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Am J Physiol Heart Circ Physiol 290: H1976-H1987, 2006. First published December 22, 2005; doi:10.1152/ajpheart.01180.2005
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Cardiac microimpedance measurement in two-dimensional models using multisite interstitial stimulation

Andrew E. Pollard1 and Roger C. Barr2

1Cardiac Rhythm Management Laboratory, Department of Biomedical Engineering, University of Alabama at Birmingham, Birmingham, Alabama; and 2Department of Biomedical Engineering Duke University, Durham, North Carolina

Submitted 8 November 2005 ; accepted in final form 19 December 2005


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
We analyzed central interstitial potential differences during multisite stimulation to assess the feasibility of using those recordings to measure cardiac microimpedances in multidimensional preparations. Because interstitial current injected and removed using electrodes with different proximities allows modulation of the portion of current crossing the membrane, we hypothesized that multisite interstitial stimulation would give rise to central interstitial potential differences that depend on intracellular and interstitial microimpedances, allowing measurement of those microimpedances. Simulations of multisite stimulation with fine and wide spacing in two-dimensional models that included dynamic membrane equations for guinea pig ventricular myocytes were performed to generate test data ({partial}{phi}o). Isotropic interstitial and intracellular microimpedances were prescribed for one set of simulations, and anisotropic microimpedances with unequal ratios (intracellular to interstitial) along and across fibers were prescribed for another set of simulations. Microimpedance measurements were then obtained by making statistical comparisons between {partial}{phi}o values and interstitial potential differences from passive bidomain simulations ({Delta}{phi}o) in which a wide range of possible microimpedances were considered. Possible microimpedances were selected at 25% increments. After demonstrating the effectiveness of the overall method with microimpedance measurements using one-dimensional test data, we showed microimpedance measurements within 25% of prescribed values in isotropic and anisotropic models. Our findings suggest that development of microfabricated devices to implement the procedure would facilitate routine measurement as a component of cardiac electrophysiological study.

bidomain modeling; virtual electrode; simulation; ventricular myocyte

MATHEMATICAL MODELING of cardiac electrical activity has provided important insights into mechanisms for arrhythmia initiation, maintenance, and termination. Since the presentation of the landmark phase II membrane equations by Luo and Rudy in 1994 (33), detailed representations of sarcolemmal currents, ion diffusion, excitation-contraction coupling, and intracellular signaling have been systematically integrated into species-, disease-, and region-dependent mathematical models for the isolated myocyte (1, 3, 36, 37, 41, 52, 59). The detail in these membrane equations is a consequence, primarily, of the extensive quantitative data available from single-cell electrophysiological studies. Strategies for careful positioning of myocytes with cellular and subcellular resolution into structural models have also been described. Resulting meshes incorporate details of the cellular architecture available from histological (22, 50, 53) and imaging (43, 49, 56) data.

Specification of intracellular electrical connections via the myoplasm and gap junctions and of interstitial connections via the collagenous cleft space remains immature by comparison. Some quantitative detail regarding gap junctional conductance is available from studies using cell pairs that remain attached after disaggregation (9, 32, 34, 48, 60, 61). In general, however, detail regarding tissue impedances on the size scale of individual myocytes, i.e., cardiac microimpedances, is unavailable. Because the data are unavailable, investigators who use mathematical modeling often prescribe microimpedances that allow simulations to replicate indirect measures of tissue status. Microimpedances prescribed to reflect action potential duration dispersion or conduction velocities do not necessarily reflect those of the tissue.

The discrepancy in detail is a consequence, in large part, of the lack of a straightforward method for microimpedance measurement that can be integrated into electrophysiological studies as a routine component. Experiments to obtain these data are technically challenging. They require a sequence of transmembrane potential (Vm) recordings in close proximity to a stimulating electrode to measure the electrotonic decay in Vm from which intracellular and interstitial microimpedances are derived. The approach is largely impractical for multidimensional preparations, because accurate positioning of the fragile electrodes is problematic. Furthermore, analytic frameworks for interpreting the Vm decay are closely related to the one-dimensional continuous core-conductor model, so their applicability to tissue preparations is limited. An alternate method involving four-electrode tissue impedance measurements has been used extensively to demonstrate changes accompanying the development of ischemia (7, 11, 54). However, decomposition of tissue impedance into its intracellular and interstitial components is complicated.

