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Implications of macroscopic source strength on cardiac cellular activation models
Implications of Macroscopic Source Strength on Cardiac Cellular Activation Models
R. Plonsey*, PhD, and A. van Oosteromt, PhD
Abstract: This paper compares several cellular-level models of cardiac activation according to the sources that are generated and their macroscopic fields. Since macroscopic field patterns and strengths are well documented, by comparing the models of cardiac activation, the more controversial microscopic processes can be evaluated. The results show that it is most likely that both macroscopic and microscopic activation of cardiac tissue are uniform processes, and in a direction that is transverse to the fiber axis. Key words: double layers, tissue resistivity, linear cable theory, anisotropy.
not been fully evaluated because of limited, direct, experimental information. In this paper the sources that are generated by cardiac activation are described. Based on these sources one can calculate the magnitude of the macroscopic electric potential fields that would arise both within and outside the heart; these can be compared with the relatively well documented macroscopic data. This approach was, in fact, followed by one of the authors (RP) a number of years ago,’ but with equivocal results. This paper brings the cardiac activation approach up to date by including new activation hypotheses and utilizing improved models consistent with current knowledge of cardiac histology and electrophysiology. It is shown that the source description required to account for normal macroscopic electric potential measurements imposes strong requirements on cellular level behavior. An assessment of the implications for both normal and abnormal activation based on source-strength requirements is a goal of the present investigation. An earlier, shorter treatment of this material has been published as a conference paper.2
Because of the syncytial nature of cardiac tissue, activation may proceed in essentially any direction. Indeed, experiments with ectopic stimuli show ellipsoidal-like propagation away from the stimulation site. This demonstrates first, that propagation can take place in all directions, and second, that the velocity is anisotropic (the highest value lies in the direction of the fiber axis). A considerable interest is attached to the activation process that is taking place at the cellular level since this can contribute to an understanding of cardiac arrhythmias. Experimental study is hampered by the small size of the cells and the damage to the preparation that can arise from the insertion of macro- and microelectrodes. As a consequence, a number of competing ideas about cardiac activation on a cellular level exist, but have
* From the Department of Biomedical Engineering, Duke University, Durham, North Carolina, and f the Laboratory of Medical Physics and Biophysics, University of Nijmegen, The Netherlands. Reprint requests: R. Plonsey, Department of Biomedical Engineering, Duke University, Durham, NC 27706.
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Cardiac
Vol. 24 No. 2 April 1991
Activation
Macroscopic Description A description of the macroscopic activation of the heart has been obtained experimentally by placing a large number of multipoint needle electrodes (plunge electrodes) into the heart. The time of arrival of the activation wave at each electrode is registered by a characteristic intrinsic defection in the wave. I
I
Fig. 1. An example
I
I
I
I
I
From the position of all electrodes showing synchronous activation, an (isochronous) surface can be estimated (interpolated), which describes the wavefront of rhe advancing wave at that instant. A graphical description of sequential isochronous surfaces (isochrones) fully characterizes cardiac activation. The earliest such study was performed by Scher et al. using a canine model,3 while a similar study of the human heart was performed by Durrer et a1.4 In Figure 1 a portion of the results of a more recent study by Scher and Spach is reproduced.’
7
of activation isochrones measured in the canine heart. Two heart sections of two different dogs are shown, which are part of a more comprehensive study by Scher and Spach.’ (Redrawn from Sche?). Note that the activation wave, once propagation is well established, lies roughly parallel to the endo- and epicardial surface. Activation sequence is initiated at the region shown in black; successive isochrones are at 5 ms intervals. The activation sequence is determined from the intrinsic deflection recorded by point electrodes that lie on the needles (visible as straight lines on the figure). The circles show the location of Purkinje fibers.
Cardiac
Cellular
A casual inspection of activation isochrones, such as those shown in Figure 1, demonstrates that propagation can proceed readily in all directions. This electrophysiological property has long been understood and is the basis for characterizing cardiac tissue as syncytial. However, if the activation pattern arising from an ectopic stimulus is examined, then, rather than being spherical, the pattern is seen to be ellipsoidal, hence reflecting anisotropy. In fact, the propagation velocity along the fiber axis is roughly three times greater than in a cross-fiber direction.6 Since the direction of cardiac fibers is essentially parallel to the endocardium and epicardium,7 an examination of Figure 1 leads to the conclusion that the macroscopic apparent progression of wavefronts is in the cross-fiber direction. (This is not surprising given that this is the “slow” direction). The initiation of ventricular excitation along broad endocardial surfaces by the conduction system enhances this behavior.’ The emphasis in this paper will be on modeling potentials arising from a uniform plane-propagating wave proceeding in a cross-fiber direction in healthy cardiac tissue.
Microscopic
Hypotheses
Uniform (Continuous) Cross-fiber Model. In electrophysiology, the simplest example of activation (propagation) is that involving a single axon lying in an unbounded volume conductor. This is the model developed by Hodgkin and Huxley’ and further extended and implemented by numerous authors. ‘O One can immediately apply this work to that of a multicellular (three-dimensional) tissue when the latter consists of a bundle of parallel fibers that lies in oil. In this case the behavior of each fiber is identical and described by the same linear-core conductor model’ I as applies to the isolated fiber, except that for the unbounded case in the expression for the transmembrane current the axial resistance is ri (unit: fLm - ’ ) . Under bounded conditions the interstitial resistance, rp, must be included in this expression by replacing ri with (r; + re). As noted above, propagation in cardiac tissue generally takes place across, rather than along fibers. Thus, the com-
’ In the initial moments following an ectopic stimulus the anisotropic velocity as well as the underlying anisotropic conductivities enter the source description both with regard to direction and magnitude.’ However, for normal propagation under macroscopic conditions, the results are nearly planar wavefronts that are oriented almost entirely in the cross-fiber direction, In this paper, only normal activation is considered so that anisotropy is merely latent and not actually expressed.
Activation
Models
l
Plonsey
and van Oosterom
101
plexities of cellular structure in particular, including intercellular junctions, may have to be considered. On the other hand, at some level of approximation, one can treat the structure as if it were continuous (syncytial), and the fibers constituting the tissue as if they were continuous, uniform, and supporting propagation in the direction in which it occurs. TO the extent that this is valid, a “uniform cable” treatment may be applied to the propagating cardiac wave on a microscopic level. This constitutes one possible cellular model, where both microscopic and macroscopic propagation proceeds in a cross-fiber direction. Other possibilities, to be considered in this paper, are as described below.
Microscopic Axial-Macroscopic Transverse. Clerc” describes transverse propagation as following a path established by the end-to-end intercellular (gap) junctions. This is illustrated in Clerc’s figure reproduced in Figure 2. Assuming activation to progress this way one would conclude that the activation of the interior portion of each cell is initiated from its ends. If macroscopic propagation is uniform, as in the normal case, then interior cellular activation would necessarily be initiated from both ends, more or less synchronously. The idea that cellular activation is axial is explicitly contained in the paper by Geselowitz et a1.13 Here it is assumed that activation of each cell is essentially axial regardless of the macroscopic propagation direction. Thus, in this model, even though macroscopic activation is transverse, cellular activation nevertheless is assumed to proceed in a longitudinal direction.
Zig-zag Cellular Propagation. Propagation on a cellular level has been investigated both experimentally and with models by Spach, Dolber, and HeidIage, “L’ 5 They describe a “zig-zag” path arising mainly from the presence of nonconducting septa. These affect side-to-side gap junctional coupling (not considered by Clerc), which are believed responsible for transverse propagation. The progressive loss with age of side-to-side junctions is described as leading to a zig-zag course of activation. The conduction pathway is seen as irregular and longer due to this geometry. Figure 3 is taken from Spach et al. I4 and shows a zig-zag transverse path. A further elaboration on the zig-zag form of propagation is described by Spach et aI.15 in the case of longitudinal propagation of premature action potentials. The hypothesis is advanced that propagation shifts from longitudinal to dissociated and transverse since the latter is safer, though slower.
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I 8
f
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I I L-r.
;
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:
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1
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1
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4
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Fig. 2. This figure emphasizes the presence of end-to-end intracellular junctional resistance and its role in facilitating
progagation in the transverse direction (redrawn from Clerc’*). (A) Physical structure. (B) Transverse current flow. The junctional resistance is rj while the cytoplasmic resistance is r,. Nonuniform Transverse. A final hypothesis examined here assumes ceIIuIar activation to be transverse to the fiber axis, but nonuniform. The nonuniformity arises from the presence of the discrete intercellular junctions. This model could be thought of as an extension of the work of Rudy and Quani6 and Henriquez and Plonsey,” which, however, apply to axial propagation. In these papers it is shown that within each cell activation may take place at an increasingly high velocity for increasingly elevated values of junctional resistance. In this case the propagation velocity may be determined mainly by the delay arising in crossing the junctional resistance. Thus, while an a cellular level propagation is in the same direction as it is macroscopically, it is nonuniform in a temporal sense.
In this paper each of the aforementioned rnicroscopic models will be considered with respect to the macroscopic saurces they establish. The extracellular fields produced by those sources will then be contrasted and compared to those measured macroscopically.
Quantitative Analysis of the Source Models This section develops expressions for the tative description of macroscopic sources from each of the four cellular microscopic considered. The resulting expressions will be
quantiarising models utilized
Cardiac
Cellular
--
Activation
Models
Plonsey
l
-----
--
---
200pm
I
-
JC
_I
-Fig. 3. In this figure, the “sawtooth”
to the long axis of the
--
--
--
_-_-_-
--
-curve denotes
103
--
---
--
and van Oosterom
-___
----
the irregular zig-zag course of excitation
spread in a direction
transverse
fibers (redrawn from Spach14).
in a subsequent section for evaluating the aforementioned cardiac activation hypotheses through an examination of the macroscopic potential distributions that are generated. Since this material is mainly drawn from the literature, this summary is brief. In this analysis the volume conductor will be assumed to be unbounded, that is, only so-called primary sources will be considered. A finite boundary is readily included through the introduction of secondary sourcesL8 lying on that boundary. (If the boundary is with air, then the secondary sources insure that the total internal field has a zero normal derivative over the bounding surface, a condition that can be used to solve for the secondary sources). The field generated by the secondary sources must be added to those arising from the primary sources. Activation of a single cell lying in a volume conductor of infinite extent has been shown” to result in an equivalent double layer source located in the membrane surface and oriented normal to that surface. Designating the source as k, then z? = (@jai - @&)6,
(1)
where ii is the dimensionless unit outward normal from the membrane, the subscripts e and i denote, respectively, evaluation at the outside and inside of the membrane, and 2 has the units of A.m.m-* (ie, a current dipole sz4rface density since A is amperes and m is meters). For a multicellular preparation equation 1 can be applied to each cell; the resulting sources all lie in an unbounded uniform medium. The potential field generated by these sources can be evaluated by summing the free-space field of each
element (ie, a surface integral over all membranes). Alternatively, with respect to extracellular space, the sources can be averaged and approximated by a continuous volume source density function 1 In view of the unbounded “free-space” environment, the volume conductor field, @, can be readily found by superposition. Since any elementary source (ie, j( ?) dv) at source point T sets up a “free-space” dipole field (ie, $V ( l/R) dv), where R is the length of T’ T, a vector pointing from a source at T to an arbitrary fieldpoint T’, and V( l/R) is the unit dipole field, then the field established at T’ is a+‘)
=
$& j- .i.v P
(l/R)
dv,
(2)
where the integral in equation 2 is taken over all cells present. Further details are found in Plonsey” and van Oosterom.”
