Annals of Biomedical Engineering, Vol. 28, pp. 244–252, 2000 Printed in the USA. All rights reserved.
0090-6964/2000/28共3兲/244/9/$15.00 Copyright © 2000 Biomedical Engineering Society
Influence of the Electric Axis of Stimulation on the Induced Transmembrane Potentials in Ellipsoidal Bidomain Heart EMILIA ENTCHEVA Department of Biomedical Engineering, The Johns Hopkins University School of Medicine, Baltimore, MD (Received 15 March 1999; accepted 3 January 2000)
when the cells 共or fibers兲 are perpendicular to the stimulating field. Despite some quantitative discrepancies between the calculated11 and the measured22 transmembrane potentials induced in frog myocytes by parallel versus perpendicular fields, the ellipsoidal idealization proved to be of practical importance. Considering the complexity of the cardiac muscle structure12,13,20 it is more difficult to accept a similar approximation on a whole heart scale. Currently, the Auckland heart13 is the only ventricular model available incorporating detailed geometric and structural description 共statistically representing a small sample of about 20 canine hearts兲. Case-specific analysis using realistic heart models might become trivial in the near future as new experimental techniques develop10,17,19 to simultaneously describe cardiac geometry and fiber orientation. For this study, however, we decided to use a simplified geometry, an ellipsoidal bidomain heart,4,5 with a detailed representation of the fiber structure and tissue properties. Such a model gives us the ability to separately examine the contribution of fiber curvature, fiber rotation, and conductivity properties in the context of the problem at hand. This study was motivated by preliminary experimental observations by Wikswo and Lin24 of the epicardial polarization in a rabbit heart stimulated by uniform electric field under a variable angle. The x,y,z components of the integrated epicardial image of the prompt response 共the shock-induced polarization at the end of the electric shock兲 were examined. The above study did not find significant deviations of the polarization axis from the axis of the applied electric field when the angle of stimulation was changed. Based on these findings, the authors concluded that the particular fiber architecture might not affect the prompt response significantly. Hereby, we model the above study using an ellipsoidal anisotropic bidomain heart with transmurally rotating fibers, placed in a uniform electric field under a variable angle. We explore numerically the influence of the electric axis of stimulation on the resulting polarization of the heart.
Abstract—This theoretical study was provoked by and designed to interpret, complement and extend the implications of recent experimental observations by Wikswo and Lin 共PACE, 21:940, 1998兲 on the epicardial surface of rabbit hearts. Using a macroscopic bidomain representation of the cardiac structure and the finite element method, we model the response of the heart to uniform electric fields applied under different angles. To overcome intra- and interspecies differences in the geometric and structural characteristics of the cardiac muscle, the analysis is conducted for an idealized ellipsoidal heart. Although idealized, this heart model incorporates important structural features, i.e., fiber curvature, transmural fiber rotation, and unequal anisotropy for the intra- and extracellular domains. This study shows that regions of maximum polarization of opposite sign may develop along an axis, significantly deviating from the axis of the applied electric field. The polarization evoked inside the ventricular wall seems to be a major contributor to this phenomenon. Nonperiodic structural inhomogeneities on multicellular level 共endocardial ‘‘trabeculation’’ in our model兲 result in local unaligned polarization dipoles weakening the magnitude of the global polarization dipole and reducing its deviation from the axis of stimulation. Our results might be helpful in improving current understanding of defibrillation mechanisms. © 2000 Biomedical Engineering Society. 关S0090-6964共00兲00703-7兴 Keywords—Anisotropy, Cardiac, Computer simulation, Defibrillation, Fiber curvature, Fiber rotation, Finite element method.
