Download Timing of Depolarization and Contraction in the Paced

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Timing of Depolarization and Contraction in the Paced Canine
Left Ventricle: Model and Experiment
ROY C.P. KERCKHOFFS, PH.D.,∗ OWEN P. FARIS, PH.D.,† PETER H.M. BOVENDEERD, PH.D.,∗
FRITS W. PRINZEN, PH.D.,‡ KAREL SMITS, M.SC.,¶ ELLIOT R. McVEIGH, PH.D.,†
and THEO ARTS, PH.D.∗ ,§
From the ∗ Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands; †Laboratory of
Cardiac Energetics, National Institutes of Health, National Heart, Lung, and Blood Institute, Bethesda, Maryland, USA; ‡Department of
Physiology, Maastricht University, Maastricht, The Netherlands; ¶Department of Lead Modeling, Medtronic Bakken Research Center,
Maastricht, The Netherlands; and §Department of Biophysics, Maastricht University, Maastricht, The Netherlands
Modeling the Paced LV. Introduction: For efficient pump function, contraction of the heart should be
as synchronous as possible. Ventricular pacing induces asynchrony of depolarization and contraction. The
degree of asynchrony depends on the position of the pacing electrode. The aim of this study was to extend
an existing numerical model of electromechanics in the left ventricle (LV) to the application of ventricular
pacing. With the model, the relation between pacing site and patterns of depolarization and contraction
was investigated.
Methods and Results: The LV was approximated by a thick-walled ellipsoid with a realistic myofiber
orientation. Propagation of the depolarization wave was described by the eikonal-diffusion equation, in
which five parameters play a role: myocardial and subendocardial velocity of wave propagation along the
myofiber cm and ce ; myocardial and subendocardial anisotropy am and ae ; and parameter k, describing
the influence of wave curvature on wave velocity. Parameters cm , ae , and k were taken from literature.
Parameters am and ce were estimated by fitting the model to experimental data, obtained by pacing the
canine left ventricular free wall (LVFW). The best fit was found with cm = 0.75 m/s, ce = 1.3 m/s, am =
2.5, ae = 1.5, and k = 2.1 × 10−4 m2 /s. With these parameter settings, for right ventricular apex (RVA)
pacing, the depolarization times were realistically simulated as also shown by the wavefronts and the time
needed to activate the LVFW. The moment of depolarization was used to initiate myofiber contraction in a
model of LV mechanics. For both pacing situations, mid-wall circumferential strains and onset of myofiber
shortening were obtained.
Conclusion: With a relatively simple model setup, simulated depolarization timing patterns agreed with
measurements for pacing at the LVFW and RVA in an LV. Myocardial cross-fiber wave velocity is estimated
to be 0.40 times the velocity along the myofiber direction (0.75 m/s). Subendocardial wave velocity is about
1.7 times faster than in the rest of the myocardium, but about 3 times slower than as found in Purkinje fibers.
Furthermore, model and experiment agreed in the following respects. (1) Ventricular pacing decreased both
systolic pressure and ejection fraction relative to natural sinus rhythm. (2) In early depolarized regions,
early shortening was observed in the isovolumic contraction phase; in late depolarized regions, myofibers
were stretched in this phase. Maps showing timing of onset of shortening were similar to previously measured maps in which wave velocity of contraction appeared similar to that of depolarization. (J Cardiovasc
Electrophysiol, Vol. 14, pp. S188-S195, October 2003, Suppl.)
eikonal-diffusion equation, electromechanics, finite elements
Introduction
In the normal heart, the depolarization wave propagates
through the AV node to the His bundle, through the right
and left bundle branches into a network of fast-conducting
Purkinje fibers (3–4 m/s) near the endocardium.1 At the
Purkinje-muscular junctions (PMJs), the depolarization wave
enters the ventricular myocardium, where propagation is
much slower (0.6–1.0 m/s).2-5 From this moment on, the
wave propagates mainly from endocardium to epicardium.