One possible method to overcome the limitations of these traditional approaches is multisite interstitial stimulation (42). Stimulation with electrodes separated over a distance that is small with respect to the space constant establishes a central interstitial potential difference that reflects interstitial current flow (38). Then stimulation with more widely spaced electrodes establishes a lower interstitial potential difference, because the applied current redistributes between interstitial and intracellular compartments. We recently showed that interstitial potential differences during multisite stimulation in one-dimensional Luo-Rudy dynamic (LRd) membrane equation (23, 24, 33, 41) simulations could be fit to the analytic response with the assumption of a core-conductor geometry. That fit allowed us to obtain highly accurate intracellular and interstitial microimpedance measurements. Here, we present a new approach that allows use in multidimensional preparations. Instead of fitting interstitial potential differences to an analytic response, we make statistical comparisons with interstitial potential differences from passive simulations in which a wide range of possible microimpedances are considered. Because we focus on two-dimensional models, our results have direct applicability to cardiac cell monolayer studies with preparations seeded using standard techniques that yield isotropic cultures (2, 10, 12, 25, 26) and to studies with patterned growth that yield anisotropic monolayers (14–16, 44). With modest adaptation, however, we believe the approach will also be applicable to in vivo and in vitro tissue preparations.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Bidomain modeling with LRd membrane equations. To assess the feasibility of the new approach, we first completed a set of LRd simulations using two-dimensional bidomain models to obtain interstitial potential difference ({partial}{phi}o) values during multisite interstitial stimulation. These {partial}{phi}o values were treated as test data. For those simulations, intracellular and interstitial compartments were coupled to one another via cell membrane (39, 46) using the governing equation

Formula 1(1)
where Im is transmembrane current density, go represents specific interstitial bidomain conductivities in the x and y directions, {partial}{phi}o is the interstitial potential, {sigma}o is the stimulus current density, gi represents specific intracellular bidomain conductivities in the x and y directions, and Vm is transmembrane potential. Domain coupling was then described by

Formula 2(2)
where beta is the ratio of membrane surface to element volume [6,350/cm following Giles and Imaizumi (17)], Cm is specific membrane capacitance (1 µF/cm2 as nominal for biological tissue), and Iion is total transmembrane current density resulting from ion channels, pumps, and exchangers as described by the LRd membrane equations.

Model myocytes. Individual myocytes within the models included nodes that were separated from one another at fixed-space steps of {Delta}x = {Delta}y = 12.5 µm in arrangements analogous to that shown in the circuit diagram in Fig. 1A. With that spacing, eight individual segments were located along each 100-µm-long LRd myocyte and two individual segments were located across myocytes separated laterally by 25 µm.


Figure 1
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  		Fig. 1. A: schematic diagram including circuit components used to represent intracellular, interstitial, and membrane current flow paths for an individual myocyte in the simulations. Dashed line represents a myocyte boundary. B: arrangement of recording ({circ}) and stimulating (bullet) sites in the fine-spacing regions. Electrodes A and D were used for stimulation. Electrodes B and C were used for central recording. Interstitial current was fixed for simulations with fine (electrodes i–ix) spacing. C: arrangement of stimulating (bullet) sites in the wide-spacing (electrodes x–xv) region.

 
Microimpedance assignment for isotropic models. Responses to multisite interstitial stimulation were assessed in models in which the intracellular and interstitial microimpedances in the x and y directions from Fig. 1A were assumed to be isotropic or anisotropic. We viewed the assignment of isotropic microimpedances as necessary, because cardiac electrophysiological studies that make use of cultured monolayers typically assume an isotropic two-dimensional arrangement of myocytes. As described by Jongsma and van Rijn (26), standard monolayers cultured from rat neonatal ventricular myocytes are isotropic, because no preferential direction for myocyte arrangement results from seeding. The isotropic microimpedances we prescribed were determined from the data of Jongsma and van Rijn. The intracellular microimpedance was derived from a specific intracellular resistivity of 500 {Omega}·cm (=2.0 mS/cm) reduced to gi,x = gi,y = 1.60 mS/cm in recognition of the report of Polimeni et al. (40) that ~80% of tissue is intracellular. The interstitial microimpedance was derived from an extracellular resistivity of 70 {Omega}·cm (=14.3 mS/cm) reduced to go,x = go,y = 2.86 mS/cm to reflect the remaining 20% of tissue that is interstitial.

Microimpedance assignment for anisotropic models. We viewed the assignment of anisotropic microimpedances as necessary, separate from analyses with isotropic models, because tissue in vivo is anisotropic. Furthermore, as described by Fast et al. (14), deposition of collagen on coverslips followed by fine brushing of the collagen coat in advance of myocyte seeding produces an adhesion matrix with parallel alignment of cultured cells. Such anisotropic cellular arrangements establish lateral and longitudinal contacts with neighboring cells that closely approximate contact distributions in adult canine ventricular myocardium (27). For the anistropic model, assuming fibers oriented in the x direction of Fig. 1A, we prescribed values of go,x = 3.17 mS/cm and gi,x = 4.82 mS/cm. This assignment followed the measurements of Kl’eber and Rieger (29) in perfused rabbit papillary muscles after they accounted for intracellular and interstitial volume fractions (40). As reported by Clerc (6), go,y = 1.17 mS/cm and gi,y = 0.51 mS/cm were then derived from directional conductivity ratios of 2.7 (go,x/go,y) and 9.4 (gi,x/gi,y), respectively.

