Electrical Impedance Spectroscopy imaging of the thigh using current excitation frequencies in the mid-β frequency dispersion range.
Jose Antonio Gutierrez-Gnecchi1, Miguel Angel Mendoza-Mendoza1, Carlos Eduardo Guillen-Nepita1, Daniel Lorias-Espinoza2, Adriana del Carmen Tellez-Anguiano1. 1 Instituto Tecnológico de Morelia, Departamento de Ingeniería Electrónica, Avenida Tecnológico 1500, Col. Lomas de Santiaguito, Morelia, Michoacán, México. C. P. 58120. Telephone: (52) 4433121570 ext 270, Email:
[email protected]. 2 CINVESTAV/IPN, Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, C.P. 07360 México, D.F., México. Telephone: (52) 5557473800. Email:
[email protected]
Abstract. Despite the profuse activity in development of electrical tomography imaging systems during the last three decades, qualitative image reconstruction methods are chosen for practical applications because quantitative methods may fail to converge when imaging heterogeneous media. However, quantitative information is of great interest for clinical diagnosis purposes. The authors present thigh images using qualitative (Modified Back-Projection: MBP) and quantitative (Modified Newton-Raphson: MNR) methods using current excitation signals in the α and β dispersion frequencies. Qualitative results indicate that as frequency increases it is possible to differentiate anatomical features. Quantitative images are in agreement with qualitative results although some small anatomical features remain undetected when the surrounding impedance is predominant. Nevertheless, increasing the current excitation frequency favours the quantitative reconstruction process. The advantage is that numerical impedance data can be derived from quantitative images and has important implications for in-vivo clinical diagnosis. Keywords: Electrical Impedance Tomography, Electrical Impedance Spectroscopy, Biompedance, Digital Signal Processing.
1
Introduction
Biompedance measurements stand out as a useful tool for monitoring tissue state [1], function and alterations. The importance of bioimpedance measurements has been recognized for over a century. For instance cardiac output [2], blood flow assessment [3], blood pressure [4], body and water composition [5] represent just a few of the current non-invasive clinical uses of biompedance measurements. Moreover, the success of ionizing-radiation based imaging methods, led to the proposal of soft-field
adfa, p. 1, 2011. © Springer-Verlag Berlin Heidelberg 2011
imaging methods based on impedance measurements. In particular, Electrical Impedance Tomography (EIT) are suited for imaging conductive media. EIT clinical reports have been published since the early 80s [6]: lung ventilation [7], brain activity [8] and cancer diagnosis [9] are amongst the reported applications. 1.1
Electrical Impedance Tomography fundamentals
In a similar manner to other tomographic techniques, in electrical impedance tomography imaging the goal is to obtain projections that can be used to reconstitute the internal properties (impedance) of the object studied. The two-dimensional EIT measurement technique uses a number of electrodes (normally 8, 16, 32 electrodes) separated equidistantly around the periphery of the object in a single plane (Fig. 1A).
Fig. 1. A) EIT principle of operation and B) block diagram of a typical EIT system.
A typical EIT acquisition system can be divided in three main parts: the data acquisition system, the electrode array and the image reconstruction and data acquisition controller (Fig. 1B). The premise is to apply a controlled alternating current through the electrode array and measure the voltages that develop in the periphery. Various current injection techniques have been proposed to interrogate the cross-section impedance properties. The adjacent electrode method uses a pair of consecutive electrodes for current injection. The resulting voltages around the periphery are measured selecting pairs of the remaining electrodes. Once all measurements have been obtained in all the relevant combinations (i. e. excluding redundant measurements), the current in applied using the next pair of consecutive electrodes, shifting one position and the voltage measurement process is repeated until all possible combinations have been tested. The number of measurements, Nm, obtained using the adjacent-electrode method is given by (1): 𝑁𝑚 =
𝑛(𝑛−3) 2
(1)
where n is the number of electrodes. Thus a 16-electrode array yields a data frame of 104 measurements using the adjacent electrode method.