This study was supported financially by the Medtronic Bakken Research
Center Maastricht, Maastricht, The Netherlands.
Address for correspondence: Peter H.M. Bovendeerd, Ph.D., Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
Fax: 31-40-2447355; E-mail: [email protected]
doi: 10.1046/j.1540.8167.90310.x
After 40 to 50 msec, the whole myocardium has been depolarized.6 Upon depolarization, cross-bridge formation in the
myofibers is initiated. The combined stress development in
all myofibers leads to an increase in ventricular pressure, and
finally blood is expelled from the ventricular cavity.
With ventricular pacing, depolarization of the left ventricle (LV) differs from regular sinus rhythm.7-9 Near the pacing
site, depolarization propagates slowly. Further away, propagation appears faster. During ventricular pacing, depolarization takes more time than during regular sinus rhythm. As a
result, ventricular contraction is spread out over a longer period and is more inhomogeneous. The resulting asynchrony
of myofiber contraction affects pump function.7 In the long
term, myocardial tissue structure changes10 and may even
contribute to the development of heart failure.11,12 Therefore, investigators are searching for better sites of pacing for
optimal pump function. Positioning of the pacing electrode
by trial and error is cumbersome. In assessing various pacing
Kerckhoffs et al.
sites for minimal mechanical asynchrony, realistic mathematical models of cardiac electromechanics in the ventricles are
likely to be useful tools.
The aim of this study was to extend an existing numerical model of electromechanics in the LV13 to the application
of ventricular pacing. As a compromise between accuracy
and computation time, we have chosen to describe the propagation of the depolarization wave through the myocardium
by the eikonal-diffusion equation.14 In this setup, the model
needs only a few parameters values to be known. For instance,
myocardial tissue is anisotropic, causing the wave to travel
faster along the myofiber direction than perpendicular to it.
Furthermore, in the subendocardium, the wave is propagating
faster than in the rest of the myocardium. The related parameters, being velocities of wave propagation perpendicular to
the myofibers and along the myofibers in the subendocardial
layers, respectively, were estimated by fitting the model to
experiments15 in which the heart was paced at the lateral free
wall of the left ventricle (LVFW). Next, the predictive quality of the model was assessed prospectively by comparing
the predicted sequence of depolarization for right ventricular
apex (RVA) pacing with experimental observations. In further evaluation of the model, mid-wall circumferential strain
and onset of shortening during LVFW and RVA pacing were
compared with experimentally determined values.
Materials and Methods
Experiments
Maps of timing of depolarization at the epicardium were
obtained after pacing at the LVFW and RVA in a dog, as
described previously.15 In brief, socks with 128 electrodes
were placed over the ventricular epicardium of anesthetized
dogs. Bipolar epicardial pacing electrodes were placed at the
RVA and LVFW. Epicardial recordings were obtained at an
acquisition rate of 1,000 Hz. After the electrical data were obtained, the animals were euthanized and the hearts excised.
The hearts were filled with vinyl polysiloxane in order to
maintain an end-diastolic shape and the sock electrode locations were recorded using a three-dimensional digitizer.
Unipolar voltage readings from each electrode were averaged over approximately 20 heartbeats. Depolarization time
was determined as the steepest downstroke of the electrode
voltage reading.
Modeling the Paced LV
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Figure 1. Ellipsoidal geometry of the left ventricle, with superimposed myofiber orientation. A: Epicardial view. B: Close-up of endocardium and
several myofibers showing the transmural myofiber rotation.
Parameter c (m/s) represents the velocity of the depolarization wave along the myofiber direction. Parameter k (m2 /s)
determines the influence of wavefront curvature on wave
velocity. The ratio of k/c represents a characteristic radius
of wave curvature, below which wave propagation velocity
severely depends on wave curvature. Dimensionless, transversely isotropic tensor M describes anisotropy of wave propagation. The principal direction with the largest eigenvalue
coincides with myofiber orientation, having the eigenvalue
normalized to 1. Both other principal values, which are related to the transverse principal directions, are set equal to
a−2 . The value of a represents the ratio of longitudinal to
transverse velocity of wave propagation. Wave velocity c and
the anisotropy ratio a may vary across the myocardial wall.