Axial Fiber(s) As stated in the introductory section, the interest of this paper lies with cross-fiber propagation of cardiac depolarization. However, this is based on the well-studied case of propagation along uniform fibers (ie, axial propagation). Consequently, we first review some of the relations that have been established for axial propagation. Uniform Axial Fiber(s). Equation 1 applies to a cell of any shape, but when the cell is cylindrical and the field along the membrane depends only on
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the axial coordinate, it may be transformed equivalent current dipole volume source given by
j(z) =
& (@jUj -
into an density
extracellular field can be found from the integral described in equation 2, and this comes out
Qeue)iiz.
(6) In equation 3, z is the axial variable (of T), ii, is a dimensionless unit vector along the z axis, and j signifies the current dipole moment per unit volume (unit: A.m.mp3). jcompletely fills the fiber, and in any cross-section, is uniform. Since this an equivalent source, it correctly evaluates fields in a restricted region only-in this case the extracellular space. The source described by equation 3, as pointed out earlier, lies in a uniform unbounded extracellular medium of conductivity ue, and hence set up “freespace” fields. The volume conductor field is, accordingly, found by substituting the source described in equation 3 into the source-field expression of equation 2, where the integration is now taken throughout the intracellular domain of the fiber. For a multifibered preparation the sources can be found by application of equation 3 to each fiber. Note that, again, the resulting sources all lie in a uniform unbounded medium so that equation 2 applies once more except that the volume integration extends through all fibers. While, in equation 3, a possible interaction between fibers appears to have been ignored, this in fact enters the evaluation of the actual membrane fields Qi, Qe at each fiber. Except near the periphery of the bundle, where there are many axial fibers of similar or nearly similar size, the linear core conductor model” can be applied, and equation 3 can be simplified. That is, @i and (DCcan be evaluated in terms of the transmembrane potential V,, where =
”
- re = (ri + rr) ‘,’
(D,(?‘)
= L
re
4~ ( re + r;)
d
v 132
(V,,,)&,-V(M)
dv,
where V, is chosen (defined) to be uniform over any cross-section (normal to the direction of propagation) while V denotes integration over the entire muscle space. This last modification was introduced because cardiac cells occupy approximately 80% of the total space, therefore, it is convenient to consider the sources as if they occupy the entire volume; to compensate for this, the results must be multiplied by the cell density fraction, and such a fraction is contained in equation 7. For field points within the tissue (ie, in the interstitial space), equation 7 simply returns equation 5. In particular, from equation 5, we note that the difference of potential in the interstitial space on opposite sides of an activation wave, V,,,, is the change in transmembrane potential V, from rest to peak (plateau) times the resistance divider ratio, that is, e
(ri + r,) ‘,’
j-
(7)
VWCX”P = AV,,, *,
li
qli
where the subscript n denotes the direction of propagation (we will use z for axial and x for transverse to the fiber -orientation). The volume V, designates the actual space occupied by the cells. If equations 4 and 5 are applied, assuming the applicability of the linear core-conductor model, then equation 6 can be simplified by
I
= (Vpeak -
Vm,) --?._
r, + rj’
(8) (5)
and where ri and re are the intracellular and interstitial axial resistance per unit length. When equations 4 and 5 are applicable, the source is seen to depend solely on the spatial derivative of the transmembrane action potential. In the case of uniform propagation, this can be determined from knowledge of the temporal action potential (which to a first approximation does not change greatly within a preparation) and the local conduction velocity. The sources arising from plane wave propagation along the fiber axis based on the above simplifications are expressed by equations 3,4, and 5. The
For field points outside the heart, for instance, at the torso surface, the potential from activation sources at the depolarization wavefront within the heart (ignoring secondary sources from the torso boundary and any other inhomogeneities), based on the assumption of a uniform double-layer, evaluates to
where An is the total solid angle subtended by the entire activation wavefront at the field point. Some additional details on equations 8 and 9 and their derivation from equation 6 are found, for example, in Plonsey et al.” and van Oosterom.”
Cardiac
Axial Fiber(s) With Intercellular
Cellular
Junctions.
The equations 1, 2, and 3 are directly applicable to a single skeletal muscle fiber or a bundle of such fibers. For cardiac muscle, a modification may be necessary because of the presence of intercellular junctions. From the theory underlying equations 1 and 3, these equations may be applied to the cell membrane between junctions; the junctional region, however, requires special attention. The pertinent part of this region can be idealized as a superposition of connection tubes containing the intracellular media and separated by a wall from the extracellular space. One can apply equation 3 to each tube. Since the effective source strength depends on the intracellular volume, and since the volume of all tubes is very small compared to the cell volume, one expects the net junctional source to be much less than the cellular source. A quantitative consideration of these factors shows that, indeed, the junctional source is negligible.2o The effect of junctions on the overall sources in a multicellular cardiac preparation appears to be mainly indirect, caused by the influence of the junctions on the membrane fields G$, @c over the body of each cell. This effect will be small, assuming low resistance junctional values that recent experiments suggest.*’ Under abnormal conditions, such as ischemia, decoupling results, and the effect of the junctional resistance on, @i,
(10)
where rim is the myoplasmic resistance per unit length (this could be evaluated from pi/( ITU’), where a is the cell radius and pi the myoplasmic resistivity), while Rj is the junctional resistance per cell whose length is 1. For a cylindrical bundle of parallel cardiac muscle fibers.lying in a conducting medium, since the junctional resistance is to be neglected (or simply included in the assigned intracellular resistance), the result is a structure similar to that of a bundle of skeletal muscle fibers (ie, uniform and axially con-
Activation
Models
l
Plonsey
and van Oosterom
105
tinuous). The applicability of the linear core conductor model, and correspondingly equations 4 and 5, would seem assured for fibers near the center of the bundle, but not for those at the periphery (where re = 0). A mathematical study of this preparation shows that, in fact, the linear core conductor model can be applied over essentially the entire cross-section for bundles of perhaps 1 cm in diameter.23 Probably the left ventricular free wall satisfies this requirement .24,’ 9 Resistances and Resistivities. In the analysis to follow, we will consider predominantly planar propagation, and we will study the use of equations 8 and 9. Equation 8 clearly requires the knowledge of the resistivities r, and r,. As stated before, these have their origin in the linear cable theory, both representing resistances per unit length. In applying the basic notions of linear cable theory to a three-dimensional model of propagating wavefronts, one has to introduce appropriate expressions for these resistivities. To this end, different approaches and notations have appeared in the literature. To avoid confusion we devote some space here to introducing and explaining the various resistances and resistivities that will be used in the sequel of this paper, aimed at application in equation 8. Throughout this paper R will denote a resistor, (unit fl), r will represent a resistance per unit length (unit 1R;rn~ ’ ) and p will denote a resistivity (unit 1R.v~). We first consider a unit area S of the wavefront (cross-section of tissue). For the propagation in the fiber direction the intracellular resistance per unit length follows from
(11) with pi the microscopic resistivity of the intracellular space (myoplasm), and f the fraction of tissue volume occupied by intracellular space. This corresponds to a much used model, in which the cells are represented by closely packed cylinders (eg, Weidmann25; Clerc”). (An alternative expression for ri follows from explicitly considering an individual fiber with radius u. As noted before, this reads r, = pi/(lTu’).)
For r,, the corresponding
Ie = (1 -
expression is Pe f)S.
(12)
When inserting these expressions into equation 8, the actual value of S (cross-sectional area of an entire bundle) is irrelevant since S drops out of the potential divider ratio. This has led to the -introduction of a longitudinal intracellular resistivity
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Journal of Electrocardiology
Vol. 24 No. 2 April 1991 (13)
the dimension of which is the same as that of pi (ie, n.m). For the extracellular compartment the corresponding variable pe’ follows from I_ Pe _-
Pe 1 -f’
(14)
In situations where the spatial variation of the potentials is large with respect to cell dimensions (or rather the space constant of the process) the intracellular and extracellular compartment may be considered in parallel. The longitudinal overall, gross tissue resistivity p’ in this case is related to pe’ and pi’ as PitPer
p’ = Pi’
+
(15)
Pe’
In several studies this resistivity p’ has been measured directly.26,27 Furthermore, direct measurements of the intracellular potential jump 18% I, associated with the depolarization process and the accompanying extracellular, 1A@< 1have provided values for the ratio
(16) as follows directly from equations 4, 5, and 8. In a similar way equation 8 can now be expressed as I
Vwave=
PC pe’ + pi
AV, = p
AV,.
(17)
1+P
The values for the resistivities like pi’ and pe’ reported in the literature have been computed from measured values of p’ and measured resistivity ratio’s, that is, either by combining equations 15 and 16 (eg, Weidmann25), or equations ( 15) and ( 17) (eg, Roberts et a1.26). Note that the computed values of pe’ and pi’ are very sensitive to the assumed value off since this value is of the order of 0.75. For propagation in the transverse (ie, cross-fiber) direction similar variables pi’, pk, and p’ have been proposed, although a simple relationship with cellular morphology (similar to the ones expressed in Plonsey et al. I1 and Clerc”) is no longer directly obvious. These variables can be used for relating wavefront strength V,,, to the jump in the transmembrane potential A V, in a similar manner to that expressed in Henriquez et al.” Uniform (Continuous)
Cross-fiber
Model
For cross-fiber propagation in a multifibered cardac preparation, we shall adopt, as a reference condition, a uniform equivalent fiber bundle, similar to
the axial propagation as described in the previous subsection. This is illustrated in Figure 4. In Figure 4A the cardiac muscle cells are shown in cross-section. Relevant to cross-fiber propagation are intercellular gap junctions, as noted in Figure 4A. Their presence is documented by Sommer and Scherer2’ to be relatively dense. These provide both axial and lateral coupling. Assuming their resistance to be small, as discussed above, and therefore neglecting their presence in favor of a continuum (ie, a syncytium) forms the basis for the simplified structure of Figure 4B. An electrical description of the excitable tissue described in Figure 4B is shown in Figure 4C. The latter is, again, the linear core conductor model, which now may be interpreted as representing a unit crosssectional area of tissue (relative to the direction of the propagating wave). In Figure 4C, pi and pe represent the averaged intracellular and interstitial resistance per unit length in the cross fiber direction as defined in the section on axial fibers. Accordingly, the potentials at field point i’, generated by a wavefront progressing in the cross-fiber direction, can also be found from equation 9, after insertion of the appropriate resistance or resistivity values in expressions like equation 8 or 17. This treatment of cardiac tissue as a continuum (syncytium) in the cross-fiber direction is a specific example of the approach to plane wave propagation in any arbitrary direction in the cardiac muscle, as introduced by Muler and Markin.29 This work forms the basis of the bi-domain approach to cardiac muscle, as can be found in the work of Tung3’ and Miller and Geselowitz,40 and discussed in Plonsey.3 ‘ An explicit evaluation of sources arising in the cross-fiber direction, and leading to equation 7 is given in van Oosterom.”
