## Abstract

The conductance catheter technique could be improved by determining instantaneous parallel conductance (G_{P}), which is known to be time varying, and by including a time-varying calibration factor in Baan's equation [α(*t*)]. We have recently proposed solutions to the problems of both time-varying G_{P} and time-varying α, which we term “admittance” and “Wei's equation,” respectively. We validate both our solutions in mice, compared with the currently accepted methods of hypertonic saline (HS) to determine G_{P} and Baan's equation calibrated with both stroke volume (SV) and cuvette. We performed simultaneous echocardiography in closed-chest mice (*n* = 8) as a reference for left ventricular (LV) volume and demonstrate that an off-center position for the miniaturized pressure-volume (PV) catheter in the LV generates end-systolic and diastolic volumes calculated by admittance with less error (*P* < 0.03) (−2.49 ± 15.33 μl error) compared with those same parameters calculated by SV calibrated conductance (35.89 ± 73.22 μl error) and by cuvette calibrated conductance (−7.53 ± 16.23 μl ES and −29.10 ± 31.53 μl ED error). To utilize the admittance approach, myocardial permittivity (ε_{m}) and conductivity (σ_{m}) were calculated in additional mice (*n* = 7), and those results are used in this calculation. In aortic banded mice (*n* = 6), increased myocardial permittivity was measured (11,844 ± 2,700 control, 21,267 ± 8,005 banded, *P* < 0.05), demonstrating that muscle properties vary with disease state. Volume error calculated with respect to echo did not significantly change in aortic banded mice (6.74 ± 13.06 μl, *P* = not significant). Increased inotropy in response to intravenous dobutamine was detected with greater sensitivity with the admittance technique compared with traditional conductance [4.9 ± 1.4 to 12.5 ± 6.6 mmHg/μl Wei's equation (*P* < 0.05), 3.3 ± 1.2 to 8.8 ± 5.1 mmHg/μl using Baan's equation (*P* = not significant)]. New theory and method for instantaneous G_{P} removal, as well as application of Wei's equation, are presented and validated in vivo in mice. We conclude that, for closed-chest mice, admittance (dynamic G_{P}) and Wei's equation (dynamic α) provide more accurate volumes than traditional conductance, are more sensitive to inotropic changes, eliminate the need for hypertonic saline, and can be accurately extended to aortic banded mice.

- conductance catheter
- parallel conductance
- mouse
- left ventricular
- admittance
- α
- pressure volume relationship

the conductance technique was first proposed by Baan 28 years ago to provide an invasive instantaneous left ventricular (LV) volume signal to generate LV pressure-volume (PV) relations and hemodynamic indexes from the PV plane (1). The original theory as proposed relates conductance to volume through a simple equation based on stroke volume (SV), resistivity of blood, and the length between the voltage contacts of the catheter:
_{P} is the muscle conductance in parallel (S) and assumed to be constant.

There are two major criticisms of this method. The first is that α, the calibration factor in Baan's equation, is not a constant but instead changes dynamically as the moving heart wall changes the shape of the applied electric field (29). The second is that, as the LV contracts and fills, the cardiac muscle moves closer to and farther away from the catheter, so subtracting a constant as in *Eq. 1* probably does not adequately compensate for the parallel muscle conductance (29). The latter problem is exacerbated by the aortic valve being off center relative to the midline of the LV chamber in mammalian hearts. This forces the miniaturized PV catheter to be closer to the septum than the midline (Fig. 1*A*) (23). Additionally, preload reduction used to generate load-independent measures of contractility is anticipated to further exacerbate any error from the currently practiced constant measure of parallel conductance. Thus there is a need for a dynamic measurement technique to quantify both G_{P} and α (29).

The conductance technique has historically been used in large animals and humans. More recently, in 1998, the technique was extended to the murine heart to characterize gene-altered mouse models (11, 23). Due to the small size of the mouse heart, newer α calibration methods have been developed. For example, cuvettes of known volume are used to determine a linear conductance-to-volume relationship that is often used as the primary calibration method (3, 5, 13, 22, 32). Additionally, α can be calculated as a constant ratio of conductance SV to a known standard for SV (usually an aortic flow probe or echo) (4, 9, 15, 16, 19, 25, 27). Most of these publications also use the hypertonic saline method to determine the parallel conductance value (G_{P}) as a constant. However, the historical sources of error outlined above still exist in these studies. There is a need for newer solutions to solve these two long-standing sources of error.

We have recently proposed solutions to both dynamically changing parallel conductance G_{P} (29) and dynamically changing α (30), which we term “admittance” and “Wei's equation,” respectively. Neither solution has been compared with the currently accepted standards (i.e., cuvette-calibrated or SV-calibrated use of Baan's equation and hypertonic saline) to date either by ourselves or by others (21, 23). By using echocardiography as the standard of comparison for true LV volume, we now compare our proposed solutions with Baan's original technique in the intact murine heart.

## MATERIALS AND METHODS

#### Instrumentation overview.

Custom hardware has been developed by our group to produce a complex admittance signal (both magnitude and phase) compatible with any size tetrapolar conductance catheter. For the present study, we focused on murine-sized hearts.

