are characterized by (1) (2) and are the intracellular and interrespectively. Here, and the intracellular and stitial potentials, respectively, interstitial conductivity tensors, respectively, the surface-tovolume ratio of the cardiac cells, the transmembrane current the intracellular stimulacoupling the two domains, and tion current used to start excitation at the selected endocardium passes from intracellular points. The transmembrane current to interstitial space and is given by the sum of the capacitive and ionic currents (3) F cm is the specific membrane capacitance where and is the transmembrane potential. The ionic is represented by the aforementioned Luo–Rudy current phase 1 membrane model [7], with its action-potential duration modified to match those of human ventricular cells. The combination of (1)–(3) together with the simplifying approximation of equal anisotropy in the intracellular and interstitial spaces results in the desired reaction-diffusion equation
(4) The assumption of equal anisotropy reduces the bidomain problem to a simpler monodomain one, with only the spatial being sought via the solution and temporal distribution of of (4). B. Conductivity Parameters Fiber directions rotate counterclockwise from epicardium to endocardium in the Lorange–Gulrajani model and are
determined analytically with a modification of the Beyar and Sideman [16] equation. We used a total transmural rotation of containing the 120 . Due to the rotation, the diffusion tensor interstitial conductivities varies from point to point. Interstitial and , were conductivities along and across cardiac fibers, taken to be 6.0 and 1.5 mS/cm, respectively, for an anisotropy ratio of 4. The parameter was taken as 0.5 so that intracellular conductivities were half the above interstitial values. All four conductivities were in the ballpark of measured conductivity values [17]–[19]. Moreover, using the equations given by Roth [20], the theoretically calculated ratio of wavefront velocities in the longitudinal and transverse fiber directions (which is equal to the ratio of longitudinal to transverse length constants, ) was 2. In adjusting for equal anisotropy, it is important to keep this ratio within measured experimental limits in order to obtain simulated wavefronts that will mimic observed wavefronts in the heart. Roth [20] estimates this ratio to be 2.2 0.5 from the experimental data given in [17]–[19], and our theoretical ratio falls within this range. For convenience, these and other numerical values used are summarized in Table I. C. Modifications to the Luo–Rudy Phase 1 Model In order to shorten the action-potential duration of the Luo–Rudy phase 1 model to correspond to those of human and responsible for ventricular cells, the time constants the kinetics of the slow inward current in the Luo–Rudy model were reduced by multiplying them by a factor of 0.5. The resultant action-potential waveform was used to represent endocardial as well as epicardial cells in our heart model and has a duration (at 70-mV repolarization) of 237 ms (Fig. 1). To represent the longer action-potential durations of and were kept the M-cells, however, the time constants at their Luo–Rudy values. The action-potential waveform of these M cells is also shown in Fig. 1 and its duration is 346 ms. By way of comparison, Drouin et al. [9] have measured the mean action-potential durations of endocardial, epicardial, and M-cells in humans at a pacing cycle of 1000 ms (60 beats/min) as 330, 351, and 439 ms, respectively. Our shorter action-potential durations correspond to a more normal heart rate of approximately 72 beats/min. Admittedly, the simple adjustment of the Luo–Rudy model described above only reproduces
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Fig. 1. Action-potential waveforms used to represent endocardial and epicardial cells (A), M cells (B), and “ischemic” reduced action-potential duration (APD) cells (C). Waveform (B) is also identical to that generated by the original Luo–Rudy phase 1 membrane model.
action-potential durations. It does not reproduce the more subtle variation in action-potential waveform, in particular between epicardial and endocardial cells, caused by the transient [21] that is absent in the Luo–Rudy model. outward current Transverse and longitudinal sections of the heart model are depicted in Fig. 2 and show the M-cell layer sandwiched in between endocardial and epicardial cells. D. Solution of the Reaction-Diffusion Equation Finite differences were used to solve (4) over the irregular ventricular domain. An isolated heart was assumed so that a boundary condition of zero normal current at the epicardium and endocardium was utilized. The temporal derivative of was replaced with a forward Euler approximation. A Rush–Larsen integration scheme [22], and the use of table lookups for the needed activation and inactivation variables, permitted rapid determination of the Luo–Rudy ionic currents. Approximation of the diffusion current, given by the first term on the right-hand side in (4), was done by an algorithm described by Saleheen and Ng [23]. A time step of s was used during the action-potential upstroke; this was increased to s during the plateau. The crossover to the slower time step occurred when all model points had terminated their upstrokes. The value of 1200 cm chosen for was selected to control the total excitation time, which was approximately 90 ms for normal activation of the ventricles. This is smaller than the usual value of cm used in cardiac tissue simulations [24]. The smaller value for , by increasing in (4), compensates for a slower propagation velocity due to, first, a membrane model that is not a perfect match for human ventricular cells and, second, a spatial discretization that is also not ideal. Incidentally, Huiskamp in his whole-heart simulations used a much smaller value of 400 cm for , perhaps also to keep activation times realistic.
