© 1999 American Heart Association, Inc.
Original Contribution |
A Novel Mechanism of Anode-Break Stimulation Predicted by Bidomain Modeling
From the Department of Biomedical Engineering (R.R.) and the Section of Molecular and Cellular Cardiology (G.F.T., E.M.), Department of Medicine, Johns Hopkins University School of Medicine, Baltimore, Md. The current address for Dr Ranjan is Health Sciences and Technology, Harvard Medical School, Boston, MA 02115.
Correspondence to Eduardo Marbán, MD, PhD, Section of Molecular and Cellular Cardiology, 844 Ross Bldg, Johns Hopkins University School of Medicine, 720 Rutland Ave, Baltimore, MD 21205. E-mail marban{at}welchlink.welch.jhu.edu
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Abstract |
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Abstract—Anodal stimulation by external pacemakers has been explained on the basis of bidomain models of cardiac tissue. Bidomain models predict that anodal stimuli will hyperpolarize the underlying tissue while adjacent regions become depolarized (virtual cathodes), initiating excitation. We investigated the contribution of active cellular properties to anode-break stimulation. A bidomain model was implemented in which each cell contained realistic ionic currents, including those recruited by hyperpolarization. Simulations reveal that anode-break excitation can originate at the site of stimulation itself and not only from adjacent regions of induced depolarization. The threshold for initiating excitation at the site of stimulation is lower than that for stimulation initiating from adjacent depolarized regions. Thus, incorporation of active cellular properties into a bidomain model predicts a novel mechanism for anode-break stimulation of the heart. The results will improve our understanding of anodal pacing and its risks and benefits in patients.
Key Words: pacemaker • excitation • quantitative modeling • anisotropy
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Introduction |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionReferences |
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Experimentally, the mammalian heart can be activated by cathodal or anodal stimuli at both the make and break of the pulse.1 2 3 In fact, in early diastole, anodal-break stimulation has the lowest threshold,3 a phenomenon that may potentiate the risk of anodal pacing–induced ventricular fibrillation.4 In a bipolar pacing system, activation occurs from the lowest-threshold electrode, and at close coupling intervals, this may be the anode, increasing the risk of an arrhythmia, even during bipolar pacing. The mechanism of anodal stimulation remains controversial. The most plausible explanation emerges from bidomain models of cardiac tissue.5 6 Bidomain models, based on passive properties of the cardiac tissue, assume that the ratios of electrical conductivity along the fiber direction and perpendicular to it (anisotropy) differ in the intracellular and extracellular domains.7 The difference in anisotropy generates a complex distribution of transmembrane potential in response to anodal stimulation8 : the tissue directly underlying the stimulating electrode hyperpolarizes, while neighboring regions depolarize and function as "virtual cathodes."6 The bidomain model predicts that excitation starts at the virtual cathodes and spreads to cover the remaining tissue.9 10
Recently, we identified an active mechanism for anode-break stimulation at the cellular level. Recruitment of ionic currents during anodally induced hyperpolarization enabled excitation on termination of the stimulus ("anode-break" response).11 12 Modification of an existing action potential model to include the ionic currents at hyperpolarized potentials reproduced the results. We thus incorporated the enhanced cellular model into a newly implemented bidomain model to explore the role of these currents in anode-break excitation.
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Materials and Methods |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionReferences |
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A 2-dimensional network model was implemented. For a bidomain model, the intracellular (Vi) and extracellular (Vo) potential obey the following equations8 :
 | (1) |
 | (2) |
i and
o are conductivity tensors in the
intracellular and extracellular domains, accounting for tissue
anisotropy.
For active membrane, the current per unit volume
(Im) is given by
 | (3) |
 | (4) |
lists the parameters that were used
in the model. To solve for the transmembrane potential using the
bidomain formulation, current flow was restricted to be either along or
transverse to the fiber direction. This routine simplification fixes
the crossterms of the conductivity tensor at
zero.8 14
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Combining Equations 2
, and 3 gives a parabolic
equation:
 | (5) |
was solved for
Vm in the next time step using Euler's
method with the current Vo. Combining
Equations 1, 2, and 4 gives an elliptical
equation:
 | (6) |
was solved for
Vo in the next time step with the new
Vm using the alternating direction
implicit method.15 A time step of 5 µs and a space
step of 30 µm were used. A 2-dimensional sheet measuring 5
mm by 2.5 mm was used. At the tissue boundary, no flux boundary
condition was used. The stimulating electrode is 0.5 mm by
0.25 mm in size and injects current in the extracellular domain.