The block diagram of the entire system is shown in Fig. 1 based on our previous design (7). A 20-kHz voltage sine wave drives a voltage-to-current amplifier, which is used to produce the stimulation current for *electrodes 1* and *4. Leads 2* and *3* are fed into an instrumentation amplifier (AD 624, Analog Devices, Norwood, MA), which measures the potential difference created by the induced electrical field in the blood pool. The signal is then filtered and rectified, and the inverse is taken by a multiplier chip (AD 734, Analog Devices) to transform the impedance signal to an admittance signal. This output is taken as the magnitude of the admittance signal *Y*.

We designed and developed a phase angle (θ) detection system, using discrete analog components, that determines the phase angle between two input sinusoidal signals, which is also shown in the block diagram in Fig. 1. A simplified diagram to explain how phase (θ) is measured is shown in Fig. 2. Figure 2 simplifies the measurement system to its most basic building blocks, the two electrodes, and a uniform field distribution (as is assumed by Baan's equation). The basis of the admittance technique relies on a measurable phase difference (θ) due to the presence of myocardium between the input current and the output voltage (Fig. 2*C*), whereas there is no measurable phase angle in blood alone (Fig. 2*A* and Ref. 24). However, both muscle and blood have measurable electrical conductivity that affects the output signal magnitude (the admittance magnitude is simply the ratio of input current to output voltage). The relative fraction of muscle and blood in the measurement determines the value of the admittance magnitude and phase angle (Fig. 2*B*). Separation of myocardium and blood is possible during the measurement. Traditionally, the phase of the admittance has been overlooked as a source of information, and the admittance magnitude has been mistaken for the conductance. In this study, the admittance technique defines the conductance as the real part of the complex admittance G = real{*Y*} = *Y* cos(θ), whereas the conductance technique defines it as G = *Y* (only a correct assumption if there is no measurable phase angle).

#### Murine studies.

The Institutional Animal Care and Use Committees at the University of Texas Health Science Center at San Antonio and at the University of Texas at Austin approved all experiments. A total of 38 mice were studied. The background strain was C57BlkS/J, ages 3–11 mo. Mice were placed on a heated, temperature-controlled operating table for small animals (Vestavia Scientific, Birmingham, AL). Experiments were performed at a murine body temperature of 37°C.

#### Admittance (Wei's equation), hypertonic saline (Baan's equation) comparison to echo.

Mice were anesthetized by administration of 1–2% isoflurane and were allowed to breathe spontaneously with 100% supplemental O_{2}. The right carotid artery was entered, and a tetrapolar micro-manometer catheter (Scisense, London, Ontario, Canada) was advanced into the LV of the intact beating mouse heart. The right jugular vein was cannulated for later administration of 10 μl of 3% hypertonic saline to determine steady-state parallel conductance as previously described (6). A total of *n* = 8 mice were studied. The position of the tetrapolar catheter in the LV was guided by simultaneous imaging with a transthoracic echocardiogram (VisualSonics, Toronto, Canada). The initial position for placement of the tetrapolar catheter was the off-center location (see Fig. 3*B*). In the off-center position, instantaneous LV PV relations were monitored to assure that a physiological loop was obtained before data were aquired for both the admittance and conductance raw signals. Final acceptance of the catheter position was based on both echocardiography and the appearance of the loop. Data acquisition consisted of simultaneous LV pressure, conductance (no phase) stimulated at 20 kHz, complex admittance (magnitude and phase) also at 20 kHz, and echocardiographic images plus ECG for later calculation of end-diastolic volume (EDV), end-systolic volume (ESV), and SV. An operating frequency of 20–25 kHz maximizes the observable muscle signal (24). All data were sampled at 1 kHz.

Subsequently, the tetrapolar catheter was re-positioned to be in the center of the LV cavity (see Fig. 3*A*), with final positioning confirmed by both echocardiography and a physiological LV PV loop. Data acquisition for simultaneous LV pressure, conductance, admittance, and echocardiographic images plus ECG for later calculation of EDV, ESV, and SV were repeated. A 10-μl bolus of 3% hypertonic saline was administered *n* = 3 times per mouse via the right jugular vein for later determination of steady-state parallel conductance, as previously described (6).

#### Admittance (Wei's equation), hypertonic saline (Baan's equation) comparison during inferior vena cava occlusion.

An additional group of mice (*n* = 4) were studied. Mice were anesthetized by administration of 1–2% isoflurane, intubated, and mechanically ventilated at 100% O_{2} with a rodent ventilator set at 150 breaths/min. The heart was exposed via an anterior thoracotomy. An apical stab was made in the heart with a 30-G needle, and the tetrapolar micro-manometer catheter was advanced retrograde into the LV along the long axis with the proximal electrode just within the myocardial wall of the apex. The inferior vena cava (IVC) was isolated immediately below the diaphragm for transient occlusion.

Baseline LV pressure-conductance loops and pressure-admittance loops were acquired. Data were subsequently acquired during transient occlusion of the IVC. A small animal blood flow meter (T106, Transonic Systems, Ithaca, NY) was used with a 1.5-mm Transonic flow probe (MA1.5PSL) placed on the ascending thoracic aorta, and simultaneous LV conductance, admittance, pressure and aortic flow were recorded. The flow probe was used to determine the SV for calibration of the final volume signal in this study instead of echo.