Fig. 2. Transverse (top) and longitudinal (bottom) sections of the heart model showing the M cell layer sandwiched between endocardial and epicardial cell layers.
It is worthwhile to do a rough stability analysis of the finite-difference solution to (4). This equation is a parabolic equation whose simpler form is the ordinary one-dimensional (1-D) diffusion equation (5) Using a finite-difference approximation (with spatial interval ) and a forward Euler approximation for the temporal derivative (with time increment ), if the diffusion coefficient is constant, then the numerical solution will be stable as long as [25] (6)
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In three dimensions, once again with a constant diffusion tensor whose principal directions are oriented along the , , and finite-difference axes, (5) becomes (7) and condition (6) becomes (8) , , where the spatial increment in all directions is and are the three elements of the diagonal diffusion tensor. and If we compare (7) with (4), and assume that the interstitial conductivity tensor is diagonal and invariant, then applying (8) we get (9) where and are the interstitial conductivities across and along fibers. Substituting our numerical values (see Table I), this yields the condition s
(10)
This is only a ballpark figure for the stability condition since it does not consider the rotating fiber directions, so that not only is the conductivity tensor varying but also its principal axes are not along the , , and directions. Nevertheless, given our values of s and s, it does suggest that our solution is stable. We did not observe any hint of instability in our solutions at the spatial and temporal discretizations used. A second important consideration is that of accuracy. Wu and Zipes [26], in simulations of 1-D propagation using the cable equation and the Luo–Rudy phase 1 membrane model, have shown that at spatial discretizations of 0.01 cm there are errors of 13.75% in the maximum rate of rise of the action-potential upstroke, 11.36% in the time constant of the foot of the action-potential upstroke, 3.23% in conduction velocity and 0% in action-potential duration (see their Fig. 3). Larger errors than these are therefore to be expected with our spatial discretization of 0.025 cm. These errors must be carefully analyzed, since moving to a finer spatial discretization would entail excessive computer memory usage and prohibitively long computer run times. To a large extent, we have compensated for the expected decrease in conduction velocity with a reduction in the surface-to-volume ratio so as to ensure physiologically correct excitation times for the model. The question now remains as to how our poorer spatial discretization will impact the action-potential upstroke and duration, and whether errors in the action-potential waveform will compromise our ECG simulations. Two sets of simulations were run, at different spatial discretizations, in a cubic block of myocardial tissue of dimensions 1 1 2 cm. Fibers were assumed oriented along the longer dimension. A planar wavefront was launched first along, and then transverse to, these fibers by exciting all points on a block face perpendicular to the direction of propagation. The first simulation employed the coarse spatial and temporal discretizations used by us in the heart model, namely cm s, with cm . The observed lonand
2
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Fig. 3. Action-potential upstrokes in the 1 cm 1 cm 2 cm block with spatial discretizations of 0.025 cm (coarse) and 0.01 cm (fine), during (a) longitudinal propagation and (b) transverse propagation.
