Journal of Electrocardiology

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A Possible Subcellular Structure Based on the Macroscopic Cardiac Source

Robert Plonsey, PhD, and Roger C. Barr, PhD to myoplasm resistance to be negligible. In support of this assumption we utilize the .0052@m2/pm3 mean gap junction surface density reported by Luke et al.’ For a cell volume of 40,000/p,m3 (eg, radius = lo/ km, length = 120/l.~m) and with a mean channel density of 13,000/pm2 as reported by Page,2 we find 2.7 x lo6 channels per cell. Assuming 50 pS/channe13 gives a net impedance between a cell and its neighbors of 7.4 Kfl. Hoyt et a1.4 estimate that each cell connects to nine neighboring cells, and if the junctions are all similar then they have a resistance of 67 KC! each. If the effective intracellular resistivity is 500 CIcm, then for conduction in the axial direction the myoplasmic resistance is 1.9 Ma, and thus the junctional resistance represents only 3.5% of the total. For the transverse direction, recognizing anisotropy, the (cellular) intracellular resistance of course must be calculated using its transverse component. As described below, we expect this resistivity to be nine times that in the axial direction, namely 4500 ficm. So the myoplasmic resistance across the cell is 4500/( 120 X 10w4) = 375 KR. In this case the relative junctional to myoplasmic resistance is 18% and, while not exactly negligible, is small. An observed axial to transverse conduction velocity ratio of 3 : 1 has been noted experimentally,5 and this requires that the reciprocal of the sum of the corresponding intracellular plus interstitial resistances should be in the ratio of 32/1 = 9/l. Assuming the intracellular to be the more dominant factor, then the need for an anisotropy ratio of 9 : 1 in specific intracellular resistance is required. Since the junctional resistance is expected to play a negligible role in determining the tissue resistance, how can one account for the expected nine-fold ratio of transverse to axial resistance? This is explained by another speculative idea. We suggest that the intracellular space within each cell is anisotropic, a condition that is due to the presence of myofibrils6 These constitute perhaps 85% of the intracellular volume, and if the

A deeper understanding regarding the process of cardiac activation requires an investigation at a cellular level. Possibly only in this way can we understand the factors that contribute to malfunction such as that leading to arrhythmia. However, measurements on a cellular or subcellular level are often difficult or impossible to perform. An alternative is to postulate a structural model based on the best information available and then examine the predictions of the model at a macroscopic scale where measurements are possible. For cardiac muscle structure and electrophysiology, even on a macroscopic scale, wide ranges in some data can be observed. For example, for averaged intercellular and interstitial conductances along and across fibers there is no agreement among the very few papers in the literature. Another uncertainty surrounds the size of the junctional resistance, and whether a parallel capacitance should be included. Where these parameters are involved, a range of models may be admissible justified by the range of macroscopic data. In this report we describe a model that is admittedly speculative. It incorporates a number of unusual features: ( 1) low junctional resistance, (2) intracellular anisotropy, and (3) continuous transverse propagation at a microscopic scale. We think it is worth considering, however, because it fits some data particularly well, while not being ruled out entirely by other considerations. Even if not correct in its entirety, one might wish to introduce some parts into a more complex model.

Low Junctional Resistance We make the controversial assumption that the junctional resistance is sufficiently small compared From the Department of Biomedical Engineering, Durham, North Carolina.

Duke University,

Reprint requests: Dr. Robert Plonsey, 136 Engineering, Duke University, Durham, NC 27706.

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Subcellular

region of the conducting fluid is confined to narrow strands and ribbons the resulting anisotropy might support a wide range of transverse to axial conductivity ratios. For example, if the cytosol has a resistivity of say 100 ficm, then an 80% occlusion of the space by nonconducting myofibrils would raise the axial conductivity to 500 ficm. That the packing of myofibrils affects intracellular resistance properties has never been substantiated experimentally, but seems possible from geometric considerations.* Direct measurements of intercellular resistance in cell pair preparations have reported values in the 2 MLRrange.’ However, one could argue that because of the process in which the cell pairs are separated from the remaining tissue, junctions could be decoupled, so that an abnormally low number is present in isolated pairs. Other approaches to the experimental determination of junctional resistance have been indirect. In perhaps the best of these, namely the work of Chapman and Fry’ (where all relevant factors were measured by the investigators), the total axial resistance per unit length is expressed as the sum of two components, namely rmyO + rj”,,t/i, where r,,, is the myoplasmic resistance per unit length, rjunct is the cell-to-cell junctional resistance, and 1 is the cell length. The total resistance and the myoplasmic resistivity, pmyo (from which rmyo = pmyo/(rra2) is determined), are separately measured. The junctional resistance is then found by subtraction. However the method of obtaining pmyo assumes the intracellular space of a cardiac cell to be uniform and isotropic, an interpretation that we are suggesting could be incorrect and significantly affect their conclusions. It should also be noted that values as low as 0.2 Mfz have been reported for the intercellular resistance of isolated cell pairs. lo