#### Dobutamine studies.

A study was performed in *n* = 6 mice (C57BlkS/J, female, body weight of 23 ± 2.5 g, 8.7 ± 3.0 mo) to determine the relative sensitivity of the admittance and conductance techniques to a contractility change induced by dobutamine infusion. Mice were anesthetized by administration of urethane (1,000 mg/kg IP) and etomidate (25 mg/kg IP). Complex admittance (magnitude and phase) and conductance (magnitude only) with LV pressure were obtained at steady state and during occlusion of the IVC at both baseline and during administration of 5 μg·kg^{−1}·min^{−1} of dobutamine through the jugular vein. Data acquisition for dobutamine was started 5 min after the initiation of the intravenous infusion. Volume data were calibrated using flow probe-derived SV at the end of each experiment, and subsequently converted to LV volume using both Baan's equation and Wei's equation. Parallel conductance G_{P} for Baan's equation was calculated as the constant mean value of the admittance-derived parallel conductance G_{P}(*t*) to avoid blood conductivity changes due to the injection of hypertonic saline.

#### Determination of epicardial permittivity and conductivity.

A custom-designed epicardial probe (25) was applied to the surface of the intact beating open-chest murine heart of an additional *n* = 7 mice, and the stimulus current was generated using the instrumentation described above. Real-time *Y* and θ were measured, and myocardial permittivity (ε_{m}) and conductivity (σ_{m}) were calculated as described previously (24, 25). Briefly, the surface probe “cell constant” *k* (m^{−1}), determined in saline of known electrical conductivity, is used to calculate: σ_{m} = *k*·G_{m} and ε_{m} = *k*·C_{m}, where G_{m} = Re{*Ȳ*}/ω, C_{m} = Im{*Ȳ*}/ω, and ω = 2πf = 126 krad/s.

#### Aortic banded mice.

To investigate the impact of LV hypertrophy on myocardial σ_{m} and ε_{m,} *n* = 13 mice were studied (C57BlkS/J, female, age 5.1 ± 0.3 mo) where *n* = 6 mice underwent aortic banding for 1 wk, and *n* = 7 served as controls. Epicardial permittivity and conductivity were determined as described above. The echo study described above in *Admittance (Wei's equation), Hypertonic saline (Baan's equation) comparison to echo* was repeated in these banded mice to determine the accuracy of the admittance technique with respect to echo. Traditional parallel conductance measurements were not repeated in this study to eliminate the conductivity change that occurs as a result of injection of hypertonic saline as a source of error.

#### Data analysis: hypertonic saline technique.

The instantaneously changing LV blood conductance in response to hypertonic saline injection was evaluated to determine the constant parallel conductance as described in Nielsen et al. (21). Briefly, the relationship between the end-systolic conductance G_{ES} and the end-diastolic conductance G_{ED} can be described as G_{ES} = *m* × G_{ED} + *b*. The point where the linear fit intersects the line of identity is the point where G_{ES} = G_{ED}, implying that the signal is completely derived from the myocardium. This single point (G_{P}) was used as an estimate of constant G_{P} for the entire experiment.

Traditional analysis based on Baan's equation was used to convert measured conductance into volume as previously described for intact murine hearts (11). The constant α term is calculated to force the volume difference resulting from Baan's equation to be the same as an independently measured SV (30), i.e.,
_{B-ED} and G_{B-ES} are the blood conductances at end diastole and end systole, respectively, calculated by subtracting G_{P} from the conductance measured at end diastole and end systole. In the closed-chest murine study, SV was computed using echo data in closed-chest mice, and the same SV was applied to calibrate both Baan's equation and Wei's equation. Additionally, we performed a cuvette calibration and found that α = 1.01 for the miniaturized PV catheter. Because in Baan's original description of α, and in many published studies (20) α is assumed to be 1, we calculated PV loops three ways with *1*) α = 1, *2*) α(G_{P}) determined using *Eq. 6*, and *3*) α from SV calibration (as in *Eq. 2*).

#### Admittance technique: calibration.

The catheter and instrument both produce an additional phase shift because they introduce additional capacitance to the volume measurement. The key to separation of muscle and blood lies in the phase measurement, so a calibration is performed to determine the phase contribution from these sources of artifact. The calibration performed allows quantification of both the catheter/instrument-related phase and their effects on admittance magnitude.

The admittance system is calibrated by measuring the admittance magnitude and phase angle of both the tetrapolar and epicardial catheters immersed in varying conductivity saline solutions (1,000, 2,000, 4,000, 8,000, 10,000, and 12,000 μS/cm). Saline is chosen for the calibration because it will not produce a measureable phase shift at this frequency. The range of conductivity of solutions is determined by including the smallest and largest values of conductivity expected in the in vivo mouse heart, which ranges from 1,600 μS/cm for myocardium to ∼10,000 μS/cm for blood. The admittance measurement is a parallel combination of the two. Thus the admittance system phase calibration is designed to cover this entire range. The containers for the saline solutions are much greater in diameter than the distance between the voltage electrodes so that the electrical field produced by the current is essentially unconstrained by the sides of the container. The values measured for admittance magnitude and phase are therefore independent of the shape of the saline container and only dependent on the catheter and measurement hardware.