Interstitial stimulating electrode locations. Nodes within each model were identified as recording or stimulation sites to obtain the {partial}{phi}o values. Figure 1B shows sites positioned along the x-axis. The central recording electrodes, denoted B and C, were separated from one another by 25 µm to ensure that a sufficient number of stimulating electrodes could be positioned within five myocytes laid end-to-end. A set of nine nodes, located to the left of electrode B and denoted Ai-Aix, was used for interstitial current injection. The separation between the center of the recording pair and each of the current injection electrodes, denoted p, varied from 37.5 to 237.5 µm in 25-µm steps. Similarly, a set of current removal electrodes (Di-Dix) was positioned with separation from the center of the recording pair (q) at 37.5–237.5 µm, such that 20 total electrodes made up the fine-spacing region in each model. Figure 1C shows sites selected for current injection (Ax-Axv) and removal (Dx-Dxv) in a wide-spacing region with 400-µm steps between electrodes. Positions for electrodes A and D used in different LRd simulations are summarized in Table 1. All simulations were completed with the electrodes oriented as shown in Fig. 1, B and C.


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  		Table 1. Stimulating electrode combinations

 
Computational sequence. Full discretization of Eqs. 1 and 2 resulted in sparse linear systems that were solved using the method of conjugate gradients as described in our earlier report (42). In all, 200 segments were oriented along the x- and y-axes, such that models measured 5 x 5 mm. Time steps were fixed at 2 µs to ensure stable and accurate solutions during action potential depolarization and interstitial stimulation. Action potential propagation was initiated by transmembrane current injection (300 µA/cm2, 2-ms duration) in a 4 x 4 group of nodes in a model corner. Stimuli for the multisite tests were applied 10 ms after depolarization wavefront expansion across each model was complete to ensure action potential plateau stimulation, which limited the possibility of inducing an active response. For that stimulation, interstitial current ({xi}o = 5,000 µA/cm2, 10-ms duration) was injected into one node in the model (Ai-Axv from Table 1) and removed from another node in the model (Di-Dxv from Table 1).

Microimpedance measurements. Once {partial}{phi}o values were available, we made statistical comparisons with interstitial potential differences from passive bidomain simulations ({Delta}{phi}o), in which wide ranges of possible microimpedances were considered. To make these comparisons, we developed a table-"look-up" procedure that we term SCAT, because it involves simulations with passive bidomain models (S), comparison of alternative microimpedance choices (CA), and tabulation for measurements within defined error bounds (T). Large numbers of passive bidomain simulations were completed with iterative solution of Eq. 1 using

Formula 3(3)
in place of Eq. 2. In Eq. 3, Rm is membrane microimpedance (in k{Omega}·cm2). We emphasize that Rm was not prescribed in the LRd simulations but, instead, resulted from combination of the available sarcolemmal channels during the action potential plateau when interstitial stimuli were applied. Therefore, its intrinsic value was identified as a part of the measurement process. Possible microimpedances were treated as SCAT table inputs, in which {Delta}{phi}o values with different stimulating electrode combinations were treated as SCAT table outputs. To compare alternative microimpedances, we then calculated the statistical variance ({sigma}2) between {Delta}{phi}o and {partial}{phi}o for each stimulating electrode combination. Differences (Y) were quantified using

Formula 4(4)
where j denotes stimulating electrode combinations used for comparison with N total combinations. SCAT tables were then searched to identify entries at which Y was minimized. As {Delta}{phi}o and {partial}{phi}o values approached one another in table entries, improved matches between the SCAT table and prescribed microimpedances were found.

Analysis steps. To complete the microimpedance measurements, we first analyzed the relation between stimulating electrode separation and {partial}{phi}o to select a set of combinations to be used for the analysis. Upper- and lower-bound values for the microimpedances were defined, and passive bidomain simulations using those microimpedances were completed to build SCAT tables. In assembling those tables, we made no attempt to ensure that the microimpedances prescribed for the LRd simulations to generate test data were included. As a consequence, microimpedances within each SCAT table were reviewed to identify the entries with the lowest percent errors from the prescribed microimpedances, inasmuch as these reflected the most accurate measurements available with this analysis. With the use of Eq. 4, Y values were calculated and SCAT table entries with the lowest overall Y values were identified. Then, {Delta}{phi}o percent differences, defined as ({Delta}{phi}o – {partial}{phi}o)/{partial}{phi}o with each stimulating electrode combination, were reviewed to assess whether additional SCAT table entries were necessary for the analysis.


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 GRANTS
 REFERENCES
 
Stimulating electrode selections and one-dimensional test data. To assess the effectiveness of the approach before application to the two-dimensional models, we first considered data from our earlier report (42) that focused on the one-dimensional fiber. For this test, we used {partial}{phi}o with the Ai-Di stimulating electrode combination as one of the Y components. Our earlier report showed that relatively little of the current injected and removed over such a fine spatial scale crossed the membrane with the Ai-Di stimulating electrode combination. Similarly, we used {partial}{phi}o with the Axv-Dxv stimulating electrode combination as a second Y component. Wide electrode separation led to a different response, in which intracellular and interstitial current equilibration was complete. Because the SCAT table approach requires that the number of table outputs match or exceed the number of table inputs, we used {partial}{phi}o recorded with the Aix-Dix stimulating electrode combination as the third Y component. Spacing with this combination was at the boundary between the fine and wide regions.