1.2
EIT Mathematical basic Principles
The fundamental principles governing the EIT imaging process can be derived from Maxwell’s equation and three basic simplifying assumptions [10]: 1) Quasi-static conditions hold. Therefore, Ohm’s law may be applied: 𝚥⃗ = 𝜎𝐸�⃗
(2)
where j is the current density (A m ), σ is the conductivity (S m ) and E is the electric field (V m-1). -2
-1
2) There are no electrical current sources or sinks within the bounded volume. ∇ ∙ 𝚥⃗ = 0
(3)
3) The conductivity distribution, σ, within the bounded volume is isotropic. Thus the electric field density, E, as a function of potential 𝜙 (4); 𝐸�⃗ = ∇𝜙
(4)
∇ ∙ (σ∇ϕ) = 0
(5)
∇2 𝜙 = 0
(6)
Combining equations (2), (3) and (4), yields Poisson’s equation (5):
If the conductivity distribution is considered homogenous, equation (5) can be further simplified to Laplace’s equation (6):
A unique solution for (6) can be sufficient boundary conditions are specified 𝜙=0 𝜕𝜙
𝜎� � = 0 𝜕𝜂 𝜕𝜙
𝜎� � = 𝑗 𝜕𝜂
𝑎𝑡 𝑒𝑎𝑟𝑡ℎ
𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒𝑠 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒
(7)
However, equation (6) is complicated and there is no analytical solution for an arbitrary conductivity distribution. EIT image reconstruction requires solving the forward and inverse problems.
1.3
EIT Forward problem
Since the applied current is known and the resulting voltages around the object are measured, solving the forward problem consists of finding the potential and current density functions inside the object. A common approach consists of discretizing the domain using a Finite Element Model to simplify the solution of (6) into a linear set of equations to find a piece-wise approximation to the exact solution (Fig. 2).
Fig. 2. Domain discretization using a Finite Element Model
For the general element, e, an approximation is sought in such way that the outside of the element (8): 𝜙 𝑒 (𝑥, 𝑦) = 0
𝑒 = 1, … , 𝐸
(8)
where E is the total number of elements. The approximated solution can be expressed as (9): Φ(𝑥, 𝑦) = ∑𝑒 𝜙 𝑒 (𝑥, 𝑦)
(9)
𝜙 𝑒 (𝑥, 𝑦) = 𝑁1 (𝑥, 𝑦)𝜙1 + 𝑁2 (𝑥, 𝑦)𝜙2 + 𝑁3 (𝑥, 𝑦)𝜙3
(10)
𝜙(𝑥, 𝑦) = 𝐴 + 𝐵𝑥 + 𝐶𝑦
(11)
���⃗ = 𝐶⃗ [𝑌] ∙ Φ
(12)
The next step consists of choosing the interpolation function to represent the element approximation 𝜙 𝑒 (𝑥, 𝑦) in terms of the unknown node potentials. Usually polynomials are employed as interpolation functions since they are easy to manipulate both algebraically and computationally. Assuming that the element e has s modes, the approximation function in element e is given by:
The shape functions, Ni, are obtained bearing in mind that the potential within each element is expressed as:
It can be shown that the result can be expressed in matrix notation as (12):
The Y conductance matrix is an n x n sized matrix where n is the number of nodes. This matrix is symmetrical, positive definite, double centred and sparse. Φ is an n-
sized vector which represents the voltage at n points inside and on the boundary of the vessel. C is an n-sized vector which contains the applied current on the boundary. 1.4
Qualitative image reconstruction
In the 70s, Geselowitz [11] demonstrated that a change in conductivity distribution inside the test region can be determined from data obtained at the boundary. Later in the 80s, Kim et al. [12] and Murai and Kagawa [13] defined the forward problem of relating a conductivity distribution to resulting transfer impedance by using the sensitivity theorem of Geselowitz. Geselowitz compensation theorem states that if the conductivity distribution inside the object changes from σ(x,y) to σ(x,y) + Δσ(x,y) the change in mutual impedance Δz throughout the volume is given by: ∆𝑧 = −∆𝜎∫𝜐
∇𝜙 ∇(𝜎+∆𝜎) 𝐼𝜓
𝐼𝜙
𝑑𝜐
(13)
