AJP - Heart Calcium Transients and Cell-Sarcomere
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Am J Physiol Heart Circ Physiol 287: H2402-H2411, 2004. First published July 29, 2004; doi:10.1152/ajpheart.00289.2004
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Feasibility of cardiac microimpedance measurement using multisite interstitial stimulation

Andrew E. Pollard,1 William M. Smith,1 and Roger C. Barr2

1Department of Biomedical Engineering, Cardiac Rhythm Management Laboratory, University of Alabama at Birmingham, Birmingham, Alabama 35294; and 2Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708

Submitted 23 March 2004 ; accepted in final form 21 July 2004


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
This study was designed to test the hypothesis that analyses of central interstitial potential differences recorded during multisite stimulation with a set of interstitial electrodes provide sufficient data for accurate measurement of cardiac microimpedances. On theoretical grounds, interstitial current injected and removed using electrodes in close proximity does not cross the membrane, whereas equilibration of intracellular and interstitial potentials occurs distant from electrodes widely separated. Multisite interstitial stimulation should therefore give rise to interstitial potential differences recorded centrally that depend on intracellular and interstitial microimpedances, allowing independent measurement. Simulations of multisite stimulation with fine (25 µm) and wide (400 µm) spacing in one-dimensional models that included Luo-Rudy dynamic membrane equations were performed. Constant interstitial and intracellular microimpedances were prescribed for initial analyses. Discrete myoplasmic and gap-junctional components were prescribed intracellularly in later simulations. With constant microimpedances, multisite stimulation using 29 total electrode combinations allowed interstitial and intracellular microimpedance measurements at errors of 0.30% and 0.34%, respectively, with errors of 0.05% and 0.40% achieved using 6 combinations and 10 total electrodes. With discrete myoplasmic and junctional components, comparable accuracy was maintained following adjustments to the junctions to reflect uncoupling. This allowed uncoupling to be quantified as relative increases in total junctional resistance. Our findings suggest development of microfabricated devices to implement the procedure would facilitate routine measurement as a component of cardiac electrophysiological study.

bidomain modeling; gap junctions; ventricular myocyte

CELLULAR UNCOUPLING is broadly invoked as a mechanism for the initiation and maintenance of cardiac arrhythmias. Often, uncoupling as a term is used in a qualitative sense without precise definition. While patch clamping is an experimental tool that provides detailed information regarding sarcolemmal and intracellular ion channel behavior (22), and biochemical assay protocols provide detailed information regarding gap junction protein kinetics (25), no such acceptable and sufficient strategy for measuring tissue impedances on the size scale of individual myocytes, i.e., cardiac microimpedances, is available. Traditional measurement strategies are either technically impractical or inherently flawed. Indirect measures of tissue status, such as action potential duration dispersion (4, 46), conduction velocity changes (2, 21, 13), and notches in electrograms (39, 40) provide indirect evidence that cellular uncoupling in disease states is marked. Ultimately, microimpedance measurements must be integrated with careful ion channel and gap junction kinetics modeling (33, 38, 44) for development of a detailed theoretical understanding of cardiac electrophysiological phenomena. Most impedances from intact tissue have been measured with the use of cable theory in one-dimensional preparations. Subthreshold current injection is used to induce a transmembrane potential (Vm) distribution whose electrotonic decay depends on the space constant of the tissue ({lambda}). Distant from the stimulation site (at multiple {lambda}), potentials recorded intracellularly and interstitially equilibrate. Relating the interstitial potential gradient measured centrally to the current injected and removed at the preparation ends provides a total impedance measurement that can then be separated into intracellular and interstitial components after central recordings of the respective potentials (42). With an alternate, four-electrode configuration (34), current injected and removed at two outer electrodes establishes an interstitial potential difference recorded at two inner electrodes that allows an effective impedance measurement. Application of cable theory requires Vm recording with glass microelectrodes during stimulation, which is problematic because electrode arrangement is technically challenging. Impedance measurements derived from four-electrode experiments, which do not require Vm measurement, have historically been flawed due to the comingling of intracellular and interstitial components of the tissue into the measurements (25).

In this study, we describe a microimpedance measurement strategy that requires no Vm measurement and allows separation of interstitial from myoplasmic or junctional effects. The strategy assumes access to the interstitium at multiple sites, with stimulating electrodes arranged on opposite sides of a central recording pair. To assess the feasibility of the strategy, we assembled one-dimensional models that included interstitial and intracellular components coupled via active membrane, as described by the Luo-Rudy dynamic (LRD) membrane equations (15, 16, 21, 27) for guinea pig ventricular myocytes and simulated responses to interstitial current injection and removal from different sites. For initial simulations, constant interstitial and intracellular microimpedances were prescribed to consider the suitability of the strategy under experimental conditions consistent with cable theory. In later simulations, intracellular components were divided into myoplasmic and gap-junctional contributions to quantify uncoupling, which is thought to be a process resulting from preferential changes in junctional microimpedances. Highly accurate microimpedance measurements were obtained in both sets of simulations, suggesting the development of devices for implementation of the measurement procedure would allow determination as a routine component of cardiac electrophysiological studies.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Bidomain modeling. All simulations were completed using one-dimensional models of cardiac tissue represented as a bidomain with intracellular and interstitial portions coupled to one another via cell membrane (24, 32). For a given direction (x)

(1)
where Im is the transmembrane current density, gox the specific interstitial bidomain conductivity, {varphi}o the interstitial potential, {zeta}o the stimulus current density, gix the specific intracellular bidomain conductivity, and Vm the transmembrane potential. Domain coupling was then described by

(2)
with {beta} the ratio of membrane surface to element volume, Cm the specific membrane capacitance, and Iion the total transmembrane current density resulting from ion channels, pumps, and exchangers, as described by the LRD membrane equations. To discretize Eq. 1, we used {Delta}x = 5 µm space steps such that 20 individual segments were located inside single LRD myocytes of 100 µm length as shown in the schematic of Fig. 1A. The intracellular volume (Vi) for each segment measured 3.80 x 10–6 cm2 based on LRD myocytes of 22 µm diameter.