#### Theory for dynamic parallel conductance removal using admittance.

Instead of using hypertonic saline injection to determine the parallel conductance, we introduce the concept of a complex measurement of admittance (*Y*). Our method makes use of the native capacitive properties unique to myocardium to identify its contribution to the measured signal so that it can be removed in real time without the need for hypertonic saline injection. The basis of measuring admittance (magnitude and phase) rather than conductance (magnitude only) is that at frequencies around 20 kHz, it has been shown that blood is purely resistive and has no measurable capacitance, but myocardium has both capacitive and resistive properties (Fig. 2) (24, 25, 29). This fact allows separation of the admittance of the myocardium from the combined admittance signal using electric field theory.

For a vector electric field *E* in homogeneous tissue, the conductance and capacitance between the electrodes that establish the field are given by:
*I* is current (A), V is potential (V), σ is electrical conductivity (S/m), F is the field geometry factor (m), C is capacitance (F), and Q is charge (C). The integration is from one electrode to the other along a vector pathway(*L*), and the surface (*S*) encloses all of the current from the source electrode. For homogeneous tissue, the measured conductance and capacitance are related by a simple ratio: G = Cσ/ε. This is the central principle of our method. The important ratio of σ/ε is determined by a surface probe measurement, which is described in detail in Ref. 24.

Briefly, the imaginary part of the admittance is defined as Im{*Y*} = *Y* sin(θ) = ωC_{myocardium} + ωC_{catheter}. Therefore, once the (purely imaginary) catheter contribution has been determined and subtracted through calibration in saline, C_{myocardium} = *Y* sin(θ)/ω. The real part of the admittance is Re{*Y*} = *Y* cos(θ) = G_{blood} + G_{myocardium}, implying that G_{blood} = *Y* cos(θ) − C_{myocardium}σ/ε. Based on these equations, we can determine the instantaneous values of parallel conductance and blood conductance.

#### Wei's conductance to volume equation.

As described previously by Wei et al. (30), the relationship between conductance and volume should include a nonconstant α term in any size heart because of the dynamically changing field shape. Wei's equation for converting conductance to volume is
_{B}) is a new time-varying expression for α, and G_{B} is the blood conductance (G − G_{P}) calculated as described above.

Wei describes α as dependent on the blood conductance G_{B}

The larger positive solution for γ is used in all calculations. G_{B-ED}, G_{B-ES}, and SV are determined during steady-state conditions, so although γ is constant, α(G_{B}) is dependent on G_{B}, which changes as the heart beats, contrary to the constant α in Baan's equation.

#### Echocardiographic calculations: EDV and ESV.

End diastole was defined as coincident with the peak of the R wave on the ECG, and end systole was defined as minimum LV volume. A long axis view was taken for both end systole and end diastole, and the volume measured was determined using the cardiac analysis package from the Vevo 770 software (VisualSonics, Toronto, Canada). We measured the long axis, and a single short axis within a long axis view (see Fig. 3, *A1* and *B1*), and assumed the second short axis to be of equal length. Volume was then computed using the prolate ellipse method. SV was determined as the difference between EDV and ESV and was used in the calculation of both conductance and admittance-derived LV volume.

#### Center and off-center catheter positions.

Both center and off-center positions were studied since placement of the miniaturized PV catheter across the aortic valve in vivo forces catheter placement near the septum and off the true LV center (Fig. 1*A*; Ref. 23). Thus, without echo guidance, the off-center position is most often used by investigators.

To allow for comparison between a centered position and an off-center catheter position, the distance between the LV free-wall (LVFW) and the center of the catheter was measured, as was the distance between the center of the catheter and the septum (SEP). An example of these measurements is displayed in Fig. 3, *A2* and *B2*. The percentage deviation from the center was calculated as

In this calculation, a value of 100% indicates that the catheter position coincides with the inner myocardial wall, and a value of 0% indicates that the catheter is in the center of the ventricle. Catheter positions that are symmetric around the long axis result in the same percentage deviation from the center.

#### Statistical analysis.

EDV and ESV derived by Wei's equation using admittance G_{P}(*t*) were compared with echo data for EDV and ESV at both the centered and off-center positions using a two-sample right-tailed Student's *t*-test. All calculations were performed using Matlab software (The Mathworks, Natick MA). EDV and ESV derived by Baan's equation using hypertonic saline G_{P} were also compared with echo data using the same methods at both the centered and off-center positions. These analyses are also presented as Bland-Altman plots. The enhancement in contractility in response to dobutamine between baseline and drug infusion for both Baan's equation (SV calibration of α) and Wei's equation using admittance G_{P}(*t*) were compared with Student's *t*-test, and the alteration in myocardial properties between control and aortic-banded mice were also compared with Student's *t*-test.

## RESULTS

#### Non-banded myocardial properties measurement.

The conductivity and permittivity of myocardium were measured in a separate experiment in *n* = 7 mice in an open-chest preparation as described in Ref. 24. All values are reported as means ± SD. The relative permittivity of myocardium derived from these measurements at 20 kHz was ε_{m} = (11,844 ± 2,700)·ε_{0} F/m, and the electrical conductivity of myocardium was measured at σ_{m} = 0.160 ± 0.046 S/m. These results were used in calculations related to the admittance-conductance-echocardiography comparisons.