In the model, we distinguished between the velocity cm in
the myocardium and a velocity ce at the subendocardium
(Fig. 2 and Table 1). Anisotropy at the subendocardium
ae was allowed to be different from anisotropy in the
myocardium am .
The LV was assumed electrically insulated. Depolarization was started at t = 0 at the pacing regions LVFW or RVA
(Fig. 2).
Simulations
Design of the model of wave propagation
The LV wall at end-diastole was represented by a thickwalled truncated prolate ellipsoid.13 Volumes of LV wall and
cavity were 140 and 80 mL, respectively. The distribution
of helix and transverse (or imbrication) angles of myofiber
orientation was realistic, as described previously16 (Fig. 1).
The helix angle varied nonlinearly from 70◦ at the subendocardium to −50◦ at the subepicardium. The mid-wall transverse angle was on the order of −20◦ near the apex and 10◦
at the base.
The moment of depolarization tdep within the wall was
determined by solving the eikonal-diffusion equation for the
· tdep ):
gradient of tdep (∇
dep − k ∇
· (M · ∇t
dep ) = 1.
dep · M · ∇t
c ∇t
(1)
Figure 2. Distribution of depolarization wave velocities ce and cm at the
subendocardium and myocardium, respectively, and of anisotropy ratios ae
and am . The left ventricular subepicardium at the septal side represents the
subendocardium of the right ventricle. LVFW = location for pacing at the
left ventricular free wall; RVA = location for pacing at the right ventricular
apex.
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Journal of Cardiovascular Electrophysiology
Vol. 14, No. 10, Supplement, October 2003
TABLE 1
Parameter Values in the Model and Reported in the Literature
Parameter
Model Value
Literature
Value
Reference
Source
cm (m/s)
ce (m/s)
am
ae
k (m2 /s)
0.75
0.75, 1.3, 1.8
2.0, 2.5, 3.0
1.5
2.1 × 10−4
0.6-1.0
1.2
2.1-3.3
1.5
2.1 × 10−4
2,∗ 3, 4, 5
2∗
2,∗ 3, 4, 5
2∗
21
All reported measurements were done in the left ventricle, except for the
measurement in the right ventricle denoted by the asterisk (∗ ).
cm = wave velocity parallel to the myofibers in the myocardium; ce =
wave velocity parallel to the myofibers in the subendocardium (see also
Fig. 2); am = anisotropy ratio of depolarization wave velocities parallel and
perpendicular to the myofibers in the myocardium; ae = anisotropy ratio of
depolarization wave velocities parallel and perpendicular to the myofibers
in the subendocardium; k = diffusion constant that determines the influence
of wavefront curvature on wave velocity.
Modeling mechanical properties of cardiac tissue
Wall mechanics were determined from solving equations
of force equilibrium. Myocardial material was considered
anisotropic, nonlinearly elastic, and time dependent, as reported previously13 (Fig. 3). Local active force developed
at the moment of depolarization. LV pressure was determined from the interaction of the LV with an aortic input
impedance.17 For LVFW and RVA pacing, a complete cardiac cycle was simulated. To compare the pacing simulations
with that of normal sinus activation, for control a simulation
was performed with synchronous activation of the LV.
Numerical implementation
The equations were solved using the finite element package SEPRAN (SEPRA, Leidschendam, The Netherlands) on
a 64-bit Origin 200 computer (SGI, Mountain View, CA,
USA), using a single processor at 225 MHz on a UNIX platform.