(Cellular) Intracellular Anisotropy The following is a review of an argument for the assumption of (cellular) intracellular anisotropy. This suggestion comes from considering the intracellular structure, such as depicted in Figure 1 (which illustrates the density of myofibrils found in muscle tissue). Morphometric studies described in Eisen*For a hexagonal array of myofibrils the analysis of Gielen’ (of the extracellular conductivity of a hexagonal array of skeletal muscle fibers) can be applied. This shows that even with very dense packing, the maximum ratio of transverse to longitudinal resistance is 2 : 1. So a 9 : 1 ratio woufd appear to require at feast some intracellular regions in which there is a departure from a regular array of elements.

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berg” estimate the myofibrils of the human Vastus Lateralis to occupy 85% of the intracellular volume, though it is unclear if this space is entirely nonconducting. The possible contribution of myotibrils to the effective intracellular resistance was first suggested by Katz. l2 Cole13 measured the resistivity of extruded squid axoplasm as 1.4 times that of the extracellular medium. In the absence of other data and assuming a similar ratio of extracellular to intracellular ionic composition, we take this ratio as valid for muscle. The recent measurements of Kleber and Riegger’* using an in vivo papillary muscle preparation give the ratio of extracellular to intracellular longitudinal resistance as 1.2. We also have the ratio of the extracellular to intracellular cross-sectional area of 0.33 from Polimeni. ’ 5 Now for cylindrical fibers re = pe/ A,, while ri = pi/(kAi), where A, is the interstitial cross-sectional area, Ai is the intracellular cross-sectional area, and k is the fraction of area actually available for conduction. Dividing re by ri and solving for k we obtain k

=

A&pi &ripe

From the aforementioned values k evaluates to 0.55 so that around 45% of intracellular space is nonconducting (which we interpret as due to nonconducting filaments). Clerc5 evaluates r&i = 0.278 (rather than 1.2), and that value would lead to an estimate that 87% of the intracellular space is nonconducting, but the preparation of Kleber and Riegger inspires more confidence.

Continuous Transverse Propagation

If one accepts the presence of relatively low junctional resistances, as described above, then the myocardium can be considered to be a continuum (a syncytium), In this case, plane wave propagation in any direction should be uniform. A quantitative evaluation of the electrical sources under these circumstances has been described in two previous paper+” and will only be summarized here. Experimental descriptions of normal activation of the left ventricular free wa1118 shows, essentially, a propagating wave in the endocardial to epicardial direction. We assume that this can be characterized as a plane wave in the outward, hence cross-fiber, direction. Figure 2 shows a cross-section transverse

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-

Glycogen

-

Fibril Mitochondrion

-Sarcolemma Sarcoplasmic4 reticulum

-(T

Tubule system)

Fig. 1. Intracellular space of a skeletal muscle fiber. The illustration describes the dense myofibrillar array and is the basis for an assumption of cellular intracellular anisotropv. 1 1 From Hoyle: How is muscle turned on and off? Scientific American 222:84, 1970. With permission.

to the fiber axis and successive isochrones, according to the previous activation hypothesis. A single-cell element of narrow width is described here and in greater detail in Figure 3. The source associated with this element arises from the membrane surfaces 1 and 2 is given by l9 ;i, = (@iUi - @&e)ii (2) where Cpiand @‘eare the potentials just inside and outside the membrane, oi and ue are the intracellular and extracellular microscopic conductivities, and ii is a unit vector in the direction of the outward surface normal. The source is described as a double layer lying in surfaces 1 and 2 having identical source densities. A similar result is expected from each cell element in Figure 2 lying between the same isochrones (separated by Ax). The total field is found by integrating all such source contributions, namely (3)

* = (47Fu,)-l

;is.V( l/r) dS.

