An improved data acquisition method for electrical impedance ...

INSTITUTE OF PHYSICS PUBLISHING

PHYSIOLOGICAL MEASUREMENT

Physiol. Meas. 22 (2001) 31–38

www.iop.org/Journals/pm

PII: S0967-3334(01)20128-2

An improved data acquisition method for electrical impedance tomography Todd E Kerner, Alex Hartov, K Sunshine Osterman, Christine DeLorenzo and Keith D Paulsen Thayer School of Engineering, Dartmouth College, Hanover, NH, USA

Received 12 December 2000 Abstract Isaacson, Cheney and Seager have demonstrated that simultaneously applying trigonometric patterns of current to a circular electrode array optimizes the sensitivity of EIT to inner structure. We have found that it is less desirable to measure voltage at an electrode that also applies a current due to variable contact impedance. In order to preserve the optimum sensitivity while minimizing the effect of electrode artefacts, we have devised an approach where we sequentially apply a current between each individual electrode and a separate, fixed ground while measuring voltages at all other electrodes for each consecutive current impulse. By adding weighted sums of both the applied currents and corresponding measured voltages from individual passes, we can synthesize trigonometric patterns of any spatial frequency. Since only one of the electrodes in any given acquired data set is used as a source, this approach significantly dilutes the effect of contact impedance on the resulting voltage measurements. We present simulated data showing the equivalency between the synthesized and actual trigonometric excitation patterns. In addition, we report experimental data, both in vitro and in vivo, that show improved results using this data acquisition technique. Keywords: EIT, electrical impedance, imaging, data acquisition, synthesized trigonometric patterns

1. Introduction Electrical impedance tomography may be useful for identifying specific lesions in the human body since numerous studies have shown characteristic electrical conductivity and permittivity for different tissues (Gabriel et al 1996, Jossinet 1996). We have constructed an EIT system that is capable of applying arbitrary excitation patterns in either the V-mode or the I-mode (Kerner et al 2000). In the V-mode, the system drives the load with a constant amplitude sinusoidal voltage and measures the magnitude and phase of the current that results. Conversely, in the I-mode, the hardware applies a constant amplitude sinusoidal current to the load and measures the magnitude and phase of the resulting voltage. In either mode, the system can apply and/or 0967-3334/01/010031+08$30.00

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Figure 1. Simple demonstration of measurement errors due to the unknown contact impedance r in both the I-mode and the V-mode. R represents the object being imaged. Only a single channel is shown, but this effect occurs at all electrodes which simultaneously drive and measure voltage and current.

measure voltage or current between any combination of electrodes either simultaneously or sequentially. The primary excitation patterns which we have applied are the trigonometric patterns (V-mode and I-mode) which are defined as ⎧ L ⎪ ⎪ cos(Kθ ) K = 1, 2, . . . , ⎨ 2 % & (1) applied signal = A∗ L L ⎪ ⎪ θ K = + 1, . . . , L − 1 ⎩ sin K − 2 2

where A is the maximum signal amplitude (in volts or amps), L is the total number of electrodes arranged about the phantom (16 here), θ is the angle of each electrode about the centre of the phantom and K is the spatial frequency of the applied pattern (Kerner et al 2000). The high spatial frequencies enable the system to detect objects near the periphery while low spatial frequencies enable it to sense objects near the centre. Unfortunately, in either the V-mode or I-mode, it is not ideal to make a measurement at an actively excited electrode because of the voltage drop pursuant to the unknown and potentially variable contact impedance. When the trigonometric patterns are applied, all 16 electrodes are used for excitation and measurement, so the system is prone to electrode artefact errors in both cases. In the I-mode, the measured voltage is inaccurate due to the contribution of an unknown voltage drop across the electrode-body contact layer (see equation (2)). Conversely, in the V-mode the measured current is also inaccurate for essentially the same reason (see equation (3)). The circuit equivalents of V-mode and I-mode illustrated in figure 1 yield the following relationships which show that the measured quantity is altered relative to the applied signal by the contact impedance, r, in either case: Vmeas = Iapplied (R + r) Imeas =

Vapplied . (R + r)

(2) (3)

To avoid this problem in the I-mode, one could apply current between the combinations of opposite and/or adjacent electrodes and measure the voltage elsewhere. Since we are modelling a linear, time-invariant system, the principle of superposition can be employed. The sum of the resulting voltages and currents from multiple excitation patterns applied at different times should equal the set of voltages and currents that would result if all those excitation patterns were applied at once. Hence, we synthesize the trigonometric patterns by the weighted summation of unit impulse currents applied sequentially between each electrode and a fixed ground electrode as illustrated schematically in figure 2. We call this new excitation pattern the ‘synthesized trigonometric pattern’ or STP for short.