Macroscopic Transverse
Axial-microscopic
According to Clerc” propagation in the transverse direction proceeds through coupled resistances at the ends of cells. This process is illustrated in Figure 2. A consequence of excitation of cells at their ends only must be that propagation within each cell proceeds in the longitudinal direction (with a collision somewhere in the interior). For an essentially uniform transverse wave, then, roughly, both ends will be excited synchronously and the collision site will occur at the center of the cell. In this model the cellular dipole sources are directed along the cell axis, with those at the “left” opposite those on the “right”. As a consequence, the sources will have a quadru-
Cardiac
Cellular
Activation
Models
l
Plonsey
and van Oosterom
107
gap junction ,~~~~~intracellular ‘~~~~ space .-.y ;..:.. l
reAx
reAx
reAx
C Fig. 4. (A) A conceptual cross-sectional view of a 3-D cardiac tissue preparation showing individual fibers and their nexal junctions. The fiber axes are normal to the page. Plane wave propagation is assumed to proceed from left to right (normal to the fiber axes). (B) For low resistance intercellular (nexal) junctions, propagation from left to right behaves as if the cardiac tissue structure consisted of cylindrical fibers oriented from left to right as shown. These fibers are coupled, laterally, (intracellularly), as shown. (C) The electrical network that is associated with transverse plane wave propagation, for low resistance intercellular junctional resistance, is described here. This is the linear-core-conductor model and can be derived from either (A) or (B).
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polar component, at the expense of an otherwise dipolar component, and this must result in a severely reduced extracellular field. Furthermore, its direction will be primarily tangential rather than radial.
forVW,, and
surface potentials), and partial or complete cancellation of transverse components. If we assume a mean inclination of the zig-zag angle to be CX,, then in place of equation 8 we have VWave= cos %n( Vpeak
Zig-zag Propagation
-
vrest)
gif COS* CX, +
This type of conduction is proposed by Spach et al. r4,15 However it is not described in great enough detail to permit a quantitative analysis. Based on Figure 3 we may consider the effect of propagation in a direction that is both normal to the fiber axis and also inclined at an angle (Yto the radial direction (ie, the direction normal to the free wall). This is described in Figure 5 where a is the aforementioned angle of inclination of the local wave. Assuming junctional resistances to be small, the effect will be to generate the same dipole sources as in the uniform reference model, where 01 = 0; however the dipole direction will be at an angle (Y # 0 to the radial. A consequence will be a loss of dipole strength in the outward direction (ie, that direction which accounts
gifCOS2
CL, +
gilsin* (Y,
x
,lJi’sin* (Y,
+g,‘COS*
Ci,
+
g,‘sin* CX,’ (18)
where gi’ = l/pi’, gef = l/p,‘, etc.29
Nonuniform Transverse In one-dimensional simulations, when cellular decoupling reaches a high level, then the phase velocity within the cell will increase, while the propagation velocity itself decreases (as a result of a propagation delay across the intercellular junction). The source density within the cell is given by equation 7 and the gradient of V, can be expressed as avi?l -= an
av,,, dt = _1 av, -0 at ’ at dn
(19)
where 0 is the cellular phase velocity. Clearly as the decoupling increases and 8 increases, the source strength decreases. The picture is actually further complicated because it becomes no longer possible to obtain spatial behavior from temporal behavior (ie, in the absence of uniform propagation the wave equation is not satisfied). But the qualitative conclusion can be drawn that the source-strength is diminished with increasing decoupling.
radial direction
Evaluation
fl
of the Four Models
Based on the analysis presented in the previous section we now present an evaluation of each of the four cardiac source models considered. The differences in the depth at which each of these models could be analyzed will be reflected in this evaluation.
endocardium
The Axial Situation
Fig. 5. Geometry associated with zig-zag propagation, which makes an angle of cx with the radial direction. In this example it is assumed that there is no component of propagation in or out of the page (no longitudinal component).
Several studies dealing with along-fiber propagation have been reported, the analysis being based on the approaches described in the section on uniform cross-fiber models. Fiber bundles or superfused cardiac fiber strands of different animals and heart segments have been used. In spite of the fact that this represents the easier of the two extreme types of propagation (along- vs cross-fiber), a wide range of values, both for the actual measured ones and for
/
epicardium
Cardiac
Table
1. Linear Cable Parameters
Cellular
as Measured
Models
Pr
mV
P
252
_
-
-
188 160 340 852 383
152 125 199 213 174
96 100 100 99 _
24 21.5 46 74 _
0.33 0.33 0.86 2.92 _
252
I16
5 1.5
1.08
f
(ih?l)
(R.cm)
_
_
-
-
_
WeidmannL5 Clerc’L Roberts”’ Robens35 Epstein3’
470 402 -
51 48 _ -
0.75 0.70 _ -
624 575 360 291 319
KleberS3
166
63
0.75
221
Uniform (Continuous)
Cross-fiber
Model
In this section an evaluation of source and field of a uniform cross-fiber model is considered. This model is described in Figure 4 and its basic source analysis given in the sections on axial fibers and uniform cross-fiber models. Our present objective is to examine the correspondence between the field strength predicted by the model (ie, equation 9) with actual measurements. We consider two cases, the first for fields lying within the heart muscle (ie, application of equation 8), and the second where the field is at the torso surface (ie, application of equation 9).
Cardiac Field Points Lying Intramurally. There is a substantial literature describing interstitial intramural potentials arising from a propagating wavefront. From these one finds essential agreement Table 2. Linear Cable Parameters Pi
(R.cm)
as Measured
.f
(ATim)
(&m,
98.7
that the magnitude of the potential across this wave, v WilVe,is around 40 mV. 19,24,34,35 It is this potential that is evaluated in equation 8. To compare the predictions of equation 8 with the aforementioned 40 mV value, knowledge of the cross-fiber intracellular and interstitial resistances (pit and pet) is required. In contrast to the along-fiber case, relatively few data have been reported for the cross-fiber case. In Table 2 we list these few data, as obtained from Clerc” and from Roberts and Scher,26 which are the only sources for data on transverse intracellular cardiac resistance. From the data in Table 2, we note that. using resistances from Roberts and Scher35 gives values of VWUYCin close agreement with experiment, while Clerc gives results that are rather low. In a critique, van Oosterom points out that the model on which Clerc interprets his experimental results fails to include-in the transverse direction-the fact that no fully developed (planar) wavefront couId have materialized within the width of the bundle studied.36 This is clearly evident from Clerc’s Figure 4, where, according to the conditions required for the application of the linear cable approach, the intracellular and the extracellular waveforms should have been proportioned throughout time, and clearly these are not. As a consequence, the anisotropy in the resistivity values as computed from this experimentally determined value of p is probably a severe overestimation. On the other hand, some modest support for the data of Roberts and Scher3’ comes from the experiments of Epstein and Foster,32 where, at least for extracellular resistances, which can be compared, by Various Authors:
I
Pe
(R.cm)
109
mV
(O.cm)
those computed based on the model, can be found. In Table 1 we present a survey of the results of several of these studies. Note that all of these pertain to the axial (ie, longitudinal) situation. Interstitial resistance values are included for Epstein and Foster32 for skeletal muscle (on the assumption that it should correspond reasonably well with cardiac muscle, since the structural contributions are about the same). In addition, axial resistances found from the perfused papillary muscle preparation of Kleber and Riegger33 are also included.
and van Oosterom
Longitudinal Propagation I Vwuw A v,
I
(R.cm) Rush”
Plonsey
l
by Various Authors:
Pi’
Pe
Pi
Activation
f (R;m)
Transverse I
Am, ~
Propagation
Av, I
mV
VWIl”II mV
B
-
-
-
-
-
563
-
-
_
Clerc’* Robert?
3620
127 -
0.70 -
Robens’
1 _
_
_
5181 3800 1677 -
424 750 1247 1250
392 625 715 _
98 100 100 -
8 16 43 _
0.09 0.19
Rush”
Epstein3’
0.43 -
110
Journal of Electrocardiology
Vol. 24 No. 2 April 1991
there is somewhat better agreement in contrast to the somewhat poorer agreement with Clerc. The data in Tables 1 and 2 include the intriguing possibility that V,,, is nearly the same in both the longitudinal and transverse directions (namely, approximately 50 mV from the recent work of Kleber, and approximately 40 mV based on Roberts and Scher and others, as noted above). Such a condition requires that oe’/oi’ = oefloif or, namely, an equal anisotropy ratio. When this occurs, as can be confirmed from equation 18, V,,, becomes independent of the direction of propagation CLThis means that for any activation wave, even if the direction of propagation of some segments is not transverse to the fiber axis, as we have been assuming here, the source strength is still the same. Thus a uniform double-layer source can, in this way, be justified under more general conditions. The implication is that while anisotropy would have an important effect on the activation shape (eg, elliptical for an ectopic site) the source density is nevertheless essentially uniform. Cardiac Field Points Lying at Body Surface.
Equation 9 can be used to calculate electrocardiographic potentials at the body surface if the activation pattern is known or can be inferred. Based on the activation isochrone geometry and the position on the torso at which potentials are desired, Aln can be evaluated. While a’, can now be found using equation 9, it should be noted that this is a primary-source field (ie, the field that would be found if the sources in the heart lay in a uniform homogeneous volume conductor of infinite extent). The various inhomogeneities must also be taken into account, and this can be done (actually or conceptually) by the addition of secondary sources” situated at surfaces separating regions of different conductivity (blood cavity, heart muscle, lungs, and air). This is essentially the procedure (boundary-element method) followed by van Oosterom and Huiskamp,37 who chose V,,, = 40 mV,
uhearb/ung
=
5,
.!m,od~uh~an
=
3,
and,
of
course, Uair = 0. The simulations of van Oosterom and Huiskamp are based on reasonable physiological activation sequences (actually, these were found from measured body surface potentials through an inverse procedure3’), and their forward simulations are in excellent quantitative agreement with the measured electrocardiograms. The outcome lends additional support to the choice of V,,, = 40 mV associated with the cellular process. Independent simulations of the magnetocardiogram were also performed based on this same data leading, again, to excellent agreement with measurements.39 Since the simulations noted above are affected by
the choice of the assumed conductivity ratios and geometry as well as of V,,,, one should examine the degree of certainty with regard to the former variables. The sensitivity of the simulation to the assumed values of conductivity was studied by van Oosterom and Huiskamp.37 They showed that inclusion of inhomogeneities was essential to obtain a good quantitative iit to the measured date. Most important is inclusion of accurate body shape and ventricular blood cavity. These two factors were quantified by van Oosterom and Huiskamp using MRI cross-sectional scans. Furthermore, blood conductivity and heart muscle conductivity are generally well established quantities. Consequently, one can conclude that the magnitude of VW,, is confirmed in these simulations with very good confidence. Microscopic Transverse
Axial-macroscopic
As stated below, according to Clerc12 propagation in the transverse direction proceeds through coupled resistances at the ends of cells. In fairness to Clerc, it should be noted that at the time of his paper the evidence for lateral junctions was poorly documented. The fact that, for an essentially uniform transverse wave, the resulting sources will be exactly quadrupolar (as stated earlier), results in a too severly reduced field. Furthermore, since its direction would be primarily tangential rather than radial, this model of the sources representing the depolarization process could not be comparable with quantitative measurements of VW,,, and surface electrocardiograms. The assumption that activation of each cell is longitudinal, even while macroscopic activity proceeds in another direction (ie, in the cross-fiber direction that we have been assuming) is explicitly made by Geselowitz and Miller. l3 In spite of several reasonable results that this leads to, one can conclude from the above source analysis that the model is incompatible with the strength and orientation needed to generate observed extracellular fields. Zig-zag Cellular
Propagation
The analysis of this model can be based on equation 18. For a,,, = 7r/6, a value that seems reasonable based on Spach et al., then V,,,, = 0.866 x 100 X 0.59 = 5 1 mV, based on Table 2 (choosing the Roberts and Scher resistivity data). While the resulting ECG would have an increased amplitude the increase is too small to be statistically significant. So this type of zig-zag propagation is not ruled out.