INTRODUCTION Previous research on the cellular,22 multicellular,8 and tissue block7 level has been motivated by the search for a preferential axis of stimulation in the heart. The ellipsoid was the shape of choice for an idealized representation of cardiac cells in a theoretical study by Klee and Plonsey11 on the effects of the cell shape and the electric axis of stimulation. Their predictions were experimentally tested by Tung et al.,22 who confirmed previous findings that the threshold for field stimulation is greater Address correspondence to Emilia Entcheva, Ph.D. Department of Biomedical Engineering, Johns Hopkins University School of Medicine, 720 Rutland Ave., Rm. 703 Traylor Bldg., Baltimore, MD 21205. Electronic mail:
[email protected]
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Ellipsoidal Bidomain Heart in Electric Field
245
METHODS Model, Mesh, Solution The potential distribution in the steady-state bidomain heart model is governed by Equations 共1ab兲.9 The monodomain, Laplace, equation applies in the bath and blood medium 共Eq. 1c兲; ⵜ• 共 e ⵜ⌽ e 兲 ⫽⫺
ⵜ• 共 i ⵜ⌽ i 兲 ⫽
 共 ⌽ i ⫺⌽ e 兲 , Rm
 共 ⌽ i ⫺⌽ e 兲 , Rm
ⵜ• 共 ⵜ⌽ 兲 ⫽0.
共1a兲
共1b兲 共1c兲
For the myocardium, the subscripts 共i, e兲 refer to the intracellular and extracellular domains. In Eq. 共1c兲 the isotropic conductivity, 关 Sm⫺1 兴 , and electric potentials, ⌽ 关 V兴 , in the blood or bath describe the material properties and the electric field for the volume conductor outside the heart. The following parameters are used in Eqs. 共1a兲 and 共1b兲: myocardial conductivity tensors in the global coordinate system, 关 Sm⫺1 兴 , potentials, ⌽ 关 V兴 , transmembrane current density per unit volume, I m 关 Am⫺3 兴 , surface-to-volume ratio, 关m⫺1兴, and membrane resistance times unit area, R m 关 ⍀ m2 兴 . The transmembrane potential, V m , is obtained as the difference between the intracellular and extracellular potential, (⌽ i ⫺⌽ e ). The transformation of the conductivity tensors from the local coordinate system associated with a fiber to the global coordinate system of the ellipsoidal heart is done by three consecutive rotations using Euler angles.5 The finite element method in its Galerkin formulation is used to solve for the potentials.2,16 The domain is tessellated in linear tetrahedral elements and the equations are enforced on each element. We use rowindexed sparse matrix storage format, and a preconditioned biconjugate gradient method for the solution of the resulting linear system of equations, as reported in previous studies.4,5 The boundary conditions for the problem concern the normal components of the extra- and intracellular current, as well as the extracellular potential at the surface of the heart, ⍀: 共 e ⵜ⌽ e 兲 nˆ 兩 ⍀ ⫽ 共 ⵜ⌽ 兲 nˆ 兩 ⍀ ,
共2a兲
共 i ⵜ⌽ i 兲 nˆ 兩 ⍀ ⫽0,
共2b兲
⌽ e 兩 ⍀ ⫽⌽ 兩 ⍀ .
共2c兲
FIGURE 1. Model geometry: rotated ellipsoidal heart in a cylindrical bath. „a… A cross section of the cylinder „in the xy plane at z Ä0… is shown. The surface triangulation for the FEM mesh is visible. The coordinate system is shown on the right-hand side. The electric field is applied in the x direction through plate electrodes on the top and bottom of the cylinder. The heart is rotated clockwise with respect to the x axis „in this particular geometry the heart is rotated on 60°…. „b… A schematic explaining fiber rotation. Shown are the outline of the heart shape and one parallel. At a given point P, the epicardial fiber „running along the parallel… and the endocardial fiber „rotated on 120° counterclockwise… are schematically presented with a thicker line. The arrow shows the direction of fiber rotation.
