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1.1 Overview

Electrical Impedance Tomography (EIT) is the common name to those imaging modalities used to determine the conductivity distribution within an object. In general, the EIT process can be broken down into three steps: 1) the application of currents within the object of interest, 2) measurement of these currents‟ effects, and 3) reconstruction of the conductivity distribution using this measurement data. In conventional EIT, currents are injected into an object through surface electrodes, peripheral voltages are measured, and then an impedance image is reconstructed from these voltages. The maximum number of electrodes that can be placed on the object‟s surface physically limits the number of independent measurements. Also, peripheral voltages are less sensitive to conductivity perturbations in the interior regions of the object. Hence, reconstructed images suffer from poor spatial resolution and non-uniform sensitivity. In this project, Magnetic Resonance Imaging (MRI) will be used to observe the effects of applied currents throughout an entire object -- not just on the periphery. Sinusoidal alternating current within the frequency range of 100Hz to 10KHz will be injected into the object. MRI will measure the component of the current-generated magnetic field that is parallel to the main MRI static field. This data, along with MRI‟s more conventional proton density, T1 weighted, and T2 weighted information, will be used to reconstruct the conductivity distribution within an object. Use of high resolution MRI instead of a limited number of surface voltages measurements will yield better resolution. Use of measurement data throughout the entire object will result in uniform sensitivity. Specifically, EIT capabilities will be integrated into UCI‟s 4T MRI system. This project will investigate the proposed method first in two dimensions, then in 3D phantoms of increasing complexity, with an overall goal of resolving 5mm3 conductivity

perturbations. To this end, two major components of this imaging modality must be developed:

1. High-resolution MRI measurements of the magnetic fields generated by injected alternating currents in complex 3-D objects, obtained in a clinically reasonable time scale. (Data Acquisition) 2. Optimized reconstruction algorithms for determining conductivity distributions from MRI data, focusing on the Finite Element Method and use of the sensitivity matrix. (Data Processing)

1.2 Significance

The ability to acquire high resolution impedance images obtained at various frequencies will allow for investigations that will lead to a better understanding of various physical and biological processes. In addition to potential new findings, conductivity measurements of a person have already been shown to be of direct clinical benefit. It is possible to differentiate between different tissue types using EIT that would be difficult to distinguish using other modalities. For example, muscle and blood tissues have significantly different conductivities, but have nearly the same x-ray attenuation coefficients [1]. Of particular significance, the conductivity of many tumors, including malignant breast tumors, differ from the conductivity of surrounding normal tissue [2]. Hence, EIT can provide a relatively safe and inexpensive method of detecting and localizing tumors. Preliminary studies on EIT breast imaging [3,4] indicate that this modality could eventually be used in conjunction with, or even replace traditional x-ray mammography. Numerous other clinical applications of EIT, such as the imaging of respirator function [5], have been reported [6,7]. Accurate knowledge of an object‟s impedance distribution allows for better calculation of internal field and current distributions caused by external phenomena. Such information can be used to improve the design and application of applied current devices, such as those used in cardiac defibrillation and tumor electrochemo-therapy [8,9]. It would also allow for better estimates on the physiological effects of

electromagnetic radiation, such as in predicting the effects of prolonged exposure to nearby power lines. Conductivity plays a crucial role in the inverse problems of EEG, MEG, ECG, and other electrophysiological diagnostic methods. Localization of electric and magnetic sources from surface measurements requires knowledge of an object‟s conductivity distribution, which is often times based on generalized models. EIT can provide objectspecific conductivity information, resulting in more accurate solutions [10,11].

1.3 History

In 1983, Barber and Brown reported the first in vivo tomographic impedance images [1, 12]. Direct current injection through surface electrodes was the original method of current application, and remains the clinical standard. For the Shefield strategy [13], the most widely used, 16 electrodes are placed around a slice of the object, and various combinations of 2 of these electrodes are used to inject currents. The remaining 14 electrodes measure peripheral voltages, which are used to reconstruct the impedance image. To avoid the use of a physical limited of number electrodes, Ahlfors and Ilmoniemi proposed the technique of Magnetic Impedance Tomography (MIT) in 1992 [14], whereby the magnetic fields produced by an internal current distribution are measured outside the object using SQUIDS. While this allows for an increase in the number of measurements, this method still suffers from non-uniform sensitivity and poor interior resolution. The idea of using MRI to measure current-generated magnetic fields within the interior of an object was suggested by Ider, Muftuler, and Birgul in 1995 as a means of achieving higher resolution and uniform sensitivity [15]. The techniques applicable to measurements of this kind were first proposed for DC in the context of current density of imaging by Pesikan et al. [16]. Later, the method was expanded to measure RF magnetic fields at the resonance frequency of protons within the MRI system [17]. In 1996, Muftuler reported measurement of AC-generated magnetic fields in the audio frequency