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  		Fig. 1. A: schematic diagram of a single myocyte within the one-dimensional bidomain model. Boxes show membrane elements separating interstitial from intracellular components. Intracellularly, junctions separated myoplasmic elements at myocyte ends in simulations using discrete junctional and myoplasmic components. B: arrangement of recording ({circ}) and stimulating ({bullet}) sites in the fine (top) and wide (bottom) spacing regions. Electrodes A and D were used for stimulation. Electrodes B and C were used for central recording. Interstitial current (Io) was fixed for simulations with fine (electrodes i-ix) or wide (electrodes x-xv) spacing. Ai-Axv, current injection electrodes; Di-Dxv, current removal electrodes. C: transmembrane (top) and interstitial (bottom) potentials from the model center in one simulation. S2 shows timing for current injection and removal. The interstitial potential was referenced to that at the model end for display.

 
Constant microimpedances. For all simulations, we prescribed constant gox values of 3.17 mS/cm to reflect the specific extracellular resistivity (Ro) reported by Kléber and Rieger (17) of 63 {Omega}·cm (=15.85 mS/cm) in perfused rabbit papillary muscles. Reduction from 15.85 to 3.17 mS/cm was a consequence of scaling by an interstitial volume fraction (1-f) of 0.2 (26), consistent with a bidomain representation of tissue structure (30). For simulations with constant microimpedances, we prescribed gix of 4.82 mS/cm to all intracellular components. This value was also selected to reflect Kléber and Rieger (17), who reported an intracellular specific resistivity (Ri) of 166 {Omega}·cm (=6.02 mS/cm). Reduction from 6.02 to 4.82 mS/cm was a consequence of scaling by an intracellular volume fraction (f) of 0.8 (26). In Eq. 2, {beta} was set to 6,350/cm to reflect rabbit left ventricular epicardial myocytes (11) and Cm was set to 1 µF/cm2 to reflect nominal values for biological tissue.

Discrete myoplasmic and junctional components. For simulations in which we considered myoplasmic and gap junctional components as separate contributors to intracellular microimpedances, we used

(3)
as the basis for gix assignments, with Rmyo a specific myoplasmic resistivity, Rj as the total junctional resistance between adjacent myocytes, Send as the membrane surface area for LRD myocyte ends (3.8 x 10–6 cm2), and L as the LRD myocyte length. Here, we recognized reported estimates for the relative contributions of myoplasmic and junctional components to effective axial resistance at 10% and 90%, respectively (28). We assumed a nominal Rj of 0.4 M{Omega}, which resulted in a junctional contribution to effective Ri of 152 {Omega}·cm. We then prescribed an Rmyo of 16.9 {Omega}·cm such that effective Ri was 169 {Omega}·cm and Rmyo was 10% of effective Ri. That Ri was comparable to Kléber and Rieger (17). Our nominal Rj was below reported values of 1.7 and 5.7 M{Omega} for rat (43) and rabbit (41) ventricular cell pairs, respectively. However, a lower value for myocytes coupled in tissue is likely because the disaggregation process for establishing cell pairs available for electrophysiological study causes some gap junction internalization (35). In these simulations, gix values of 47.62 mS/cm were prescribed at myocardial components to reflect the absence of gap junctions, whereas gix values of 0.26 mS/cm and below were prescribed at junctional components to reflect the inclusion of Rj over the {Delta}x = 5 µm segments. Rj increments in successive simulations from 0.4 M{Omega} (nominal) to 4.0 M{Omega} modeled a 10-fold range.

Multisite stimulation. As shown in Fig. 1B, models were built from 50 LRD myocytes laid end to end. This resulted in 1,000 total intracellular nodes and 1,000 total interstitial nodes. This spacing was sufficiently fine to allow interstitial potential differences ({Delta}{varphi}o) to be recorded on a size scale below that of an individual myocyte. Full discretization resulted in a sparse linear system that we solved using the method of conjugate gradients as described in our earlier study (7), with the refinement that all Vm and {varphi}o terms in Eq. 1 were written at time t + {Delta}t. Time steps were fixed at {Delta}t = 2 µs to ensure stable and accurate solutions during action potential depolarization and interstitial stimulation.

Multisite stimulation was achieved by completing different simulations with adjustments in locations for interstitial current (Io) injection (at position A) and removal (at position D) with recording electrodes fixed (at positions B and C). Separations between position A and the midpoint of the recording pair and between position D and the midpoint of the recording pair were denoted p and q, respectively. Locations for positions A and D were adjusted over nine positions (positions i-ix) in a fine spacing region, limited to five total myocytes, and over six positions (positions x-xv) in a wide spacing region that extended from the boundaries of the fine spacing region toward the model ends. With fine spacing, locations for positions A and D, denoted Ai-Aix and Di-Dix, were selected at 25-µm increments from the recording electrodes. With wide spacing, locations for positions A and D, denoted Ax-Axv and Dx-Dxv, were selected at 400-µm increments. Values for p and q with Ai-Axv and Di-Dxv are summarized in Table 1. Tests in which the A and D electrodes were moved symmetrically from the recording electrodes were combined with asymmetric tests, in which the Ai electrode was fixed, whereas the D electrode was moved from positions Di to Dxv.


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  		Table 1. Stimulating electrodes

 
Stimulation protocol. For all simulations, action potential propagation was initiated by transmembrane current injection (300 µA/cm2, 2 ms duration) at four nodes on one model end. Activation sequences were therefore initiated with basic drive stimuli that just exceeded diastolic threshold. Interstitial current ({zeta}o = 5,000 µA/cm2, 10 ms duration) was injected and removed 25 ms later. Total current injected via electrode A therefore measured Io = {beta}{zeta}oVi{Delta}x(1 – f)/f = 15.1 nA. The stimulus latency ensured action potential plateau stimulation, which limited the possibility of inducing an active response. Figure 1C shows Vm (top) and {varphi}o (bottom) recorded from a central site in the model during one simulation. We used a 10-ms duration for the plateau pulse to ensure steady-state responses. {Delta}{varphi}o was then recorded as the potential difference between two central sites separated by 25 µm. Here, we recognized that such separation is practical based on examples provided by Hofer et al. (13), who recorded electrograms from superfused guinea pig papillary muscles using 15 x 15 µm microfabricated electrodes spaced 50 µm apart.