SCAT table assembly. To assemble the SCAT table for the initial test, we completed individual passive simulations in which unique combinations of go,x, gi,x, and Rm were used to determine {Delta}{phi}o values with the Ai-Di, Aix-Dix, and Axv-Dxv stimulating electrode combinations. For the first SCAT table entry, bounding values of go,x = 10.00 mS/cm, gi,x = 20.00 mS/cm, and Rm = 0.50 k{Omega}·cm2 were prescribed, because we recognized them as being outside the range suggested from the experimental literature. Subsequent SCAT table entries were identified after 25% reductions in gi,x from values at the preceding table entries. When gi,x reached a lower bound, well below values suggested from the experimental literature (0.011 mS/cm), go,x was similarly decremented and gi,x was returned to its upper-bound value. When go,x reached a lower bound (0.006 mS/cm), Rm was incremented and gi,x and go,x were returned to upper-bound values. The process was repeated until the SCAT table included entries spanning an Rm up to 88.68 k{Omega}·cm2. In total, 27·27·19(=13,851) table entries were assembled from 41,533 separate simulations. Assembly of the table required 12.9 h of computational time on a 1.44-GHz Pentium III Linux-based computer server (model 1400SC, Dell, Austin, TX).

Statistical comparisons. To appreciate the wide range of Y values between {Delta}{phi}o and {partial}{phi}o with the selected stimulating electrode combinations, we plotted all 13,851 Y values against the SCAT table entries (Fig. 2A). We present data in this way to further clarify the SCAT table assembly process. Within Fig. 2A, individual Y values are collected into 19 groups, which are separated by vertical dashed lines. Each group includes all Y values for one membrane microimpedance, with the Rm values designated above the groups. Y values varied widely within each group, indicating many combinations of gi,x, go,x, and Rm that were poor matches for the underlying parameters. Some matches were excellent, however, and microimpedances of gi,x = 4.75 mS/cm and go,x = 3.16 mS/cm at Rm = 6.66 k{Omega}·cm2 were identified at the SCAT table entry with the lowest overall Y value. These matches to the prescribed gi,x and go,x were 1.00 and 0.34%, respectively, reflecting a measurement accuracy comparable to that achieved using the analytic approach in our earlier report (42). Table search to identify this entry required only 1–2 s of computational time.


Figure 2
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  		Fig. 2. A: Y differences for all input microimpedances in the 1-dimensional SCAT table. Dashed vertical lines mark boundaries between entries in the SCAT table at which membrane microimpedance (Rm) was adjusted. B: differences between {partial}{phi}o and {Delta}{phi}o with Ai-Di stimulating electrode combination and interstitial microimpedance adjustments. go,x, Specific interstitial bidomain conductivity in the x direction.

 
Comparison of alternatives. The superiority of this combination of microimpedances was evident in comparisons with alternatives from the SCAT table. Prescribed values for the intracellular and interstitial microimpedances and the {partial}{phi}o values recorded using the selected stimulating electrode combinations, as well as Y values, the SCAT table microimpedances, and the {Delta}{phi}o percent difference for each stimulating electrode combination, are shown in Table 2. Only those entries at the 5 smallest of the 13,851 available Y values are shown. At the lowest Y value (0.000291), the difference between {Delta}{phi}o and {partial}{phi}o with the Ai-Di stimulating electrode combination was only 0.30%. Here, we report the {Delta}{phi}o percent difference, as opposed to {Delta}{phi}o – {partial}{phi}o, to reflect variations in {partial}{phi}o magnitudes with the different stimulating electrode combinations. Low {Delta}{phi}o percent differences of 0.05 and 1.80% were also observed with the Aix-Dix and Axv-Dxv stimulating electrode combinations.


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  		Table 2. SCAT table entries at Y minima for one-dimensional data

 
At the next three Y values (0.000953, 0.001776, and 0.001858), adjustments in Rm established larger {Delta}{phi}o percent differences, although gi,x = 4.75 mS/cm and go,x = 3.16 mS/cm were consistently identified. Adjustment in gi,x or go,x was not apparent until Y = 0.001986, when gi,x = 6.33 mS/cm was identified, reflecting a measurement error of 31.39%. Collectively, these observations suggested that SCAT table analysis would effectively identify close matches to the prescribed microimpedances, if such matches were available in the SCAT table. The relatively low {Delta}{phi}o percent difference for the Ai-Di stimulating electrode combination with all entries shown in Table 2 was related to the close match between the prescribed go,x and the go,x used for these passive bidomain simulations. As a consequence, we next sought to establish whether go,x measurement based solely on matches between {partial}{phi}o and {Delta}{phi}o with this stimulating electrode combination was practical. Figure 2B shows the differences between {partial}{phi}o and {Delta}{phi}o for the Ai-Di stimulating electrode combination with go,x adjusted from 2.66 to 3.66 mS/cm in 0.01-mS/cm steps. This range was selected, because it bounded the prescribed go,x. The difference was lowest at 3.17 mS/cm, which was the value we prescribed in our original report (42).