The solution of (13) is identical to that of the inverse problem in EIT whereby the change in conductivity distribution σ+ Δσ is to be obtained from a given change in mutual impedance, Δz, taken at the boundary of the object resulting from the applied current distribution. In general the inverse problem of EIT image reconstruction can be approached by inverting the forward transformation matrix. However, the matrix inversion problem is ill-conditioned and non-linear. This requires the use of iterative methods. A comparison of several iterative approaches has been made by Yorkey et al. [14]. The large amount of data storage and computational time required for iterative image reconstruction lead to the development of alternative methods such as the adaptive current method suggested by Gisser et al. [15-16]. However, by addressing the problem of imaging small changes in conductivity distribution [17], the current distribution within the object can be considered unchanged. If the change of conductivity is small it can be shown that the small changes in Δz is governed by the integral (14): ∫𝜐
∇𝜙 ∇ψ 𝐼𝜙 𝐼𝜓
𝑑𝜐
(14)
which is termed the sensitivity coefficient.. For a discretized volume (14) correspond to the sensitivity coefficient matrix S. A number of researchers have addressed the linearized EIT image reconstruction problem via analogy with the method of filtered backprojection. Barber et al. [18] employed the backprojection of normalised changes in boundary potential. Kotre [|19-20] used a sensitivity coefficient weighted backprojection approach with the inverse matrix approximated by a weighted transpose of the sensitivity coefficient matrix for single-step reconstruction. Abdullah et al. [22] modified Kotre’s preferred algorithm which assumes orthogonality of the sensitivity matrix such that the sensitivity matrix S: 𝑆𝑇𝑆 = 𝐼
(15)
To minimise the problems associated with this assumption, S is multiplied by a diagonal weighting matrix composed with the absolute values of the rows of the matrix
S prior to transposition. The resulting pixels grey levels, g, constituting the image are determined from (16): 𝑔⃗ = 𝑆𝑤𝑇 log �
𝜐𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
𝜐ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠
�
(16)
This algorithm is used throughout this thesis for qualitative image reconstruction. A Skyline storage scheme was introduced [22] and finally included data acquisition features to control the data gathering process. 1.5
Quantitative image reconstruction
Yorkey [14] demonstrated the usefulness of an iterative image reconstruction algorithm (Modified Newton-Raphson), and the implications for process monitoring and model validation led to develop the method further. Abdullah [23] re-derived Yorkey’s original MNR-based algorithm to enable it to work with real data. The method uses FEM (like the weighted backprojection algorithm) to replace the Laplace equation with a set of linear algebraic equations. These are solved to form an n dimensional vector ν: 𝑘,𝑘+1 = 𝑓(𝜌1 , 𝜌1 , … , 𝜌𝑚 , 𝐼𝑖 , 𝐼𝑖+1 ) 𝜐𝑖,𝑖+1 1≤𝑖≤𝜌 𝑓𝑜𝑟 𝑖 + 2 ≤ 𝑘 ≤ 𝑁 − 1, 𝑖 = 1 𝑖 + 2 ≤ 𝑘 ≤ 𝑁, 𝑖 ≠ 1
(17)
which represents the differential voltage collected at both the k and k+1th electrodes resulting from a current applied between the i and i+1th pair. This vector contains n independent data sets from a mesh containing N boundary electrodes and P projection angles. In order to obtain a complete set of independent measurements, p, current patterns are applied and their corresponding boundary voltages calculated, to form an nth dimensional column vector νcal and its equivalent measured counterparts are νmea. The objective function is found from the sum squared difference between νmea and νcal, [23]: 1
𝜙 = [𝜐𝑐𝑎𝑙 − 𝜐𝑚𝑒𝑎 ]𝑇 [𝜐𝑐𝑎𝑙 − 𝜐𝑚𝑒𝑎 ] 2
(18)
so that the desired update function, can be shown to be (19): ′ ]𝑇 [𝜐 ′ ]}−1 [𝜐 ′ ]𝑇 [𝜐 (𝜌 − 𝜌𝑘 ) = −{[𝜐𝑐𝑎𝑙 𝑐𝑎𝑙 − 𝜐𝑚𝑒𝑎 ] 𝑐𝑎𝑙 𝑐𝑎𝑙
(19)
Yorkey demonstrated that iterative algorithms (MNR) were superior to other image reconstruction methods such as the backprojection method when it comes to noisefree data. However, in realistic situation where the measured boundary voltages are bound to contain more than 1% RMS error, stability problems occur. In this work this algorithm was used for off-line image reconstruction as means of comparison with the weighted back projection method.