gitudinal and transverse velocities were 78.6 and 31.9 cm/s for a ratio of 2.46. This ratio is in reasonable agreement with the expected theoretical value of 2. Moreover, the discrepancy can be explained by the fixed spatial segmentation in both longitudinal and transverse directions. Wu and Zipes [26] have shown that the simulated velocity of propagation diminishes with an increase in the relative spatial segmentation (spatial segmentation divided by the length constant). Since the transverse length constant is smaller than the longitudinal one, the relative spatial segmentation is larger for transverse propagation, leading to a greater decrease in the simulated transverse propagation velocity. This increases the longitudinal to transverse propagation velocity ratio to a value greater than the theoretically expected ratio of 2. The temporal distributions of the longitudinally propagating and transversely propagating action-potential upstrokes of this first set of simulations were as shown in Fig. 3(a) and (b), respectively. Next, keeping the same, the spatial and temporal discretizations were reduced to cm, and s. Longitudinal and transverse velocities were now 84.6 and 40.3 cm/s for a ratio of 2.1, and the action-potential upstrokes are also shown in Fig. 3. The increase in longitudinal and transverse velocities is expected,
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as is the better correspondence of the ratio to the theoretically expected value of 2. Apart from a slight ringing, the temporal upstrokes of longitudinally propagating action potentials with the improved discretization are close to those with the discretization we employed. Major discrepancies with the coarser discretization were noted, however, for the transversely propagating action potential. These were a longer foot and a larger ringing artifact at the end of the upstroke [Fig. 3(b)]. Since the predominant direction of propagation in normal excitation of the heart is transverse to the fibers, the effect of these discrepancies on the ECG merits immediate discussion. Given that the ECG is proportional to the spatial gradient of the transmembrane potential, the major determinants of the ECG are the velocity and spatial pathways of the excitation. Due to the spatial gradient operation, similar alterations in action-potential upstroke at every model point should affect the ECG (and by extension the QRST integral map) to a lesser extent. Thus, the errors should be small as long as the conduction velocity and excitation pattern are not affected. Finally, action-potential durations in the block between simulations effected with coarse and fine discretizations differed by at most 1 ms, indicating little difficulty with repolarization simulations. E. Computer Implementation Simulations were generally run using 16 of the 64 R12000 processors of the Silicon Graphics Origin 2000 computer. The clock speed of this machine is 400 MHz. The simulation program consisted of three main steps, executed in sequence at each time step. These steps were the calculation of the ionic current, the diffusion current, and finally the transmembrane potential. Each of these three main steps was parallelized, thereby creating parallel threads within the steps, with each thread taking care of a particular spatial region of the heart model and sent to a separate processor. The number of processors to be used is assigned ahead of time. Normal excitation of the heart model using 16 processors took just over 14 h (50603 s), with depolarization taking just over 6 h (21695 s ) and repolarization the balance. Normal excitation simulations were also run with a reduced number of processors (1, 2, 4, and 8, respectively), in order to obtain an idea of the speedup attained by using multiple processors. Ideally, doubling the number of processors should half the run time. The times using 1, 2, 4, 8, 16 processors scaled according to the ratio 1, 0.54, 0.37, 0.22, 0.14, instead of the theoretical 1, 0.5, 0.25, 0.125, 0.0625. The increased values over the theoretical minimum reflect the lost time due to unbalanced load sharing and data passing among the processors. No particular effort was made to optimize the code and memory allocation in order to minimize this lost time. F. Calculation of Torso Surface Potentials and Simulation Protocols Calculation of the ECG and the body surface potential map (BSPM) was done by using the spatial gradient of the transmembrane potential distribution determined from (4) to calculate elemental current dipoles at each model point. This assumes the approximation of an isotropic myocardium, with both intra-
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cellular and interstitial milieus being treated as homogeneous and isotropic, and it results in dipoles that are everywhere normal to the excitation wavefront. Thus, myocardial anisotropy is ignored when calculating the surface potentials. The elemental dipoles were combined by heart region to result in 58 time-varying current dipoles, that, in conjunction with an inhomogeneous human torso model, were then used to calculate the torso surface potentials employing a standard integral equation approach [27]. The torso model consisted of lungs, intraventricular blood masses, and a skeletal muscle layer. The latter was an approximately equivalent isotropic representation of an anisotropic skeletal muscle layer, obtained as suggested by McFee and Rush [28]. Model parameters were initially adjusted to result in normal activation isochrones, as well as a normal ECG (both QRS and T complexes) and BSPM. The QRST integral map was also determined. Next, two ectopic beats were simulated, starting from the left and right ventricular free walls, respectively. QRST integral maps were computed for these ectopic beats to verify the relative invariance of QRST integral maps with a change in activation sequence. To see the effect of large-scale changes in action-potential duration, we repeated the normal excitation with the M-cells eliminated, in other words, with no gradient in action-potential durations. An essentially flat QRST distribution is expected in this case as QRS and T integrals should cancel in every lead. A final set of two simulations reintroduced the M-cells, but also introduced pathologic “ischemic” regions of smaller duration action potentials in the right and left ventricle, respectively. These reduced-duration action potentials were and of the obtained by multiplying the time constants by a factor of 0.25, and their waveform slow inward current is also plotted in Fig. 1. This last set of simulations was to see how sensitive the QRST integral maps would be to the size of isolated regions of locally reduced action-potential durations in the right and left ventricle. III. RESULTS A. Normal Excitation of the Heart Model The model heart was excited, initially using the same start times and start points on the endocardium as in the Lorange–Gulrajani model. Isochrones, ECGs, and BSPMs were all calculated. It was found that start times on the right ventricular endocardium had to be advanced 4 ms over the Lorange–Gulrajani values for an optimum ECG. Fig. 4 shows the activation isochrones following this change. An overall comparison of the isochrones revealed close correspondence with the isochrones described by Durrer et al. in the human heart [29]. Fig. 5 shows the corresponding ECG and is within normal limits. Note the largely upright T waves due to the M cells introduced in the model. Fig. 6 plots the BSPMs during QRS and T waves. These too are normal, with the evolution of positive and negative extrema as described by Taccardi et al. [30]. B. Electrotonic Modulation of Action-Potential Durations Fig. 1 indicates that the intrinsic difference in action-potential durations between isolated endocardial (or epicardial) cells and
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Fig. 5. Normal ECG simulated by the model. P waves are absent as the model does not include the atria.