#### Hemodynamic and echocardiographic parameters for admittance-conductance comparison.

For *n* = 8 mice, the body weight ranged from 19 to 30 g (mean = 26.5 ± 3.5 g), heart weight ranged from 85 to 152 mg (mean = 123.8 ± 21.2 mg), and LV weight ranged from 67 to 107 mg (mean = 90.8 ± 14.6 mg). The mean LV peak systolic pressure was 102 ± 10 mmHg, and the heart rate was 407 ± 28 beats/min. The mean echo EDV was 40.9 ± 12.1 μl, and the mean echo ESV was 14.1 ± 5.5 μl. The echocardiographic SV used in the calculation of both alpha (α) and gamma (γ) had a mean value of 26.8 ± 7.2 μl.

The parallel conductance calculated by the hypertonic saline method was 476 ± 182 μS and by the admittance technique ranged dynamically from 552 ± 79 μS at end-systole to 346 ± 67 μS at end-diastole. The mean off-center α was 0.22 ± 0.15, as determined with echo SV, and it was 1.01 with cuvette. The mean γ was 1,350 ± 389 μS. The location of the catheter position relative to the wall is shown in Table 1. The center technique was closer to the true midline than the off-center position (*P* < 0.0001) based on the M-mode echo measurements. Typical examples of echo center and off-center catheter positions are shown in Fig. 3.

#### Wei's equation using admittance is more accurate than Baan's equation using HS and either SV or cuvette calibration with off-center catheter placement.

The off-center derived PV loops from Wei's equation, Baan's equation with SV calibration (α = 0.22 ± 0.15), and Baan's equation with cuvette calibration (α = 1.01) for all *n* = 8 mice are shown in Fig. 4. As is visually apparent in five of the eight mice, the SV-calibrated conductance-derived PV loops are to the right of echo (*mice 2*, *4*, *5*, *7*, *8*), and the cuvette calibrated conductance loops are to the left of echo (*mice 1–3*, *6*, *7*). In contrast, all of the Wei's equation with admittance loops are similar to the echo-derived volumes. The end-diastolic and end-systolic volumes derived using Wei's equation [α(G_{B}) ≈ 0.7–0.9] show significantly less absolute error (−2.49 ± 15.33 μl error ES and ED) than the same quantities calculated using SV-calibrated conductance (α ≈ 0.1–0.3) (35.89 ± 73.22 μl error ES and ED) or cuvette-calibrated conductance (α = 1) (−29.10 ± 31.53 μl error ED, and −7.53 ± 16.23 μl error ES), taking the echo volumes as the standard of comparison (*P* < 0.03).

These results are shown in graphical format as Bland-Altman plots in Fig. 5*A*. The Bland-Altman plot shows that most of the error lies in the low repeatability (high standard deviation) of measurements when relying on Baan's equation with SV calibration. A consistently small absolute volume and SV are observable in results from Baan's equation with cuvette calibration, which is consistent with observations in the literature (14, 21). These results show that, when the catheter is not placed in an optimal, i.e., central position, admittance provides an advantage in accuracy.

#### Wei's equation using admittance is as accurate as Baan's equation using HS and SV or cuvette calibration with centered catheter placement.

For the same eight mice, when the catheter was shifted to the central position and the same measurements were taken, the same trend was observed: every Wei's equation using admittance PV loop was a smaller volume than those derived from Baan's equation α = 0.22 ± 0.15 with HS and larger than every PV loop from Baan's equation α = 1.01 using cuvette calibration (Fig. 5). However, the difference in error between Wei's equation and echo measurement, and between both calibration methods for Baan's equation and echo, were not found to be statistically significant in either case with one exception: in the end-diastolic measurements from cuvette calibration, Wei's equation shows lower error (*P* < 0.03). The closeness of these three results may be due to the removal of equivalent amounts of parallel conductance when the tetrapolar catheter is centered with echo guidance, where the average G_{P} using hypertonic saline is 476 ± 182 μS vs. admittance's average G_{P}(*t*) of 448 ± 60 μS. The mean of an instantaneously varying parallel conductance and an example real-time signal are shown in Fig. 6*A*. It is interesting to note that even though the difference among the results for the center position are not statistically significant, they still have much lower variance, as evidenced by the Bland-Altman plot in Fig. 5.

#### Banded mice.

In an additional *n* = 13 mice, the impact of LV hypertrophy (aortic banding for 1 wk, *n* = 6) on myocardial properties was determined compared with controls (*n* = 7). Heart weight (86 ± 4 mg control, 104 ± 24 mg banded; *P* = not significant) and LV weight (70 ± 7 mg control vs. 86 ± 24 mg banded; *P* = not significant) tended to increase with banding but not significantly due to the large standard deviation in the banded group. In banded mice, ε_{m} = (21,267 ± 8,005)·ε_{0}F/m, a significant increase in myocardial properties (11,844 ± 2,700 to 21,267 ± 8,005)·ε_{0}F/m occurred compared with control (*P* < 0.05). The myocardial conductivity showed no significant change between banded σ_{m} = 0.200 ± .080 S/m and control σ_{m} = 0.160 ± 0.046 S/m (*P* = not significant) mice.