The geometry used in this study follows the experimental setting by Wikswo and Lin.24 An ellipsoidal heart is placed in an insulated cylinder with an axis along the x axis of the global coordinate system, diameter 100 mm and height 100 mm, filled with Tyrode solution. The dimensions of the heart ellipsoid approximate those of an average rabbit heart: long semiaxis a epi⫽22 mm, a endo ⫽12 mm, and index of ellipticity 共the ratio of the major to minor semiaxis兲, k⫽1.6. For some simulations a spherical heart, k⫽1, with the same long semiaxes is used. The conditions of a uniform electric field of 10 V/cm are accomplished by applying constant voltage at the top and bottom plates of the cylinder, much bigger in size than the dimensions of the heart. The surface area of each of the electrodes is about 80 cm2. A structured finite element mesh is constructed for this problem, with gradual rotation of the axis of symmetry between the cylinder and the heart surface in clockwise direction 关Fig. 1共a兲兴. The tetrahedral finite element mesh has the following characteristics: number of nodes 120 100, number of tetrahedral elements 705 600 共of them 273 600
246
EMILIA ENTCHEVA TABLE 1. Model parameters. Parameter
Value
Description
Source
el et il it Rm 
0.625 Sm⫺1 0.236 Sm⫺1 0.174 Sm⫺1 0.019 Sm⫺1 0.91 ⍀ m2 140 000 m⫺1
longitudinal extracellular conductivity transverse extracellular conductivity longitudinal intracellular conductivity transverse intracellular conductivity membrane resistance times unit area membrane surface area to tissue volume ratio blood conductivity bath (tyrode solution) conductivity
1
0.6 Sm⫺1 2.0 Sm⫺1
blood bath
in the myocardium兲. Mesh generation, finite element calculations and postprocessing of the data were performed using custom-designed software written in C programming language. Tissue Properties, ‘‘Trabeculation’’ All model parameters, including the conductivity values for the anisotropic myocardium, the blood cavity, and the surrounding bath, are listed in Table 1. The anisotropy ratios of the intracellular and interstitial space for the listed conductivity values by Clerc2 are 9.2 and 2.6, respectively. Cardiac fibers are modeled as forming concentric shells, parallel to the surface. The fibers rotate transmurally around the vector normal to the epicardial surface at each point, and the angle of rotation, ␣, is a function of wall depth. In this idealized model we have chosen a linear function to govern the fiber rotation. Figure 1共b兲 illustrates schematically the fiber rotation. The specific epicardial fiber orientation, ␣ epi 共measured with respect to the equator兲, and the total angle of rotation throughout the wall, ␣ tot , are listed in Table 2. Several cases are considered, summarized in Table 2. AR is the standard case of ellipsoidal anisotropic heart with fiber rotation. Fiber rotation is assigned starting from the epicardium with fibers along the parallels 共the lines running parallel to the equator兲, i.e., ␣ epi⫽0°, and completing a 120° ( ␣ tot) linear counterclockwise rotation TABLE 2. Influence of electric axis of stimulation: cases considered. Case
k
Anisotropy
␣ epi
␣ tot
Specifics
AR ANR I trAR sAR
1.6 1.6 1.6 1.6 1.0
unequal unequal isotropic unequal unequal
0° 90° 0° 0° 0°
120° 0° 0° 120° 120°
none none none ‘‘trabeculated’’ none
Note:AR⫽anisotropic ellipsoidal heart with fiber rotation; ANR ⫽anisotropic ellipsoidal heart without fiber rotation; I⫽isotropic ellipsoidal heart; trAR⫽AR with trabeculation; sAR⫽anisotropic spherical heart with fiber rotation.
1 1 1 23 23
21 21
into the depth of the wall. Case ANR is an ellipsoidal heart without fiber rotation. Fibers are running along the meridians 共the longitudinal lines connecting the two poles兲, i.e., ␣⫽90°. Case I is an isotropic heart, with conductivity equal to iso⫽0.144 Sm⫺1. In some of the simulations the endocardial surface of an anisotropic ellipsoidal heart with fiber rotation is ‘‘trabeculated’’ 共case trAR兲, e.g., the nonhomogeneous structure of the endocardium due to trabecular muscles is taken into account. This is accomplished by randomly 共probability 0.5兲 assigning myocardial or blood conductivity to the tissue in up to 2 mm depth from the endocardial surface. Since the randomization is done by elements, and the average element side length in that region is ranging from 400 to 800 m, we expect to have clusters of 5–20 or more cells.