Cardiac Cellular Activation Zig-zag propagation could also occur in a way that introduces axial cellular activation. In this case the effect on sources depends critically on the cellular processes, and realistic details must be examined. To the extent that a substantial component of membrane depolarization is associated with longitudinal conduction, which, as we have seen, does not contribute sources that are productive in terms of body surface potentials and V,,,, then such a model may be ruled out.
Nonuniform
Transverse
This situation has been studied for longitudinal propagation with a rigorous one-dimensional model by Quan and Rudy,22 who supply an evaluation of the decrease in amplitude with rising junctional resistance. They show a 40% decrease in extracellular potential magnitude for a tenfold increase in junctional resistance. Even a hundredfold increase in junctional resistance leaves an extracellular potential magnitude that is 20% of control. Consequently, it would not seem that one can distinguish a discontinuous transverse propagation model from a continuous one solely on the basis of a measurement of V waveor from the EGG. One should also note that the above considerations apply, strictly, to a one-dimensional strand of cardiac cells oriented longitudinally. If a multifibered three-dimensional tissue is considered, it is not clear what behavior will be seen. For one, because of the many staggered junctions that are present, it is possible that the overall behavior is less discontinuous. On the other hand, since propagation is transverse, the effective cell length between junctions will be an order of magnitude shorter than for the simulated longitudinal orientation.
Models
l
Plonsey
and van Oosterom
111
Using currently available data it would appear that cardiac tissue is activated transverse to the fiber axis and, furthermore, such activation occurs not only macroscopically but also on a cellular level. Thus, in spite of the possibilities for complex pathways, including cellular activation in a longitudinal direction, activation would appear to take place essentially uniformly across each cell membrane but in a transverse direction. It is important to note that while this model assumes the maximum production of outward sources that are possible, these are never too large with respect to the size of the macroscopic field that is required. Any other model results in either too modest a change, and therefore cannot be ruled out, or a very large decrease, which is unacceptable. The type of zig-zag propagation described by Spach et al. is consistent with the requirements of a macroscopic source as long as deviation from this main activation is not large.
Acknowledgments The authors are indebted to Drs. J. Sommer and P. Dolber of the Department of Pathology at Duke University for helpful assistance with regard to the 3-D structure of cardiac tissue and for sharing (J.S.) some recent unpublished studies on which Figure 4A is based. The author (RP) thanks Prof. H. K. Boom for making available resources of the Division of Biomedical Engineering at the University of Twente, where this work was done, while on sabbatical leave from Duke University, and for his helpful discussions.
References Conclusions The main goal of this paper is, again, to illustrate how a cellular model can be evaluated by examining the sources that result, and then comparing the simulated field amplitude with that obtained in macroscopic measurements. As shown above, all models depend in one way or another on pPt and pif (or pp’ and pi’). At present, the large differences in the values reported in the literature hamper an unequivocal conclusion on the various models considered. In particular the transverse data seem to be uncertain. The basic measurements of CIerc’2 need to be repeated, while avoiding the pitfalls in that approach.36
Plonsey R: An evaluation of several cardiac activation models. J Electrocardiol, 7:237, 1974 Plonsey R, van Oosterom A: A cellular activation model based on macroscopic fields. In Sideman S, Beyar R, Kleber A (eds): Cardiac electrophysiology, circulation, and transport. Kluwer Academic Publishers, Norwell, MA, 1991 Scher A, Young, AC: Ventricular depolarization and the genesis of the QRS. Ann NY Acad Sci 65:768, 1957 Durrer D, van Dam RT, Freud GE et al: Total excitation of the isolated human heart. Circulation, 41:899, 1970 Scher A, Spach MS: Cardiac depolarization and repolarization and the electrocardiogram. p. 357. In Berne RM (ed): Handbook of physiology, section 2: the cardiovascular system; volume 1. The Heart. Am. Physiol. Sot., Bethesda, MD, 1979
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6. Spach MS, Miller WT, Miller-Jones E et al: Extracellular potentials related to intracellular action potentials during impulse conduction in anisotropic canine cardiac muscle. Circ Res 45: 188, 1979 7. Streeter Jr DD, Spotnitz HM, Pate1 DP et al: Fiber orientation in the canine left ventricle during diastole and systole. Circ Res 24:339, 1969 8. Colli-Franzone P, Guerri L, Viganotti C et al: Potential fields generated by oblique dipole layers modeling excitation wavefronts in the anisotropic myocardium: comparison with potential fields elicited by paced dog hearts in a volume conductor. Circ Res 5 1:330, 1982 9. Hodgkin AL, Huxley AF: A quantitative description of the membrane current and its application to conduction and excitation in nerve. J Physiol 117:500, 1952 10. Cronin J: Mathematical Aspects of Hodgkin-Huxley Neural Theory. Cambridge University Press, Cambridge, 1987 11. Plonsey R, Barr RC: Bioelectricity: a quantitative approach. Plenum Press, New York, 1988 12. Clerc L: Directional differences of impulse spread in trabecular muscle from mammalian heart. J Physiol 255:335, 1976 13. Geselowitz DB, Miller WT: Active electric properties of cardiac muscle. Bioelectromagnetics 3: 127, 1982 14. Spach MS, Dolber PC: Relating extracellular potentials and their derivatives to anisotropic propagation at a microscopic level in human cardiac muscle. Circ Res 58:356, 1986 15. Spach MS, Dolber PC, Heidlage JF: Influence of the passive anisotropic properties on directional differences in propagation following modification of the sodium conductance in human atria1 muscle. Circ Res 62:811, 1988 16. Rudy Y, Quan WL: A model study of the effects of the discrete cellular structure on electrical propagation in cardiac tissue. Circ Res 61:815, 1987 17. Henriquez CS, Plonsey R: Effect of resistive discontinuities on wave-shape and velocity in a single cardiac fiber. Med Biol Eng Comput 25:428, 1987 18. Plonsey R: The formulation of bioelectric source-field relationships in terms of surface discontinuities. J Franklin Inst 297:3 17, 1974. (Also reprinted in Pilkington T, Plonsey R: Engineering contributions to biophysical electrocardiography. IEEE Press, New York, 1982) 19. van Qosterom A: Cell models: macroscopic source descriptions. p. 155. In Macfarlane PW, Lawrie ‘ITV (eds): Comprehensive electrocardiology. Vol. I. Pergamon Press, Oxford, 1989 20. Plonsey R, Barr RC: Effect of junctional resistance on source-strength in a linear cable. AM Biomed Eng 13:95, 1985 21. Metzger P, Weingart R: Electric current flow in cell pairs isolated from adult rat hearts. J Physiol 366: 177, 1985
22. Quan W, Rudy Y: The reflection of discontinuous propagation of myocardial excitation in extracellular potentials. Personal communication 23. Hemiques CS, Trayanova N, Plonsey R: Potential and current distributions in a cylindrical bundle of cardiac tissue. Biophys J 53:907, 1988 24. van Oosterom A: Cardiac potential distributions. PhD thesis, University of Amsterdam, Amsterdam, 1978 25. S. Weidmann: Electrical constants of trabecular muscle from mammalian heart. J Physio12 10: 104 1, 1970 26. Roberts DE, Hersch LT, Scher AM: Influence of cardiac fiber orientation on wavefront voltage, conduction velocity, and tissue resistivity in the dog. Circ Res 44:701, 1979 27. Rush S, Abildskov JA, McFee RS: Resistivity of body tissues at low frequencies. Circ Res 7~262, 1963 28. Sommer J, Scherer B: Geometry of cell and bundle appositions in cardiac muscle: light microscopy estimation. Am J Physiol 248:h792, 1985 29. Muler AL, Markin VS: Electrical properties of anisotropic nerve-muscle syncytia-II spread of flat front of excitation. Biophysics 22~536, 1977 30. Tung L: A bidomain model for describing ischemic myocardial d-c potentials. PhD thesis, MIT, Cambridge, MA, 1978 31. Plonsey R: The use of the bidomain model for the study of excitable media. Lectures on mathematics in the life sciences 21:123, 1989 32. Epstein BR, Foster KR: Anisotropy in the dielectric properties of skeletal muscle. Med Biol Eng Comput 21:51, 1983 33. Kleber AG, Riegger CB: Electrical constants of arterially perfused rabbit papillary muscle. J Physiol 385:307, 1987 34. Corbin LV, Scher AM: The canine heart as an electrocardiographic generator. Circ Res 41: 58, 1977 35. Roberts DE, Scher AM: Effect of tissue anisotropy on extracellular potential fields in canine myocardium in situ. Circ Res 50:342, 1982 36. van Oosterom A: Anisotropy and the double layer concept. p. 91. In Macfarlane PW (ed): Progress in electrocardiology. Pitman, Tunbridge Wells, 1979 37. van Oosterom A, Huiskamp GJM: The effect of torso inhomogeneities on body surface potentials quantified by using tailored geometry. J Electrocardiol 22:53, 1989 38. Huiskamp GJM, van Oosterom A: The depolarization sequence of the human heart surface computed from measured body surface potentials. IEEE Trans Biomed Eng BME-35:1047, 1988 39. van Oosterom A, Oostendorp TF, Huiskamp GJM, ter Brake HJM: The magnetocardiogram derived from electrocardiographic data. Circ Res 67: 1503, 1990 40. Miller WT III, Geselowitz DB: Simulation studies of the electrocardiogram. I. The normal heart. Circ Res 43:301, 1978
R. Plonsey*, PhD, and A. van Oosteromt, PhD
Abstract: This paper compares several cellular-level models of cardiac activation according to the sources that are generated and their macroscopic fields. Since macroscopic field patterns and strengths are well documented, by comparing the models of cardiac activation, the more controversial microscopic processes can be evaluated. The results show that it is most likely that both macroscopic and microscopic activation of cardiac tissue are uniform processes, and in a direction that is transverse to the fiber axis. Key words: double layers, tissue resistivity, linear cable theory, anisotropy.