The eikonal-diffusion equation was discretized with 8noded Galerkin-type hexahedral elements (44,064 elements,
46,987 degrees of freedom) having trilinear interpolation and
a mean spatial resolution of 1.4 mm.
The equations related to mechanics were discretized with
27-noded hexahedral elements with triquadratic interpolation. The LV wall was subdivided into 108 elements, with
3,213 degrees of freedom.
Simulations and data analysis
The myocardial wave velocity cm and subendocardial
anisotropy ratio ae were fixed at values of 0.75 m/s and
1.5, respectively (Table 1). Parameter k was set to 2.1 ×
10−4 m2 /s. For the subendocardium, three values for wave
velocity ce were used (Table 1). In addition, three values for
the anisotropy factor am in the myocardium were applied.
Thus, simulations were obtained for all nine combinations of
the latter values while pacing from a position on the LVFW
(Fig. 2). The calculated maps of epicardial depolarization
were compared with the experimentally measured map for
LVFW pacing.
The following criteria were used for comparison in order
to find the best combination of parameters:
r The moment of breakthrough tdep,b (msec), defined as the
r
moment at which the apparent epicardial wave velocity of
depolarization increased in a stepwise fashion. This was
quantified by filtering maps of epicardial depolarization
with a second derivative Laplacian with smoothing distance about 4% of image size.
The maximum time tdep,max (msec) needed to completely
depolarize the epicardium of the LVFW.
The root mean squared (RMS) difference between experiment and simulations for tdep,b and tdep,max was computed,
according to:
rms = ((tdep,b )2 + (tdep,max )2 )/2
(2)
where tdep,b and tdep,max represent the difference in tdep,b
and tdep,max , respectively, in experiment and simulation.
Next, keeping the thus found parameter setting, a simulation was performed when pacing from the RVA. The results
of the simulation were compared to the map as measured with
RVA pacing, focusing on maximum epicardial depolarization
time.
The maps of depolarization timing for LVFW pacing that
best matched the experiments and the map for RVA pacing
were used as input in the simulation of mechanics.
Complete cardiac cycles were computed for RVA and
LVFW pacing. To facilitate comparison between simulations
and experiments,18 mid-wall circumferential strain was computed. At the mid-wall, myofibers are oriented almost completely circumferential. Furthermore, prestretch was defined
as extra lengthening of mid-wall myofibers that takes place
in the isovolumic contraction phase after normal diastole.
Figure 3. Characteristics of the model of cardiac mechanics. A: Stress (kPa) of passive material parallel
(solid line) and perpendicular (dotted line) to the myofiber for biaxial stretching as a function of stretch
ratio. B: Active myofiber stress as a function of time
for isometric twitches at sarcomere lengths of 1.6, 1.9,
and 2.2 µm. C: Relation between sarcomere shortening
velocity and myofiber force F, normalized to maximum
force F 0 .
Kerckhoffs et al.
Modeling the Paced LV
S191
Timing of mechanics was characterized by the moment of
onset of circumferential shortening in the mid-wall.18
Results
Experiments
Figure 4A shows a map of the measured moments of depolarization on the epicardial surface of the heart during LVFW
pacing. The inner circle represents the border of the region
of the measurement, which is approximately the equatorial
region. Epicardial isochrones in the early depolarized region
resembled quasi-ellipses with a long-to short-axis ratio of
about 2.5. Isochrones initially were close together, indicating
Figure 5. Timing of depolarization (in milliseconds) in the left ventricle
for a simulation of left ventricular free-wall pacing. Parameter values were
cm = 0.75 m/s, am = 2.5, ce = 1.3 m/s, ae = 1.5, k = 2.1 × 10−4 m2 /s. A part
of the anterior free wall is removed, showing the septal endocardium, and
a cross-section of the mid-septum. Note that the long axes of the ellipsoidal
wavefronts in the early depolarized region are aligned with the epicardial
myofiber orientation (Fig. 1).
slow apparent propagation, whereas this distance increased
after tdep,b = 59 msec (breakthrough). At the RV free wall, the
wave converged toward the base. Maximum epicardial depolarization time of the LVFW tdep,max was 111 msec, located
posteriorly at the base.