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Figure 2. Simple demonstration of the synthesized approach: the ground node is the upper right electrode. The first three circles have a unit current applied between a different side electrode and the same ground node. The fourth circle is an overlay of the signals from the first three excitations. The lines inside the circles are not physical in nature. They serve merely to graphically illustrate the concept of linear superposition of the signals from the individual electrode excitations as being equivalent to simultaneous excitation of all three positions.

The entire synthesis process can be summarized mathematically in the group of equations (4). Matrix U represents either the applied unit impulse currents or the corresponding measured voltages while matrix T contains the desired trigonometric patterns. The cosine T expression (left) is valid for k = 1, 2, . . . , L/2 while the sine T (right) is valid for k = L/2 + 1, . . . , L − 1. ⎛ ⎞ ⎛ ⎞ 1 0 0 ... 0 1 ⎜0 1 0 ... 0⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜0 0 1 ... 0⎟ ⎜ ⎟ ⎜ 3 ⎟ /L) ⎟ U=⎜ T = cos(2π k ⎜0 0 1 ... 0⎟ ⎜ . ⎟ ⎜. . . . ⎟ ⎝ .. ⎠ ⎝ .. .. .. .. 0 ⎠ L−1 0 0 0 ... 1 ⎛ ⎞ 1 ⎜ 2 ⎟ ⎜ ⎟ 3 ⎟ T = sin(2π[k − L/2] ⎜ S = WU W = U\T + null(U). (4) ⎜ . ⎟ /L) ⎝ .. ⎠ L−1 Matrix S represents the STPs where the weighting function is contained in matrix W. Since MATLAB is used for computation here, we use the MATLAB notation ‘\’ indicating left matrix division. If U is square, then W is the solution to UW = T computed by Gaussian elimination. If the system is under-specified, then W is a solution to UW = T in the least squares sense. The function ‘null(U)’ returns the orthonormal basis for the null space of U found by singular value decomposition. The complete minimum norm solution is found by adding the solution to UW = T to an arbitrary vector from the null space. 2. Simulations In order to demonstrate the equivalence of the synthesized trigonometric patterns and the actual ones and the reversibility of the voltage and current as the driving and measured signals, a set of simulations were performed in a noise-free environment. These equivalences between synthesized and simultaneous data as well as I-mode and V-mode reconstructions under ideal conditions allow us to argue in section 5 that trigonometric patterns must also be optimal for the V-mode in a circularly symmetric geometry even though no formal mathematical proof has been developed to date. The demonstration of reversibility of voltage and current as driving

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Figure 3. The σ (left image) and ε (right image) material distribution used in the simulation. The σ image (left) has a colourmap depicting electrical conductivity in units of S m−1 while the ε image (right) has a colourmap depicting the relative electrical permittivity, which is unitless. The material values were chosen to be close to actual physiological values in arm tissue at 125 kHz. The vertical and horizontal axes are in metres.

(a)

(b)

(c) Figure 4. The σ and ε images from the simulations at 125 kHz. The σ images (left) have a colour map depicting electrical conductivity in units of S m−1 while the ε images (right) have a colour map depicting relative electrical permittivity, which is unitless. The vertical and horizontal axes are in metres. (a) Actual trigonometric excitation pattern (I-mode). (b) Synthesized trigonometric excitation pattern (I-mode). (c) Actual trigonometric excitation pattern (V-mode).