not been fully evaluated because of limited, direct, experimental information. In this paper the sources that are generated by cardiac activation are described. Based on these sources one can calculate the magnitude of the macroscopic electric potential fields that would arise both within and outside the heart; these can be compared with the relatively well documented macroscopic data. This approach was, in fact, followed by one of the authors (RP) a number of years ago,’ but with equivocal results. This paper brings the cardiac activation approach up to date by including new activation hypotheses and utilizing improved models consistent with current knowledge of cardiac histology and electrophysiology. It is shown that the source description required to account for normal macroscopic electric potential measurements imposes strong requirements on cellular level behavior. An assessment of the implications for both normal and abnormal activation based on source-strength requirements is a goal of the present investigation. An earlier, shorter treatment of this material has been published as a conference paper.2
Because of the syncytial nature of cardiac tissue, activation may proceed in essentially any direction. Indeed, experiments with ectopic stimuli show ellipsoidal-like propagation away from the stimulation site. This demonstrates first, that propagation can take place in all directions, and second, that the velocity is anisotropic (the highest value lies in the direction of the fiber axis). A considerable interest is attached to the activation process that is taking place at the cellular level since this can contribute to an understanding of cardiac arrhythmias. Experimental study is hampered by the small size of the cells and the damage to the preparation that can arise from the insertion of macro- and microelectrodes. As a consequence, a number of competing ideas about cardiac activation on a cellular level exist, but have
* From the Department of Biomedical Engineering, Duke University, Durham, North Carolina, and f the Laboratory of Medical Physics and Biophysics, University of Nijmegen, The Netherlands. Reprint requests: R. Plonsey, Department of Biomedical Engineering, Duke University, Durham, NC 27706.
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Cardiac
Vol. 24 No. 2 April 1991
Activation
Macroscopic Description A description of the macroscopic activation of the heart has been obtained experimentally by placing a large number of multipoint needle electrodes (plunge electrodes) into the heart. The time of arrival of the activation wave at each electrode is registered by a characteristic intrinsic defection in the wave. I
I
Fig. 1. An example
I
I
I
I
I
From the position of all electrodes showing synchronous activation, an (isochronous) surface can be estimated (interpolated), which describes the wavefront of rhe advancing wave at that instant. A graphical description of sequential isochronous surfaces (isochrones) fully characterizes cardiac activation. The earliest such study was performed by Scher et al. using a canine model,3 while a similar study of the human heart was performed by Durrer et a1.4 In Figure 1 a portion of the results of a more recent study by Scher and Spach is reproduced.’
7
of activation isochrones measured in the canine heart. Two heart sections of two different dogs are shown, which are part of a more comprehensive study by Scher and Spach.’ (Redrawn from Sche?). Note that the activation wave, once propagation is well established, lies roughly parallel to the endo- and epicardial surface. Activation sequence is initiated at the region shown in black; successive isochrones are at 5 ms intervals. The activation sequence is determined from the intrinsic deflection recorded by point electrodes that lie on the needles (visible as straight lines on the figure). The circles show the location of Purkinje fibers.
Cardiac
Cellular
A casual inspection of activation isochrones, such as those shown in Figure 1, demonstrates that propagation can proceed readily in all directions. This electrophysiological property has long been understood and is the basis for characterizing cardiac tissue as syncytial. However, if the activation pattern arising from an ectopic stimulus is examined, then, rather than being spherical, the pattern is seen to be ellipsoidal, hence reflecting anisotropy. In fact, the propagation velocity along the fiber axis is roughly three times greater than in a cross-fiber direction.6 Since the direction of cardiac fibers is essentially parallel to the endocardium and epicardium,7 an examination of Figure 1 leads to the conclusion that the macroscopic apparent progression of wavefronts is in the cross-fiber direction. (This is not surprising given that this is the “slow” direction). The initiation of ventricular excitation along broad endocardial surfaces by the conduction system enhances this behavior.’ The emphasis in this paper will be on modeling potentials arising from a uniform plane-propagating wave proceeding in a cross-fiber direction in healthy cardiac tissue.
Microscopic
Hypotheses
Uniform (Continuous) Cross-fiber Model. In electrophysiology, the simplest example of activation (propagation) is that involving a single axon lying in an unbounded volume conductor. This is the model developed by Hodgkin and Huxley’ and further extended and implemented by numerous authors. ‘O One can immediately apply this work to that of a multicellular (three-dimensional) tissue when the latter consists of a bundle of parallel fibers that lies in oil. In this case the behavior of each fiber is identical and described by the same linear-core conductor model’ I as applies to the isolated fiber, except that for the unbounded case in the expression for the transmembrane current the axial resistance is ri (unit: fLm - ’ ) . Under bounded conditions the interstitial resistance, rp, must be included in this expression by replacing ri with (r; + re). As noted above, propagation in cardiac tissue generally takes place across, rather than along fibers. Thus, the com-
’ In the initial moments following an ectopic stimulus the anisotropic velocity as well as the underlying anisotropic conductivities enter the source description both with regard to direction and magnitude.’ However, for normal propagation under macroscopic conditions, the results are nearly planar wavefronts that are oriented almost entirely in the cross-fiber direction, In this paper, only normal activation is considered so that anisotropy is merely latent and not actually expressed.
Activation
Models
l
Plonsey
and van Oosterom
101
plexities of cellular structure in particular, including intercellular junctions, may have to be considered. On the other hand, at some level of approximation, one can treat the structure as if it were continuous (syncytial), and the fibers constituting the tissue as if they were continuous, uniform, and supporting propagation in the direction in which it occurs. TO the extent that this is valid, a “uniform cable” treatment may be applied to the propagating cardiac wave on a microscopic level. This constitutes one possible cellular model, where both microscopic and macroscopic propagation proceeds in a cross-fiber direction. Other possibilities, to be considered in this paper, are as described below.
Microscopic Axial-Macroscopic Transverse. Clerc” describes transverse propagation as following a path established by the end-to-end intercellular (gap) junctions. This is illustrated in Clerc’s figure reproduced in Figure 2. Assuming activation to progress this way one would conclude that the activation of the interior portion of each cell is initiated from its ends. If macroscopic propagation is uniform, as in the normal case, then interior cellular activation would necessarily be initiated from both ends, more or less synchronously. The idea that cellular activation is axial is explicitly contained in the paper by Geselowitz et a1.13 Here it is assumed that activation of each cell is essentially axial regardless of the macroscopic propagation direction. Thus, in this model, even though macroscopic activation is transverse, cellular activation nevertheless is assumed to proceed in a longitudinal direction.
Zig-zag Cellular Propagation. Propagation on a cellular level has been investigated both experimentally and with models by Spach, Dolber, and HeidIage, “L’ 5 They describe a “zig-zag” path arising mainly from the presence of nonconducting septa. These affect side-to-side gap junctional coupling (not considered by Clerc), which are believed responsible for transverse propagation. The progressive loss with age of side-to-side junctions is described as leading to a zig-zag course of activation. The conduction pathway is seen as irregular and longer due to this geometry. Figure 3 is taken from Spach et al. I4 and shows a zig-zag transverse path. A further elaboration on the zig-zag form of propagation is described by Spach et aI.15 in the case of longitudinal propagation of premature action potentials. The hypothesis is advanced that propagation shifts from longitudinal to dissociated and transverse since the latter is safer, though slower.
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Journal of Electrocardiology
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I 8
f
I’
I I
I’
I
:
wT : :
I
I I
II 1 I I
I I L-r.
;
_---__-
:
I
I
1
I
I
1
I
I t
I
I
4
:
I
!
I
I
I I
t
I
;
i
t
:
I
I I
I
I
I
’
I
t
I
I
I
I I -we
42th
L
I
Fig. 2. This figure emphasizes the presence of end-to-end intracellular junctional resistance and its role in facilitating
progagation in the transverse direction (redrawn from Clerc’*). (A) Physical structure. (B) Transverse current flow. The junctional resistance is rj while the cytoplasmic resistance is r,. Nonuniform Transverse. A final hypothesis examined here assumes ceIIuIar activation to be transverse to the fiber axis, but nonuniform. The nonuniformity arises from the presence of the discrete intercellular junctions. This model could be thought of as an extension of the work of Rudy and Quani6 and Henriquez and Plonsey,” which, however, apply to axial propagation. In these papers it is shown that within each cell activation may take place at an increasingly high velocity for increasingly elevated values of junctional resistance. In this case the propagation velocity may be determined mainly by the delay arising in crossing the junctional resistance. Thus, while an a cellular level propagation is in the same direction as it is macroscopically, it is nonuniform in a temporal sense.
In this paper each of the aforementioned rnicroscopic models will be considered with respect to the macroscopic saurces they establish. The extracellular fields produced by those sources will then be contrasted and compared to those measured macroscopically.
Quantitative Analysis of the Source Models This section develops expressions for the tative description of macroscopic sources from each of the four cellular microscopic considered. The resulting expressions will be
quantiarising models utilized
Cardiac
Cellular
--
Activation
Models
Plonsey
l
-----
--
---
200pm
I
-
JC
_I
-Fig. 3. In this figure, the “sawtooth”
to the long axis of the
--
--
--
_-_-_-
--
-curve denotes
103
--
---
--
and van Oosterom
-___
----
the irregular zig-zag course of excitation
spread in a direction
transverse
fibers (redrawn from Spach14).
in a subsequent section for evaluating the aforementioned cardiac activation hypotheses through an examination of the macroscopic potential distributions that are generated. Since this material is mainly drawn from the literature, this summary is brief. In this analysis the volume conductor will be assumed to be unbounded, that is, only so-called primary sources will be considered. A finite boundary is readily included through the introduction of secondary sourcesL8 lying on that boundary. (If the boundary is with air, then the secondary sources insure that the total internal field has a zero normal derivative over the bounding surface, a condition that can be used to solve for the secondary sources). The field generated by the secondary sources must be added to those arising from the primary sources. Activation of a single cell lying in a volume conductor of infinite extent has been shown” to result in an equivalent double layer source located in the membrane surface and oriented normal to that surface. Designating the source as k, then z? = (@jai - @&)6,
(1)
where ii is the dimensionless unit outward normal from the membrane, the subscripts e and i denote, respectively, evaluation at the outside and inside of the membrane, and 2 has the units of A.m.m-* (ie, a current dipole sz4rface density since A is amperes and m is meters). For a multicellular preparation equation 1 can be applied to each cell; the resulting sources all lie in an unbounded uniform medium. The potential field generated by these sources can be evaluated by summing the free-space field of each
element (ie, a surface integral over all membranes). Alternatively, with respect to extracellular space, the sources can be averaged and approximated by a continuous volume source density function 1 In view of the unbounded “free-space” environment, the volume conductor field, @, can be readily found by superposition. Since any elementary source (ie, j( ?) dv) at source point T sets up a “free-space” dipole field (ie, $V ( l/R) dv), where R is the length of T’ T, a vector pointing from a source at T to an arbitrary fieldpoint T’, and V( l/R) is the unit dipole field, then the field established at T’ is a+‘)
=
$& j- .i.v P
(l/R)
dv,
(2)
where the integral in equation 2 is taken over all cells present. Further details are found in Plonsey” and van Oosterom.”