For the *n* = 6 banded mice, the mean LV peak systolic pressure was 157 ± 30 mmHg and the heart rate was 471 ± 67 beats/min. The mean echo EDV was 33.7 ± 11.0 μl, and the mean ESV was 12.9 ± 4.8 μl. The echocardiographic SV used in the calculation of gamma (γ) had a mean value of 20.8 ± 6.3 μl. The overall volume error compared with the echo standard was 6.74 ± 13.06 μl, which is comparable to the error in the non-banded mice (−2.49 ± 15.33 μl) and less than the error for traditional conductance (35.89 ± 73.22 μl).

#### IVC occlusions show dynamic removal of a nonconstant parallel conductance.

In additional mice (*n* = 4), IVC occlusions were performed to demonstrate the ability of the admittance technique to dynamically separate blood conductance and myocardium conductance. Figure 6*A* shows the typical separation of the blood and myocardial components during IVC occlusion in a single mouse using conductance and admittance techniques, and all four mice showed the same phenomenon. The observation from this figure, which represents a departure from traditional conductance theory, is that, during an IVC occlusion, the amount of myocardium in the sensing field increases while the blood component of conductance still decreases and dominates the total conductance signal. Traditional conductance methods assume a constant value for parallel conductance, which is demonstrated to be an inaccurate assumption by these data.

Preload reduction is commonly utilized to generate measures of LV function such as the end-systolic elastance (E_{es}), diastolic chamber compliance, and others. We hypothesized that calculated values of E_{es} and other parameters would be affected by the change from traditional conductance with Baan's equation to admittance with Wei's equation. Thus we analyzed the impact of these two techniques on calculated hemodynamic parameters for an identical transient occlusion of the IVC from a single mouse. Results are shown in Fig. 6*B*. Despite the similarity in conductances in Fig. 6*C*, the appearance of the loops in Fig. 6*B* is significantly different.

Additionally, there was an increase in measures of contractility using the admittance technique. E_{es} rose from 3.4 to 15.4 mmHg/μl; maximum elastance rose from 12 to 25 mmHg/μl, preload recruitable stroke work rose from 52 to 95 mmHg/μl, conductance to admittance, respectively. Furthermore, there was an increase in chamber stiffness from 0.061 s^{−1} with conductance to 0.076 s^{−1} by admittance.

#### Wei's equation using admittance is more sensitive to the detection of inotropic stimulation.

The hemodynamic response of *n* = 6 mice was determined at baseline and following 5 min of steady-state infusion of 5 μg·kg^{−1}·min^{−1} of intravenous dobutamine. Heart rate increased from 384 ± 47 to 523 ± 75 beats/min (*P* < 0.05), respectively. P_{max} was unchanged from 97 ± 13 to 98 ± 11 mmHg, respectively (*P* = not significant). dP/d*t*_{max} increased from 5,821 ± 1,802 to 12,309 ± 3,832 mmHg/s, respectively (*P* < 0.01), and dP/d*t*_{min} decreased from −5,703 ± 1,491 to −7,505 ± 825 mmHg/s, respectively (*P* < 0.05).

To evaluate the impact on parameters derived using Wei's equation and Baan's equation (both using SV calibration) during the identical IVCO, linear E_{es} and dP/d*t*-EDV were determined in these same mice. E_{es} increased from 4.9 ± 1.4 to 12.5 ± 6.6 mmHg/μl using Wei's equation (*P* < 0.05) and from 3.3 ± 1.2 to 8.8 ± 5.1 mmHg/μl using Baan's equation (*P* = not significant). A representative example is shown in Fig. 7. Similarly, the dP/d*t*-EDV relationship increased from 158 ± 127 to 542 ± 320 mmHg·s^{−1}·μl^{−1} using admittance (*P* < 0.05) and from 117 ± 78 to 263 ± 123 mmHg·s^{−1}·μl^{−1} using Baan's equation (*P* = not significant).

## DISCUSSION

The current study compares the V(*t*) determined by Baan's equation using G_{P} from HS with SV or cuvette calibration to V(*t*) determined by Wei's equation using admittance. We have demonstrated *1*) more accurate volumes with Wei's equation using admittance when the catheter is in the off-center position than Baan's equation using SV or cuvette calibration, *2*) that the parallel conductance, G_{P}, varies between end-systole and end-diastole, and even more significantly during IVC occlusion, *3*) the advantages of using a dynamically changing α, which is dependent on G_{B}(*t*), *4*) improved sensitivity in the detection of inotropic stimulation with Wei's equation and G_{P}(*t*) calculated using admittance, compared with Baan's equation and G_{P} calculated using HS with SV calibration of α, *5*) that the myocardial properties change with LV hypertrophy, and *6*) that the improved accuracy of admittance in the determination of LV volumes extends to mice with aortic banding.