Fig. 4. Transverse (top) and longitudinal (bottom) sections depicting the activation isochrones generated for normal activation of the ventricles by the model. Calibration color bar for the isochrones is given on the right.
M cells is 109 ms. During excitation of the heart model, however, this intrinsic difference was diminished by the electrotonic coupling introduced between adjacent cells by (4). Fig. 7(a) plots the intrinsic distribution of action-potential durations of model cells across a section of the left ventricle and the same distribution when the cells are coupled during activation. Similar curves are shown for the right ventricle in Fig. 7(b). A continuum of action-potential durations exists during activation. For the thinner right ventricle, in particular, note the greatly diminished duration of M cells and the greatly enhanced duration of endocardial and epicardial cells. Interestingly, even with no intrinsic difference in action-potential duration across the walls (no M-cells), there still existed a gradient of action-potential durations [dashed traces in Fig. 7(a) and (b)]. C. QRST Integral Maps Fig. 8 shows the QRST integral maps for normal activation as well as for each of the two ectopic activation sequences. The QRST integral map for normal activation of the heart model agrees with clinically measured QRST integral maps of normals [11]. The similarity of QRST integral maps with completely different activation sequences is striking, especially
considering the fact that due to the larger activation times of the ectopic sequences the end of QRS impinges on the beginning of the T wave (see associated ECG above each ectopic integral map). Differences between the ectopic and normal QRST integral maps are plotted below each of the ectopic maps. These difference maps indicate maximum deviations of 7 mV-ms. Most of these deviations may be attributed to induced differences in action-potential durations caused by the differences in electrotonic coupling that accompany the different activation sequences. Next, QRST integral maps for cases in which the activation sequence was left unaltered but intrinsic action-potential durations were changed are shown in Fig. 9. The first column (map a2) shows the QRST integral map when all model cells were identical (no M cells) and is essentially zero everywhere. The small values of the extrema present in this map may be attributed to induced gradients in action potential across the ventricular walls (see Fig. 7). The ECG above this map shows the inversion of the T wave that results under these circumstances. The next two columns in Fig. 9 show simulations for local regions of reduced action-potential duration, first in the right ventricle and next in the left ventricle. The activation sequences remained unaltered from normal, as evidenced by the unchanged QRS complexes and only minor T wave changes (see ECG leads in Fig. 9). The QRST integral maps in both cases are plotted, as is the difference between these maps and the normal QRST integral map. Visually, it is extremely difficult to distinguish the
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Fig. 6. BSPMs corresponding to normal activation of the model. Left half of each map represents the anterior torso, and right half the back. Region of positive potentials is shaded gray. Contour interval is 100 V, and the heavier trace indicates the zero isocontour. Positive and negative extrema are marked with and signs, respectively, and their magnitudes indicated below each map. Time of each map following ventricular excitation is indicated below each map as well as marked on the ECG tracing at the top.
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affected regions from the QRST integral maps. Nor is it possible to do so from the BSPMs (not shown). The only indication is provided by the difference maps which show regions of positivity in body surface locations overlying the affected region (corresponding to the more familiar ST segment elevation seen in ECG leads) and regions of negativity in locations opposite to the affected region. Furthermore, the size of the affected regions was chosen to just result in QRST integral differences
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of 14 mV-ms, or twice the values shown earlier as resulting simply due to induced differences in action-potential duration secondary to altered activation sequences. It shows that nearly eight times more cells need to be affected in the deeper left ventricular wall than in the proximal right ventricle (350 302 versus 44171) for the same degree of manifestation in the difference maps. The number of affected cells in the left and right ventricle represent, respectively, just 2.79% and 0.35%
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is easy to accomplish via computer simulation. Earlier simulations of the modulation of action-potential duration via electrotonic interactions had only been accomplished for a 1-D strand [33] and a 2-D sheet [34]. We note that a slight modulation also occurs even if intrinsically all action-potential waveforms are identical (Fig. 7, dashed traces). The QRST integral map simulations verify that these map patterns are, for the most part, unaffected even by major alterations in activation sequence but do change with alterations in action-potential duration. While the overall QRST integral map pattern may not be very sensitive to alterations in actionpotential duration [35], the changes in these integral maps as manifested in the difference maps are uniquely sensitive to these alterations in duration. A very small percentage of cells needs to be affected to cause these changes. An idea of the ventricular region affected can also be gleaned from the difference maps, corroborating a similar recent finding by Tyˇsler et al. [36]. We, however, provide a quantitative idea of the extent of the action-potential duration alterations needed in order to be reflected at the body surface.