Now, suppose that all six surfaces of the cell in Figure 3 carried the same uniform double layer described in equation 2. In this case it is well known that no extracellular field results. Consequently, the generated field can also be found from the negative of such hypothetical sources on sides 3, 4, 5, and 6. The source on sides 3 and 4, however, must be exactly canceled by adjoining cell elements, being evaluated by identical quantities arising from the uniform propagation. However the contribution from side 5 and 6 when summed with adjoining elements in the direction of propagation do not cancel. The sum is a residual that depends on the rate of change in field quantities in the direction of propagation. In the limit Ax * 0 the resulting dipole source volume density is given by ?v = d(@iUi - @cU,)/dXZ, (4) The dipole source density described in equation 4 fills the region occupied by the cells. However, we

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Fig. 2. Cross-section of cardiac muscle fiber bundle whose axis is in the z direction. A uniform plane wave is propagating in the transverse x direction and sequential isochrones are labeled 1, 2, 3, 4, 5. From Plonsey and van Oosterom.” With permission.

can assume that it fills all of the tissue space (for purpose of calculating external fields) by correcting for the larger volume of the latter. If the volume fraction occupied by the cells is f, then a uniform source density that applies to the entire tissue volume is given by equation 4 times f. Consistent with the continuum assumption is the application of the linear core conductor model that links intracellular and interstitial potentials with the transmembrane potential, Vm.20,21 We have (5) and

@i = ri/(ri + re)Vm

Q’, = -r,/(ri + re)Vm (6) where ri and re are resistances per unit axial length for a unit cross-section of the propagating wave. Substituting equations 5 and 6 into equation 4 and evaluating the extracellular field from these sources results in (7)

Qc(P) = 1/(47r)(rJ(ri +

Fig. 3. Volume element of cell identified in Figure 1 by cross-hatching. The element height, y, is that of the cell; the upper membrane surface is 1 and the lower is 2. The distance Ax is the separation between isochrones at elapsed time At = A x/velocity apart. The extent A z in the axial direction is arbitrary since we assume no variation of field quantities in this direction. From Plonsey and van Oosterom. l 6 With permission.

W

I

”

a/ax(V,)Zi;V(

l/r) dV

where V is the total volume, and the factor f, discussed above, has contributed to the coefficient on the right hand side of equation 7. In anisotropic syncytial tissue the values of re and ri depend on the direction of propagation, as described in Muler and Markin. For field points at a distance that is large compared with the axial extent of the source wave, the latter

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can be considered an idealized (infinitely thin) double layer. Its source strength is found by integrating equation 7 with respect to x across the wave. The result is simply the peak value of transmembrane potential minus the resting value (ie, Vpeak - V,,,,). Under these conditions equation 7 becomes simply (8)

@‘e(P) =

(Vpeak

-

v,,t)(rJ(ri

+

r,))MW4

where A fi is the solid angle subtended by the activating wave at the field point P.

Discussion The double-layer

source strength is given by the 8. For Vpeak - V,,, a Vahe Of 100 mV would be consistent with most measurements of transmembrane potential. The resistance ratio, on the other hand, is given in only two published papers, Clerc5 and Roberts and Scher,22 the last reference replacing an earlier one. Using the latter, the coefficient in equation 8 comes out 43 mV. Based on a value of 40 mV, and essentially using equation 8, van Oosterom and Huiskamp23 compute body surface potentials that are in excellent agreement with their measurements. (These calculations assume the ratio of heart to lung conductivity of 5, blood to lung conductivity of 3, and geometry determined by magnetic resonance imaging. Miller and Geselowitz24 also find that a double-layer source strength of 40 mV is needed to give an amplitude of simulated body surface potential waveforms that correlates with measurement. In addition to these confirmations there are reported measurements of intramural bipolar signal strengths of lo-30 mV from a number of papers.25-27 These results would seem to rule out cellular process other than a continuous activation in the outward direction, since a diminished source strength would otherwise result. One such alternate process is where local excitation is thought to proceed along rather than across each fiber. This expectation is suggested if one assumes that activation of each cell occurs only at its ends. The problem with such a model is that if the gradient of transmembrane potential lies only parallel to the cell axis no outward source strength arises. In fact, since the sources tend to be set up along both positive and negative axial directions, the result is a cancellation of dipole components leaving only the smaller quadrupole source. In the two-dimensional model of Leon and Roberge,28 precisely such activation patterns result, thereby raising questions concerning the structural assumptions of the model. Another assumption arising from the alternate model structure in which intercellular conduction COefflCient