Global quantitative measures for the prompt response Polarization Dipole Moments. Here we define ‘‘dipole moment,’’ dm, as the integral directional value of the transmembrane potential over a given surface or a volume, ⍀,24 normalized by the respective number of points for that particular domain, N ⍀ : N
dm⫽
1 ⍀ V 共 s ⫺s ¯ 兲 , s→x,y,z. N ⍀ j⫽0 m j
兺
共3兲
The dipole component in the z direction 共the axis of rotation of the heart兲 was expected and found to be negligible 共at least an order of magnitude different from the smaller, y component兲. Thus the magnitude, A dm , and the angle, P dm , of the polarization dipole are calculated as follows: A dm ⫽ 冑共 dm x 2 ⫹dm y 2 兲 , P dm ⫽arctg
dm y . dm x
共4a兲 共4b兲
Ellipsoidal Bidomain Heart in Electric Field
247
FIGURE 3. Transmembrane polarization patterns on the midmyocardial surface for an ellipsoidal anisotropic heart „AR… subjected to a uniform field orthogonal to its axis of symmetry „90°…. Left: front view, right: a view from the pole. Scale for the potentials is Á0.05 V. Red represents depolarized areas, blue—hyperpolarized areas. FIGURE 2. Transmembrane polarization patterns in an ellipsoidal anisotropic heart with fiber rotation „AR… when the electric axis of stimulation varies. The heart is sectioned at z Ä0 level in the xy plane, with the front half removed. The view includes a transverse section of the wall plus a half of the endocardial surface „behind the section plane…. For convenience, in the presentation the orientation of the heart is kept constant while the direction of the field varies as indicated by the arrows. The angle between the electric axis of stimulation and the long axis of the heart is: first row ˆ0°, 15°, 30°, 45°‰ and second row ˆ60°, 75°, 90°‰. Scale for the potentials is Á0.05 V. Red represents depolarized areas, blue–hyperpolarized areas.
We choose the polarization dipole moment as an adequate global descriptor of the shock-induced polarization, due to the fact that it conveys positional information. The electric field in this study is applied along the x axis; the heart is rotated in the xy plane, around the z axis. Changes in the y-dipole moment 共and, if any, in the z-dipole moment兲 will be indicative of a polarization dipole developing along an axis different from the electric axis of stimulation.
stant of decay. In the second case, the field is perpendicular to the fibers only in the z⫽0 plane, i.e., longitudinal components of the conductivity tensor will contribute to the space constant. Given that the conductivity tensors are defined over volume elements, rather than at grid points 共in this z⫽0 plane兲, the effective space constant 共even for z⫽0兲 for the second case will have a higher value than the one in the first case. Moreover, the two cases are different in terms of changes in the orientation of the fibers with respect to the field lines. Consequently, a gradient in conductivity values with respect to the field 共contributing to the induced polarization5,21兲 is absent in the first case, and present in the second case. The ‘‘bulk’’ polarization, away from the tissue boundaries, forms due to tissue anisotropy and fiber curvature. The singularity and the massive branching at the poles in our model result in an interesting transformation of the ‘‘bulk’’ polarization when the field becomes more orthogonal to the long axis of the heart. Abutting regions
RESULTS Polarization Patterns Figure 2 shows transmembrane potential distribution in an anisotropic ellipsoidal heart with fiber rotation 共AR兲 for uniform electric field applied under different angles: 0°, 15°, 30°, 45°, 60°, 75°, 90°. As the electric axis of stimulation varies, changes are observed in the ‘‘border’’ polarization due to current redistribution at the bath-tissue and blood-tissue interface. For the particular epicardial fiber orientation in our model 共fibers at 0° with respect to the parallels兲, the extent of the ‘‘border’’ polarization at the epicardium is different for stimulation along the long axis of the heart and for orthogonal stimulation. In the first case, the field is strictly perpendicular to the fibers at all surface points, i.e., their conductivity in the transverse direction will determine the space con-
FIGURE 4. Transmembrane polarization „TMP… isopotential lines on the midmyocardial level for an ellipsoidal anisotropic heart „AR… rotated with respect to the electric axis of stimulation. The TMP isopotential lines are drawn at À0.05 „dark blue…, 0.0 „green…, ¿0.05 V „red…. The angles of rotation of the heart with respect to the electric axis of stimulation are ˆ0°, 45°, 90°‰.