range [18]. This method is the one investigated in this project, and explained in further detail in later sections. Over the years, variations in data processing for EIT have also been reported. Barber and Brown‟s original work used backprojection [11]. Since then, numerous groups have tried other algorithms, such as Wexler‟s iteration [19], constrained iteration by the conjugate gradient method [20], and Newton‟s one-step error reconstructor [21]. Of particular important to this project use of the Finite Element Method (FEM) in EIT, first proposed by Murai and Kawaga in 1985 [22]. This method will be investigated in detail and is explained further in later sections.


Preliminary studies were carried out in 2-D to verify and gain familiarity with the reported technique of AC magnetic field measurements [18] and data processing utilizing the sensitivity matrix.

2.1 Theory

The MRI pulse sequence used in the preliminary studies is given in Figure **. In the absence of the applied AC, it is the same sequence used by Maudsley et al. to calculate magnetic field inhomogeneities [23]. A 90-degree RF (RF90) pulse is applied in the presence of a slice selection gradient. This combination rotates the magnetic moment of protons within a selected slice along the z direction into the x-y plane. Spatial gradients Gx and Gy are then applied for a duration Tg. The NMR signal is then refocused by a 180-degree RF pulse (RF180), and the resulting spin echo signal is recorded. The sine wave AC that generates the time varying magnetic field to be measured is triggered just after RF180 and before data collection. In the presence of this time varying (AC) magnetic field b(x,y) cos(t+2), Muftuler showed that the spin echo signal becomes [18]:

S(u, v, t )   m(x, y) exp(i1 ) exp{i(xu  yv)}exp{i[b(x, y) sin(t   2 ) / ]}dxdy .
Where m(x,y) is a measure of the density of protons,  is the proton gyromagnetic ratio constant, and u and v are „spatial frequencies‟ defined as u = * Gx Tg and v = * Gy *Tg. T2 decay and static magnetic field inhomogeneities have been neglected for simplicity. For each scan (TR), the values of u and v are fixed, and a time signal is obtained for a point in (u,v) space. The sequence is repeated for different values of (u,v) by adjusting Gx and Gy until an entire block of (u,v,t) has been sampled, yielding a 3-D data matrix. For each value of t, a 2-D (u,v) matrix is extracted. A 2-D Fourier transform (FT2{}) with respect to u and v is applied to each 2-D matrix, transforming the data from the frequency domain to the spatial domain:
s( x , y, t )  F21{S(u , v, t )}

 m( x, y) exp( i1 ) exp{ i[b( x, y) sin(t   2 ) / ]}  m( x, y) exp( i1 ) exp{ i m f ( x, y) sin(t   2 )}

where mf(x,y) =  b(x,y) / . To determine the z-component of the magnetic fields generated by the AC, the Linear Matrix Equation Formulation (LMEF) can be applied [18]. This first involves linearizing the 2-D spatial time signal by taking the logarithm:
ln s(u, v, t )  ln m( x, y)  i 1  i m f ( x, y) sin(t   2 )

The imaginary part defines a new time signal f(x,y) for each voxel.
f (u, v, t )  1  m f ( x, y) sin(t   2 )  1  m f (x, y) cos 2 sin t  m f (x, y) sin 2 cos t

For each (x,y), the time variation of f(x,y,t) forms a set of linear equations:

cos(t )   f ( x , y, t )  1 sin(t ) 1   f ( x , y,2t )  1 sin(2t ) cos(Nt )      m f ( x, y) cos  2                m f ( x , y) sin  2    f ( x , y, Nt ) 1 sin(2t ) cos(Nt )

f  Ax
x can be solved in the least squares sense:

AT  f  AT  A  f
x  ( A T  A) 1  A  f

The AC magnetic field can be determined from x(3) and x(4):
 2  arctan[x (2) / x (3)]
m f ( x, y)  x (3) / cos  2
b( x , y)   m f ( x , y) /    x (3)  cos  2

Determination of an unknown conductivity distribution within a 2-D object in the x-y plane given the z-component of the magnetic field is defined as the inverse problem. Since the relationship between conductivity and the magnetic field generated by currents within the object is nonlinear, an analytical solution in general does not exist. To solve the problem, a linear approximation will be made, then an iterative technique applied with the goal of converging towards a meaningful solution. As the approximation, a linear relationship between a change in the conductivity distribution  and the resulting change in the (z-component) magnetic field B is assumed, such that:

B  Sσ

where S is called the sensitivity matrix. Given n conductivity elements (ie: n finite elements) and m measurement points,  and B are column vectors given by:

 Δσ1  Δσ       Δσ n   
 B1  B       B m   

where i is the conductivity of element i, and Bj is the magnetic field at point j. S is a mXn matrix. The sensitivity matrix component Sij is the change in magnetic field Bi at point i with respect to a change in the conductivity of element j. Hence, (17) can be written as:
 B1   1  B1   B 2       1  B m       B  m  1  B1  2 B 2  2  B m  2 B1   n   B 2   1      n         n    B m     n   

Note that in general, Sij is not a constant, but depends on the (initial) conductivity distribution , where i is the conductivity of element i. For a given column j of S, the changes in the magnetic field Bi are with respect to a change in the conductivity of a single element j. Solution of the forward problem (determining the field given the conductivity distribution) provides a direct means of calculating the elements of this column:

1. For an initial conductivity distribution , solve for the magnetic field distribution B using the Finite Element Method, where Bi is the field at measurement point i. 2. Change the conductivity of element j by . 3. Solve for the new magnetic field distribution B’ using the FEM. 4. Approximate Sij as (Bi‟-Bi)/.

Once the elements of a given column have been computed, the procedure can be repeated, changing the conductivity of different element to determine another column, until all the elements have been calculated. With a procedure for finding S, the linear approximation can be used to solve the inverse problem. Given a B and computing S, a solution  can be found. If S was a square matrix, then simple matrix inverse could be used to solve the problem.
σ  S 1B

However, in general S is not square. A generalized matrix inversion technique (PenroseMoore) will be used to find the pseduoinverse St. Using singular value decomposition (SVD), S is written as:
S  UV T

where U and V are square orthogonal matrices and  is a diagonal mXn matrix, where m is the number of measurement points and the number of conductivity elements. The inverse of an orthogonal matrix is equal to the transpose of the matrix.
U 1  U T V 1  V T

The inverse of a diagonal matrix is found by inverting its elements i, which are called the singular values. (27)

0 1     Σ 1    n    0 


1   1  0  

 1 n

 0     

The psudoinverse St is given by: (28)
S t  (UΣΣ T ) 1  U 1 Σ 1 (V T ) 1  U T Σ 1 V .

In constructing St, if any of the singular values i are close to zero, the corresponding values composing -1 will be extremely large and create errors in the resulting ~ conductivity distribution change σ . In this case, S is referred to as an ill-conditioned matrix. Singular values below some predefined value t must be truncated by replacing them with zero. The optimal truncation level varies for different objects. ~ The change in conductivity σ can be obtained through matrix multiplication as:
~ σ  S t B

~ where σ is a minimum-norm least squares solution. Recall that S, and hence St, is
calculated based on an initial conductivity distribution i. Ideally, this initial distribution should be equal to the actual (unknown) distribution. An iterative procedure can be utilized to determine the conductivity distribution 0 of an unknown object.

1. The magnetic field distribution B0 within the object is measured. 2. An initial conductivity distribution i is assumed, possibly based on an informed estimate of the actual (unknown) conductivity distribution.

3. The magnetic field distribution Bi produced by the conductivity distribution i is calculated using the FEM. 4. The sensitivity matrix S is calculated based on the conductivity distribution i. ~ 5. With B = B – B and σ = -  , solve for  using (24).
0 i 0 i 0

6. Repeated steps 3-5 with 0 becoming the new i. The iteration is continued until the resulting i‟s converge towards stable solution.

2.2 Experiments

IN PROGRESS - (Still writing this section.)


3.1 Data Acquisition

The preliminary studies were all done in 2-D. However, objects to be imaged in a clinical setting, such as the human body, are inherently three-dimensional. Several 3-D studies have been performed to measure the magnetic fields generated by DC [23,24], but no known studies of 3-D AC field measurements exist. A portion of this project will be devoted to investigating the AC measuring methodology in 3-D. A main complication in moving from 2-D to 3-D objects is that injected currents have additional room to spread out, resulting in lower current densities. This dispersal of currents results in a weaker magnetic field density. Also, current densities are not confined to the x-y plane, meaning that the resulting magnetic fields are no longer „focused‟ into just a z-component, but are three-dimensional as well. Hence, not only is the magnitude of the field smaller, but its measurable z-component may be even less. This project will determine the feasibility of measuring these smaller fields, and find the minimum injected current limits necessary for accurate impedance imaging. Simulations will first be performed, using the FEM to calculate the magnetic field