Only one quadrant was modeled to reduce computational time. The
simulations were done on an INDIGO R10000 Silicon Graphics
Workstation. |
Results |
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A resistive bidomain model was implemented first to test the validity of the computation method used in the present study. The results of the passive model generated in the present study (not shown; see Reference 1111 ) mimic those obtained with numerical methods.8 The model with unequal anisotropy produced a dog-bone distribution of the transmembrane potential typical of bidomain models. The dog-bone distribution disappeared when the anisotropy ratio was made equal in the 2 domains.8
Having validated the computational method, we next implemented the
active model. Figure 1 shows the
transmembrane potential distribution as a function of x and
y, calculated using the modified Luo-Rudy model at 8 time
points during and after delivery of a 0.3-mA, 10-ms anodal stimulation.
The lower left-hand corner of the sheet is stimulated. Anodal
stimulation results in the establishment of virtual cathodes in the
convexity of the hyperpolarized region. The depolarization induced in
the virtual cathodes suffices to initiate excitation, which then
propagates to cover the entire tissue.
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Next, we explored the predictions of the model at lower stimulation
strengths. By analogy to our previous cellular work,12 we
reasoned that recruitment of active membrane properties might suffice
to produce an anode-break response at stimulation thresholds lower than
those resulting from the virtual cathode. Figure 2 shows the response to a 0.15-mA, 10-ms
anodal stimulus. Once again, virtual cathodes are induced, but now the
extent of depolarization is not enough to initiate excitation. On
termination of the stimulus, the depolarization decays toward resting
levels (10- and 15-ms time frames). Nevertheless, the tissue underneath
the stimulating electrode had been hyperpolarized sufficiently to
elicit an anode-break response. The excitation in this case initiates
directly underneath the stimulating electrode and then propagates
throughout the tissue. No such excitation was observed in bidomain
simulations using the original Luo-Rudy action potential model.
Reassuringly, the response to cathodal stimulation was identical for
both the modified and unmodified action potential models (not
shown).
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Given that the extent of depolarization induced in the virtual cathode
depends on the relative anisotropy ratios in the intracellular and
extracellular domains, the relative contribution of the 2 mechanisms
for anodal excitation would also be expected to depend on these values.
For Figures 1 and 2, we used an extracellular anisotropy
ratio of 2.5 and an intracellular ratio of 7.5. These values are
representative of ratios reported for cardiac tissue
and used in previous bidomain models (ranging from 1.5 to 4 in the
extracellular domain and from 5.8 to 11.7
intracellularly).8 12 16 17 To examine the role played by
the disparity in intracellular and extracellular anisotropy ratios on
stimulation threshold, we varied the ratios in the 2 domains. Figure 3A shows the anodal stimulation threshold
determined using a bidomain model with a fixed intracellular anisotropy
ratio of 8 and a varying extracellular ratio. For extracellular ratios
of 1 to 2, anodal excitation originates at the virtual cathodes and is
an anode-make stimulation. For extracellular anisotropy ratios of >2,
the new mechanism of anodal stimulation appears: excitation starts
directly underneath the stimulating electrode, as an anode-break
response. When it is present, the excitation threshold for the
novel anode-break mechanism is lower than that for the conventional
mechanism of stimulation from the induced virtual cathodes. Figure 3B shows the anodal stimulation threshold determined using a
bidomain model with a fixed extracellular anisotropy ratio of 2.5 and a
varying intracellular anisotropy ratio. Once again, for a range of
intracellular anisotropy ratios (<10), excitation occurs at the site
of stimulation itself at lower thresholds before the virtual cathode
becomes the site of excitation at higher stimulus strengths.
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Discussion |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionReferences |
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Routine bidomain models explain anodal stimulation of cardiac tissue on the basis of the different anisotropy ratios in the intracellular and extracellular domains of cardiac tissue.5 In the present study, we investigated the consequences of introducing an ionic membrane model that can actively generate an anode-break response into a bidomain model. Implementation with an unmodified Luo-Rudy phase I action potential model predicts that virtual cathodes are established in response to anodal stimulation, consistent with earlier findings.9 Inclusion of the Ranjan et al12 refinements revealed a new mechanism of anode-break stimulation (Figures 2
and 3).
At lower stimulus strengths, the depolarization induced at the virtual
cathodes is insufficient to initiate a response, but it is strong
enough to yield an anode-break response in the hyperpolarized regions
directly underlying the electrode. This excitation then propagates to
other regions of the tissue. On increasing the stimulus strength in
this model, the depolarization induced in the virtual cathodes
suffices to induce excitation. This mechanism predominates at higher
stimulus strengths, because it occurs during the stimulus itself
(Figure 1).