#### Dynamic parallel conductance calculation.

In an early study by Lankford et al., parallel conductance was demonstrated to be a constant between end-diastole and end-systole (18). Historically, this study was the theoretical basis for the acceptance of hypertonic saline injection determination of G_{P} in the literature. However, myocardium has both a conductive and a dynamic capacitive component (25, 30, 31). This concept is not reflected in the assumptions or conclusion of Lankford's study, nor in any traditional conductance technique where phase is not measured (1, 2, 8, 10, 11, 21, 26). Mathematically speaking, the G_{P} calculated using admittance is a closer approximation to the true value of G_{P} than traditional conductance because it takes into account the complex nature of myocardium *Y*_{P} = G_{P} + jωC_{P}, and therefore includes measurable changes during the cardiac cycle. The current results (Fig. 6*A*) and our previous study (30) imply that parallel conductance is changing even between end-diastole and end-systole, as predicted by electric field theory (31).

#### Calculation of the gain term α.

In the early conductance literature, the value of α was not calculated. The term α was assumed to be equal to 1, as described by Baan et al. (2) and more recently by Uemura et al. (28). The purpose of the term α is to calibrate the SV of the resulting conductance signal to match a standard of comparison (usually a flow probe or echo). However, because α is dependent on the field geometry, and field geometry is constantly changing throughout the cardiac cycle, α is not a constant. As shown in Fig. 8 during an IVC occlusion in a single mouse, the value of α is different depending on which calibration method is used. For example, using Wei's equation, the dynamic nature of α is evident both between systole and diastole, and down the IVC occlusion ramp. In contrast, both other calibration procedures using flow or echo calibration (α = 0.79 for centered position) and the volume cuvette method (α = 1) recommended by manufacturers of the currently used commercial mouse conductance systems use a constant value for α. The impact of using a variable α is that the volume equation will more closely model the changes that the field sees, as opposed to assuming a constant shape of the electric field.

#### Limitations of Cuvette Calibration.

Review of the literature (122 publications) revealed that 63% of murine conductance studies utilize the volume cuvette as the calibration method of choice to directly convert voltage to absolute volume (20). However, recent studies (14, 21) have shown that cuvette-calibrated left-ventricular volumes are consistently smaller than those derived with a standard such as MRI, confirmed as well in the present study. Volume cuvettes have an electrically insulating boundary at the myocardial wall, which cramps the field causing an artificially high value of α (implying a small SV). This high value of α causes small resultant volumes and small resultant SVs (see Fig. 4).

#### Limitations of a constant α.

A correction for the SV error introduced by the volume cuvette is to force the final SV to be the same as an independent measure. However, forcing only the SV to the independently verified value in an off-center position will reduce the value of α, increasing the “gain factor” (α^{−1}) of the volume equation. The benefit to the conductance technique is an accurate SV, but the detriment is an inflation of all volumes. As shown in Fig. 4, although all SV are more physiological when α = 0.22 ± 0.15, in *mice 2*, *7*, and *8*, there is an inflation of volumes to an unphysiological value. Although Fig. 4 was generated from the off-center position of the miniaturized PV catheter, this is often the location of catheter in the heart without echo guidance, since the aortic valve is off-center relative to the middle of the LV chamber.

#### Dynamically changing α (Wei's equation).

The advantage of using a dynamic value for α is that more realistic volumes are obtained. As shown in Fig. 4, incorporating a dynamic value for α provided absolute volumes closest to the echo standard. Wei's equation provides a nonlinear conductance to volume relationship, which has been noted in some recent papers (21). However, the most significant improvement is that Wei's α(G_{B}) is a function of blood conductance G_{B} (see *Eq. 6*) and γ. The derived value, γ, represents the value of conductance at saturation (where the catheter is placed in an arbitrarily large pool of conductive solution), so the expression α(G_{B}) incorporates the changing geometry of the electric field as the heart beats. This advantage of the γ formulation will be particularly important when changes in loading conditions, such as IVC occlusion, are performed.

#### Wei's equation using admittance G_{P}(t) improves volume estimates for off-center placement of the PV catheter.

One assumption of Baan's equation is that the tetrapolar catheter is placed in the center of the left ventricle. However, the off center location of the aortic valve relative to the mid-LV chamber in mammalian hearts forces the miniaturized PV catheter to be closer to the septum (Fig. 1*A*; Ref. 23). Only the use of an independent imaging modality, such as echo, will allow a miniaturized PV catheter to be moved closer to the true center of the LV. Echocardiography, however, is not routinely used to center the conductance catheter. In the present study, despite the use of echo guidance, the mean central placement was actually 14 ± 9% at end-diastole, where 0% was true center and 100% is the endocardium or septum. In addition, as the heart moves, one cannot assume that the catheter position stays at the midline. Some investigators estimate that the catheter shifts by 50–70% of the ventricular radius (17). Thus placement of the tetrapolar catheter in an off-center position is a common source of error.

In the present study, the off-center tetrapolar catheter Wei's equation using admittance G_{P}(*t*) show statistically less error than traditional conductance measurements taken simultaneously in a closed-chest mouse, with simultaneous high-frequency echocardiography used as the standard for true LV volume. One explanation is that admittance provides a measure of instantaneous parallel conductance, whereas all previous methods cannot. This is particularly evident in Fig. 6*A* during occlusion of the IVC, where the myocardial component of the admittance signal is shown to vary both between end-diastole and end-systole, as well as during the occlusion ramp.