in eqUatiOn

occurs solely at the ends of a cell is that each cell activates synchronously, leading to a membrane action potential. As noted by Geselowitz and Miller, macroscopic propagation can thus be in the outward direction even though microscopic activation is essentially synchronous.29,30 This model is described in Figure 4A where the potential variation over each cell is shown to be constant, while there are step changes from cell to cell. This contrasts with the continuum model also shown (dotted curve). The relative source density associated with each model is found by applying equation 7, plotted in Figure 4B. Since the sources whose densities are evaluated by equation 7 lie in the cellular crosssection, their total magnitude is proportional to that cross-sectional area. For the continuum the average cross-section is roughly the volume fraction of the cells themselves, perhaps 0.75 to the tissue crosssection. But for the discontinuous profile, the crosssection is literally that of the junctions at which the discontinuities must lie and hence is very small (this is discussed in detail by Plonsey and Barr3’ for the longitudinal case). While the net relative source for each distribution plotted in Figure 4B is the same (namely Vp& - V,,,, as discussed in connection with equation 8), the additional area factor makes the total source of the continuum several orders of magnitude greater than the discontinuous one. And, in fact, the derivation of equation 8 assumed that the source lay in the entire tissue (it would not have generated the expected body surface potentials found experimentally otherwise). The idea that cardiac cells can be modeled as long cylinders with a single interconnection at each end, as suggested above and described graphically in Clerc,5 is not consistent with current anatomical beliefs. A more appropriate description should include the stepped surfaces and multiple connections to numerous adjoining cells. (As noted earlier Hoyt et a1.4 state that each cell connects to nine neighbors.) In addition to longitudinal junctions, there are lateral junctions (and because of the stepped geometry many junctions serve to facilitate transverse as well as longitudinal current flow). Consequently, while discontinuities may arise at these junctions, their large numbers and their spatial distribution would tend to produce continuous behavior. While the above discussion has emphasized the case where the entire V,, is accounted for by discontinuities at the junctions, intermediate values of rj will result in a potential variation lying between the two extremes shown in Figure 4. A one-dimensional study by Rudy and Quan3’ may be pertinent to this point. For axial propagation they show only a 20% loss of source strength for an increase of the intercel-

Subcellular

Fig. 4. (A) The variation of V,(x), assumed in the model described herein (based on low resistance intercellular junctions and hence continuous propagation) is shown by the dotted curve. The solid curve results when it is assumed that each cell fires synchronously (no potential variation within each cell) so that transmembrane potential variations arise discontinuously between cells. The curve describes the behavior along path c-c. (Note that the actual extent of the rising phase of the action potential should extend several hundred pm and hence encompass perhaps 30 or so fiber cross-sections. Its extent was reduced to permit the use of the enlarged region that was introduced in Figure 2.) (B) Derivatives of each V, curve in A evaluate the relative source density of each, according to equation 7. To get the absolute source density, the aforementioned derivative must be multiplied by the coefficient in equation 7 and by the corresponding crosssectional area of the cells.

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A

lular resistance from 0.5 MCl to 2.5 MR. So, while the source amplitude seems nicely confirmed assuming a very low junctional resistance, in view of the impreciseness of the data and the relative insensitivity just noted, one cannot rule out the presence of a non-negligible junctional resistance (provided that propagation is still fundamentally continuous). Furthermore, in either case, it would seem that the effect of the junctions can be neglected when determining the cardiac sources.

%

Conclusions We have shown that a continuous activation model on a cellular level generates a magnitude and orientation of source consistent with body surface potential observations. The achievement of a continuous activation seems to also require the absence of large junctional resistances. It is also suggested that the observed tissue anisotropy, which has seemed to depend on the junctional resistances, could also be

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explained (at least in part) by cellular intracellular anisotropy brought about by the myofibrillar structure. Because the reduction of source strength by increasing junctional resistance may be modest, and because much data is not well established, the results do not rule out non-negligible junctional resistance provided the outward propagation is still primarily continuous. The questions raised here will be resolved by more detailed model studies on a cellular level, and on additional measurements of cardiac parameters, both macroscopic and microscopic, and both anatomic and electrophysiologic.