Fig. 7. Plots of action-potential duration across (a) left and (b) right ventricular walls. Intrinsic isolated cell durations, as well as the coupled durations during activation, are plotted. Also shown are the coupled durations when no M cells are present. Distances are expressed as a percentage from the endocardium (0%) to the epicardium (100%).
of the total number of ventricular model cells (approximately 12.5 million) and give an idea of the sensitivity of changes in body surface QRST integral maps to local variations in action-potential duration. IV. DISCUSSION We demonstrate here the use of a multiprocessor computer in large-scale heart model simulations. The ECG and BSPM simulations were close to those recorded in normals. This represents a more acid test of the simulation process than just looking at isochrones, since the entire chain from model activation to equivalent dipoles, and finally to surface potentials, is validated. Even more interesting is the demonstration of the reduction in the intrinsic action-potential duration across the ventricular wall due to electrotonic coupling, particularly for the thinner right ventricle. Experimental curves by Yan et al. [31], obtained in a portion of dog myocardium, showing the continuous variation of action-potential duration across the ventricular wall, are similar to the curves of Fig. 7. This group, as well as others [32], has noted the effect of electrotonic coupling in reducing the in vitro difference in action-potential durations between M cells and endocardial/epicardial cells to much smaller in situ values. Verification of this electrotonic coupling hypothesis in the whole heart is difficult experimentally, but as evidenced by our results,
The limitations of our current model stem largely from the lack of sufficient computer power, hence the need to compensate for insufficient spatial discretization by the use of a reduced surface-to-volume ratio . Our tests with the tissue block were done, however, to ensure that this drawback did not significantly affect our simulated body surface potential distributions. The advent of more powerful parallel machines will no doubt improve the accuracy of whole heart simulations to the precision currently used in tissue simulations [24], [37], [38]. Some obvious improvements to our model are the inclusion of the atria, a fractal tree representation for the His–Purkinje conduction system, the incorporation of measured cardiac fiber directions, and an even more up-to-date membrane representation than the Luo–Rudy phase 1 equations. In particular, it is important to incorporate the subtle waveform differences between endocardial, epicardial, and M cells by employing a membrane model that includes the transient outward current. Another improvement would be an algorithm to track the activation wavefront in space and time so as to know where and when to switch from the smaller time step used for depolarization to the larger time step used for repolarization. Our criterion of just making the switch after the entire heart is depolarized is only applicable for the simulation of a single cardiac beat. When simulating arrhythmias, however, multiple disparate wavefronts from consecutive beats may coexist at any given time instant, and a more sophisticated space-time wavefront tracking algorithm would be of help. One such adaptive tracking algorithm was proposed recently by Cherry et al. [39], and the incorporation of such an algorithm into the model would permit knowing where and when to make the switch to a larger time step. The use of finite volume discretizations as opposed to finite difference ones may also be particularly appropriate for complicated heart geometries [40]–[42]. With the greater availability of 256-processor machines, many other improvements, until now limited by insufficient
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Fig. 8. QRST integral maps for (a) normal activation, (b) left, and (c) right ventricular ectopic excitation. Representative ECG lead is indicated above each map. Difference between ectopic and normal QRST integral maps is plotted below each of the ectopic maps.
Fig. 9. QRST integral maps for normal activation with (a1) and without (a2) M-cells, and with cells of reduced action-potential duration in the (b) right ventricle and (c) left ventricle. Difference maps with respect to normal action-potential durations are also shown for the latter two cases of reduced action-potential durations. Anatomic extent of these reduced durations is depicted by the dark regions in the insets showing the right and left ventricles (44171 cells or 0.35% of the total ventricular myocardium for the right ventricle and 350 302 cells or 2.79% for the left ventricle).