Microimpedance measurements. For comparison with {Delta}{varphi}o measured from the simulations, we used the analytic solution resulting from a core-conductor assumption

(4)
because that assumption is equivalent to the one-dimensional bidomain. In Eq. 4, {Delta}0 is the gradient as p and q approach 0, and kx is the interstitial to intracellular conductivity ratio, i.e., gox/gix. For a given set of {Delta}{varphi}o recordings, we determined {Delta}0, kx, and {lambda} by nonlinear least-squares analysis over the selected range of (p,q) values. Specifically, we used the lmdif routine from the minpack collection of software modules available on netlib (10). That routine minimizes the sum of the squares of a set of nonlinear functions by a modification of the Levenberg-Marquardt algorithm with the Jacobian calculated by a forward difference approximation. Iteration was continued until the relative error in the sum of squares or the relative errors between theoretical and measured {Delta}{varphi}o reached a tolerance of 10–10. We supplied an initial guess of {Delta}0 = {Delta}{varphi}o/s{Delta}x, using the {Delta}{varphi}o measured with the Ai-Di combination because this was the finest stimulating electrode separation tested. Here, s was the number of {Delta}x steps between the recording electrodes. For kx, the initial {Delta}o was combined with {Delta}{infty} = {Delta}{varphi}o/s{Delta}x using the {Delta}{varphi}o measured with the Axv-Dxv combination because this was the widest stimulating electrode separation tested, and

(5)
for {lambda}, we used an initial estimate of 500 µm. Once kx, {lambda}, and {Delta}o were determined from the nonlinear least-squares procedure, we measured gox from

(6)
and

(7)
An equivalent procedure using the applied Io instead of {zeta}o is to measure gox from

(8)
and gix from Eq. 7.

Addition of lead noise. The individual {Delta}{varphi}o recordings obtained in practice are likely to include noise. We therefore considered the feasibility of the microimpedance measurement procedure in the presence of noise that would result from the separation between each recording electrode and the inputs to a high-quality analog differential amplifier used to measure {Delta}{varphi}o. To represent the case with an amplification stage adjacent to the recording electrodes, random noise that ranged between 0.1 and 5.0 mV peak-to-peak was added to the {Delta}{varphi}o recorded during selected simulations with multisite stimulation. To represent the case with an amplification stage distant from the recording electrodes, random noise of the same magnitude was added to the unipolar potentials recorded at electrodes B and C before {Delta}{varphi}o determination. For each level of added noise, we determined the average {Delta}{varphi}o and the standard deviation for {Delta}{varphi}o over the final 1 ms of recording. Average {Delta}{varphi}o values were used in the nonlinear least-squares analysis. Standard deviations were used to determine signal-to-noise ratios to relate microimpedance measurement errors to added lead noise.

Gap-junctional uncoupling. In simulations that included discrete myoplasmic and junctional components, we quantified the magnitude of uncoupling as a relative increase from the nominal total junctional resistance. Assuming 90% of the effective axial resistance as junctional and the intracellular volume fraction of 0.8, nominal Rj (Rj,nom) was expressed in terms of nominal gix (gix,nom)

(9)
Then, under conditions where the myoplasmic contribution remains unchanged as gap junctions uncouple

(10)
Uncoupling was then quantified using f of 0.8 from

(11)
which had the practical advantage that values for L and Send were not necessary for the measurement, provided they changed little as gap-junctional contributions increased.


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 GRANTS
 REFERENCES
 
Stimulation with fine and wide spacing. As expected, stimulation with electrodes located in the fine and wide spacing regions induced different responses. With fine spacing, relatively small {varphi}o changes induced small Vm changes near the stimulating electrodes, but a relatively large {Delta}{varphi}o between the recording electrodes, consistent with the concept that the injected current remained interstitial. With wide spacing, relatively large {varphi}o changes induced large Vm near the stimulating electrodes but a smaller {Delta}{varphi}o between the recording electrodes, consistent with redistribution of injected current between the intracellular and interstitial domains. Figure 2A shows {varphi}o responses to stimulation with the Ai-Di electrode combination and the Axv-Dxv electrode combination over the extent of the full model (Fig. 2A, left) and over the central five myocytes in the model (Fig. 2A, right). With the Ai-Di combination, the resulting {varphi}o changes were limited to tissue near the stimulating electrodes, whereas with the Axv-Dxv combination, much larger {varphi}o magnitudes were observed. Figure 2B shows Vm responses from these same simulations. Stimulation with Axv-Dxv caused pronounced membrane depolarization (to {approx}250 mV) and hyperpolarization (to {approx}–150 mV) near the model ends. Vm changes near the recording electrodes were considerably smaller. Stimulation with Ai-Di induced more modest depolarization (to {approx}55 mV) and hyperpolarization (to {approx}20 mV). Vm changes were confined to the region near the recording electrodes. Figure 2C, left, shows {Delta}{varphi}o responses during stimulation with Ai-Di and Axv-Dxv combinations. Stimulation for 10 ms provided sufficient time for development of steady-state responses with both combinations, as evidenced by the constant {Delta}{varphi}o values at the ends of the pulses. Figure 2C, right, shows {Delta}{varphi}o over all 29 stimulating electrode combinations. Asymmetric stimulation with the A electrode fixed and the D electrode moved (Ai-Di to Ai-Dxv) induced smaller {Delta}{varphi}o values than symmetric stimulation with the A and D electrodes both moved (Ai-Di to Axv-Dxv). From the 29 total combinations, 15 {Delta}{varphi}o values were collected during symmetric stimulation and 14 {Delta}{varphi}o values were collected during asymmetric stimulation.



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  		Fig. 2. A: interstitial potential distributions in response to stimulation with fine (thick line, Ai-Di) and wide (thin line, Axv-Dxv) spacing using constant interstitial and intracellular microimpedances. Left, potentials in the full model. Right, potentials over five central myocytes. B: transmembrane potential distributions. C, left: interstitial potential differences recorded centrally during stimulation with fine (thick line, Ai-Di) and wide (thin line, Axv-Dxv) spacing. Right, interstitial potential differences at the end of interstitial stimulation pulses with symmetric (Ai-Di to Axv-Dxv) and asymmetric (Ai-Di to Ai-Dxv) arrangement of the stimulating electrodes. For symmetric arrangements, the horizontal axis denotes the positions for current injection (p) and removal (q) electrodes. For asymmetric arrangements, the horizontal axis denotes the positions for q only because p was fixed at the Ai location.