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EMILIA ENTCHEVA
of opposite polarity occur at the poles due to virtual ending of the fibers. In the presence of transmural rotation, they seem to form a characteristic spiral 共Fig. 3兲. While the model might be unrealistic considering the fiber arrangement at the base of the heart, it is likely to resemble the fiber configuration at the apex. In addition, the rotation of the fibers leads to a twist in the zero line throughout the ellipsoid as the orientation of the field deviates from the axis of symmetry 共Fig. 4兲. This figure presents a better view of the three-dimensional spatial distribution of the polarization patterns and their complex change with variations in the axis of the applied field. Dipole Moments Figure 5 presents the x and y components of the polarization dipoles in an AR heart for seven different orientations of the applied field between 0° and 90°. Three dipoles are differentiated: endocardial, epicardial, and a global, whole heart dipole. The existence of a small, changing y component in all three cases is dem-
FIGURE 6. The effect of anisotropy on the polarization dipoles in ellipsoidal heart. Normalized magnitude and angle of the polarization dipole in an ellipsoidal heart „AR, ANR, I… when the electric axis of stimulation varies ˆ0°, 15°, 30°, 45°, 60°, 75°, 90°‰. Left column—normalized dipole magnitude †V cm‡, logarithmic scale; right column—dipole angle with respect to the axis of stimulation. The direction of the endocardial dipole is inverted. For AR, the whole heart and the endocardial dipole angles are close and difficult to distinguish.
FIGURE 5. Polarization dipole moments in ellipsoidal anisotropic heart when the electric axis of stimulation varies. The field is applied under the following angles: ˆ0°, 15°, 30°, 45°, 60°, 75°, 90°‰. „a… Dipole components „dmx, dmy… on the epicardial surface; „b… dipole components „dmx, dmy… on the endocardial surface; „c… dipole components „dmx, dmy… for the whole myocardial mass. Note: dmx is along the electric axis of stimulation; dmy—orthogonal to the electric axis of stimulation.
onstrated. This component of the polarization dipole indicates deviation of the axis of polarization from the axis of stimulation. The effect of anisotropy and fiber rotation on the polarization dipoles is examined in Fig. 6. In the isotropic case 共I兲, a small magnitude whole heart dipole develops which deviates not more than 8° from the axis of stimulation. The reason for this lies in the endocardial and epicardial polarization dipoles, similar in magnitude and opposite in direction, as well as in the lack of ‘‘bulk’’ polarization. Anisotropy-related ‘‘bulk’’ polarization is most likely responsible for the deviation of the whole heart dipole on more than 20° from the axis of stimulation in the anisotropic cases 共AR and ANR兲. The magnitude of the whole heart polarization dipole and its
Ellipsoidal Bidomain Heart in Electric Field
249
FIGURE 8. The effect of endocardial ‘‘trabeculation’’ on the polarization dipoles. Dipole normalized magnitude „left column, logarithmic scale… and dipole angle „right column…, with respect to the electric axis of stimulation: „a… endocardium; „b… whole ventricular mass.
eliminated, and the amplitude of the resulting dipole is extremely reduced 关Fig. 8共a兲兴. Conversely, a slight increase is observed in the magnitude and the deviation of the whole heart polarization dipole from the electric axis of stimulation 关Fig. 8共b兲兴. FIGURE 7. The effect of heart geometry on the polarization dipoles. Polarization dipoles „angle… for a spherical anisotropic heart, sAR, and ellipsoidal „ k Ä1.6… anisotropic heart, AR, when the electric axis of stimulation varies ˆ0°, 15°, 30°, 45°, 60°, 75°, 90°‰.