computing power and memory, become feasible. The most important of these would be to use the separate bidomain equations (1) and (2), without their combination on the basis of the assumption of equal anisotropy. This will permit direct computation of the interstitial potentials in the heart as well as the transmembrane potentials. Epicardial potential distributions
in an isolated heart are then automatically obtained. Penland et al. [42] have described the methodology for these bidomain simulations, but only describe simulations in tissue blocks and a myocardial wedge. Further improvement would necessitate introducing the boundary conditions coupling the bidomain heart to the monodomain torso to calculate the torso potentials
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in a single computation [43], [44], as opposed to the two-step approach of first exciting the isolated heart and then using equivalent dipoles to compute the torso potentials. Initial computations of this sort may be found in Lines et al. [44], who describe a fully coupled simulation with a three-dimensional heart model implanted in a torso model. Such simulations will become routine with greater use of parallel computation. ACKNOWLEDGMENT The authors thank the Réseau québécois de calcul de haute performance (RQCHP) for its support and for providing free time on its 64-processor Silicon Graphics Origin 2000 computer at the Université de Montréal. They greatly appreciate the help rendered by J. Richer and B. Lorazo of the RQCHP and by A. Bleau of the Institute of Biomedical Engineering, Université de Montréal. They also wish to thank the two anonymous reviewers for their many valuable suggestions that have helped in greatly improving the paper. REFERENCES [1] M. Lorange and R. M. Gulrajani, “A computer heart model incorporating anisotropic propagation. I. Model construction and simulation of normal activation,” J. Electrocardiol., vol. 26, pp. 245–261, 1993. [2] L. J. Leon and B. M. Horáˇcek, “Computer model of excitation and recovery in the anisotropic myocardium. I. Rectangular and cubic arrays of excitable elements,” J. Electrocardiol., vol. 24, pp. 1–15, 1991. [3] O. Berenfeld and S. Abboud, “Simulation of cardiac activity and the ECG using a heart model with a reaction-diffusion action potential,” Med. Eng. Phys., vol. 18, pp. 615–625, 1996. [4] A. V. Panfilov, “Modeling of re-entrant patterns in an anatomical model of the heart,” in Computational Biology of the Heart, A. V. Panfilov and A. V. Holden, Eds. Chichester, U.K.: Wiley, 1997, pp. 259–276. [5] O. Berenfeld and J. Jalife, “Purkinje-muscle reentry as a mechanism of polymorphic ventricular arrhythmias in a 3-dimensional model of the ventricles,” Circ. Res., vol. 82, pp. 1063–1077, 1998. [6] G. Huiskamp, “Simulation of depolarization in a membrane-equationsbased model of the anisotropic ventricle,” IEEE Trans. Biomed. Eng., vol. 45, pp. 847–855, July 1998. [7] C. Luo and Y. Rudy, “A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction,” Circ. Res., vol. 68, pp. 1501–1526, 1991. [8] S. Sicouri and C. Antzelevitch, “A subpopulation of cells with unique electrophysiological properties in the deep subepicardium of the canine ventricle: The M cell,” Circ. Res., vol. 68, pp. 1729–1741, 1991. [9] E. Drouin, F. Charpentier, C. Gauthier, K. Laurent, and H. Le Marec, “Electrophysiologic characteristics of cells spanning the left ventricular wall of human heart: Evidence for presence of M cells,” J. Amer. Coll. Cardiol., vol. 26, pp. 185–192, 1995. [10] J. A. Abildskov, M. J. Burgess, P. M. Urie, R. J. Lux, and R. F. Wyatt, “The unidentified information content of the electrocardiogram,” Circ. Res., vol. 40, pp. 3–7, 1977. [11] M. J. Gardner, T. J. Montague, C. S. Armstrong, B. M. Horacek, and E. R. Smith, “Vulnerability to ventricular arrhythmia: Assessment by mapping of body surface potential,” Circulation, vol. 73, pp. 684–692, 1986. [12] M.-C. Trudel, R. M. Gulrajani, and L. J. Leon. Simulation of propagation in a realistic-geometry computer heart model with parallel processing. presented at 23rd Annu. Int. Conf. IEEE Eng. Med. Biol. Soc. [CD-ROM], 2001 [13] R. M. Gulrajani, M.-C. Trudel, and L. J. Leon, “A membrane-based computer heart model employing parallel processing,” in Proc. 3rd Int. Symp. Noninvasive Functional Source Imaging Within the Human Heart and Brain, Biomedizinische Technik, vol. 46, 2001, pp. 20–22. [14] M.-C. Trudel, R. M. Gulrajani, and L. J. Leon, “Electrotonic coupling reduces action potential duration gradients in the ventricle: A simulation study,” in Proc. XXIXth Int. Congr. Electrocardiol. 4th Int. Conf. Bioelectromagnetism, Int. J. Bioelectromagnetism, vol. 4, 2002, pp. 55–56.