 
Microimpedance measurement with multisite stimulation. To assess the measurement procedure under conditions that we recognized as the most likely to succeed, our initial analysis focused on multisite stimulation with constant gox and gix, i.e., no gap-junctional contributions. With the use of all 29 stimulating electrode combinations, we measured gox = 3.16 mS/cm and gix = 4.80 mS/cm. These initial measurements differed from the prescribed values of gox = 3.17 mS/cm and gix = 4.82 mS/cm by 0.30% and 0.34%, respectively. For comparison, use of {Delta}{varphi}o measured with Ai-Di stimulation alone instead of {Delta}0 resulted in an equivalent gox of 3.33 mS/cm. This gox differed from the prescribed gox by 5.21%. Multisite stimulation therefore facilitated a refined measurement relative to that available using a standard four-electrode technique with fine spacing because the additional combinations improved {Delta}o determination. That refinement, in combination with accurate kx determination, led to a gix measurement of comparable quality to the gox measurement.

Limiting stimulating electrode combinations. All 29 stimulating electrode combinations were not required to measure gox and gix values within 1% of prescribed gox and gix. A mix of combinations from the fine and wide spacing regions was needed, however. To assess influences of combinations from the fine and wide spacing regions on measurement errors, we repeated the nonlinear least-squares analyses using different combinations for 4 of the 15 {Delta}{varphi}o values recorded during symmetric stimulation in different analyses. Here, four {Delta}{varphi}o values were required for algorithm convergence. The importance of combinations from the fine spacing region is highlighted in Fig. 3A, which shows percent errors using {Delta}{varphi}o values from combinations fixed near the recording pair (Ai-Di), near the fine-wide spacing region boundary (Aix-Dix), and near the model ends (Axv-Dxv) in analyses with each remaining {Delta}{varphi}o during symmetric stimulation used as the fourth value for the nonlinear least squares algorithm. The horizontal axis of Fig. 3A denotes p = q for this fourth value. Errors in gox and gix when the fourth value was taken from the fine spacing region were all <2%. Larger errors were observed using {Delta}{varphi}o values from the wide spacing region, suggesting emphasis be placed on combinations within the fine spacing region. The importance of including at least one combination from the wide spacing region is highlighted in Fig. 3B, which shows percent errors using {Delta}{varphi}o values from combinations fixed at Ai-Di, Aii-Dii, and Aiv-Div in analyses with the fourth value from the other available symmetric stimulating electrode combinations. Errors in gix <10% were only observed with five combinations tested. All five combinations were in the wide spacing region, and all were located toward the model's ends, as opposed to the boundary with the fine spacing region.



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  		Fig. 3. A: percent errors in interstitial ({bullet}) and intracellular ({circ}) microimpedances from prescribed values in analyses with four stimulating electrode combinations. The Ai-Di, Aix-Dix, and Axv-Dxv combinations were used for each analysis. The horizontal axis shows the (P = q) location for the fourth combination. The hatched gray area marked F shows the fine spacing region. The unfilled area marked W shows the wide spacing region. Selected stimulating electrode combinations are marked. B: percent errors with the Ai-Di, Aii-Dii, and Aiv-Div combinations used for all analyses. The horizontal axis shows the (P = q) location for the fourth combination.

 
Errors in gox and gix using {Delta}{varphi}o values measured with Ai-Di, Aii-Dii, Aiv-Div, and Axv-Dxv stimulation were only 0.03% and 0.42%, respectively. Recognizing this high accuracy was achieved with 10 total electrodes, including 2 for recording and 8 for stimulation (4 per side), we limited subsequent analyses to symmetric and asymmetric combinations with this electrode configuration. For reference, including the asymmetric combinations of Ai-Dii and Ai-Div such that six total combinations were used for microimpedance measurements resulted in comparable errors in gox and gix at 0.05% and 0.40%, respectively.

Microimpedance measurements with lead noise. The addition of lead noise had relatively little influence on microimpedance measurement accuracy under conditions representing amplification close to the recording electrodes themselves. Figure 4A shows {Delta}{varphi}o with the Ai-Di and Axv-Dxv stimulating electrode combinations after 0.2 mV (Fig. 4A, left) and 2.0 mV (Fig. 4A, right) random noise was added to the recordings. The signal-to-noise ratio with 0.2 mV added noise, determined as the ratio of the standard deviation to the average {Delta}{varphi}o with the Axv-Dxv stimulating electrode combination, measured 167. The signal-to-noise ratio with 2.0 mV added noise measured 8.9. Figure 4B shows percent errors for gox and gix with signal-to-noise ratio at different levels of added noise. All errors were <1%. Under conditions representing amplification more distant from the recording electrodes, the microimpedance errors were larger. Figure 4C shows {Delta}{varphi}o recordings with 0.2 mV (left) and 2.0 mV (right) random noise added to the unipolar potentials from electrodes B and C before {Delta}{varphi}o determination. Signal-to-noise ratios under these conditions measured 122 and 6.0, respectively. Figure 4D shows the percent errors for gox and gix with signal-to-noise ratio. Microimpedance errors <1% were only observed with signal-to-noise ratios >20.



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  		Fig. 4. A: interstitial potential differences recorded centrally during stimulation with fine (Ai-Di) and wide (Axv-Dxv) spacing and random noise added to the potential difference recordings. Numbers above the recordings indicate signal-to-noise ratios with different levels of peak noise. B: percent errors in interstitial ({bullet}) and intracellular ({circ}) microimpedances from prescribed values in analyses with the stimulating electrode combinations Ai-Di, Aii-Dii, Ai-Dii, Aiv-Div, Ai-Div, and Axv-Dxv. Errors are displayed as functions of signal-to-noise ratio. C: interstitial potential differences recorded centrally during stimulation with fine (Ai-Di) and wide (Axv-Dxv) spacing after lead noise was added to the unipolar potentials and (D) microimpedance errors displayed as functions of signal-to-noise ratio from these recordings.