Isotropic model stimulating electrode selections. A first step in measuring microimpedances in the two-dimensional isotropic model using this overall approach was identification of stimulating electrode combinations for use in SCAT table assembly. This step was necessary, because current flow paths in the two-dimensional model necessarily differed from those under core-conductor assumptions. Those differences influenced {partial}{phi}o values during multisite stimulation. To appreciate those differences, we show Vm recorded at electrode locations Ai, B, C, and Di from simulations with the two-dimensional isotropic model, in which the model was stimulated using the Ai-Di and Axv-Dxv electrode combinations in Fig. 3A. Responses under core-conductor assumptions, determined in separate one-dimensional simulations using only those nodes located on the x-axis from the isotropic model, are included. Records from these simulations (dashed lines) are denoted Ai-Di(1D) and Axv-Dxv(1D) throughout Fig. 3. Action potential propagation into these locations occurred slightly earlier in the one- than in the two-dimensional model, with the differences in propagation delay being a consequence of wavefront curvature. The timing for interstitial stimulation, however, was maintained for all simulations. In response to Ai-Di stimulation, Vm hyperpolarization was observed at the Ai and B electrode locations, and Vm depolarization was observed at the Di and C electrode locations, as expected. Because injected current was confined to the restricted interstitial space along the x-axis in the one-dimensional model, the magnitudes for Vm responses to Ai-Di stimulation exceeded the magnitudes for the corresponding responses in the two-dimensional model. Axv-Dxv stimulation caused no change in Vm in the one- or two-dimensional model because of the wide separation between the recording and stimulating electrodes. Although limited influence on Vm during Axv-Dxv stimulation was observed, there were marked differences in {partial}{phi}o between the one- and two-dimensional simulations. Figure 3B shows {partial}{phi}o during interstitial stimulation of both models using the Ai-Di and Axv-Dxv combinations. An asymptotic {partial}{phi}o that measured 67% of the {partial}{phi}o with the Ai-Di combination was observed with the separation of stimulating electrodes to the Axv-Dxv combination in the one-dimensional model. By comparison, {partial}{phi}o with the Axv-Dxv combination in the two-dimensional model was negligible. This discrepancy suggested that use of the Axv-Dxv combination would have limited value in SCAT table analysis. Figure 3C shows {partial}{phi}o for all stimulating electrode combinations, with the two-dimensional model as a function of electrode separation from the center of the recording pair. The decay in {partial}{phi}o with increasing electrode separation was pronounced. As a consequence, we selected the Ai-Di, Aiv-Div, and Aix-Dix stimulating electrode combinations for SCAT table assembly in an attempt to resolve the observed decay in {partial}{phi}o.


Figure 3
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  		Fig. 3. A: transmembrane potential (Vm) from Luo-Rudy dynamic (LRd) simulations with the 2-dimensional isotropic model and a 1-dimensional model built using nodes along the x-axis. Records from nodes Ai, B, C, and Di are shown. B: {partial}{phi}o during stimulation with Ai-Di and Axv-Dxv electrodes from these same simulations. C: {partial}{phi}o at the end of the stimulus pulse with all stimulating electrode combinations in the isotropic LRd simulations shown as a function of electrode separation from the center of the recording pair.

 
Isotropic SCAT table. Once these combinations were selected, we built a SCAT table using 25% adjustments over bounds of 0.056–10.00 mS/cm for gi,x, 0.056–10.00 mS/cm for go,x, and 0.50–49.89 k{Omega}·cm2 for Rm. This SCAT table included 6,137 entries. Assembly required 426 h (17.7 days) of total computational time. Figure 4A shows Y values plotted against entry number and grouped by the designated Rm values. As in the one-dimensional analysis, a wide range in Y values resulted from the large number of possible microimpedances. Most choices reflected poor matches to the underlying parameters. The five SCAT table entries with lowest overall Y values from these comparisons are shown in Table 3. The closest available match to the prescribed gi,x of 1.60 mS/cm was 1.78 mS/cm (11.25%), whereas the closest available match to the prescribed go,x of 2.86 mS/cm was 3.16 mS/cm (10.48%). Consistent with the one-dimensional analysis, the closest available match for go,x was identified at all the SCAT table entries shown. In contrast to the one-dimensional analysis, however, gi,x varied widely, and the closest match available was not identified at the lowest overall Y value. Furthermore, {Delta}{phi}o percent differences were relatively large with the Ai-Di and Aiv-Div stimulating electrode combinations, indicating that improved SCAT table microimpedances were likely available.