[15] W. T. Miller III and D. B. Geselowitz, “Simulation studies of the electrocardiogram. I. The normal heart,” Circ. Res., vol. 43, pp. 301–315, 1978. [16] R. Beyar and S. Sideman, “A computer study of the left ventricular performance based on fiber architecture, sarcomere dynamics, and transmural electrical propagation velocity,” Circ. Res., vol. 55, pp. 358–375, 1984. [17] L. Clerc, “Directional differences of impulse spread in trabecular muscle from mammalian heart,” J. Physiol., vol. 255, pp. 335–346, 1976. [18] D. E. Roberts, L. T. Hersh, and A. M. Scher, “Influence of cardiac fiber orientation on wavefront voltage, conduction velocity, and tissue resistivity in the dog,” Circ. Res., vol. 44, pp. 701–712, 1979. [19] D. E. Roberts and A. M. Scher, “Effects of tissue anisotropy on extracellular potential fields in canine myocardium in situ,” Circ. Res., vol. 50, pp. 342–351, 1982. [20] B. J. Roth, “Electrical conductivity values used with the bidomain model of cardiac tissue,” IEEE Trans. Biomed. Eng., vol. 44, pp. 326–328, Apr. 1997. [21] S. H. Litovsky and C. Antzelevitch, “Transient outward current prominent in canine ventricular epicardium but not endocardium,” Circ. Res., vol. 62, pp. 116–126, 1988. [22] S. Rush and H. Larsen, “A practical algorithm for solving dynamic membrane equations,” IEEE Trans. Biomed. Eng., vol. BME-25, pp. 389–392, 1978. [23] H. I. Saleheen and K. T. Ng, “A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues,” IEEE Trans. Biomed. Eng., vol. 45, pp. 15–25, Jan. 1998. [24] A. Muzikant and C. S. Henriquez, “Bipolar stimulation of a three-dimensional bidomain incorporating rotational anisotropy,” IEEE Trans. Biomed. Eng., vol. 45, pp. 449–462, Apr. 1998. [25] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing, 2nd ed. Cambridge, MA: Cambridge Univ. Press, 1992, p. 838. [26] J. Wu and D. P. Zipes, “Effects of spatial segmentation in the continuous model of excitation propagation in cardiac muscle,” J. Cardiovasc. Electrophysiol., vol. 10, pp. 965–972, 1999. [27] R. C. Barr, T. C. Pilkington, J. P. Boineau, and M. S. Spach, “Determining surface potentials from current dipoles with application to electrocardiography,” IEEE Trans. Biomed. Eng., vol. BME-13, pp. 88–92, 1966. [28] R. McFee and S. Rush, “Qualitative effects of thoracic resistivity variations on the interpretation of electrocardiograms: The low resistance surface layer,” Amer. Heart J., vol. 76, pp. 48–61, 1968. [29] D. Durrer, R. T. van Dam, G. E. Freud, M. J. Janse, F. L. Meijler, and R. C. Arzbaecher, “Total excitation of the isolated human heart,” Circulation, vol. 41, pp. 899–912, 1970. [30] B. Taccardi, L. De Ambroggi, and C. Viganotti, “Body-surface mapping of heart potentials,” in The Theoretical Basis of Electrocardiology, C. V. Nelson and D. B. Geselowitz, Eds. Oxford, U.K.: Clarendon, 1976, pp. 436–466. [31] G.-X. Yan, W. Shimizu, and C. Antzelevitch, “Characteristics and distribution of M cells in arterially perfused canine left ventricular wedge preparations,” Circulation, vol. 98, pp. 1921–1927, 1998. [32] E. P. Anyukhovsky, E. A. Sosunov, R. Z. Gainullin, and M. R. Rosen, “The controversial M cell,” J. Cardiovasc. Electrophysiol., vol. 10, pp. 244–260, 1999. [33] R. W. Joyner, “Modulation of repolarization by electrotonic interactions,” Jap. Heart J., vol. 27, no. Suppl., pp. 167–183, 1986. [34] M. D. Lesh, M. Pring, and J. F. Spear, “Cellular uncoupling can unmask dispersion of action potential duration in ventricular myocardium. A computer modeling study,” Circ. Res., vol. 65, pp. 1426–1440, 1989. [35] J. E. Burnes, R. N. Ghanem, A. L. Waldo, and Y. Rudy, “Imaging dispersion of myocardial repolarization, I. Comparison of body-surface and epicardial measures,” Circulation, vol. 104, pp. 1299–1305, 2001. [36] M. Tyˇsler, V. Szathmáry, and M. Turzová, “Model study of assessment of local heart repolarization changes by several ECG methods,” Int. J. Bioelectromagnetism, vol. 5, pp. 252–253, 2003. [37] C. S. Henriquez and A. A. Papazoglou, “Using computer models to understand the roles of tissue structure and membrane dynamics in arrhythmogenesis,” Proc. IEEE, vol. 84, pp. 334–354, Mar. 1996. [38] E. J. Vigmond, F. Aguel, and N. A. Trayanova, “Computational techniques for solving the bidomain equations in three dimensions,” IEEE Trans. Biomed. Eng., vol. 49, pp. 1260–1269, Nov. 2002. [39] E. M. Cherry, H. S. Greenside, and C. S. Henriquez, “A space-time adaptive method for simulating complex cardiac dynamics,” Phys. Rev. Lett., vol. 84, pp. 1343–1346, 2000.