 
Discrete myoplasmic and junctional components. Highly accurate microimpedance measurements were also obtained in simulations in which myoplasmic and gap-junctional components were treated as discrete entities. That accuracy was achieved over the 10-fold range for total junctional resistance between adjacent myocytes that we examined. The measurement procedure was therefore effective under conditions where action potential propagation appeared continuous, and as that propagation became discontinuous. Figure 5A, left, shows action potentials recorded at positions p = 47.5 µm and p = 52.5 µm, which were located on opposite sides of a junctional component with an Rj of 0.4 M{Omega}, in advance of interstitial stimulation. Figure 5A, right, shows Vm responses during Ai-Di and Axv-Dxv stimulation. The waveforms suggested that action potential propagation was effectively continuous because smooth transitions from rest into the upstroke and from the peak into the plateau phase were observed. The conduction delay across the junctional component was negligible. Similarly, the responses to Ai-Di and Axv-Dxv stimulation were comparable to those observed in simulations with constant microimpedances prescribed between all nodes in the model, i.e., Fig. 2B. Figure 5B shows action potentials (left) recorded from the same sites alongside Vm responses to interstitial stimulation (right) with an Rj of 4.0 M{Omega}. The higher Rj established discontinuous action potential propagation, as transitions from rest into the upstroke were considerably longer than with the lower Rj, and pronounced hyperpolarization in the transition from the peak to the plateau phase was present. The conduction delay across the junctional component was pronounced. In addition, the junctional components altered the responses to interstitial stimulation by establishing "sawtooth" patterns, in which nodes adjacent to the junctional components were alternately hyperpolarized and depolarized. Despite this dramatic influence of the junctional components as Rj increased, errors in effective microimpedance measurements were comparable to those obtained with constant microimpedances. Figure 5C shows the percent errors in gox (solid circles) and gix (open circles) using {Delta}{varphi}o values from the six stimulating electrode combinations identified previously. Errors in both impedances were <3% at all Rj values examined. All gox errors were <0.5%. With accurate effective gix measurements obtained as gap-junctional uncoupling developed, it was also possible to quantify uncoupling in terms of relative increases in Rj with high accuracy. Figure 5D shows percent errors in Rj/Rj,nom obtained from Eq. 11 for the full set of simulations in which discrete myoplasmic and gap junctional components were included in the models. All measured ratios were within 0.5% of the prescribed ratios.



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  		Fig. 5. A: action potential upstrokes (left) and spatial distributions of transmembrane potential after stimulation (right) in a model with junctional resistance components at Rj = 0.4 M{Omega}. B: action potentials (left) and spatial distributions of transmembrane potential (right) in a model with junctional components at Rj = 4.0 M{Omega}. C: percent errors in interstitial ({bullet}) and effective intracellular ({circ}) microimpedances after Rj adjustments shown on the horizontal axis. D: percent errors in nominal Rj (Rj,nom) as determined from changes in effective intracellular microimpedances after Rj adjustments shown on the horizontal axis.

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
GRANTS
 REFERENCES
 
The present study demonstrates that, under conditions analogous to Weidmann (42) and Clerc (8), highly accurate cardiac microimpedance measurements for a given direction can be obtained from analyses of interstitial potential gradients in response to multisite interstitial stimulation. Although we used one-dimensional LRD simulations, we note that Weidmann (42) and Clerc (8) each assumed a core-conductor representation in which injected current divided between intracellular and interstitial compartments and implemented experimental steps to support that assumption, consistent with the theoretical basis in our analyses. A noteworthy aspect of the approach we present is the stable and independent measurement of gox and of gix. Because no intracellular access is required, Vm recording is unnecessary. This feature is significant in that it will facilitate device development to implement the measurement procedure for use in a much wider range of preparations and situations than has, to date, been practical.

Fine electrode spacing. Accurate microimpedance measurements were a consequence, primarily, of our ability to model electrode arrangements with known positions and fine and wide spacing. Comparable dimensions for electrode arrays have recently been described. For example, Witkowski et al. (45) assembled silver-silver chloride arrays with 25-µm wires at center-to-center separations of 65 µm and found minimal local distortion of {varphi}o associated with the electrode arrangement. An approach involving microfabrication also seems plausible, as the precision and consistency in silicon-based arrays used for neural electrophysiological studies (14, 47) are likely better than that available for hand-assembled arrays. In this regard, we note that Hofer et al. (13) demonstrated high-quality electrograms recorded from superfused guinea pig papillary muscles using fabricated 15 x 15 µm electrodes spaced 50 µm apart. Collectively, these studies suggest an arrangement analogous to the one we modeled is technically feasible and potentially straightforward to achieve.

Interstitial microimpedance. Highly accurate gox measurements were obtained here because fine spacing allowed precise four-electrode impedance measurements. This observation is consistent with theoretical descriptions by Plonsey and Barr (23), who analyzed interstitial potential distributions resulting from current injection and removal in an idealized anisotropic bidomain with passive components and equal anisotropic conductivity ratios in all directions. Those investigators showed development of intracellular current depended critically on stimulating electrode separation. With separation at 0.2 {lambda}, no intracellular current was established because the spatial distribution of transmembrane potential was equal and opposite to the distribution of interstitial potential. A relatively modest intracellular current developed with electrode separation to 2.0 {lambda}, consistent with observations from cable theory that multiple {lambda} are necessary for equilibration of interstitial and intracellular potentials. In the present study, we made no attempt to determine intrinsic {lambda} because we recognized membrane resistance in the action potential plateau differs from resting membrane resistance (48). Membrane resistance measurement would therefore have required detailed analyses of instantaneous current-voltage curves in companion simulations with isolated LRD myocytes (48), complicated by the dependence of that membrane resistance on Vm at different points in the model. However, the {lambda} value recovered with the nonlinear least-squares analysis in the simulations with constant micro-impedances using all stimulating electrode combinations was 449 µm. Therefore, stimulating electrode combinations Ai-Di, Aii-Dii, and Aiv-Div were all within this {lambda}. Refinement associated with mutiple combinations was likely responsible for our observation that the gox measurement obtained with multisite stimulation (error of 0.30%) was superior to the four-electrode gox measurement (error of 5.21%) obtained using only the Ai-Di combination. More broadly, we note that little or no information is available regarding gox differences between species and regions of the heart, changes with development from neonate to adult and senescent preparations, or responses to diseases and pharmacological interventions. Routine determination using multisite stimulation could provide such detail, which would serve to elucidate contributions of interstitial remodeling to cardiac electrical activity that are presently ignored.