2
Data acquisition system (DAQ)
As with all measurement systems the sensor is probably the most critical part. Practical considerations place a number of constraints. Ideally it should be robust, non-intrusive and require minimum maintenance or calibration. On the other hand, the wide dynamic range of measurement conditions occurring in clinical applications requires the DAQ to be flexible enough to obtain measurements with optimum signalto-noise ratio. Consequently, a wide range of current injection amplitudes, wide bandwidth and the capability of connecting a large number of electrodes are all necessary features. Although 'flexibility' is the key term considered when designing EIT systems, the strict safety requirements of clinical instrumentation constraint, accordingly, biomedical EIT data collection systems. Fig. 3 shows the schematic diagram of the data acquisition developed for EIT biomedical applications.
Fig. 3. Block diagram of the data acquisition system. A) The host computer interfaces through B) an isolated USB circuit. C) Direct Digital Synthesizer (DDS). D) 250 MHz buffer. E) Array of 16 voltage controlled current sources and F) 16 electrodes. The measured voltages are collected through G) a multiplexer array, H) signal conditioned and I) digitized for image reconstruction. J) A microcontroller synchronizes the data acquisition process. K) The DAQ is powered by an isolated DC-DC converter.
The voltage signal is generated through a Direct Digital Synthesizer (DDS) which allows a wide operation bandwidth. The voltage controlled current source (VCCS) stage is critical for the correct operation of the entire system. A common approach is to use analogue multiplexer to select the current excitation electrode pair. However, the use of multiplexers reduces the useful bandwidth. Here a multiplexerless VCCS based on operational transconductance amplifiers is used to allow measurements in the β dispersion range, as described elsewhere [24]. A fast approach for measuring the voltages that develop in the periphery due to current excitation, involves using a modulation-demodulation scheme. Here a 1-billion samples per second analogue to digital converter is used to digitize the measured signals in the frequency range proposed. A microcontroller unit synchronizes the data acquisition process. The entire
data acquisition system is powered through an isolated DC-DC converter to reduce leakage current issues and comply with safety directives.
3
Experimental setup
Being an iterative method, quantitative image reconstruction requires a good SNR to benefit the convergence process. However, in medical applications the maximum auxiliary current allowed into the patient is restricted in magnitude and as a function of frequency. Thus the maximum current is restricted by hardware (Fig. 4A) to be 1mA at 200 kHz and onwards. In addition to limiting the amplitude of the excitation current, the isolated power supply and isolated USB bus interface connections between the digital electronics and the PC contribute to the safety of the equipment. The system was set to generate a 0.5 mA current signal well within safety considerations. The test trials where conducted on a volunteer’s thigh, after exercising during 5 minutes to increase the blood content. A set of 16 2” x 1” stainless-steel electrodes were located equidistantly in the thigh for imaging a single plane (Fig. 4B).
Fig. 4. A) Close-up of the 16 VCCS used for EIT imaging. B) Set of 16 electrodes attached to the thigh.