This mechanism of anode-break stimulation at lower stimulus strengths
is different from the mechanisms proposed earlier, even though it can
be argued that in both cases the site of stimulation is the
hyperpolarized region of tissue underlying the electrode. Based on the
mechanism proposed by Roth,9 the depolarization induced in
the virtual cathode diffuses to the adjoining hyperpolarized region and
excites it, resulting in anode-break excitation. In the new mechanism
proposed in the present study, the diffusion of charges from the
depolarized virtual cathodes is not enough to cause excitation in the
hyperpolarized regions. As shown in Figure 2 (15-ms time frame),
the entire tissue returns to resting membrane potential levels after
the end of the stimulus. But the hyperpolarization
induced in the region underlying the electrode is enough to recruit the
hyperpolarization-activated current,
If, and cause excitation in these areas
independent of diffusion of charges from adjoining areas. This
excitation then proceeds to cover the remaining tissue. At higher
stimulus strengths, the previously proposed mechanism of anode-break
stimulation seems more plausible6 : that is, diffusion
of charge from the virtual cathode occurs at a faster rate and induces
excitation in the hyperpolarized regions before anode-break stimulation
under the electrode.
The range of stimulus strengths that elicits anode-break excitation at
the site of stimulation depends on the anisotropy ratios assumed in the
model (Figure 3). The degree of
hyperpolarization underneath the excitation
electrode does not primarily depend on the anisotropy ratios, but the
depolarization induced in the virtual cathodes does depend on the
disparity of anisotropy ratios in the 2 domains. The higher the
disparity in the ratios, the greater the depolarization in the virtual
cathodes. As a result, when the disparity in ratios is greater (at
extracellular anisotropies of 1 to 2 in Figure 3A and
intracellular anisotropy ratio of >10 in Figure 3B), the
depolarization induced at the virtual cathode is high enough for anodal
excitation at stimulus strengths below those needed for anode-break
stimulation. This window of stimulation with lower thresholds assumes
significance after considering the fact that the threshold for
anodal-break excitation, as reported by existing bidomain models, is
about 3 to 5 times higher than that determined
experimentally.10 At the most realistic anisotropy ratios
(corresponding to 
3 in Figure 3A), the anode-break mechanism
is quite prominent. Nevertheless, experiments in cardiac
tissue4 will be important in establishing the relative
importance of these 2 mechanisms.
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Acknowledgments |
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This work was supported by NIH R37 HL36957.
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Footnotes |
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This manuscript was sent to Michael R. Rosen, Consulting Editor, for review by expert referees, editorial decision, and final disposition.
Received July 17, 1998; accepted November 3, 1998.
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References |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionReferences |
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1. Cranefield PF, Hoffman BF, Siebens AA. Anodal excitation of cardiac muscle. Am J Physiol. 1957:190:383–390.
2. Brooks CM, Hoffman BF, Suckling EE. In: Excitability of the Heart. New York: Grune & Stratton; 1955:82–123.
3. Dekker E. Direct current make and break thresholds for pacemaker electrodes on the canine ventricle. Circ Res. 1970:27:811–823.
4. Preston TA. Anodal stimulation as a cause of pacemaker-induced ventricular fibrillation. Am Heart J. 1973:86:366–372.
5. Roth BJ. Mechanisms for electrical stimulation of excitable tissue. Crit Rev Biomed Eng. 1994:22:253–305.
6. Wikswo JP Jr, Lin SF, Abbas RA. Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation. Biophys J. 1995:69:2195–2210.
7. Roth BJ. How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle. J Math Biol. 1992:30:633–646.
8. Sepulveda NG, Roth BJ, Wikswo JP Jr. Current injection into a two-dimensional anisotropic bidomain. Biophys J. 1989:55:987–999.
9. Roth BJ. A mathematical model of make and break electrical stimulation of cardiac tissue by a unipolar anode or cathode. IEEE Trans Biomed Eng. 1995:42:1174–1184.
10. Roth BJ. Strength-interval curves for cardiac tissue predicted using the bidomain model. J Cardiovasc Electrophysiol. 1996:77:722–737.
11. Ranjan R. Mechanism of Anodal Stimulation in Cardiac Tissue [dissertation]. Baltimore, Md: Johns Hopkins University; 1997.
12. Ranjan R, Chiamvimonvat N, Thakor NV, Tomaselli GF, Marban E. Mechanism of anode break stimulation in the heart. Biophys J. 1998:74:1850–1863.
13. Luo CH, Rudy Y. A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res. 1991:68:1501–1526.
14. Roth BJ, Wikswo JP Jr. Electrical stimulation of cardiac tissue: a bidomain model with active membrane properties. IEEE Trans Biomed Eng. 1994:41:232–240.
15. Press WH, Teukolasky SA, Vetterling WT, Flannery BP. In: Numerical Recipes in FORTRAN. New York: Cambridge University Press; 1992:701–744.
16. Clerc L. Directional differences of impulse spread in trabecular muscle from mammalian heart. J Physiol (Lond). 1976:255:335–346.
17. Roberts DE, Scher AM. Effect of tissue anisotropy on extracellular potential fields in canine myocardium in situ. Circ Res. 1982:50:342–351.
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