The Ellipticity Effect
deviation from the axis of stimulation reach their maxima in the anisotropic case without fiber rotation 共ANR兲. The case with 120° transmural fiber rotation, AR, produces intermediate results when compared to I and ANR. Figure 7 examines the effect of the heart shape by comparing the angle of the polarization dipoles for a spherical and ellipsoidal heart 共sAR and AR兲. The deviation of the dipoles from the axis of stimulation is very similar for these two cases. The largest differences are observed on the endocardial level, about 10°. Finally, the ‘‘trabeculation’’ of the heart surface profoundly affects the characteristics of the resulting dipoles. In the endocardium, the deviation of the polarization dipole from the axis of stimulation is almost
If a uniform electric field is applied to an isotropic homogeneous body, we expect the most affected regions to be those where the field lines are normal to the surface, and the least affected regions to be those where the field lines are tangential to the surface. It has to be pointed out that a simple ellipsoidal geometry will be rather sensitive to the location of the minimum and maximum of the resulting polarization as the index of ellipticity varies. Unlike the sphere, the ellipsoid exhibits angle differences between the radius vector and the vector normal to the surface at a given point, for all points except those at azimuthal angles 0° and 90° 关Fig. 9共a兲兴. This naturally is expected to lead to an axis of the resulting polarization deviating from the electrical axis of stimulation in an ellipsoid, but not in a sphere 关Fig. 9共b兲兴. First, the angle of the radius vector, , can be expressed as follows:
DISCUSSION
250
EMILIA ENTCHEVA
reality and in our simulations, these ideal conditions are not completely satisfied. More specifically, distortions in the field lines due to the presence of the ellipsoid are not excluded; and an inhomogeneity is introduced by the blood cavity at the core of the ellipsoid. Nevertheless, in the simulations presented in this study, we show that the generic curve based in Eq. 共7兲, Fig. 9共b兲, is qualitatively valid for the shock-induced polarization in a complex fibrous structure such as the ellipsoidal bidomain heart. In particular, we demonstrate how the anisotropy, fiber rotation, and structural discontinuities modulate the resulting polarization. Geometry, Anisotropy, and Structural Inhomogeneities
FIGURE 9. The radius vector and the vector normal to the surface in an ellipsoid „a… The angle, ␦, representing the difference between the radius vector and the vector normal to the surface in an ellipsoid „angle for the normal n …. In a sphere no such difference exists. „b… Expected difference between the axis of stimulation and the axis of the resulting dipole „maximum polarization regions with opposite polarity… in an isotropic ellipsoid. Three indices of ellipticity are shown: 0.8; 1.6; 2.0.
r 冑y 2 ⫹z 2 tan共 兲 ⫽ ⫽ , x x
r⫽ 冑y 2 ⫹z 2 .