Figure 4
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  		Fig. 4. A: Y differences for the 2-dimensional isotropic model. B: differences between {partial}{phi}o and {Delta}{phi}o with the Ai-Di stimulating electrode combination and interstitial microimpedance adjustments ({blacksquare}).

 

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  		Table 3. SCAT table entries for isotropic model

 
Isotropic interstitial microimpedance refinement. To test for improvements, we completed additional simulations in which refinements in go,x were considered. Here, we expected that by minimizing the difference between {partial}{phi}o and {Delta}{phi}o using the Ai-Di stimulating electrode combination, identification of the prescribed go,x as occurred in the one-dimensional analysis would result. We therefore fixed gi,x at 10.0 mS/cm and Rm at 37.415 k{Omega}·cm2, because these values were associated with minimum Y in the initial SCAT table, adjusted go,x from 2.3 to 4.2 mS/cm in 0.1-mS/cm steps, and completed passive bidomain simulations with the Ai-Di stimulating electrode combination. Figure 4B shows differences between {partial}{phi}o and {Delta}{phi}o for these simulations. Because this difference was minimized at go,x = 2.9 mS/cm, we completed additional simulations in which refinement in 0.01-mS/cm steps was performed over a range of 2.81–2.99 mS/cm. Overall, the difference was minimized at go,x = 2.87 mS/cm, which differed from the prescribed go,x by only 0.34%. Fixing go,x to 2.87 mS/cm, we then refined the SCAT table using the gi,x and Rm values from the initial SCAT table. The SCAT table entries with the five lowest overall Y values after refinement are shown in Table 3. By using the go,x measured independently, differences between {partial}{phi}o and {Delta}{phi}o declined markedly. At the lowest Y, the SCAT table gi,x of 1.78 mS/cm was identified.

Anisotropic model. Intracellular and interstitial microimpedances of comparable accuracy were also obtained with the anisotropic model. Figure 5A shows the {partial}{phi}o values. As with the isotropic model, these {partial}{phi}o values reduced to magnitudes that we anticipated would be difficult to resolve experimentally with stimulating electrode combinations that included wide spacing. For SCAT table analysis, we considered the widest separation to be the Ax-Dx stimulating electrode combination. Because SCAT table analysis requires that the number of {partial}{phi}o values match or exceed the number of individual microimpedances, {partial}{phi}o values with the Ai-Di, Aii-Dii, Aiii-Diii, and Av-Dv stimulating electrode combinations were also included. The {partial}{phi}o values with these stimulating electrode combinations are shown in Fig. 5A. Once these combinations were selected, we built a SCAT table with 25% adjustments over 1.90–4.50 mS/cm for go,x, 2.74–6.50 mS/cm for gi,x, 1.12–2.00 mS/cm for go,y, 0.56–1.00 mS/cm for gi,y, and 3.00–7.11 k{Omega}·cm2 for Rm. Generation of the table, which contained 1,024 entries, required 153 h (6.4 days) of computational time. Figure 5B shows all Y values plotted against SCAT table entry number. The five SCAT table entries with the lowest overall Y are shown in Table 4. The closest matches available in the SCAT table were 4.88 mS/cm (1.25%) for gi,x, 3.38 mS/cm (6.62%) for go,x, 0.56 mS/cm (9.80%) for gi,y, and 1.12 mS/cm (4.27%) for go,y. At the lowest overall Y, microimpedances of 4.88, 3.38, 0.42, and 1.12 mS/cm were identified for gi,x, go,x, gi,y, and go,xy, respectively. Although the identified gi,y was not the closest match available, it differed from the prescribed gi,y by only 17.65%. Therefore, all values were within the 25% steps used for SCAT table assembly. Furthermore, relatively low {Delta}{phi}o percent differences were observed at this entry, supporting the likelihood that these parameters matched those prescribed for generation of the {partial}{phi}o test data.


Figure 5
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  		Fig. 5. A: {partial}{phi}o at the end of the stimulus pulse with all stimulating electrode combinations in the anisotropic LRd simulations. B: Y differences for the anisotropic model.

 

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  		Table 4. SCAT table entries for anisotropic model