References 1. Luke RA, Beyer EC, Hoyt RH, Saffitz JE: Quantitative analysis of intercellular connections by immunohistochemistry of the cardiac gap junction protein connexin43. Circ Res 65: 1450, 1989 2. Page E: Cardiac gap junctions. p. 1020. In Fozzard HA et al (eds): The heart and cardiovascular system. 2nd ed. Raven Press, New York, 1992 3. Burt JM, Spray DC: Single channel events and gating behavior of the cardiac gap junction channel. Proc Nat1 Acad Sci 85:3431, 1988 4. Hoyt RH, Cohen ML, Safftz JE: Distribution and three-dimensional structure of intercellular junctions in canine myocardium. Circ Res 64:563, 1989 5. Clerc L: Directional differences of impulse spread in trabecular muscle from mammalian heart. J Physiol 255:335, 1976 6. Plonsey R: Effect of intracellular anisotropy on electrical source determination in a muscle fibre. Med Biol Eng Comput 28:312, 1990 7. Gielen FLH, Cruts HEP, Albers BA et al: Model of electrical conductivity of skeletal muscle based on tissue structure. Med Biol Eng Comput 24:34, 1986 8. Weingart R, Mauer P: Cell-to-cell coupling studied in isolated ventricular cell pairs. Experientia 43: 1091, 1987 9. Chapman RA, Fry CH: An analysis of the cable properties of frog ventricular myocardium. J Physiol 283: 263, 1978 10. White RL, Carvalho AC, Spray DC et al: Gap junctional conductance between isolated cell pairs of ventricular myocytes from rat. Biophysiol J 4 1:2 17a. 1983 11. Eisenberg BR: Quantitative ultrastructure of mammalian skeletal muscle. Section 10, skeletal muscle. p. 73. In Handbook of physiology. American Physiological Society, Bethesda, Maryland, 1983 12. Katz B: The electrical properties of the muscle fiber membrane. Proc R Sot B 135:506, 1948 13. Cole KS: Resistivity of axoplasm. I. Resistivity of extruded squid axoplasm. J Gen Physiol 66: 133, 1975 14. Kleber AG, Riegger CB: Electrical constants of arterially perfused rabbit papillary muscle. J Physiol 385: 307, 1986 15. Polimeni PI: Extracellular space and ionic distribution in rat ventricle. Am J Physiol 227:676, 1974

16. Plonsey R, van Oosterom A: A cellular activation model based on macroscopic fields. p. 3. In Sideman S, Beyar R, Kleber AG (eds) : Cardiac electrophysiology, circulation, and transport. Kluwer Academic, Boston, MA, 1991 17. Plonsey R, van Oosterom A: Implications of macroscopic source strength on cellular activation models. J Electrocardiol 24:99, 1991 18. Scher A, Spach MS: Cardiac depolarization and repolarization and the electrocardiogram. p. 357. In Beme RM (ed): Handbook of physiology. Section 2. Vol. 1. American Physiological Society, Bethesda, 1979 19. 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, Ploney R: Engineering contributions to biophysical electrocardiography. IEEE Press, New York, 1982) 20. Plonsey R, Barr RC: Bioelectricity: a quantitative approach. Plenum Press, New York, 1988 21. Muler AL, Markin VS: Electrical properties of anisotropic nerve-muscle syncytia-II: spread of flat front of excitation. Biophysics 22:536, 1977 22. Roberts DE, Scher AM: Effect of tissue anisotropy on extracellular potential fields in canine myocardium in situ. Circ Res 50:342, 1982 23. van Oosterom A, Huiskamp GJM: The effect of torso inhomogeneities on body surface potentials quantified by using tailored geometry. J Electrocardiol 22:53, 1989 24. Miller WT III, Geselowitz DB: Simulation studies of the electrocardiogram. I. The normal heart. Circ Res 43:301, 1978 25. Vander Ark CR, Reynolds EW Jr: An experimental study of propagated electrical activity in the canine heart. Circ Res 26:451, 1970 26. Durrer D, van der Tweel LH: Excitation of the left ventricular wall of the dog and goat. Ann NY Acad Sci 65: 779, 1957 27. Selvester RH, Kirk WL Jr, Pearson RB: Propagation velocities and voltage magnitudes in local segments of dog myocardium. Circ Res 27:619, 1970 28. Leon LJ, Roberge FA: Structural complexity effects on transverse propagation in a two-dimensional model of myocardium. IEEE Trans Biomed Eng 38:997, 1992 29. Geselowitz DB, Miller WT III: Active electric properties of cardiac muscle. Bioelectromagnetics 3: 127, 1982 30. Geselowitz DB: Letter to the editor. J Electrocardiol 25:75, 1992 31. Plonsey R, Barr RC: Effect of junctional resistance on source-strength in a linear cable. Ann Biomed Eng 13: 95, 1985 32. Rudy Y, Quan W: Propagation delays across cardiac gap junctions and their reflection in extracellular potentials: a simulation study. J Cardiovasc Electrophysiol 2:299, 1991 33. Genong WF: Medical physiology. Lange Medical Publications, Los Altos, CA 1971