共5兲
Second, given the equation of the ellipsoid, x 2 /a 2 ⫹(y 2 ⫹z 2 )/b 2 ⫽1, the normal to the surface is defined as nˆ (2x/a 2 ,2冑y 2 ⫹z 2 /b 2 ); thus the angle of the vector normal to the surface, n , is tan共 n 兲 ⫽
冉
冊冉 冊
冑y 2 ⫹z 2 2 冑y 2 ⫹z 2 2x 2 : ⫽k . b2 a2 x
共6兲
The difference angle, ␦, can be expressed then as a function of the angle of the vector normal to the surface, n , and the index of ellipticity, k;
␦ ⫽ n ⫺ ⫽ n ⫺arctan
冉
冊
tan共 n 兲 . k2
共7兲
The simple relation in Eq. 共7兲 is applicable for analysis of polarization dipoles induced by perfectly uniform electric field in an isotropic homogeneous ellipsoid. In
According to our analysis of the ellipticity effect 关Fig. 9共b兲兴, the axis of stimulation and the direction of the polarization dipole for the sphere should coincide, if the heart geometry is the sole factor determining shockinduced polarization. In contrast, the results of this study show that the anisotropic sphere exhibited the same trend of changes in the polarization dipoles as the ellipsoidal anisotropic heart but the deviation from the axis of stimulation was slightly smaller 共Fig. 7兲. Moreover, Fig. 6 demonstrates that changes in the tissue structural properties 共anisotropy and fiber rotation兲 can have more pronounced effect on the orientation of the induced polarization dipoles than changes in geometry alone. Going from isotropic 共I兲 to anisotropic 共ANR兲 heart increased the whole heart dipole deviation from the axis of stimulation by about 70° 共Fig. 6兲, while 60% change in ellipticity 共from k⫽1 to k⫽1.6兲 led to less than 5° difference 关Fig. 7共c兲兴. Hence, the current study presents evidence that cardiac tissue anisotropy acts in a way of modulating the apparent ellipticity in the geometry of the heart when its response to a variable electric axis of stimulation is considered. As a result a spherical heart, which if isotropic, should not exhibit any deviations of the polarization dipole from the electric axis of stimulation 关Fig. 9共b兲兴, does have a similar behavior to an ellipsoidal heart 共Fig. 7兲. Furthermore, Fig. 6 suggests that the major contributor for the offset of the resulting dipole from the axis of the applied field is the interior polarization. In the isotropic heart, the ‘‘border’’ polarization dipoles of similar magnitude and opposite sign resulted in negligible 共8°兲 deviation of the whole heart dipole from the axis of stimulation because of lack of ‘‘bulk’’ polarization. The interior or ‘‘bulk’’ polarization is caused by current redistribution away from the boundaries and is present only in the conditions of unequal anisotropy ratios in the intra- and extracellular domains. The results in Fig. 2 illustrate the complex spatial changes in the ‘‘bulk’’ polarization as the axis of stimulation varies. Fiber rotation increases the transmural TMP gradient, as pointed out in a previous study.5 From Fig. 6 it is
Ellipsoidal Bidomain Heart in Electric Field
clear that the rotation of the fibers reduces the influence of the anisotropy, thus resulting in a polarization dipole in an intermediate position with respect to the isotropic case on the one hand, and the anisotropic case with no fiber rotation, on the other hand. Both, anisotropy and fiber rotation cause a very complex polarization transmurally 共Fig. 3兲. The ‘‘zero’’ line is twisted throughout the wall 共Fig. 4兲. The gradient in the transmembrane polarization and the closeness of regions of opposite polarity change dramatically as the orientation of the applied field varies. These results imply a possibility for polarization phase singularities in the ventricular wall, caused by the shock,3 even in the conditions of a uniform electric field. The random ‘‘trabeculation’’ of the endocardial surface resulted in smaller islands of polarization, not necessarily associated with the curvature or fiber rotation induced response of the tissue. Naturally, the latter leads to a weaker dipole formed on a larger scale and a reduced anisotropy effect. For the endocardial surface, which was subjected to such inhomogeneities, the magnitude of the polarization dipole was reduced and its deviation from the axis of stimulation was almost negligible 关Fig. 8共a兲兴. On the whole heart level, the magnitude and the deviation of the polarization dipole were increased in the presence of ‘‘trabeculation’’ compared to the solid wall endocardium 关Fig. 8共b兲兴. The obvious reason for this effect