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Electrical Conductivity and Permittivity of Murine Myocardium Karthik Raghavan, John E

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Electrical Conductivity and Permittivity of Murine Myocardium Karthik Raghavan, John E 2044 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 56, NO. 8, AUGUST 2009 Electrical Conductivity and Permittivity of Murine Myocardium Karthik Raghavan, John E. Porterfield, Anil T. G. Kottam, Marc D. Feldman, Daniel Escobedo, Jonathan W. Valvano, Member, IEEE, and John A. Pearce∗, Senior Member, IEEE Abstract—A classic problem in traditional conductance mea- (5 mm length, 160 mg mass, and 40 µL LV volume) and its surement of left ventricular (LV) volume is the separation of the rapid heart rate (500–700 beats/min). Approaches such as ul- contributions of myocardium from blood. Measurement of both trasonic crystals [2], MRI [3], [4], and echocardiography [4] the magnitude and the phase of admittance allow estimation of the time-varying myocardial contribution, which provides a sub- have been used to measure instantaneous LV volume with stantial improvement by eliminating the need for hypertonic saline some degree of success. Unfortunately, all these technologies injection. We present in vivo epicardial surface probe measure- have severe limitations, particularly during dynamic maneuvers, ments of electrical properties in murine myocardium using two such as transient occlusion of the inferior vena cava or aorta, different techniques (a digital and an analog approach). These which are required to generate load-independent indexes of methods exploit the capacitive properties of the myocardium, and both methods yield similar results. The relative permittivity varies contractility. from approximately 100 000 at 2 kHz to approximately 5000 at Conductance catheter technology, as introduced by Baan 50 kHz. The electrical conductivity is approximately constant at et al. in 1981 [6], offers a more robust alternative to gener- 0.16 S/m over the same frequency range. These values can be ate instantaneous LV P –V relations in the intact murine heart. used to estimate and eliminate the time-varying myocardial con- Single-frequency conductance has been used in mice to gener- tribution from the combined signal obtained in LV conductance catheter measurements, thus yielding the blood contribution alone. ate measures of ventricular function [5], [7], [8]. However, the To study the effects of albumin on the blood conductivity, we also traditional conductance method is limited in the mouse because present electrical conductivity estimates of murine blood with and the instantaneous LV conductance signal includes both blood without typical administrations of albumin during the experiment. conductance and parallel admittance in the myocardium and, The blood conductivity is significantly altered (p < 0.0001) by ad- unless corrected, yields an overestimate of the true LV blood ministering albumin (0.941 S/m with albumin, 0.478 S/m without albumin). volume [6]. Investigators have applied the hypertonic saline technique developed for larger mammals to determine a sin- Index Terms—Admittance, blood conductivity, conductance, gle value of steady-state parallel (cardiac muscle) conductance myocardial conductivity, myocardial permittivity. and used it for the derivation of absolute LV volume [9]. The saline technique, however, is problematic in small animals such I. INTRODUCTION as mice and rats since administration of even small volumes of ETERMINATION of left ventricular (LV) pressure– hypertonic saline significantly alters both blood resistivity and D volume (P –V ) relations has provided a framework for hemodynamics (i.e., blood volume) [5], violating the framework understanding cardiac mechanics in larger experimental ani- of the governing assumptions [6]. Simultaneous measurement at mals and humans [1]. Extension of this technique to the mouse two frequencies combined with the hypertonic saline technique model has proven valuable [2]. Determination of instantaneous has been proposed by other investigators [10]–[12]. However, volume in the murine LV is difficult due to the small heart size in all cases, these methods determine only a single value of steady-state parallel conductance. Thus, conductance measure- Manuscript received April 23, 2008; revised September 12, 2008. Current ment in its present reduction to practice, in both small and large version published July 15, 2009. This work was supported in part by the Na- subjects, cannot calculate the instantaneous change in paral- tional Institutes of Health under Grant R21 HL079926 and a U.S. Department of Veterans Affairs (VA) Merit Grant (MDF). Asterisk indicates corresponding lel conductance occurring throughout the cardiac cycle as the author. LV cavity shrinks around the intracardiac electric field during K. Raghavan, J. E. Porterfield, and J. W. Valvano are with the Depart- occlusion of the inferior vena cava or at end systole. ment of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]; Measurements of the