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and measured signals also shows that our image reconstruction algorithm is not biased towards V-mode or I-mode under ideal circumstances. This is important for eliminating characteristics intrinsic to the reconstruction algorithm as the source of the image quality differences observed in the reconstructions from experimental data in section 4. To complete this simulation, an arbitrary material distribution of σ and ε was created (as shown in figure 3). Then a series of different excitation patterns were applied to this particular system. The following cases were considered: (a) actual trigonometric patterns in the I-mode, (b) synthesized trigonometric pattern in the I-mode and (c) actual trigonometric patterns in the V-mode (summarized in figure 4). Our standard forward solver calculated the measured currents or voltages for each excitation pattern. Finally, the excitation patterns and calculated measurements were supplied to the same exact reconstruction program to generate final σ and ε images (Paulsen et al 1994, 1998). From figures 4(a), 4(b) and 4(c), one can see that the trigonometric images, whether synthesized or actual, are nearly identical and match the original material distributions quite well. Figures 4(a) and 4(c) demonstrate the equivalence of the images obtained from trigonometric voltage excitation patterns versus trigonometric current excitation patterns under the idealized conditions of a noiseless environment. 3. Experimental setup 3.1. EIT A host computer executing custom software completely controls the EIT system. Each channel is individually set with 12-bit accuracy to apply a given amplitude sinusoidal voltage (1 V maximum in V-mode) or current (5 mA maximum in I-mode) to each of the 16 electrodes. Each channel can be switched to one of three modes: voltage source, current source or float. The computer controls the frequency of the signal applied to all channels, and in these experiments we picked one frequency: f0 = 125 kHz. During the experiments reported here, the trigonometric patterns were applied in V-mode with all channels actively supplying a specified voltage although related studies have also investigated trigonometric patterns in I-mode as well. Conversely, the synthesized patterns were administered in the I-mode with only the excitation electrode active (current source) while all remaining electrodes operated as passive voltage sensors. In the voltage mode, we applied a known voltage to each electrode and measured the resulting current magnitude and phase. In the current mode, we applied a known current to a single electrode, and measured the resulting magnitude and phase of the voltages at the remaining electrodes. In both cases, we used a 16-bit A/D with an effective sampling rate of 100 MHz (employing a multi-period undersampling technique) to digitize and record our measurements (Hartov et al 2000). 3.2. Electrodes All experiments listed below used 16 8 mm diameter Ag/AgCl electrodes (In Vivo Metric, Inc.) mounted on a rigid supporting ring. Each electrode could be moved radially and locked in place to accommodate objects of variable diameter. Before each experiment, the electrodes were buffed with steel wool and saline gel was applied to their surface to improve electrical contact with the material being imaged. Figure 5 illustrates the experimental setup for the agar and leg as described below.

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Figure 5. Electrode apparatus during the agar and the leg imaging experiments.

Figure 6. Absolute conductivity images (in S/m) of agar at 125 kHz with a 2.4 cm diameter air hole near 12 o’clock reconstructed from V-mode trigonometric (left) and I-mode STP (right) data; ten iterations were performed to produce each image. The x and y axes are in metres.

Figure 7. Absolute permittivity images (unitless) of the leg at 125 kHz reconstructed from V-mode trigonometric (left) and I-mode STP (right) data; ten iterations were performed for each image. The x and y axes are in metres.

3.3. Agar A 10 cm diameter, 4 cm high cylinder was made by mixing 0.9% NaCl solution, 4% agar, and 0.1% DowicilTM 95 preservative. The solution was heated for 30 min at 121 ◦ C and allowed