Intracellular microimpedance. Our finding of highly accurate gix measurements suggests opportunities to understand intracellular microimpedance changes alongside interstitial remodeling using an appropriate device to implement the measurement procedure. Such quantification would provide detail regarding disease that, to date, has only been available in the experimental ischemia literature (4). With the use of continuously perfused rabbit papillary muscle preparations (18), in which the muscles are considered one-dimensional structures, measurements of "passive cable-like properties" under tightly controlled conditions that involve the use of each preparation as its own control are possible. Required measurements are limited to a single microelectrode impalement, a surface recording from a separate electrode and a reference recording from the tissue bath. By following Buchanan et al. (2), voltage ratios measured with assumptions of planar wavefront expansion and negligible intracellular or interstitial potential gradients within resting and fully depolarized regions of the muscle are straightforward to convert into longitudinal intracellular and extracellular core conductor resistances. These assumptions obviate the need to record multiple Vm responses and have allowed documentation of impedance measurements with ischemic time course in reponse to hypoxia (29), hypocalcemia (5), and reperfusion (6). Uncoupling in this context is important, as development of phase 1b arrhythmias is highly correlated with an interval of partial uncoupling as reflected by increased tissue impedance from control (37).

Spatial frequencies. Our finding that stimulating electrode combinations from both the fine and wide spacing regions are necessary for accurate microimpedance measurements is consistent with predictions by Roth et al. (31). Those investigators studied relationships between the excitation or stimulation of a single muscle fiber embedded in a bundle of fibers and the resulting electromyogram. They solved the volume conductor problem using functions of the spatial frequency as opposed to functions of axial position because that solution facilitated electromyogram potential determination by allowing straightforward filtering via convolution of transmembrane potentials. A key assumption on which their method was based was that microscopically inhomogeneous tissue could be replaced by a macroscopically homogeneous medium whose electrical properties were described by effective electrical conductivities. They derived an effective axial conductivity that depended on interstitial (or extracellular) conductivity, intracellular conductivity, intracellular volume fraction, the space constant, and the separation between current passing electrodes for stimulation or the spatial extent of the depolarization wave front for action potential propagation. They also derived an effective radial conductivity that depended on interstitial (or extracellular) conductivity, intracellular volume fraction, fiber radius, and membrane impedance. Successful derivation of expressions for the axial and radial tissue conductivities allowed those conductivities to be incorporated into volume conductor calculations, which in turn allowed demonstration of the effects of frequency-dependent conductivities on the measured interstitial (or extracellular) potentials. By using model parameters appropriate for skeletal muscle, they showed that the magnitude of the effective axial conductivity included two plateaus. One plateau was associated with spatial frequencies much less than the space constant. This condition is analogous to our use of stimulating electrode combinations in the fine spacing region. The other plateau was associated with spatial frequencies much greater than the space constant. This condition is analogous to our use of stimulating electrode combinations in the wide spacing region. The transition between the plateaus was associated with spatial frequencies close to the space constant. Here, we note that stimulating electrode combinations located close to the boundary between the fine and wide spacing region contributed little in measurements here. Collectively, these findings suggest that information regarding the spatial frequencies of the tissue could be used effectively to select stimulating electrode combinations that will and will not contribute to accurate microimpedance measurements.

Gap-junctional uncoupling. We believe the ability of our method to quantify gap-junctional uncoupling is potentially superior to the approach used in experimental ischemia because of its ability to focus attention on the junctions themselves. The junction is recognized as the dominant contributor in the longitudinal pathway, with estimates for the myoplasmic component at 10% or less of the total (28). Support for this assumption is provided by total junctional resistance measurements in disaggregated cell pairs using simultaneous suction pipette attachments in current-clamp mode (41, 43) and optical mapping in response to field stimulation (36). Also, increased intracellular resistance in response to cytosolic calcium accumulation, intracellular acidosis and second messengers (28) correlates with gap junction kinetic responses in cell cultures (35). In this regard, we note that quantifying uncoupling in terms of junctional changes provides an alternative to approaches based on alternating current (AC) impedance. Cooklin et al. (9) stimulated guinea pig ventricular myocyte suspensions at multiple frequencies to measure intracellular impedance in the absence of junctional contributions and made comparisons with measurements obtained from in vitro trabeculae to distinguish between junctional and myoplasmic contributions. Le Guyader et al. (19) completed theoretical analyses of the ability to measure anisotropic bidomain conductivities using AC stimulation to shorten {lambda} and found that inclusion of junctional resistance in parallel with junctional capacitance in their structural representation facilitated impedance measurements. While these approaches allow determination of junctional contributions, the ability to complete such measurements in an environment where artifacts associated with recording during AC stimulation are minimized (28) has practical advantages.

Limitations. In assessing our findings, it is important to recognize certain limitations. First, we assumed stimulating and recording electrodes filled each 5-µm segment, where interstitial current was injected and removed. Electrode sizes were therefore smaller relative to center-to-center spacing than sizes used by Witkowski et al. (45). While this would likely minimize the influence of individual electrodes on the microimpedance measurements by ensuring independence of interstitial potential recordings, the noise resulting from central recording with such small surface areas is potentially problematic. As shown here, lead noise in electrodes B and C that resulted in {Delta}{varphi}o recordings with low signal-to-noise ratios caused errors in the microimpedance measurements. Second, we used relatively low magnitude interstitial stimuli, which minimized Vm changes near the recording pair with both fine and wide spacing. Whereas this provided a practical advantage in that it likely induced small changes in membrane resistance and therefore {lambda} near the recording pair, larger stimuli may be required under experimental conditions to achieve sufficient signal-to-noise ratios for application of the nonlinear least-squares analysis. Third, we tested only the measurement procedure in one-dimensional models. As a consequence, we were able to compare the effective microimpedances measured on the axis through which the stimulating and recording electrodes were aligned to prescribed microimpedances whose values were based on earlier experimental reports. In practice, that comparison will likely be more difficult because the effective microimpedances will include contributions from the anisotropic cellular architecture of the myocardium, the extracellular milieu separating the stimulating and recording electrode platform from tissue during surface measurements, and heterogeneity in cleft size, cell size, and gap junction density. The extent to which these issues limit measurement accuracy merits further study. Fourth, the measurement procedure requires knowledge of underlying cellular parameters. In particular, Eq. 8 assumes knowledge of cell size (Vi) and the ratio of cell size to total cell and cleft size (fi) to allow interstitial microimpedance measurement. Species-, disease-, and tissue-dependent values for cell size are available in the single cell literature. Information regarding the ratio of cell size to total cell and cleft size is more limited. Inaccurate specification of either parameter will contribute to inaccuracy of microimpedance measurements.