TRUDEL et al.: SIMULATION OF QRST INTEGRAL MAPS WITH MEMBRANE-BASED COMPUTER HEART MODEL
[40] D. M. Harrild and C. S. Henriquez, “A finite volume model of cardiac propagation,” Ann. Biomed. Eng., vol. 25, pp. 315–334, 1997. [41] D. M. Harrild, R. C. Penland, and C. S. Henriquez, “A flexible method for simulating cardiac conduction in three-dimensional complex geometries,” J. Electrocardiol., vol. 33, pp. 241–251, 2000. [42] R. C. Penland, D. M. Harrild, and C. S. Henriquez, “Modeling impulse propagation and extracellular potential distributions in anisotropic cardiac tissue using a finite volume element discretization,” Comput. Visual. Sci., vol. 4, pp. 215–226, 2002. [43] M. Buist and A. Pullan, “Torso coupling techniques for the forward problem of electrocardiography,” Ann. Biomed. Eng., vol. 30, pp. 1299–1312, 2002. [44] G. T. Lines, M. L. Buist, P. Grøttum, A. J. Pullan, J. Sundnes, and A. Tveito, “Mathematical models and numerical methods for the forward problem in cardiac electrophysiology,” Comput. Visual. Sci., vol. 5, no. 4, pp. 215–239, 2003.
Marie-Claude Trudel was born in Montreal, QC, Canada, in 1974. She received the B.S. degree in physics and M.Sc.A. in biomedical engineering from Université de Montréal, Montréal, in 1999 and 2001, respectively. Since 2001 she has been working at EBS Inc., Montreal, as a Consultant in healthcare technologies.
Bruno Dubé received the B.Sc. degree in physics and biophysics and the M.Sc.A. degree in biomedical engineering, both from the Université de Montréal, Montréal, QC, Canada, in 1980 and 1984, respectively. Since 1984, he has been a Research Engineer and Computer Analyst at Sacré-Cœur Hospital, Montreal. His research interests include the modeling of cardiac electrophysiological phenomena and digital signal processing.
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Mark Potse received the M.Sc. degree in physics and the Ph.D. degree, both from the University of Amsterdam, The Netherlands, in 1996 and 2001, respectively. From 1994 to 2002, he has been involved in research on electrocardiographic body surface mapping and intracardial mapping, with the groups of Dr. C. A. Grimbergen in the Medical Physics Department and of Dr. J. M. T. de Bakker in the Experimental Cardiology Department, Academic Medical Center, University of Amsterdam. He worked as a Postdoctoral Fellow in the same groups. Since 2002, he has been a Postdoctoral Fellow at the Institute of Biomedical Engineering, Université de Montréal, where he is currently developing a bidomain model of the human heart. His research interests include both the simulation and analysis of electrocardiographic data, and the integration of these two methodologies in the investigation of clinical and fundamental problems in electrocardiology.
Ramesh M. Gulrajani (S’68–M’72–SM’85–F’01) received the Ph.D. degree in electrical engineering from Syracuse University, Syracuse, NY, in 1973. He was with the Université de Montréal, Montréal, QC, Canada, from 1973, where he was a Professor in the Institute of Biomedical Engineering. His research interests included the forward and inverse problems of electrocardiography and electroencephalography and in computer heart models. He was the author of the graduate text Bioelectricity and Biomagnetism (New York, NY: Wiley, 1998). Dr. Gulrajani died unexpectedly on March 18, 2004.
L. Joshua Leon (M’97) received the B.S. and M.S. degrees in mathematics and the Ph.D. degree in biophysics, all from Dalhousie University, Halifax, Canada, in 1980, 1982, and 1987, respectively. He subsequently joined the Institute of Biomedical Engineering at Ecole Polytechnique de Montreal, Montreal, QC, Canada, as a Postdoctoral Fellow, in 1987, and became a Research Assistant Professor in 1989. He moved to the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada, in 2000, where he is currently a Professor and Department Head. His research into bioelectric phenomena includes basic physical mechanisms underlying cardiac arrhythmias and approaches to preventing or terminating them. He has developed a number of mathematical computer models of the heart, which have been used to examine such phenomena as the dynamics of tachycardia and fibrillation and the mechanisms of cardiac defibrillation.