In the same heart, for RVA pacing at the LVFW, the wave
converged toward the equatorial region, where the maximum
depolarization time was 113 msec (Fig. 4A). Breakthrough
occurred after 43 msec, but this was located in the RV.
Simulations
Figure 4. Results for left ventricular free-wall (LVFW) pacing (left column) and right ventricular apex (RVA) pacing (right column). A: Maps of
measured timing of epicardial depolarization. The maps are represented as
bull’s-eye maps with the apex in the center. The inner circles represent the
border of the region of the measurement, which is approximately the equatorial region. The locations of breakthrough and maximum LV depolarization
time are denoted by the cross (×) and red dot (·), respectively.·The dotted
purple line indicates the LV/RV attachment. B: Maps of timing of epicardial depolarization from the simulations. The circles encompass the regions
that were measured in the experiments. The locations of breakthrough and
maximum LV depolarization time are denoted by the cross (×) and dot (·),
respectively. The dotted purple line indicates the LV/RV attachment. C: Timing of onset of mid-wall myofiber shortening. D: Sarcomere length at the
beginning of ejection, shown on the deformed LV mesh. Parameter settings:
cm = 0.75 m/s, ce = 1.3 m/s, am = 2.5, ae = 1.5. P = posterior; RV = right
ventricular side; S = septum.
For solving the eikonal-diffusion equation, calculation
times were approximately 15 minutes. Calculation times
for solving the equations of mechanics were approximately
9 hours.
For LVFW pacing, isochrones in the early depolarized regions resembled ellipses, with their major axes aligned with
the myofiber direction (Fig. 5). The epicardial depolarization
timing patterns are represented as bull’s-eye plots in Figure 6.
Increase of am , the ratio of myofiber to cross-fiber wave velocity in the myocardium, caused the long- to short-axis ratio of
the isochrones near the pacing site to increase proportionally.
Furthermore, the time needed for occurrence of breakthrough
tdep,b , increased with am . Maximum LV epicardial depolarization time tdep,max also increased with an increase in anisotropy.
The main effect of an increase of endocardial wave velocity
ce was a proportional increase of the apparent wave velocity
at locations far from the pacing site. Increasing ce resulted
in a decrease in maximum depolarization time. Subendocardial wave velocity did not affect ellipticity of isochrones. All
waves ended at the septal base.
In comparing numerical and experimental results, one
should realize that the simulations hold for the LV epicardium
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Vol. 14, No. 10, Supplement, October 2003
Figure 6. Bull’s-eye plots of simulated maps of timing of epicardial depolarization (in milliseconds) for
left ventricular free-wall (LVFW) pacing. The purple
circles indicate a similar region in which the measurements were performed. The arrows in the bottom left
map indicate the right ventricular (RV) region. The dotted purple line indicates the LV/RV attachment. Only
the region right of the dotted line can be compared with
the measurements. The moments of breakthrough are
denoted by crosses (×). From bottom to top, subendocardial wave velocity ce is increased. From left to right,
myocardial anisotropy ratio am is increased. Parameter k, subendocardial anisotropy ratio ae , and myocardial wave velocity cm along the myofiber were fixed at
2.1 × 10−4 m2 /s, 1.5, and 0.75 m/s, respectively. In
the bottom row, subendocardial wave velocity equaled
myocardial wave velocity ce = cm = 0.75 m/s and ae
= am , implying no influence of the Purkinje system. A
= anterior; P = posterior; S = septum.
and the septal part of the RV endocardium. Thus, the part
between the RVA and the RV side of the base cannot be
used for comparison. Comparing Figures 4A and 6, the best
match was found for a subendocardial wave velocity ce =
1.3 m/s and a myocardial anisotropy am = 2.5. In this simulation, breakthrough occurred at 62 msec, and maximum
LV epicardial depolarization time was tdep,max = 112 msec.