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
GRANTS
 REFERENCES
 
The specific aim of the present study was to assess the feasibility of using multisite interstitial stimulation to measure cardiac microimpedances in two-dimensional preparations. As shown in our earlier report (42), which focused on conditions detailed by Weidmann (57), positioning a set of interstitial stimulating electrodes in close proximity to a central recording pair allowed recording of one set of {partial}{phi}o responses in which relatively little of the injected current crossed the cell membrane before removal. With wider spacing between the stimulating electrodes, the injected current equilibrated between the intracellular and interstitial volume conductors before removal, resulting in the lower {partial}{phi}o responses that reflected the redistribution. Assumption of a core-conductor geometry then allowed analytic solution for the interstitial potential gradient in terms of the stimulating electrode positions and the intracellular and interstitial microimpedances. Nonlinear least-squares analyses of {partial}{phi}o over a range of stimulating electrode combinations led to highly accurate microimpedance measurements and quantification of the gap junctional contributions to overall cellular uncoupling. The present study extends that report significantly. In particular, we removed the restrictive assumption of a core-conductor geometry as part of the measurement process. Replacing that part with the statistical approach allows for arbitrary geometric arrangements, provided {partial}{phi}o recordings are made with sufficiently fine spatial resolution and multisite stimulating electrode combinations allow resolution of {partial}{phi}o changes with adjustments in those combinations. When these criteria were met, we found that a process of assessing {partial}{phi}o to select stimulating electrode combinations for SCAT table assembly and analyzing {Delta}{phi}o percent differences allowed accurate microimpedance measurements.

Implications for microfabrication. A key component of the strategy we described is an assumption that electrode arrangements with known positions and fine spacing will be available for implementation. An approach based on microfabrication, which will allow electrodes to be positioned on the size scale of individual myocytes, therefore, seems best suited, because the available precision and consistency for sensor location are superior to hand assembly. In neural electrophysiology, silicone-based arrays that penetrate the cortex for acute studies (4, 35, 62) and remain in place for chronic recordings (21, 47) have been described. In cardiac tissue, Hofer et al. (20) and Kim et al. (28) used microfabricated arrays with superfused guinea pig papillary muscles and perfused mouse and rabbit papillary muscle preparations, respectively, to record electrograms. These reports demonstrate that high-quality recordings with fabricated electrodes are technically possible; yet the approach has not seen widespread application. Recently (58), we used microfabricated sensors separated by 75 µm to demonstrate stable surface recordings from perfused rabbit right ventricular free wall preparations. After considering design steps associated with careful signal conditioning to record high-quality unipolar electrograms, we measured interstitial potential differences necessary for gradient determination to compute transmembrane current density at signal-to-noise ratios comparable to those measured using hand-assembled arrays. Collectively, these studies suggest that an arrangement analogous to the one we modeled is technically feasible and will likely be easy to achieve.

Applicability to standard monolayers. Assuming that microfabricated electrode arrays are available, we believe the method will allow experimental measurements using cultured monolayers of cardiac cells. This preparation is one in which the discrepancy of information between the sarcolemmal ionic currents and the intracellular and interstitial currents is especially pronounced. Microimpedance data are primarily limited to measurements from cultured neonatal rat ventricular myocytes, as reported by Jongsma and van Rijn (26). These investigators identified specific intracellular resistivity in a sequence of experiments that involved measurements of diastolic Vm displacement using one glass microelectrode in response to hyperpolarizing current injected via a second glass microelectrode with known separation between the two electrodes. Monolayers were assumed to be isotropic, because no preferential direction for cell alignment resulted from seeding. The electrotonic decay in Vm was then fit to a mathematical description for the radial voltage drop over myoplasmic and intracellular resistance under conditions of negligible extracellular or interstitial resistance. Completion of these types of measurements is largely impractical as a standard component of the optical mapping studies used to identify mechanisms for failure of action potential propagation (14–16, 44), cellular-level responses to defibrillation shocks (5, 13, 18, 19), and initiation and maintenance of cardiac arrhythmias (2, 10, 12, 25). Preparation of cultured monolayers on substrates that included electrodes for multisite stimulation and central interstitial potential difference recordings would ensure the routine collection of microimpedance data.

Preparations with anisotropic microimpedances. Our finding that highly accurate microimpedances were measured using multisite interstitial stimulation in anisotropic two-dimensional models has similar implications for cell cultures prepared by patterned growth (14). Although such preparations are used less commonly than standard isotropic cell monolayers, their potential impact on understanding characteristics of action potential propagation in the heart is likely more significant because of the similarities in cell-to-cell contacts between anisotropic monolayers and ventricular myocardium (27). Microimpedance measurements from anisotropic preparations would have a profound impact, because data regarding the directional differences are so limited. Most investigators who attempt to represent the anisotropic cellular arrangement in theoretical studies rely on data obtained by Clerc (6), who mounted calf trabecular bundles in one chamber for longitudinal measurements and then moved those bundles to a second chamber for transverse measurements. The Clerc measurements are widely cited because of the unequal ratios (intracellular to interstitial) for longitudinal and transverse impedances. As demonstrated with theoretical modeling, assignment of such ratios leads to current redistribution, which causes virtual electrode formation in response to defibrillation-strength shocks (31, 51, 55). Virtual electrodes have been identified with optical mapping, and an understanding of their characteristics is important, because evidence suggesting their contribution to arrhythmia induction has been presented (8, 45). Here, we stress that experimental confirmation of the critical piece of underlying data on which this important body of research is based is largely unavailable for preparations other than in vitro calf trabeculae as reported by Clerc in 1976.