Axial Fiber(s) As stated in the introductory section, the interest of this paper lies with cross-fiber propagation of cardiac depolarization. However, this is based on the well-studied case of propagation along uniform fibers (ie, axial propagation). Consequently, we first review some of the relations that have been established for axial propagation. Uniform Axial Fiber(s). Equation 1 applies to a cell of any shape, but when the cell is cylindrical and the field along the membrane depends only on
104
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Vol. 24 No. 2 April 1991
the axial coordinate, it may be transformed equivalent current dipole volume source given by
j(z) =
& (@jUj -
into an density
extracellular field can be found from the integral described in equation 2, and this comes out
Qeue)iiz.
(6) In equation 3, z is the axial variable (of T), ii, is a dimensionless unit vector along the z axis, and j signifies the current dipole moment per unit volume (unit: A.m.mp3). jcompletely fills the fiber, and in any cross-section, is uniform. Since this an equivalent source, it correctly evaluates fields in a restricted region only-in this case the extracellular space. The source described by equation 3, as pointed out earlier, lies in a uniform unbounded extracellular medium of conductivity ue, and hence set up “freespace” fields. The volume conductor field is, accordingly, found by substituting the source described in equation 3 into the source-field expression of equation 2, where the integration is now taken throughout the intracellular domain of the fiber. For a multifibered preparation the sources can be found by application of equation 3 to each fiber. Note that, again, the resulting sources all lie in a uniform unbounded medium so that equation 2 applies once more except that the volume integration extends through all fibers. While, in equation 3, a possible interaction between fibers appears to have been ignored, this in fact enters the evaluation of the actual membrane fields Qi, Qe at each fiber. Except near the periphery of the bundle, where there are many axial fibers of similar or nearly similar size, the linear core conductor model” can be applied, and equation 3 can be simplified. That is, @i and (DCcan be evaluated in terms of the transmembrane potential V,, where =
”
- re = (ri + rr) ‘,’
(D,(?‘)
= L
re
4~ ( re + r;)
d
v 132
(V,,,)&,-V(M)
dv,
where V, is chosen (defined) to be uniform over any cross-section (normal to the direction of propagation) while V denotes integration over the entire muscle space. This last modification was introduced because cardiac cells occupy approximately 80% of the total space, therefore, it is convenient to consider the sources as if they occupy the entire volume; to compensate for this, the results must be multiplied by the cell density fraction, and such a fraction is contained in equation 7. For field points within the tissue (ie, in the interstitial space), equation 7 simply returns equation 5. In particular, from equation 5, we note that the difference of potential in the interstitial space on opposite sides of an activation wave, V,,,, is the change in transmembrane potential V, from rest to peak (plateau) times the resistance divider ratio, that is, e
(ri + r,) ‘,’
j-
(7)
VWCX”P = AV,,, *,
li
qli
where the subscript n denotes the direction of propagation (we will use z for axial and x for transverse to the fiber -orientation). The volume V, designates the actual space occupied by the cells. If equations 4 and 5 are applied, assuming the applicability of the linear core-conductor model, then equation 6 can be simplified by
I
= (Vpeak -
Vm,) --?._
r, + rj’
(8) (5)
and where ri and re are the intracellular and interstitial axial resistance per unit length. When equations 4 and 5 are applicable, the source is seen to depend solely on the spatial derivative of the transmembrane action potential. In the case of uniform propagation, this can be determined from knowledge of the temporal action potential (which to a first approximation does not change greatly within a preparation) and the local conduction velocity. The sources arising from plane wave propagation along the fiber axis based on the above simplifications are expressed by equations 3,4, and 5. The
For field points outside the heart, for instance, at the torso surface, the potential from activation sources at the depolarization wavefront within the heart (ignoring secondary sources from the torso boundary and any other inhomogeneities), based on the assumption of a uniform double-layer, evaluates to
where An is the total solid angle subtended by the entire activation wavefront at the field point. Some additional details on equations 8 and 9 and their derivation from equation 6 are found, for example, in Plonsey et al.” and van Oosterom.”
Cardiac
Axial Fiber(s) With Intercellular
Cellular
Junctions.
The equations 1, 2, and 3 are directly applicable to a single skeletal muscle fiber or a bundle of such fibers. For cardiac muscle, a modification may be necessary because of the presence of intercellular junctions. From the theory underlying equations 1 and 3, these equations may be applied to the cell membrane between junctions; the junctional region, however, requires special attention. The pertinent part of this region can be idealized as a superposition of connection tubes containing the intracellular media and separated by a wall from the extracellular space. One can apply equation 3 to each tube. Since the effective source strength depends on the intracellular volume, and since the volume of all tubes is very small compared to the cell volume, one expects the net junctional source to be much less than the cellular source. A quantitative consideration of these factors shows that, indeed, the junctional source is negligible.2o The effect of junctions on the overall sources in a multicellular cardiac preparation appears to be mainly indirect, caused by the influence of the junctions on the membrane fields G$, @c over the body of each cell. This effect will be small, assuming low resistance junctional values that recent experiments suggest.*’ Under abnormal conditions, such as ischemia, decoupling results, and the effect of the junctional resistance on, @i,
(10)
where rim is the myoplasmic resistance per unit length (this could be evaluated from pi/( ITU’), where a is the cell radius and pi the myoplasmic resistivity), while Rj is the junctional resistance per cell whose length is 1. For a cylindrical bundle of parallel cardiac muscle fibers.lying in a conducting medium, since the junctional resistance is to be neglected (or simply included in the assigned intracellular resistance), the result is a structure similar to that of a bundle of skeletal muscle fibers (ie, uniform and axially con-
Activation
Models
l
Plonsey
and van Oosterom
105
tinuous). The applicability of the linear core conductor model, and correspondingly equations 4 and 5, would seem assured for fibers near the center of the bundle, but not for those at the periphery (where re = 0). A mathematical study of this preparation shows that, in fact, the linear core conductor model can be applied over essentially the entire cross-section for bundles of perhaps 1 cm in diameter.23 Probably the left ventricular free wall satisfies this requirement .24,’ 9 Resistances and Resistivities. In the analysis to follow, we will consider predominantly planar propagation, and we will study the use of equations 8 and 9. Equation 8 clearly requires the knowledge of the resistivities r, and r,. As stated before, these have their origin in the linear cable theory, both representing resistances per unit length. In applying the basic notions of linear cable theory to a three-dimensional model of propagating wavefronts, one has to introduce appropriate expressions for these resistivities. To this end, different approaches and notations have appeared in the literature. To avoid confusion we devote some space here to introducing and explaining the various resistances and resistivities that will be used in the sequel of this paper, aimed at application in equation 8. Throughout this paper R will denote a resistor, (unit fl), r will represent a resistance per unit length (unit 1R;rn~ ’ ) and p will denote a resistivity (unit 1R.v~). We first consider a unit area S of the wavefront (cross-section of tissue). For the propagation in the fiber direction the intracellular resistance per unit length follows from
(11) with pi the microscopic resistivity of the intracellular space (myoplasm), and f the fraction of tissue volume occupied by intracellular space. This corresponds to a much used model, in which the cells are represented by closely packed cylinders (eg, Weidmann25; Clerc”). (An alternative expression for ri follows from explicitly considering an individual fiber with radius u. As noted before, this reads r, = pi/(lTu’).)
For r,, the corresponding
Ie = (1 -
expression is Pe f)S.
(12)
When inserting these expressions into equation 8, the actual value of S (cross-sectional area of an entire bundle) is irrelevant since S drops out of the potential divider ratio. This has led to the -introduction of a longitudinal intracellular resistivity
106
Journal of Electrocardiology
Vol. 24 No. 2 April 1991 (13)
the dimension of which is the same as that of pi (ie, n.m). For the extracellular compartment the corresponding variable pe’ follows from I_ Pe _-
Pe 1 -f’
(14)
In situations where the spatial variation of the potentials is large with respect to cell dimensions (or rather the space constant of the process) the intracellular and extracellular compartment may be considered in parallel. The longitudinal overall, gross tissue resistivity p’ in this case is related to pe’ and pi’ as PitPer
p’ = Pi’
+
(15)
Pe’
In several studies this resistivity p’ has been measured directly.26,27 Furthermore, direct measurements of the intracellular potential jump 18% I, associated with the depolarization process and the accompanying extracellular, 1A@< 1have provided values for the ratio
(16) as follows directly from equations 4, 5, and 8. In a similar way equation 8 can now be expressed as I
Vwave=
PC pe’ + pi
AV, = p
AV,.
(17)
1+P
The values for the resistivities like pi’ and pe’ reported in the literature have been computed from measured values of p’ and measured resistivity ratio’s, that is, either by combining equations 15 and 16 (eg, Weidmann25), or equations ( 15) and ( 17) (eg, Roberts et a1.26). Note that the computed values of pe’ and pi’ are very sensitive to the assumed value off since this value is of the order of 0.75. For propagation in the transverse (ie, cross-fiber) direction similar variables pi’, pk, and p’ have been proposed, although a simple relationship with cellular morphology (similar to the ones expressed in Plonsey et al. I1 and Clerc”) is no longer directly obvious. These variables can be used for relating wavefront strength V,,, to the jump in the transmembrane potential A V, in a similar manner to that expressed in Henriquez et al.” Uniform (Continuous)
Cross-fiber
Model
For cross-fiber propagation in a multifibered cardac preparation, we shall adopt, as a reference condition, a uniform equivalent fiber bundle, similar to
the axial propagation as described in the previous subsection. This is illustrated in Figure 4. In Figure 4A the cardiac muscle cells are shown in cross-section. Relevant to cross-fiber propagation are intercellular gap junctions, as noted in Figure 4A. Their presence is documented by Sommer and Scherer2’ to be relatively dense. These provide both axial and lateral coupling. Assuming their resistance to be small, as discussed above, and therefore neglecting their presence in favor of a continuum (ie, a syncytium) forms the basis for the simplified structure of Figure 4B. An electrical description of the excitable tissue described in Figure 4B is shown in Figure 4C. The latter is, again, the linear core conductor model, which now may be interpreted as representing a unit crosssectional area of tissue (relative to the direction of the propagating wave). In Figure 4C, pi and pe represent the averaged intracellular and interstitial resistance per unit length in the cross fiber direction as defined in the section on axial fibers. Accordingly, the potentials at field point i’, generated by a wavefront progressing in the cross-fiber direction, can also be found from equation 9, after insertion of the appropriate resistance or resistivity values in expressions like equation 8 or 17. This treatment of cardiac tissue as a continuum (syncytium) in the cross-fiber direction is a specific example of the approach to plane wave propagation in any arbitrary direction in the cardiac muscle, as introduced by Muler and Markin.29 This work forms the basis of the bi-domain approach to cardiac muscle, as can be found in the work of Tung3’ and Miller and Geselowitz,40 and discussed in Plonsey.3 ‘ An explicit evaluation of sources arising in the cross-fiber direction, and leading to equation 7 is given in van Oosterom.”