The RMS difference with the experiment was 2.2 msec (Table 2). With the same parameter settings, RVA pacing was
simulated.
Near the RVA pacing site, the long axes of the isochrones
of depolarization were aligned with the myofiber direction
(Fig. 4B). The simulation of RVA pacing matched the measured maps in the following respects: apparent wave velocity
in the posterior region was larger than anteriorly, and the wave
converged in a similar region in the LVFW. The equatorial
circle was reached with a difference of 8 msec (113 msec in
the experiment and after 121 msec in the simulation). Break-
TABLE 2
Root Mean Squared Difference (Equation 2) of Maximum Time tdep,max
Needed to Completely Depolarize the Epicardium of the Left Ventricular
Free Wall and Moment of Breakthrough tdep,b Between Experiment and
Simulation for Variations in Subendocardial Wave Velocity ce and
Myocardial Anisotropy am
through occurred at 63 msec in the LVFW. In the experiment,
this occurred at 43 msec, but because this was located in the
RV, a direct comparison could not be made.
Hemodynamics were similar in both the RVA and LVFW
simulations (Fig. 7). Pressure and ejection fraction were
smaller compared to a simulated normal heartbeat.13
The strain patterns were closely related to the pattern
of depolarization timing. In early depolarized regions, early
shortening was observed in the isovolumic contraction phase
(Fig. 4D). In the ejection phase, these myofibers were
stretched slightly, whereas in the isovolumic relaxation
phase, they were stretched dramatically (Fig. 8). In the late
depolarized regions, prestretch occurred in the isovolumic
contraction phase (Figs. 8 and 4D), whereas pronounced
shortening occurred in the ejection phase.
Because the pacing sites in both simulations were approximately opposite, late depolarized regions in the RVA
simulation became early depolarized regions in the LVFW
simulation. Thus, myofibers that were stretched in the late
depolarized regions for the RVA simulation (at the lateral
free wall) shortened early in the LVFW simulation, and vice
versa.
Timing of mid-wall myofiber shortening closely followed
the timing of epicardial depolarization (Fig. 4C).
Discussion
am
ce (m/s)
1.8
1.3
0.75
2.0
2.5
3.0
17.1
6.1
18.1
9.2
2.2
24.1
6.7
11.4
26.3
The best fit was found for the combination am = 2.5 and ce = 1.3 m/s.
With a relatively simple model setup, for pacing at the
LVFW and RVA, depolarization timing patterns and contraction were simulated successfully in the LV with a realistic
myofiber orientation. Here we discuss the extent to which
simplifying assumptions in the model might affect the obtained results.
Kerckhoffs et al.
Figure 7. Global hemodynamics from simulations for pacing at the right
ventricular apex (solid line), the left ventricular free wall (dashed line),
and a normal heartbeat (dotted line). The left panel represents, from top
to bottom, the left ventricular pressure, left ventricular cavity volume, and
aortic flow as a function of time. Dots indicate moments of opening and
closure of the valves. The right panel represents the pressure-volume loops.
Note the decreased systolic pressure and ejection fraction for the pacing
simulations compared to the simulated normal heartbeat.
Results
The model of wave propagation contains 5 parameters, 3
of which were fixed. Myocardial wave velocity cm parallel to
the myofiber was fixed at 0.75 m/s because of the small range
of reported wave velocities (Table 1) in canine myocardium.
Despite limited reports on the subendocardial anisotropy ratio,2 ae was fixed at 1.5, because we expect that the final
solution is relatively insensitive to this parameter. That is,
the subendocardial layer appears important for what happens
after breakthrough. Late in the period of depolarization, the
depolarization wave in the endocardium is directed mainly
Figure 8. Mid-wall circumferential strain as a function of time in the left ventricle (LV) for a complete
cardiac cycle, including the phases of filling, isovolumic contraction, ejection, and isovolumic relaxation.