Additionally, the use of a dynamic value for α that is dependent on the value of conductance shows a more physiological range for the absolute volume, while still matching the SV of an independent method (usually flow probe or echo). In Baan's equation, the α term is dependent only on the conductance-derived SV. In Wei's equation, however, the α(G_{B}) term is dependent on both the SV and the current values of conductance. Figure 6, *B* and *C*, demonstrates that this conclusion is only in part due to the differences in parallel conductance removal. In addition, the calculated volume ranges are quite different in Fig. 6*B*, whereas the conductances shown in Fig. 6*C* are similar, implying that the difference in the volumes is due to a difference in α as well.

As with any linear relationship, the slope α and the *y*-intercept G_{P} are the only two points of control for the calibration of the volume signal using Baan's equation. This makes the accuracy of Baan's equation dependent on obtaining the correct value of G_{P} (which magnifies problems of using a constant α) and the correct value of parallel conductance determined with hypertonic saline (which has a high relative error in the mouse) (2, 10).

The hypertonic saline estimate of parallel conductance is complicated in the mouse by *1*) changes in SV caused by preloading the heart with the volume of hypertonic saline injected, *2*) changes in blood electrical conductivity due to multiple injections (usually averaged), *3*) variability in the manner of the injection (volume, rate, etc.), and *4*) extrapolation of data to the line of identity. The variability of the hypertonic saline technique explains why in Fig. 4 some SV calibrated conductance-derived loops are close to echo standard, whereas others substantially overestimate echo standard ESVs. Specifically, there are examples in Fig. 4 where the traditional conductance-derived LV volumes are similar to echo standard (*mice 1*, *3*, *6*). However, other examples in the same figure demonstrate conductance-derived ESVs that overestimate the echo standard (*mice 2*, *4*, *5*, *7*, *8*). In *mice 1*, *3*, *6*, the hypertonic saline technique reports parallel conductance values (623 ± 65 μS), which are significantly higher (*P* < 0.01) than in *mice 2*, *4*, *5*, *7*, *8* (389 ± 114 μS). Consequently, *mice 1*, *3*, *6* have traditional conductance-calculated absolute LV volumes, specifically the ESV, closer to the truth because subtraction of a larger parallel conductance moves the volume to a more realistic range. In contrast, *mice 2*, *4*, *5*, *7*, *8* have significantly smaller parallel conductance subtracted, leaving the final calculated volumes larger than the echo standard.

#### Importance of myocardial properties measurement.

The present study demonstrates that the permittivity (ε_{m}) increases with LV hypertrophy induced by aortic banding for 1 wk, whereas the myocardial conductivity (σ_{m}) did not change significantly. It is known that larger myocytes have a greater membrane capacitance (12). The measurement of capacitance in the electrode field is increasing during LV hypertrophy, thus a higher permittivity is inferred from the measurement. In our bulk tissue measurement, the increased permittivity detected is likely a summation of this basic property. Thus it is critical to measure this σ-to-ε ratio in every group of mice with myocardial disease if our technique is to derive LV volume accurately.

### Limitations and Future Work

The ability of admittance to detect instantaneous parallel conductance should provide different measures of hemodynamic endpoints derived in the PV plane, although not having a standard measure of, for instance, end-systolic elastance (E_{es}), it is not clear how this advantage can be proven, other than the data shown in Figs. 6*B* and 7. Furthermore, our results question the accuracy of absolute values of hemodynamic parameters determined through the pressure-volume plane, such as E_{es}. Although relative changes in E_{es} are accurate, the absolute value of E_{es} determined using traditional conductance theory would be different than those derived from admittance down an IVC occlusion ramp. Admittance has the capability to detect and remove, in real time, this dynamic interaction between the electric field and myocardium as the heart shrinks in size on a beat-by-beat basis.

Although admittance has the same physical significance in larger hearts, it is not clear at this point whether the improvement in accuracy will be statistically significant in larger animals. Similar to conductance measurements, admittance scales with the size of the catheter (because of the length *L* between the voltage electrodes in Baan's equation), so larger lead spacing can be used to measure larger volumes without the concern of saturating the admittance measurement. Also, multi-electrode catheter theory is compatible with admittance calculations, since the total admittance between two or more adjacent pairs of sensing electrodes is equal to the sum of the pairs. We plan to extend our studies to larger animals in the future to explore these issues.

We conclude that, for closed-chest mice, admittance (dynamic G_{P}) and Wei's equation (dynamic α) provide more accurate volumes than traditional conductance, are more sensitive to inotropic changes, eliminate the need for hypertonic saline, and can be accurately extended to aortic banded mice.

## GRANTS

This study was supported by grants from the Veterans' Affairs Merit (M. D. Feldman), and National Heart, Lung, and Blood Institute (R21 HL-079926).

## ACKNOWLEDGMENTS

The studies presented in this paper were supported with tetrapolar micromanometer catheters supplied from Scisense, (London, Ontario, Canada). The authors would also like to thank Dr. G. Patricia Escobar for help with echo data acquisition and Jason Hansen for assistance with PV analysis.

- Copyright © 2009 the American Physiological Society