Macroscopic Transverse
Axial-microscopic
According to Clerc” propagation in the transverse direction proceeds through coupled resistances at the ends of cells. This process is illustrated in Figure 2. A consequence of excitation of cells at their ends only must be that propagation within each cell proceeds in the longitudinal direction (with a collision somewhere in the interior). For an essentially uniform transverse wave, then, roughly, both ends will be excited synchronously and the collision site will occur at the center of the cell. In this model the cellular dipole sources are directed along the cell axis, with those at the “left” opposite those on the “right”. As a consequence, the sources will have a quadru-
Cardiac
Cellular
Activation
Models
l
Plonsey
and van Oosterom
107
gap junction ,~~~~~intracellular ‘~~~~ space .-.y ;..:.. l
reAx
reAx
reAx
C Fig. 4. (A) A conceptual cross-sectional view of a 3-D cardiac tissue preparation showing individual fibers and their nexal junctions. The fiber axes are normal to the page. Plane wave propagation is assumed to proceed from left to right (normal to the fiber axes). (B) For low resistance intercellular (nexal) junctions, propagation from left to right behaves as if the cardiac tissue structure consisted of cylindrical fibers oriented from left to right as shown. These fibers are coupled, laterally, (intracellularly), as shown. (C) The electrical network that is associated with transverse plane wave propagation, for low resistance intercellular junctional resistance, is described here. This is the linear-core-conductor model and can be derived from either (A) or (B).
108
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of Electrocardiology
Vol. 24 No. 2 April 1991
polar component, at the expense of an otherwise dipolar component, and this must result in a severely reduced extracellular field. Furthermore, its direction will be primarily tangential rather than radial.
forVW,, and
surface potentials), and partial or complete cancellation of transverse components. If we assume a mean inclination of the zig-zag angle to be CX,, then in place of equation 8 we have VWave= cos %n( Vpeak
Zig-zag Propagation
-
vrest)
gif COS* CX, +
This type of conduction is proposed by Spach et al. r4,15 However it is not described in great enough detail to permit a quantitative analysis. Based on Figure 3 we may consider the effect of propagation in a direction that is both normal to the fiber axis and also inclined at an angle (Yto the radial direction (ie, the direction normal to the free wall). This is described in Figure 5 where a is the aforementioned angle of inclination of the local wave. Assuming junctional resistances to be small, the effect will be to generate the same dipole sources as in the uniform reference model, where 01 = 0; however the dipole direction will be at an angle (Y # 0 to the radial. A consequence will be a loss of dipole strength in the outward direction (ie, that direction which accounts
gifCOS2
CL, +
gilsin* (Y,
x
,lJi’sin* (Y,
+g,‘COS*
Ci,
+
g,‘sin* CX,’ (18)
where gi’ = l/pi’, gef = l/p,‘, etc.29
Nonuniform Transverse In one-dimensional simulations, when cellular decoupling reaches a high level, then the phase velocity within the cell will increase, while the propagation velocity itself decreases (as a result of a propagation delay across the intercellular junction). The source density within the cell is given by equation 7 and the gradient of V, can be expressed as avi?l -= an
av,,, dt = _1 av, -0 at ’ at dn
(19)
where 0 is the cellular phase velocity. Clearly as the decoupling increases and 8 increases, the source strength decreases. The picture is actually further complicated because it becomes no longer possible to obtain spatial behavior from temporal behavior (ie, in the absence of uniform propagation the wave equation is not satisfied). But the qualitative conclusion can be drawn that the source-strength is diminished with increasing decoupling.
radial direction
Evaluation
fl
of the Four Models
Based on the analysis presented in the previous section we now present an evaluation of each of the four cardiac source models considered. The differences in the depth at which each of these models could be analyzed will be reflected in this evaluation.
endocardium
The Axial Situation
Fig. 5. Geometry associated with zig-zag propagation, which makes an angle of cx with the radial direction. In this example it is assumed that there is no component of propagation in or out of the page (no longitudinal component).
Several studies dealing with along-fiber propagation have been reported, the analysis being based on the approaches described in the section on uniform cross-fiber models. Fiber bundles or superfused cardiac fiber strands of different animals and heart segments have been used. In spite of the fact that this represents the easier of the two extreme types of propagation (along- vs cross-fiber), a wide range of values, both for the actual measured ones and for
/
epicardium
Cardiac
Table
1. Linear Cable Parameters
Cellular
as Measured
Models
Pr
mV
P
252
_
-
-
188 160 340 852 383
152 125 199 213 174
96 100 100 99 _
24 21.5 46 74 _
0.33 0.33 0.86 2.92 _
252
I16
5 1.5
1.08
f
(ih?l)
(R.cm)
_
_
-
-
_
WeidmannL5 Clerc’L Roberts”’ Robens35 Epstein3’
470 402 -
51 48 _ -
0.75 0.70 _ -
624 575 360 291 319
KleberS3
166
63
0.75
221
Uniform (Continuous)
Cross-fiber
Model
In this section an evaluation of source and field of a uniform cross-fiber model is considered. This model is described in Figure 4 and its basic source analysis given in the sections on axial fibers and uniform cross-fiber models. Our present objective is to examine the correspondence between the field strength predicted by the model (ie, equation 9) with actual measurements. We consider two cases, the first for fields lying within the heart muscle (ie, application of equation 8), and the second where the field is at the torso surface (ie, application of equation 9).
Cardiac Field Points Lying Intramurally. There is a substantial literature describing interstitial intramural potentials arising from a propagating wavefront. From these one finds essential agreement Table 2. Linear Cable Parameters Pi
(R.cm)
as Measured
.f
(ATim)
(&m,
98.7
that the magnitude of the potential across this wave, v WilVe,is around 40 mV. 19,24,34,35 It is this potential that is evaluated in equation 8. To compare the predictions of equation 8 with the aforementioned 40 mV value, knowledge of the cross-fiber intracellular and interstitial resistances (pit and pet) is required. In contrast to the along-fiber case, relatively few data have been reported for the cross-fiber case. In Table 2 we list these few data, as obtained from Clerc” and from Roberts and Scher,26 which are the only sources for data on transverse intracellular cardiac resistance. From the data in Table 2, we note that. using resistances from Roberts and Scher35 gives values of VWUYCin close agreement with experiment, while Clerc gives results that are rather low. In a critique, van Oosterom points out that the model on which Clerc interprets his experimental results fails to include-in the transverse direction-the fact that no fully developed (planar) wavefront couId have materialized within the width of the bundle studied.36 This is clearly evident from Clerc’s Figure 4, where, according to the conditions required for the application of the linear cable approach, the intracellular and the extracellular waveforms should have been proportioned throughout time, and clearly these are not. As a consequence, the anisotropy in the resistivity values as computed from this experimentally determined value of p is probably a severe overestimation. On the other hand, some modest support for the data of Roberts and Scher3’ comes from the experiments of Epstein and Foster,32 where, at least for extracellular resistances, which can be compared, by Various Authors:
I
Pe
(R.cm)
109
mV
(O.cm)
those computed based on the model, can be found. In Table 1 we present a survey of the results of several of these studies. Note that all of these pertain to the axial (ie, longitudinal) situation. Interstitial resistance values are included for Epstein and Foster32 for skeletal muscle (on the assumption that it should correspond reasonably well with cardiac muscle, since the structural contributions are about the same). In addition, axial resistances found from the perfused papillary muscle preparation of Kleber and Riegger33 are also included.
and van Oosterom
Longitudinal Propagation I Vwuw A v,
I
(R.cm) Rush”
Plonsey
l
by Various Authors:
Pi’
Pe
Pi
Activation
f (R;m)
Transverse I
Am, ~
Propagation
Av, I
mV
VWIl”II mV
B
-
-
-
-
-
563
-
-
_
Clerc’* Robert?
3620
127 -
0.70 -
Robens’
1 _
_
_
5181 3800 1677 -
424 750 1247 1250
392 625 715 _
98 100 100 -
8 16 43 _
0.09 0.19
Rush”
Epstein3’
0.43 -
110
Journal of Electrocardiology
Vol. 24 No. 2 April 1991
there is somewhat better agreement in contrast to the somewhat poorer agreement with Clerc. The data in Tables 1 and 2 include the intriguing possibility that V,,, is nearly the same in both the longitudinal and transverse directions (namely, approximately 50 mV from the recent work of Kleber, and approximately 40 mV based on Roberts and Scher and others, as noted above). Such a condition requires that oe’/oi’ = oefloif or, namely, an equal anisotropy ratio. When this occurs, as can be confirmed from equation 18, V,,, becomes independent of the direction of propagation CLThis means that for any activation wave, even if the direction of propagation of some segments is not transverse to the fiber axis, as we have been assuming here, the source strength is still the same. Thus a uniform double-layer source can, in this way, be justified under more general conditions. The implication is that while anisotropy would have an important effect on the activation shape (eg, elliptical for an ectopic site) the source density is nevertheless essentially uniform. Cardiac Field Points Lying at Body Surface.