The top row represents strains at the base; the bottom
row represents strains near the apex of the LV. Strains
from the left to the right column progress around the
LV from the mid-septum to anterior, lateral free wall,
posterior, and back to the septum. Plots were obtained
for right ventricular apex (RVA) simulation (dark solid
line) and left ventricular free-wall (LVFW) simulation
(dark dashed line). The pacing sites for the RVA and
LVFW simulation are indicated by the asterisk (∗) and
plus sign (+), respectively. The dashed vertical lines
indicate moments of switching of phases.
Modeling the Paced LV
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from apex to base, thus coinciding approximately with the
subendocardial fiber direction. Because the transverse component of propagation is small under these circumstances,
the final solution will not be affected severely by inaccuracy of ae . The settings of the varied parameters am and ce
were determined such that the RMS difference of the moment
of breakthrough tdep,b and maximum time tdep,max , needed to
completely depolarize the epicardium of the LVFW was minimal. Parameter am was set to 2.5, predominantly on the basis
of moment of breakthrough. The other varied parameter, the
wave velocity near the endocardium, was set to 1.3 m/s, predominantly on the basis of apparent epicardial wave velocity
after breakthrough.
The current model has been successfully tuned to LVFW
pacing in one dog. Application of the resulting settings in
the model enabled a relatively accurate description of the depolarization wave with a different pacing site, located at the
RVA, in the same dog. Furthermore, we showed that ce and am
were determined reliably from experimental measurements
of epicardial depolarization time. Detailed knowledge of parameter settings requires more study and was beyond the goal
of the present study.
In a previous study in which a normal depolarization pattern during sinus rhythm was simulated,13 parameter values
were different. Subendocardial wave velocity ce was 4 m/s,
whereas it was 1.3 m/s in the present study. In the previous
study, the wave was started in four regions of earliest depolarization as measured previously.6 The Purkinje wave velocity
of 4 m/s accounted for a fast spread of depolarization. With
the current obtained value of ce , simulation of a normal depolarization pattern is possible by increasing the area of the
early depolarized regions. Thus, the model is capable of simulating normal and abnormal patterns, with an equal set of
parameters values.
Electrical and mechanical activation have been measured
previously.7,18 For practical reasons, timing of mechanical
activation was defined as the (measurable) moment of onset of circumferential shortening instead of onset of crossbridge formation. These definitions have different meanings.
This can be illustrated by considering two myofibers in which
cross-bridge formation starts simultaneously. The moment of
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Vol. 14, No. 10, Supplement, October 2003
onset of shortening for those myofibers can be very different, because it depends strongly on the force experienced by
those myofibers from the neighboring tissue. Currently, true
mechanical activation cannot be measured in the entire myocardium. Our model is a helpful tool for gaining insight into
excitation and contraction in a whole heart.
Time courses of circumferential strain in the pace simulations were similar to strains that were measured previously.18
Also, the pattern of onset of myofiber shortening was very
similar to that of depolarization. This finding agrees with experimental data18 in which moments of onset of shortening
and depolarization were found to be linearly related, with a
slope of about unity.
To model the timing of cardiac depolarization, the bidomain model19 also could be used. This yields transmembrane
and extracellular potentials as a function of time and space.
However, because of the need for a very dense mesh and small
time steps to represent the steep depolarization upstroke, on
the whole-heart level, the bidomain model is computationally very demanding. The computationally less demanding
eikonal-diffusion equation has been used previously.20,21 For
LVFW pacing, Colli-Franzone et al.20 found patterns of depolarization that are similar to our patterns. However, in their
model, two simulations are needed. The moment when the
depolarization wave reaches a PMJ (central PMJ) is determined in the first simulation, neglecting the Purkinje system.
In the second simulation, other PMJs are activated, according to the distance to the central PMJ and Purkinje velocity.