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to cool and congeal in a 1000 ml beaker over a few days. An apple corer was used to create a 2.4 cm diameter air hole in the agar. The ground node was an EKG electrode placed on the top of the agar. 3.4. Leg A 27 year old male placed his left leg in the electrode apparatus 2 cm below the knee cap (upper calf). The skin was not shaved. Although the cross-section here was not a perfect circle, the electrodes were adjusted so they were spaced as evenly as possible around the perimeter. The ground node was a 4 mm diameter Ag/AgCl electrode placed between electrodes 15 and 16. 4. Summary of results The STP images were consistently clearer than the corresponding V-mode trigonometric images in all experiments. In the absolute conductivity agar images, the distortion along the edges and central spurious noise decreased as shown below in figure 6. In figure 7 the tibia bone (at 2 o’clock) is more pronounced in the STP images of absolute permittivity obtained during leg experiments. 5. Discussion Initially, simulations were conducted in a noise-free environment in order to demonstrate the equivalence of the actual and synthesized trigonometric patterns under ideal conditions (figure 4). These simulations also showed that one can reverse the excitation and measurement data within the reconstruction algorithm and generate nearly identical images. With noise present in experiments (±0.001 mA, ±0.0004 V), reversing the data does yield different results. We found that the best images occurred with our reconstruction program if we used the data as though we applied a voltage and measured a current. Hence, we performed this reversal with all of our experimental data obtained in I-mode. Clearly, reversibility is an issue that warrants further study. Although we have not developed a formal mathematical proof, the striking similarity between the simulated images in the V-mode and I-mode suggests that the optimal excitation patterns must be trigonometric in both cases. Carrying this argument one step further would imply that if some other (non-trigonometric) excitation were found in the V-mode which was better, the data could be reversed to produce an improved I-mode image, which would contradict the formal proof of Isaacson regarding optimal driving patterns for I-mode in a circularly symmetric geometry. We synthesized trigonometric patterns from unit impulse currents to maximize sensitivity while minimizing the electrode artefacts inherent to this mode of excitation. During each pass of the impulse excitations, the voltage was measured across 15 passive electrodes and only one current carrying electrode, so this technique diluted the effect of electrode contact impedance. In the agar and leg experiments, the images obtained from the synthesized patterns were consistently clearer than the corresponding actual trigonometric images in the V-mode. In each case, the peripheral electrode artefacts were reduced and the central objects appeared sharper. Even though superimposing the measurements from single impulses may compound noise (reducing the signal-to-noise ratio relative to that achievable with the actual I-mode trigonometric excitations), the gain afforded by the synthesis technique was evident. In fact, we have found the synthesis technique to be quite useful in practice. The actual I-mode trigonometric patterns can fail if the multiple current drivers which are required are not calibrated to extremely high precision. Since our system was designed to operate over a wide

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frequency range (10 kHz to 1 MHz), our current sources cannot be calibrated that precisely. In addition, there are some inherent advantages with the impulse excitations. For example, there is an economy in hardware because a single current source can be switched to each electrode sequentially rather than having individual current sources for each electrode. Also, since the aggregate current output must be limited for safety, a higher peak can be injected between a single electrode and ground than between a system of 16 electrodes simultaneously delivering current, which may serve to improve the signal-to-noise ratio through gain in the measured signal strength. In all of our experiments performed at 125 kHz, we chose a 5 mA limit because this adhered to the safety regulations of the FDA. In general, the safe limits for maximum current injected into the human body increase linearly with log frequency from 1 mA at 10 kHz to 10 mA at 1 MHz. Below 10 kHz, the maximum safe current is fixed at 1 mA, and conversely above 1 MHz the maximum safe current is fixed at 10 mA. Finally, one can synthesize any arbitrary pattern from the impulse data once collected; thus, if some optimal or case-specific pattern is discovered later, one can synthesize it after the fact from the previously acquired measurements. Acknowledgment This work is currently supported by grants R01CA64588 and PO1CA80139 awarded by the National Cancer Institute. References Cheney M and Isaacson D 1992 Distinguishability in impedance imaging IEEE Trans. Biomed. Eng. 39 852–60 Gabriel C, Gabriel S and Corthout E 1996 The dielectric properties of biological tissues; I. Literature survey Phys. Med. Biol. 41 2231–49 Hartov A, Mazzarese R, Reiss F, Kerner T, Osterman S, Williams D and Paulsen K 2000 A multi-channel continuouslyselectable multi-frequency electrical impedance spectroscopy measurement system IEEE Trans. Biomed. Eng. 47 49–58 Isaacson D 1986 Distinguishability of conductivities by electric current computed tomography IEEE Trans. Med. Imaging. 5 91–5 Jossinet J 1996 Variability of impedivity in normal and pathological breast tissue Med. Biol. Eng. Comput. 34 346–50 Kerner T, Williams D B, Osterman K S, Reiss F R, Hartov A and Paulsen K D 2000 Electrical impedance imaging at multiple frequencies in phantoms Physiol. Meas. 21 67–77 Paulsen K D and Jiang H 1997 An enhanced electrical impedance imaging algorithm for hyperthermia applications Int. J. Hyperth. 13 459–80 Paulsen K D, Moskowitz M J and Ryan T P 1994 Temperature field estimation using electrical impedance profiling methods. I. Reconstruction algorithm and simulated results Int. J. Hyperth. 10 209–28 Seager A D, Barber D C and Brown B H 1987 Theoretical limits to sensitivity and resolution in impedance imaging Clin. Phys. Physiol. Meas. 8 (suppl A) 13–31