distributions resulting from a variety of 3D geometries and current injection levels. These calculation will then be used to compute the signal to noise ratio (SNR) to determine measurability. Once the theoretical limits have been determined, MRI measurements will be made on various 3D phantoms to be constructed, and these results compared with the simulations. Throughout this processes, using data obtained from both simulations and experiment, attempts to improve the SNR will also be made. This may include altering the MRI scanning parameters, modifying the current injection strategy and/or „cleaning up‟ hardware to reduce noise. Clinical practicability also requires the ability to image magnetic fields generated by currents in rather sophisticated objects. For humans, the imaging region may contain various tissues types, separated by complex boundaries. More importantly, the region may contain various unknowns (i.e.: tumors), so it becomes crucial that the measuring system be able to detect the effects of these unknowns. Several in vivo studies have been reported for measuring DC and RF current fields. In currently published AC research [25], along with the preliminary studies, only simple nearly homogenous phantoms were imaged. No known in vivo AC studies have been published. In this study, phantoms of increasing complexity will be built, and injected current measurements made. A particular difficultly in constructing these phantoms will be to compartmentalize different regions of varying conductivity without insulating these sections from each other. Previous phantoms were filled with (liquid) solution, which requires physically solid insulating barriers (i.e.: plastic) for separation. For future work, conductive gels may be used, but this material must be synthesized/located and tested for MRI compatibility. After complex phantoms have been constructed and tested, the measuring scheme will be tested on live animals, such as mice. The preliminary studies imaged in low resolution – a 32X32 grid with approximately a 1cm^2 pixel size. To be of clinical diagnostic benefit, high resolution conductivity images are needed, especially in the case of early tumor detection. While high-resolution work has been reported for the DC case [26], only low resolutions have been reported for AC use [25]. In this study, a goal of 5mm^3 resolution has been set. With higher resolution, there is a corresponding lower signal per voxel. In this study, the feasibility of measuring AC-generated magnetic fields with this lower SNR

will be investigated. Under higher resolution, imaging artifacts otherwise „smeared out‟ by a larger voxel size may become significant. Work will be done to reduce any artifacts. This may include hardware modifications and/or software post-processing to eliminate the effects of field inhomogeneities. Imaging high resolutions will also increase scan time, another issue that must be addressed. During the preliminary studies, the scan time for AC measurements was approximately 15 minutes. Using the same imaging sequence, a scan of a 20cm^3 object at 5mm^3 resolution would require several hours. In obtaining high-resolution 3-D impedance images, scan times need to be kept short to be clinical practical, with an upper limit of a couple of minutes. This study will investigate the use of faster alternative sequences for MRI data acquisition. One potential sequence uses a standard spin echo sequence with readout gradient, and can image AC-generated magnetic fields for frequencies greater than 2kHz. However, this method uses a low bandwidth during data acquisition, making it extremely sensitivity to inhomogeneities in the main static field and susceptible to artifacts. Work will be done to test and improve on this sequence. Another potential sequence uses oscillating gradients to decrease scan time, a technique also used in MRI spectroscopy [27]. SENSE is another fast imaging technique that has been proposed for spectroscopy [28], and may be modifiable for use in measuring AC fields.

3.2 Data Processing

The reconstruction process uses the Finite Element Method to calculate the fields produced by injected currents for a given conductivity distribution, an important step in determining the sensitivity matrix. Hence, work will be devoted towards developing FEM processing tailored for EIT. Several processing features, which were done „by hand‟ during the preliminary studies, need to be integrated into an FEM package. Voxels must be correlated between MRI data sets and FEM geometric modeling. From the MR images, object boundaries must be determined and current injection points located. The extraction of such geometric information should also be done in a systematic, preferably automated fashion.

These features must also be compatible with improved MRI data, namely 3D high resolution complex magnetic field maps. Computational speed is also a factor for clinical viability. For the Matlab algorithms used during the preliminary studies, one iteration of a 10X10 impedance map took 5 minutes. In high resolution 3D, the matrices involved will be of order 10^4 larger. Optimization of the sensitivity matrix reconstruction algorithm will be another portion of this study. The effects of MRI artifacts, noise, and different a priori about which linerization occurs will be investigated. Of particular importance is the effect of SVD truncation, which was seen to significantly affect the image quality and convergence rate during the preliminary studies. To be clinically viable, a systematic method for determining optimal truncation must be developed. Finally, the integration of partially known impedance information into the reconstruction process will be investigated. The SMF is in essence an assumption of linearity about some starting point. If a prior data can be used to better set this starting point, greatly reliability and faster convergence of solution should result. This becomes particularly beneficial as objects to be imaged become more complex. In this study, use of the standard proton density, T1 weighted, and T2 weighted MRI will be used to generate an a priori conductivity map, then tested in the reconstruction algorithms.

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