in our model is that the small-scale unaligned endocardial dipoles reduced the compensatory effect between the epi- and endocardial border polarization of opposite sign. Our results suggest that any syncytial inhomogeneities that do not exhibit a particular pattern6 will lead to the formation of local dipoles and weakening of the large-scale anisotropy-determined polarization dipole. Shock-induced polarization caused by nonperiodic inhomogeneities on the multicellular level is easily measurable8 and probably more important in defibrillation than the polarization associated with cellular discreteness 共‘‘saw tooth’’ effect兲,14,15 which is likely to be of much lower amplitude and thus difficult to register experimentally.8,25 Practical Value and Limitations of the Model The model used here is an idealization. The heart is not an ellipsoid, and the fiber architecture for this model, although based on experimental data,20 is a simplification of the complex real fiber organization. Structural inhomogeneities were only partially included in the model by random ‘‘trabeculation’’ of the endocardium. Their presence on a large scale everywhere in the ventricular mass 共relatively regularly, in the form of laminae,12 or irregularly as fiber branching and termination18 and/or changes in the intracellular volume fraction6兲 can alter the results obtained here. Tissue inhomogeneities in the epi- or sub-
251
epicardial region might have been the reason for the discrepancy between the epicardial polarization dipoles measured by Wikswo and Lin24 and the results presented here. The sensitivity of the polarization dipoles to multicellular structural inhomogeneities demonstrated in this study supports this hypothesis. However, more experimental work is needed to rule out differences due to resolution, imaging depth and other side effects. Without further research on the subject considering the pre-shock history, and the post-shock activation due to the nonlinear, active properties of the membrane, any attempt to generalize these results in the view of successful/unsuccessful defibrillation can turn into a pure speculation. The ultimate validation would come from experimental studies; thus one of the primary goals of this work is to encourage further theoretical and experimental investigation of the problem.
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Clerc, L. Directional differences of impulse spread in trabecular muscle from mammalian heart. J. Physiol. (Paris) 255:335–346, 1976. 2 ColliFranzone, P., L. Guerri, and S. Tentoni. Mathematical modeling of the excitation process in myocardial tissue: Influence of fiber rotation on wavefront propagation and potential field. Math. Biosci. 101:155–235, 1990. 3 Efimov, I. R., Y. Cheng, D. VanWagoner, T. Mazgalev, and P. Tchou. Virtual electrode-induced phase singularity: A basic mechanism of defibrillation failure. Circ. Res. 82:918– 925, 1998. 4 Entcheva, E., J. Eason, I. Efimov, Y. Cheng, R. Malkin, and F. Claydon. Virtual electrode effects in transvenous defibrillation-modulation by structure and interface: Evidence from bidomain calculations and optical mapping. J. Cardiovasc. Electrophysiol. 9:949–961, 1998. 5 Entcheva, E., N. Trayanova, and F. Claydon. Patterns of and mechanisms for shock-induced polarization in the heart: A bidomain analysis. IEEE Trans. Biomed. Eng. 46:260–271, 1999. 6 Fishler, M. G. Syncytial heterogeneity as a mechanism underlying cardiac far-field stimulation during defibrillationlevel shocks. J. Cardiovasc. Electrophysiol. 9:384–394, 1998. 7 Frazier, D. W., W. Krassowska, P.-S. Chen, P. D. Wolf, E. G. Dixon, W. M. Smith, and R. E. Ideker. Extracellular field required for excitation in three-dimensional anisotropic canine myocardium. Circ. Res. 63:147–164, 1988. 8 Gillis, A. M., V. G. Fast, S. Rohr, and A. Kleber. Spatial changes in transmembrane potential during extracellular electric shocks in cultured monolayers of neonatal rat ventricular myocytes. Circ. Res. 79:676–690, 1996. 9 Henriquez, C. S. Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21:1–77, 1993. 10 Hsu, E. W., A. Muzikant, S. Matulevicius, R. Penland, and C. Henriquez. Magnetic resonance myocardial fiber orientation mapping with direct histological correlation. Am. J. Physiol. 43:H1627–H1634, 1998. 11 Klee, M. and R. Plonsey. Stimulation of spheroidal cells-The
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