Implications for microfabrication. We believe that microfabrication will be a necessary component to accomplish measurements in real myocardial tissue and envision platforms with one- and two-dimensional arrays for implementation. While that implementation will likely be complicated by the four main limitations noted above, the most fundamental issue is precision in manufacturing and calibrating small electrode systems. The demands for these systems are likely to be more stringent than demands for traditional electrode arrays because small electrodes that remain in contact with tissue will be required. The present technology suggests operational strategies to address and overcome these limitations. Experimental validation 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 University 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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  1. Brugada J, Mont L, Boersma L, Kirchhof C, and Allessie MA. Differential effects of heptanol potassium and tetrodotoxin on reentrant ventricular tachycardia around a fixed obstacle in anisotropic myocardium. Circulation 84: 1307–1318, 1991.[Abstract/Free Full Text]
  2. Buchanan JW, Oshita S, Fujino T, and Gettes LS. A method for measurement of internal longitudinal resistance in papillary muscle. Am J Physiol Heart Circ Physiol 251: H210–H217, 1986.[Abstract/Free Full Text]
  3. Burton FL and Cobbe SM. Dispersion of ventricular repolarization and refractory period. Cardiovasc Res 50: 10–23, 2001.[Free Full Text]
  4. Cascio WE, Johnson TA, and Gettes LS. Electrophysiologic changes in ischemic ventricular myocardium. I. Influence of ionic, metabolic, and energetic changes. J Cardiovasc Electrophysiol 6: 1039–1062, 1995.[Web of Science][Medline]
  5. Cascio WE, Yan GX, and Kléber AG. Passive electrical properties, mechanical activity and extracellular potassium in arterially perfused and ischemic rabbit ventricular muscle. Effects of calcium entry blockade or hypocalcemia. Circ Res 66: 1461–1473, 1990.[Abstract/Free Full Text]
  6. Cascio WE, Yang H, Johnson TA, Muller-Borer BJ, and Lemasters JJ. Electrical properties and conduction in reperfused papillary muscle. Circ Res 89: 807–814, 2001.[Abstract/Free Full Text]
  7. Cates AW and Pollard AE. A model study of intramural dispersion of action potential duration in the canine pulmonary conus. Ann Biomed Eng 26: 567–576, 1998.[CrossRef][Web of Science][Medline]
  8. Clerc L. Directional differences of impulse spread in trabecular muscle from mammalian heart. J Physiol 255: 335–346, 1976.[Abstract/Free Full Text]
  9. Cooklin M, Wallis WRJ, Sheridan DJ, and Fry CH. Changes in cell-to-cell electrical coupling associated with left ventricular hypertrophy. Circ Res 80: 765–771, 1997.[Abstract/Free Full Text]
  10. Garbow BS, Hillstrom KE, and More JJ. Minpack Project. Argonne, IL: Argonne National Laboratory, 1996.
  11. Giles WR and Imaizumi Y. Comparison of potassium currents in rabbit atrial and ventricular cells. J Physiol 405: 123–145, 1988.[Abstract/Free Full Text]
  12. Hayashi H, Wang C, Miyauchi Y, Omichi C, Pak HN, Zhou S, Ohara T, Mandel WJ, Lin SF, Fishbein MC, Chen PS, and Karagueuzian HS. Aging-related increase to inducible atrial fibrillation in the rat model. J Cardiovasc Electrophysiol 13: 801–808, 2002.[CrossRef][Web of Science][Medline]
  13. Hofer E, Urban G, Spach MS, Schafferhofer I, Mohr G, and Platzer D. Measuring activation patterns of the heart at a microscopic size scale with thin-film sensors. Am J Physiol Heart Circ Physiol 266: H2136–H2145, 1994.[Abstract/Free Full Text]
  14. Hoogerwerf AC and Wise KD. A three-dimensional microelectrode array for chronic neural recording. IEEE Trans Biomed Eng 41: 1136–1146, 1994.[CrossRef][Web of Science][Medline]
  15. Hund TJ, Kucera JP, Otani NF, and Rudy Y. Ionic charge conservation and long-term steady state in the Luo-Rudy dynamic cell model. Biophys J 81: 3324–3331, 2001.[Web of Science][Medline]
  16. Hund TJ and Rudy Y. Determinants of excitability in cardiac myocytes: mechanistic investigation of memory effect. Biophys J 79: 3095–3104, 2000.[Web of Science][Medline]
  17. Kléber AG and Rieger CB. Electrical constants of arterially perfused rabbit papillary muscle. J Physiol 385: 307–324, 1987.[Abstract/Free Full Text]
  18. Kléber AG, Rieger CB, and Janse MJ. Electrical uncoupling and increase of extracellular resistance after induction of ischemia in isolated arterially perfused rabbit papillary muscle. Circ Res 61: 271–279, 1987.[Abstract/Free Full Text]
  19. Le Guyader P, Trelles F, and Savard P. Extracellular measurement of anisotropic bidomain myocardial conductivities. I. Theoretical analysis. Ann Biomed Eng 29: 862–877, 2001.[CrossRef][Web of Science][Medline]
  20. Lubbe WF, Podzuweit T, and Opie LH. Potential arrhythmogenic role of cyclic adenosine monophosphate (AMP) and cytosolic calcium overload: implications for prophylactic effects of beta-blockers in myocardial infarction and proarrhythmic effects of phosphodiesterase inhibitors. J Am Coll Cardiol 19: 1622–1633, 1992.[Abstract]
  21. Luo CH and Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 74: 1071–1096, 1994.[Abstract/Free Full Text]