Data frames were obtained using a 0.5mA excitation current, at 200 kHz, 1 MHz and 20 MHz. After the current injection electrode pair has been selected, 15 cycles are applied to the tissue. The impedance is calculated using the last 10 cycles. Images were reconstructed using qualitative and quantitative reconstruction methods using a 464-element mesh, and post-processed using the bilinear interpolation method on a 100 X 100 grid. All the code was written in C++, except for the contouring tracing procedure which was implemented using the AVS Toolmaster programming suite (Advanced Visual Systems).
4
Results and discussion
Fig. 5 shows a graphical summary of the test results. Row A shows a sequence of qualitative images; quantitative results are shown in row B. The thigh represents a challenge for soft-field imaging; it is complex structure including veins, arteries, skin, bone, muscle, fat and nerve tissues. At lower frequencies (200 kHz) the reconstructed images are in agreement, although very little information can be obtained.
Fig 5. Graphical summary of the results. Row A): images obtained using the qualitative method. Row B) quantitative imaging results. Apart from the muscle region with high vascularization due to exercising, EIT cannot differentiate other anatomical features at low frequencies, mainly due to the highly heterogeneous nature of the thigh. As frequency increases various effects can be observed. Although the reconstructed images are shapely distorted because a circular finite element model was used to represent a non-circular cross-section, some higherconductivity regions can be identified and are consistent with higher vascularization due to exercising, corresponding to the long head biceps femoris, semitendious and semimembranous muscles (Fig 5 Ai). At low frequencies, the similar conductivity of bone (0.01–0.06 S/m), muscle (0.04–0.14 S/cm) and fat (0.02–0.04 S/m) impedes delimiting the internal organ boundaries (Fig. 5Aii). The increased blood concentration (0.43–0.7 S/m) contributes to identify the region with higher vascularization. As frequency increases, the impedance properties of the different tissues begin to appear. At 20 MHz the reconstructed images show two other regions of higher conductivity consistent with blood flow at the location of the femoral artery and femoral vein (Fig. 5 Aii, Fig 5Bii) and the great saphenous vein (Fig. 5Aiii, Fig. 5Biii). The image reconstruction methods tested appear to be more sensitive to regions of higher conductivity. The results indicate that unless there is a significant impedance change that can be correlated to a characteristic location or property, it is difficult to differentiate tissues. Increasing the current excitation frequency contributed to identify some muscle location and vascular activity due to higher vascularization, and thus higher conductivity. In contrast, it was not enough to differentiate bone tissue (femur)
from muscle tissue (vastus lateralis). Another factor that impeded the identification of the bone-muscle boundary is the resolution of the adjacent electrode measurement used: the poorest resolution is obtained at the centre of the image. In contrast, it was shown that it is possible to extend the overall operating frequency. Thus the results suggest that there is room for improvement, by increasing the frequency of the current excitation signal.
5
Conclusions
The use of qualitative and quantitative image reconstruction methods was presented. At low frequencies, qualitative and quantitative images where morphologically similar, although both failed to differentiate most anatomical features. As frequency increased, it was possible to distinguish some predominantly conductive regions. The advantage of the quantitative method is that numerical resistivity values could be obtained in contrast with comparative from the qualitative method. Another important factor is that qualitative images could be obtained using a low amplitude current excitation signal. The results suggest that there are other clinical diagnosis applications that can benefit from EIT measurements, for instance, in breast cancer diagnosis since the reported conductivity of tumors (0.22–0.4 S/m) is ten times larger than that of fat, but ten times lower than blood. The relative permittivity of tumors is also of great interest: tumors exhibit lower permittivity that fat and muscle. Thus it is essential to maintain a measurement scheme that can be used to determine both the real and imaginary components.
6
Acknowledgements
The authors acknowledge the financial support of Secretaría de Educacion Pública through Dirección General de Educación Superior Tecnológica (SEP/DGEST) and CONACYT to carry out this work.
7
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