General utility of the SCAT table approach. The general utility of our approach is a consequence of the relatively low computational expense for any one passive simulation. That low expense allows large numbers of microimpedance combinations to be made available for the statistical comparisons. Furthermore, the addition of entries is straightforward, allowing refinements to the SCAT table when reduction in {Delta}{phi}o percent differences is desired. Analyses with SCAT tables, once assembled, are rapid, because the process involves looking up table entries. Development of the SCAT table approach was instrumental in obtaining the accurate microimpedance measurements in the present study, and we emphasize that the approach has significant advantages over alternative methods. All entries computed for these analyses are available for reuse. This has implications for future experimental studies, because assembly of electrode arrangements with dimensions considered here will allow direct use of the SCAT tables built to date. Because the SCAT tables were built from passive bidomain simulations, any adjustments in stimulus current magnitude will simply require scaling of {Delta}{phi}o values. In the event that experimental electrode arrangements differ from those we considered, any entries associated with stimulating electrode combinations that we did consider will be available to populate portions of new SCAT tables, limiting the number of combinations required for that population. The process of assembling an initial SCAT table requires minimal information regarding passive simulations that have been or need to be completed. In the event that full retabulation is necessary, the independence of the individual passive simulations suggests that determination of SCAT table entries is an excellent candidate for grid computing using hundreds or thousands of processors, because no communication overhead would be associated with that tabulation.

Limitations. In assessing our findings, it is important to recognize certain limitations. 1) In developing the mathematical approach, we made no attempt to modify the interstitial microimpedances at locations in the model where the electrodes were located; therefore, we neglected any changes to the interstitial current distribution that would result from the electrodes themselves. Knisley and Pollard (30) recently reported that placement of a 1-cm disk patterned from translucent indium tin oxide within the shock field on the epicardium of perfused rabbit heart induced changes in Vm at the boundaries of the disk and that these changes were indicated by bidomain modeling to be associated with resistive inhomogeneity in the extracellular volume conductor. Although we anticipate that the ability of the recording electrodes to induce similar effects in the proposed arrangement will be much less pronounced because of their small sizes, we nevertheless recognize the possibility that changes in Vm may result and that such changes may influence the microimpedance measurements. The availability of passive bidomain simulations in which electrodes are considered in SCAT table assembly would address this issue, although we made no attempt to include electrode effects in the analyses here. 2) Interstitial microimpedances that we prescribed were derived with an assumption that no current flowed into surrounding extracellular fluid. Accounting for an extracellular current flow path would necessarily change the magnitude of {Delta}{phi}o values. As with electrode effects, however, we note that adjustments to the passive bidomain simulations to include extracellular nodes represent a straightforward extension for SCAT table assembly. 3) We made no attempt to account for how lead noise in the {partial}{phi}o recordings influences the measured microimpedances. However, our earlier report (42) suggested that time averaging the recorded {partial}{phi}o values over late intervals during interstitial stimulation resulted in small microimpedance measurement errors associated with lead noise. We anticipate similar findings under the conditions studied here. 4) We only tested multisite stimulation during a coupling interval 10 ms after depolarization. With different coupling intervals, we anticipate that changes in {partial}{phi}o would result from alterations in membrane microimpedance. As described by Zaniboni et al. (63), who used instantaneous current-voltage curves to measure membrane microimpedance during different repolarization phases in isolated guinea pig ventricular myocytes, that microimpedance does change during repolarization.

Implications for microimpedance measurements in cardiac electrophysiological studies. We believe that microfabricated arrays will be necessary to accomplish these measurements in cell monolayers and in real myocardial tissue. In cell culture preparations, we envision myocyte seeding onto arrays following the approach used recently by Zeevi-Levin et al. (64), who cultured rat ventricular myocytes onto fabricated electrode arrays to allow continuous potential recordings during induction of hypoxia. Use of selected electrodes for stimulation and recording would allow implementation of our method for microimpedance measurements. In myocardial tissue, we envision positioning of microfabricated arrays onto tissue following the approach we recently described (58). In that report, we used microfabricated sensors with spacing on the size scale of individual myocytes that were packaged in flexible assemblies that included adjacent signal conditioning to facilitate high-quality interstitial potential difference recordings. Addition of passive stimulating electrodes will be straightforward. Although implementation will likely be complicated by the limitations noted above and the ability to align electrodes with myocardial fibers in anisotropic preparations, we note that our choice to align electrodes with myocytes here is not a requirement of the method. SCAT tables assembled from passive simulations with electrodes oriented at intermediate angles to fibers could be used for arbitrary orientations. Experimental validation, however, remains to be shown.


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This work was supported by National Heart, Lung, and Blood Institute Grants HL-50537, HL-67728, HL-67961, and HL-77607.


   FOOTNOTES
 

Address for reprint requests and other correspondence: A. E. Pollard, Cardiac Rhythm Management Laboratory, Univ. of Alabama at Birmingham, Volker Hall B140, 1670 Univ. Blvd., Birmingham, AL 35294 (e-mail: pollard{at}crml.uab.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


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