Tetrapolar Measurement of Electrical Conductivity and Thickness of Articular Cartilage

[+] Author and Article Information
J. S. Binette, M. Garon

Institute of Biomedical Engineering

P. Savard

Department of Electrical Engineering

M. D. McKee

Faculty of Dentistry, and Department of Anatomy and Cell Biology, McGill University, Montréal, Québec, Canada

M. D. Buschmann

Department of Chemical Engineering, École Polytechnique, Montréal, Québec, Canada

J Biomech Eng 126(4), 475-484 (Sep 27, 2004) (10 pages) doi:10.1115/1.1785805 History: Received June 16, 2003; Revised December 23, 2003; Online September 27, 2004
Copyright © 2004 by ASME
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Experimental configuration and model of tetrapolar conductivity measurements. a) The tested sample is represented as two isotropic homogeneous layers of infinite lateral dimensions where layer 1 (with conductivity σ1) is the articular cartilage of thickness “d,” and layer 2 (with conductivity σ2) is the subchondral bone, taken to be of infinite thickness. There are three tetrapolar electrode configurations used in this study where the current is injected using one of three possible pairs of electrodes (#1–#8, #2–#7, #3–#6) while the potential difference is always measured by the central electrode pair (#4–#5). b) A light micrograph shows the 300 μm regular spacing of the 8 electrodes of 50 μm diameter. The faint circles above the electrodes are the nylon mesh in cross-section.
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Theoretical prediction and calibration of the apparent conductivity as function of the thickness of layer 1. a) The apparent conductivity (Eq. 2) for the three tetrapolar configurations as a function of the thickness of layer 1 (d), calculated using model assumptions explained in Fig. 1 and with σ1=1 S/m2=0.2 S/m and a=300 μm. The model predicts that the apparent conductivity approaches that of layer 1 when layer 1 thickness is >2×the inter-electrode spacing (∼600 μm for equidistant configuration #3–#6) and approaches that of layer 2 when the thickness of layer 1 is ∼10×less than the inter-electrode spacing (∼30 μm for equidistant configuration #3–#6). Spacing apart the injection electrodes (using #2–#7, #1–#8) results in lower apparent conductivity due to deeper penetration of the current and therefore greater influence of the lower conductivity of layer 2. b) The apparent conductivity was measured using three different NaCl concentrations (0.03 M, 0.1 M, 0.5 M) with electrical conductivities (σ1) of (0.34 S/m, 1.04 S/m, 4.54 S/m). The distance, d, between the upper post and chamber bottom was varied to change the thickness of the layer of known conductivity, σ1, and measured apparent conductivity was normalized as σapp1 shown in the figure. The x axis in Figure 2b was shifted such that the point of contact was taken as 50 μm separation rather than 0 μm separation, due to non-parallelism of the two surfaces. The mean±sd of these three normalized conductivities (3 salt concentrations) are shown for each of the tetrapolar configurations (○ for #3–#6, ▪ for #2–#7, ▵ for #1–#8) along with the theoretical prediction calculated using Eq. 3 with σ2=0.
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Experimental apparatus for electrical conductivity measurement. a) Articular cartilage slices (2 mm wide, 12 mm long and 200-550 μm thick) were positioned on the center of the microelectrode array and slightly compressed (5 μm) under uniaxial unconfined compression geometry to assure good electrical contact. Electrical conductivity was measured using the equidistant electrode configuration with electrodes #3 and #6 injecting current. b) Electrical conductivity of disk-shaped samples (3.5 mm diameter with full-thickness articular cartilage attached to underlying bone) were measured by gluing the bone surface onto a transparent plastic plate that was then glued to a stainless steel bolt that was heavy enough (7.2 g) to ensure good contact when placing the articular cartilage surface in contact with the exposed electrodes. All three tetrapolar configurations were used to measure conductivity in this case. For both types of experiments, the test chamber was filled with 0.15 M phosphate-buffered saline (PBS) that has an electrical conductivity of 1.52 S/m. An alternating current of 1 μA amplitude was injected at frequencies in the range of 10 Hz to 1000 Hz.
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Example raw data. a) Typical measured sinusoidal voltage for the equidistant tetrapolar configuration. b) Fourier-transformed raw data shows the ratio of the measured voltage to applied current, amplitude and phase (voltage–current) and total harmonic distortion (THD) as function of frequency.
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Electrical conductivity of cartilage-only slices taken from different depths from the articular surface. Electrical conductivity of full thickness cartilage (the 11 points at 0% depth) and slices (n=28) taken from different depths was measured with the apparatus shown in Fig. 3a. No dependence of electrical conductivity with depth was apparent in this 1 to 2-year-old bovine humeral head articular cartilage (R2=0.011).
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Apparent conductivity as a function of sample thickness for the three tetrapolar configurations (example data). The apparent conductivity of disk samples were obtained with the chamber of Fig. 3b. The cartilage thickness was reduced by removing slices consecutively with a vibratome and the sample thickness (cartilage+bone) was measured with a micrometer after each slice was removed. The apparent conductivity decreases with decreasing cartilage thickness, and when the injecting electrodes are farther apart, as predicted by the model (Fig. 2).
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Histological appearance of cartilage/bone disks with and without cutting to reduce cartilage thickness. Disk samples were matched as adjacent duplicates during isolation where one of each duplicate was used for conductivity measurements as a function of cartilage thickness (Fig. 3b) and then preserved by aldehyde fixation (B, D, F) while the other was chemically fixed directly without cutting (A,C,E). Disks were washed, embedded, sectioned and stained with toluidine blue. The last cut usually arrived just above the tidemark (B), however some uncalcified cartilage could remain (D). Variations in bone porosity and the thickness of the subchondral bone plate were evident (A and E to C). With respect to Table 2, B=sample 2,D=sample 5,F=sample 9.UC=uncalcified cartilage, CC=calcified cartilage. Scale bar=300 μm.
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Comparison of cartilage thickness measured by conductivity and by a micrometer plus histology for 4 example disks. Conductivity measurements were performed on each disk using all three tetrapolar configurations at each thickness. These 3 conductivity values (Type A data) were used as inputs to a fitting routine that minimized differences between the 3 conductivity values and Eq. 2 by fixing σ2 to either 0.2 S/m (○) or 0 S/m (□) and varying σ1 and d. The resulting best fit thickness’ after each cut are shown compared to the thickness (▵) obtained by subtracting the bone thickness measured microscopically on the histological section from the total disk thickness (cartilage+bone) measured with a micrometer after removing each slice. The thickness values obtained from best fit conductivity measurements using σ2=0 S/m appeared to match histology/micrometer thickness values better than those obtained using σ2=0.2 S/m.




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