With our current model setup, only one simulation is needed.
In another study, Tomlinson et al.21 computed depolarization
times for ventricular pacing using the eikonal-diffusion equation. The Purkinje system was not included, and the maximum depolarization time was unphysiologic. They stated that
a fast conduction system is needed to obtain more realistic
simulations.
Model Setup and Limitations
In the real heart, the influence of the Purkinje system on
myocardial propagation velocity in ventricular pacing depends on the distribution and density of the PMJs, which
have been reported to be variable.6,22-24 Furthermore, it has
been reported that retrograde propagation delay is generally
shorter than anterograde. Anterograde delays from 1 msec
to infinity (in the latter case the wave could not leave the
Purkinje system) during pacing have been reported.1,25,26 In
our model, PMJ density, distribution, and propagation delay
are condensed into effective subendocardial wave velocities
of 1.3 m/s and 0.87 m/s along and perpendicular to the myofiber, respectively.
This approach seems realistic if the distribution of PMJs
is dense. Consequently, only one wavefront is formed. However, in case of a coarse distribution of PMJs,20,27,28 it also is
possible that the depolarization wave reaches a remote area of
myocardium through the Purkinje fibers before the wave that
propagates through the myocardium. In that case, so-called
secondary wavefronts can occur, which cannot be described
by our model.
The real geometry of the heart, including the RV, is more
complex than the truncated ellipsoid we used. Although the
faster RV subendocardial wave velocity is accounted for, the
model lacks the RV free wall. The RV free wall also contains
Purkinje fibers, which might affect the depolarization pattern.
For LVFW pacing, neglecting the RV will affect the depolarization pattern only far from the pacing site. In the RVA
pacing experiment, the electrodes were located at the epicardial RVA, whereas in the simulation, depolarization was
started subendocardially. Because the RV wall is thin and the
wave velocity in the septum is high, we expect that exclusion of the RV free wall hardly affects the LV depolarization
pattern.
For the LV subendocardium, a rotationally symmetric region of faster propagation was assumed. Near the His bundle
in the upper septum, however, PMJs are not present.1 This
does not seem critical for the present study, but a more detailed approach may be required for study of pacing near the
His bundle.
Myocardium is organized into laminar sheets29 four to
eight cells thick. Because of this sheet structure, macroscopic myocardial electrophysiologic properties may be orthotropic,30 in contrast to the assumed transverse isotropy.
Although there are some indications for orthotropy from numerical models,31 accurate measurements of orthotropic electrophysiologic properties have not yet been performed.
The eikonal-diffusion equation solves for depolarization
times. Simulations of pathologies, such as bundle branch
block, Wolff-Parkinson-White syndrome, and pacing, are
possible. However, pathologies related to repolarization (e.g.,
fibrillation, as in the model of Berenfeld et al.27 ) cannot be
simulated with the eikonal-diffusion equation.
Conclusion
With a relatively simple model setup using a combination of the eikonal-diffusion equation and equations of force
balance, depolarization timing and strain patterns for pacing
at the LVFW and RVA were simulated successfully in an LV
with a realistic myofiber orientation. Timing of breakthrough
and total time needed to depolarize the LV in the simulations
and experiments were similar.
Within the myocardium, cross-fiber velocity of wave propagation is estimated to be 0.4 times the velocity along the
myofiber direction (0.75 m/s). Near the endocardium, wave
propagation is about 1.7 times faster than in the rest of the
myocardium, but about 3 times slower than found in Purkinje
fibers.
With respect to mechanics, model and experiment agreed
in the following respects. (1) Both systolic pressure and ejection fraction decreased relative to natural sinus rhythm. (2)
In early depolarized regions, early shortening was observed;
in late depolarized regions, prestretch was observed.
Maps showing timing of onset of shortening were similar
to earlier reported maps in which contraction wave velocity
was similar to depolarization wave velocity.
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