Equation 9 can be used to calculate electrocardiographic potentials at the body surface if the activation pattern is known or can be inferred. Based on the activation isochrone geometry and the position on the torso at which potentials are desired, Aln can be evaluated. While a’, can now be found using equation 9, it should be noted that this is a primary-source field (ie, the field that would be found if the sources in the heart lay in a uniform homogeneous volume conductor of infinite extent). The various inhomogeneities must also be taken into account, and this can be done (actually or conceptually) by the addition of secondary sources” situated at surfaces separating regions of different conductivity (blood cavity, heart muscle, lungs, and air). This is essentially the procedure (boundary-element method) followed by van Oosterom and Huiskamp,37 who chose V,,, = 40 mV,
uhearb/ung
=
5,
.!m,od~uh~an
=
3,
and,
of
course, Uair = 0. The simulations of van Oosterom and Huiskamp are based on reasonable physiological activation sequences (actually, these were found from measured body surface potentials through an inverse procedure3’), and their forward simulations are in excellent quantitative agreement with the measured electrocardiograms. The outcome lends additional support to the choice of V,,, = 40 mV associated with the cellular process. Independent simulations of the magnetocardiogram were also performed based on this same data leading, again, to excellent agreement with measurements.39 Since the simulations noted above are affected by
the choice of the assumed conductivity ratios and geometry as well as of V,,,, one should examine the degree of certainty with regard to the former variables. The sensitivity of the simulation to the assumed values of conductivity was studied by van Oosterom and Huiskamp.37 They showed that inclusion of inhomogeneities was essential to obtain a good quantitative iit to the measured date. Most important is inclusion of accurate body shape and ventricular blood cavity. These two factors were quantified by van Oosterom and Huiskamp using MRI cross-sectional scans. Furthermore, blood conductivity and heart muscle conductivity are generally well established quantities. Consequently, one can conclude that the magnitude of VW,, is confirmed in these simulations with very good confidence. Microscopic Transverse
Axial-macroscopic
As stated below, according to Clerc12 propagation in the transverse direction proceeds through coupled resistances at the ends of cells. In fairness to Clerc, it should be noted that at the time of his paper the evidence for lateral junctions was poorly documented. The fact that, for an essentially uniform transverse wave, the resulting sources will be exactly quadrupolar (as stated earlier), results in a too severly reduced field. Furthermore, since its direction would be primarily tangential rather than radial, this model of the sources representing the depolarization process could not be comparable with quantitative measurements of VW,,, and surface electrocardiograms. The assumption that activation of each cell is longitudinal, even while macroscopic activity proceeds in another direction (ie, in the cross-fiber direction that we have been assuming) is explicitly made by Geselowitz and Miller. l3 In spite of several reasonable results that this leads to, one can conclude from the above source analysis that the model is incompatible with the strength and orientation needed to generate observed extracellular fields. Zig-zag Cellular
Propagation
The analysis of this model can be based on equation 18. For a,,, = 7r/6, a value that seems reasonable based on Spach et al., then V,,,, = 0.866 x 100 X 0.59 = 5 1 mV, based on Table 2 (choosing the Roberts and Scher resistivity data). While the resulting ECG would have an increased amplitude the increase is too small to be statistically significant. So this type of zig-zag propagation is not ruled out.
Cardiac Cellular Activation Zig-zag propagation could also occur in a way that introduces axial cellular activation. In this case the effect on sources depends critically on the cellular processes, and realistic details must be examined. To the extent that a substantial component of membrane depolarization is associated with longitudinal conduction, which, as we have seen, does not contribute sources that are productive in terms of body surface potentials and V,,,, then such a model may be ruled out.
Nonuniform
Transverse
This situation has been studied for longitudinal propagation with a rigorous one-dimensional model by Quan and Rudy,22 who supply an evaluation of the decrease in amplitude with rising junctional resistance. They show a 40% decrease in extracellular potential magnitude for a tenfold increase in junctional resistance. Even a hundredfold increase in junctional resistance leaves an extracellular potential magnitude that is 20% of control. Consequently, it would not seem that one can distinguish a discontinuous transverse propagation model from a continuous one solely on the basis of a measurement of V waveor from the EGG. One should also note that the above considerations apply, strictly, to a one-dimensional strand of cardiac cells oriented longitudinally. If a multifibered three-dimensional tissue is considered, it is not clear what behavior will be seen. For one, because of the many staggered junctions that are present, it is possible that the overall behavior is less discontinuous. On the other hand, since propagation is transverse, the effective cell length between junctions will be an order of magnitude shorter than for the simulated longitudinal orientation.
Models
l
Plonsey
and van Oosterom
111
Using currently available data it would appear that cardiac tissue is activated transverse to the fiber axis and, furthermore, such activation occurs not only macroscopically but also on a cellular level. Thus, in spite of the possibilities for complex pathways, including cellular activation in a longitudinal direction, activation would appear to take place essentially uniformly across each cell membrane but in a transverse direction. It is important to note that while this model assumes the maximum production of outward sources that are possible, these are never too large with respect to the size of the macroscopic field that is required. Any other model results in either too modest a change, and therefore cannot be ruled out, or a very large decrease, which is unacceptable. The type of zig-zag propagation described by Spach et al. is consistent with the requirements of a macroscopic source as long as deviation from this main activation is not large.
Acknowledgments The authors are indebted to Drs. J. Sommer and P. Dolber of the Department of Pathology at Duke University for helpful assistance with regard to the 3-D structure of cardiac tissue and for sharing (J.S.) some recent unpublished studies on which Figure 4A is based. The author (RP) thanks Prof. H. K. Boom for making available resources of the Division of Biomedical Engineering at the University of Twente, where this work was done, while on sabbatical leave from Duke University, and for his helpful discussions.
References Conclusions The main goal of this paper is, again, to illustrate how a cellular model can be evaluated by examining the sources that result, and then comparing the simulated field amplitude with that obtained in macroscopic measurements. As shown above, all models depend in one way or another on pPt and pif (or pp’ and pi’). At present, the large differences in the values reported in the literature hamper an unequivocal conclusion on the various models considered. In particular the transverse data seem to be uncertain. The basic measurements of CIerc’2 need to be repeated, while avoiding the pitfalls in that approach.36
Plonsey R: An evaluation of several cardiac activation models. J Electrocardiol, 7:237, 1974 Plonsey R, van Oosterom A: A cellular activation model based on macroscopic fields. In Sideman S, Beyar R, Kleber A (eds): Cardiac electrophysiology, circulation, and transport. Kluwer Academic Publishers, Norwell, MA, 1991 Scher A, Young, AC: Ventricular depolarization and the genesis of the QRS. Ann NY Acad Sci 65:768, 1957 Durrer D, van Dam RT, Freud GE et al: Total excitation of the isolated human heart. Circulation, 41:899, 1970 Scher A, Spach MS: Cardiac depolarization and repolarization and the electrocardiogram. p. 357. In Berne RM (ed): Handbook of physiology, section 2: the cardiovascular system; volume 1. The Heart. Am. Physiol. Sot., Bethesda, MD, 1979
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6. Spach MS, Miller WT, Miller-Jones E et al: Extracellular potentials related to intracellular action potentials during impulse conduction in anisotropic canine cardiac muscle. Circ Res 45: 188, 1979 7. Streeter Jr DD, Spotnitz HM, Pate1 DP et al: Fiber orientation in the canine left ventricle during diastole and systole. Circ Res 24:339, 1969 8. Colli-Franzone P, Guerri L, Viganotti C et al: Potential fields generated by oblique dipole layers modeling excitation wavefronts in the anisotropic myocardium: comparison with potential fields elicited by paced dog hearts in a volume conductor. Circ Res 5 1:330, 1982 9. Hodgkin AL, Huxley AF: A quantitative description of the membrane current and its application to conduction and excitation in nerve. J Physiol 117:500, 1952 10. Cronin J: Mathematical Aspects of Hodgkin-Huxley Neural Theory. Cambridge University Press, Cambridge, 1987 11. Plonsey R, Barr RC: Bioelectricity: a quantitative approach. Plenum Press, New York, 1988 12. Clerc L: Directional differences of impulse spread in trabecular muscle from mammalian heart. J Physiol 255:335, 1976 13. Geselowitz DB, Miller WT: Active electric properties of cardiac muscle. Bioelectromagnetics 3: 127, 1982 14. Spach MS, Dolber PC: Relating extracellular potentials and their derivatives to anisotropic propagation at a microscopic level in human cardiac muscle. Circ Res 58:356, 1986 15. Spach MS, Dolber PC, Heidlage JF: Influence of the passive anisotropic properties on directional differences in propagation following modification of the sodium conductance in human atria1 muscle. Circ Res 62:811, 1988 16. Rudy Y, Quan WL: A model study of the effects of the discrete cellular structure on electrical propagation in cardiac tissue. Circ Res 61:815, 1987 17. Henriquez CS, Plonsey R: Effect of resistive discontinuities on wave-shape and velocity in a single cardiac fiber. Med Biol Eng Comput 25:428, 1987 18. Plonsey R: The formulation of bioelectric source-field relationships in terms of surface discontinuities. J Franklin Inst 297:3 17, 1974. (Also reprinted in Pilkington T, Plonsey R: Engineering contributions to biophysical electrocardiography. IEEE Press, New York, 1982) 19. van Qosterom A: Cell models: macroscopic source descriptions. p. 155. In Macfarlane PW, Lawrie ‘ITV (eds): Comprehensive electrocardiology. Vol. I. Pergamon Press, Oxford, 1989 20. Plonsey R, Barr RC: Effect of junctional resistance on source-strength in a linear cable. AM Biomed Eng 13:95, 1985 21. Metzger P, Weingart R: Electric current flow in cell pairs isolated from adult rat hearts. J Physiol 366: 177, 1985
22. Quan W, Rudy Y: The reflection of discontinuous propagation of myocardial excitation in extracellular potentials. Personal communication 23. Hemiques CS, Trayanova N, Plonsey R: Potential and current distributions in a cylindrical bundle of cardiac tissue. Biophys J 53:907, 1988 24. van Oosterom A: Cardiac potential distributions. PhD thesis, University of Amsterdam, Amsterdam, 1978 25. S. Weidmann: Electrical constants of trabecular muscle from mammalian heart. J Physio12 10: 104 1, 1970 26. Roberts DE, Hersch LT, Scher AM: Influence of cardiac fiber orientation on wavefront voltage, conduction velocity, and tissue resistivity in the dog. Circ Res 44:701, 1979 27. Rush S, Abildskov JA, McFee RS: Resistivity of body tissues at low frequencies. Circ Res 7~262, 1963 28. Sommer J, Scherer B: Geometry of cell and bundle appositions in cardiac muscle: light microscopy estimation. Am J Physiol 248:h792, 1985 29. Muler AL, Markin VS: Electrical properties of anisotropic nerve-muscle syncytia-II spread of flat front of excitation. Biophysics 22~536, 1977 30. Tung L: A bidomain model for describing ischemic myocardial d-c potentials. PhD thesis, MIT, Cambridge, MA, 1978 31. Plonsey R: The use of the bidomain model for the study of excitable media. Lectures on mathematics in the life sciences 21:123, 1989 32. Epstein BR, Foster KR: Anisotropy in the dielectric properties of skeletal muscle. Med Biol Eng Comput 21:51, 1983 33. Kleber AG, Riegger CB: Electrical constants of arterially perfused rabbit papillary muscle. J Physiol 385:307, 1987 34. Corbin LV, Scher AM: The canine heart as an electrocardiographic generator. Circ Res 41: 58, 1977 35. Roberts DE, Scher AM: Effect of tissue anisotropy on extracellular potential fields in canine myocardium in situ. Circ Res 50:342, 1982 36. van Oosterom A: Anisotropy and the double layer concept. p. 91. In Macfarlane PW (ed): Progress in electrocardiology. Pitman, Tunbridge Wells, 1979 37. van Oosterom A, Huiskamp GJM: The effect of torso inhomogeneities on body surface potentials quantified by using tailored geometry. J Electrocardiol 22:53, 1989 38. Huiskamp GJM, van Oosterom A: The depolarization sequence of the human heart surface computed from measured body surface potentials. IEEE Trans Biomed Eng BME-35:1047, 1988 39. van Oosterom A, Oostendorp TF, Huiskamp GJM, ter Brake HJM: The magnetocardiogram derived from electrocardiographic data. Circ Res 67: 1503, 1990 40. Miller WT III, Geselowitz DB: Simulation studies of the electrocardiogram. I. The normal heart. Circ Res 43:301, 1978