  22. Marban E. Molecular approaches to arrhythmogenesis. In: Molecular Basis of Cardiovascular Disease. Philadelphia, PA: Saunders, 1999, chapt. 14, p. 313–328.
  23. Plonsey R and Barr RC. The four-electrode resistivity technique as applied to cardiac muscle. IEEE Trans Biomed Eng 7: 983–986, 1982.
  24. Plonsey R and Barr RC. Interstitial potentials and their change with depth into cardiac tissue. Biophys J 51: 547–555, 1987.[Web of Science][Medline]
  25. Plonsey RC and Barr RC. A critique of impedance measurements in cardiac tissue. Ann Biomed Eng 14: 307–322, 1986.[CrossRef][Web of Science][Medline]
  26. Polimeni PI, Williams S, and Weisman H. Application of automatic electrotonic image analyzer to the measurement of myocardial extracellular space (Abstract). Comput Biomed Res 16: 522, 1983.[CrossRef][Web of Science][Medline]
  27. Pollard AE, Cascio WE, Fast VG, and Knisley SB. Modulation of triggered activity by uncoupling in the ischemic border: a model study with phase 1b-like conditions. Cardiovasc Res 56: 381–392, 2002.[Abstract/Free Full Text]
  28. Pressler ML. Passive electrical properties of cardiac tissue. In: Cardiac Electrophysiology. From Cell to Bedside, edited by Zipes DP and Jalife J. Philadelphia, PA: Saunders, 1990, chapt. 13, p. 108–122.
  29. Riegger CB, Alperovich G, and Kleber AG. Effect of oxygen withdrawal on active and passive electrical properties of arterially perfused rabbit ventricular muscle. Circ Res 64: 532–541, 1989.[Abstract/Free Full Text]
  30. Roth BJ. Action potential propagation in a thick strand of cardiac muscle. Circ Res 68: 162–173, 1991.[Abstract/Free Full Text]
  31. Roth BJ, Gielen FLH, and Wikswo JP. Spatial and temporal frequency-dependent conductivities in volume-conductor calculations for skeletal muscle. Math Biosci 88: 159–189, 1988.[CrossRef]
  32. Roth BJ and Wikswo JP. A bidomain model for the extracellular potential and magnetic field of cardiac tissue. IEEE Trans Biomed Eng 33: 467–469, 1986.[Web of Science][Medline]
  33. Rudy Y. Cardiac conduction: an interplay between membrane and gap junction. J Electrocardiol 31: 1–5, 1998.[Web of Science][Medline]
  34. Rush S, Abildskov JA, and McFee R. Resistivity of body tissues. Circ Res 12: 40–50, 1963.[Abstract/Free Full Text]
  35. Saffitz JE, Laing JG, and Yamada KA. Connexin expression and turnover. Implications for cardiac excitability. Circ Res 86: 723–728, 2000.[Abstract/Free Full Text]
  36. Sharma V and Tung L. Theoretical and experimental study of sawtooth effect in isolated cardiac cell-pairs. J Cardiovasc Electrophysiol 12: 1164–1173, 2001.[CrossRef][Web of Science][Medline]
  37. Smith WT, Fleet WF, Johnson TA, Engle CL, and Cascio WE. The Ib phase of ventricular arrhythmias in ischemic in situ porcine heart is related to changes in cell-to-cell electrical coupling. Circulation 92: 3051–3060, 1995.[Abstract/Free Full Text]
  38. Spach MS. Mechanisms of the dynamics of reentry in a fibrillating myocardium. Developing a genes-to-rotors paradigm. Circ Res 88: 753–755, 2001.[Free Full Text]
  39. Spach MS, Miller WT, Miller-Jones E, Warren RB, and Barr RC. Extracellular potentials related to intracellular action potentials during impulse conduction in anisotropic canine cardia. Circ Res 45: 188–204, 1979.[Abstract/Free Full Text]
  40. Spear JF, Balke CW, Lesh MD, Kadish AH, Levine JL, and Moore EN. Effect of cellular uncoupling by heptanol on conduction in infarcted myocardium. Circ Res 66: 202–217, 1990.[Abstract/Free Full Text]
  41. Verheule S, Kempen MJA, Welscher P, Kwak BR, and Jongsma HJ. Characterization of gap junction channels in adult rabbit atrial and ventricular myocardium. Circ Res 80: 673–681, 1997.[Abstract/Free Full Text]
  42. Weidmann S. Electrical constants of trabecular muscle from mammalian heart. J Physiol 210: 1041–1054, 1970.[Abstract/Free Full Text]
  43. Weingart R. Electrical properties of the nexal membrane studied in rat ventricular cell pairs. J Physiol 370: 267–284, 1986.[Abstract/Free Full Text]
  44. Winslow RL, Scollan DF, Holmes A, Yung CK, Zhang J, and Jafri MS. Electrophysiological modeling of cardiac ventricular function: from cell to organ. Ann Rev Biomed Eng 2: 119–155, 2000.[CrossRef][Web of Science][Medline]
  45. Witkowski FX, Kavanagh KM, Penkoske PA, and Plonsey R. In vivo estimation of cardiac transmembrane current. Circ Res 72: 424–439, 1993.[Abstract/Free Full Text]
  46. Yan GX, Lankipalli RS, Burke JF, Musco S, and Kowey PR. Ventricular repolarization components on the electrocardiogram: cellular basis and clinical significance. J Am Coll Cardiol 42: 401–409, 2003.[Abstract/Free Full Text]
  47. Yoon TH, Hwang EJ, Shin DY, Park SI, Oh SJ, Jung SC, Shin HC, and Kim SJ. A micromachined silicon depth probe for multichannel neural recording. IEEE Trans Biomed Eng 47: 1082–1087, 2000.[CrossRef][Web of Science][Medline]
  48. Zaniboni M, Pollard AE, Yang L, and Spitzer KW. Beat-to-beat repolarization variability and its suppression by electrical coupling. Am J Physiol Heart Circ Physiol 278: H677–H687, 2000.[Abstract/Free Full Text]



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