permittivity of muscle by Gabriel john.porterfi[email protected]; [email protected]). et al. [13], [14] suggest that the relative permittivity of car- A. T. G. Kottam was with the Department of Biomedical Engineering, The diac muscle exceeds 15 000 at 20 kHz. We hypothesize that University of Texas at Austin, Austin, TX 78712 USA. He is now with Scisense, Inc., London, ON N6E 3A1, Canada (e-mail: [email protected]). the electric permittivity of muscle in vivo is so high that the M. D. Feldman and D. Escobedo are with the Department of Medicine, The admittance in the LV at frequencies in this range can be used University of Texas Health Sciences Center at San Antonio, San Antonio, TX to identify and separate the cardiac muscle component from the 78229 USA (e-mail: [email protected]; [email protected]). ∗J. A. Pearce is with the Department of Electrical and Computer Engi- combined admittance measurement. The fundamental observa- neering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: tion is that blood is semiconducting, over the same frequency [email protected]). range, so all frequency-dependent admittance is due to the mus- Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. cle component only, i.e., for a tetrapolar catheter in the LV, the Digital Object Identifier 10.1109/TBME.2009.2012401 admittance consists of two parallel components from blood and 0018-9294/$25.00 © 2009 IEEE Authorized licensed use limited to: Qualcomm. Downloaded on July 27, 2009 at 12:33 from IEEE Xplore. Restrictions apply. R AG H AVA N et al.: ELECTRICAL CONDUCTIVITY AND PERMITTIVITY OF MURINE MYOCARDIUM 2045 myocardium: Ymeas = Yb + Ym = Gb + Gm + jωCm (1) where Y is admittance (in siemens), G is conductance (in siemens), ω is the angular frequency (in radians per second), C is capacitance (in farads), “meas” refers to the measured sig- nal, “b” to the blood alone, and “m” to the myocardium. For any electric field spatial distribution E in a homogeneous medium Fig. 1. Epicardial admittivity surface probe consists of 1-mm-long four elec- I E • dA trodes spaced at 0.25 mm (typical) and 0.4 mm center to center between elec- G = = σ = σF (2a) trodes 2 and 3. Suction ports permit application of a mild vacuum to assist in V − E • dl securing the electrode. Q E • dA C = = ε = εF (2b) V − E • dl where Ym is the admittance of the myocardium (in siemens), and F is the cell constant described in (2a) and (2b). This definition where I is the current (in amperes), V the potential (in volts), σ for admittivity is clearly harmonious with Ampere’s law in point the electrical conductivity (in siemens per meter), Q the charge form (in coulombs), ε the electric permittivity (in farads per me- ter), and F is the electric field form factor, or “cell constant,” ∇×H = J + jωD =(σ + jωε) E = ψE. (4) common to both relations (in meters). The symmetry in the re- lationships of (2a) and (2b) leads to the familiar “conductance– II. METHODS capacitance” analogy. The symmetry is also the feature that In this study, we measure the electrical properties of the allows identification and elimination of the cardiac muscle from murine myocardium (permittivity ε and conductivity σ )in the combined admittance signal: if C is measured, then G m m m m order to estimate and eliminate the myocardial contribution [Y , can be calculated m (3a)] from the combined contribution [Ymeas, (1)] in the LV,thus σm Gm = Cm . (2c) yielding the blood contribution alone in real time throughout the εm cardiac cycle. An accurate value for the σm /εm ratio is required to apply this new method for eliminating the parallel admittance of cardiac A. Epicardial Surface Probe muscle. A miniature tetrapolar surface probe was applied to the epi- The dominant trend in the literature [15]–[21] following the cardial surface of the beating murine heart in vivo (Fig. 1). It original form of Baan et al.’s equation [6] is to express the elec- was custom-fabricated to our specifications by The University trical properties in the form of resistivity. However, when the of Texas Health Science Center at San Antonio (UTHSCSA), tetrapolar catheter is placed in the LV, the current path through San Antonio. The probe contains four parallel platinum elec- the myocardium flows essentially in parallel with the current trodes aligned with an intraelectrode spacing of 0.25, 0.4, and path through the blood. Therefore, it is substantially more con- 0.25 mm between electrodes 1 and 2, 2 and 3, and 3 and 4, venient to express the electrical properties in the form of elec- respectively. In the standard tetrapolar technique, electrodes 1 trical admittivity rather than impedivity. This is because over and 4 are driven with a current source and electrodes 2 and 3 the frequency range of interest, the admittance of (1) describes are used for potential measurement at negligible current (due the electrical behavior of the cardiac muscle very well, and to the high input impedance of the voltage sensing differential admittances in parallel add simply rather than combine hyper- amplifier).
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