Anisotropy In Diffusion And Electrical Conductivity .
Anisotropy in Diffusion and Electrical Conductivity Distributions of TX-151 PhantomsbyNeeta Ashok KumarA Thesis Presented in Partial Fulfillmentof the Requirements for the DegreeMaster of ScienceApproved November 2015 by theGraduate Supervisory Committee:Rosalind Sadleir, ChairVikram KodibagkarJitendran MuthuswamyARIZONA STATE UNIVERSITYDecember 2015
ABSTRACTAmong electrical properties of living tissues, the differentiation of tissues ororgans provided by electrical conductivity is superior. The pathological condition ofliving tissues is inferred from the spatial distribution of conductivity.MagneticResonance Electrical Impedance Tomography (MREIT) is a relatively new non-invasiveconductivity imaging technique. The majority of conductivity reconstruction algorithmsare suitable for isotropic conductivity distributions. However, tissues such as cardiacmuscle and white matter in the brain are highly anisotropic. Until recently, theconductivity distributions of anisotropic samples were solved using isotropic conductivityreconstruction algorithms. First and second spatial derivatives of conductivity ( σ and 2σ ) are integrated to obtain the conductivity distribution. Existing algorithms estimate ascalar conductivity instead of a tensor in anisotropic samples.Accurate determination of the spatial distribution of a conductivity tensor in ananisotropic sample necessitates the development of anisotropic conductivity tensor imagereconstruction techniques. Therefore, experimental studies investigating the effect of 2σon degree of anisotropy is necessary. The purpose of the thesis is to compare theinfluence of 2σ on the degree of anisotropy under two different orthogonal currentinjection pairs.The anisotropic property of tissues such as white matter is investigated byconstructing stable TX-151 gel layer phantoms with varying degrees of anisotropy.MREIT and Diffusion Magnetic Resonance Imaging (DWI) experiments were conductedto probe the conductivity and diffusion properties of phantoms. MREIT involved currentinjection synchronized to a spin-echo pulse sequence. Similarities and differences in thei
divergence of the vector field of σ ( 2σ) among anisotropic samples subjected to twodifferent current injection pairs were studied. DWI of anisotropic phantoms involved theapplication of diffusion-weighted magnetic field gradients with a spin-echo pulsesequence. Eigenvalues and eigenvectors of diffusion tensors were compared tocharacterize diffusion properties of anisotropic phantoms.The orientation of current injection electrode pair and degree of anisotropyinfluence the spatial distribution of 2σ. Anisotropy in conductivity is preserved in 2σsubjected to non-symmetric electric fields. Non-symmetry in electric field is observed incurrent injections parallel and perpendicular to the orientation of gel layers. The principaleigenvalue and eigenvector in the phantom with maximum anisotropy display diffusionanisotropy.ii
ACKNOWLEDGMENTSFirst and foremost, I offer my sincerest gratitude to Dr. Rosalind Sadleir forsupporting me throughout my thesis. I am grateful to Dr. Jitendran Muthuswamy, Dr.Rosalind Sadleir and Dr. Vikram Kodibagkar for serving on my defense committee.Finally, a special thanks to Ms. Laura Hawes for graduate advising. The members in theNeuro-Electricity Laboratory deserve a special thanks for providing continued supportthroughout my thesis.iii
TABLE OF CONTENTSPageLIST OF TABLES .viLIST OF FIGURES . viiCHAPTER1.INTRODUCTION. .1Purpose .22.BACKGROUND.4Previous Work.4Theoretical Considerations of MREIT.9Diffusion Tensor Imaging.183. MATERIALS AND METHODS .21Anisotropic Gel Phantom Design . 21Sample Chamber and Miter Box Design . 23Magnetic Resonance (MRI) Experiments . 25Impedance Analyzer.28Finite - Element Method.30MREIT Data Processing.33DTI Data Processing.354.RESULTS.37Diffusion Tensor Image Analysis.37MREIT Data Processing.435.DISCUSSION.56iv
CHAPTERPageDiffusion Tensor Imaging.50Magnetic Resonance Electrical Impedance 61AGLOSSARY OF TERMS.61BIMPEDANCE MEASUREMENT USING HP4192A.65CRAW DATA COLLECTED FROM BRUKER, BIOSPIN 7 T.70DPHASE UNWRAPPING AND Z-COMPONENT OF BZ.74ESPATIAL DERIVATIVES OF CONDUCTIVITY PROFILE.79FDIFFUSION TENSOR ANALYSIS.83GCONSTANT CURRENT SOURCE - POSITIVE AND NEGATIVEINJECTIONS.88HBZ FROM TRANSVERSAL CURRENT DENSITY BY BIOT-SAVARTLAW.90v
LIST OF TABLESTablePage1. Comparison of the Pros and Cons of MREIT and EIT . 82. Influence of Echo and Relaxation Time (TE, TR) on Image Contrast. . 133. Recipe for High and Low Conductivity Gels . 234. Imaging Parameters in MREIT and DTI Experiments. . 255. Current Source Parameters during MREIT Experiments. . 276. (a) Percent Decrease in SNR with Increase in Length of Diffusion-SensitizingMagnetic Field Gradients in Isotropic Voxels of Side 10.5 mm. (b) Percent Decrease inSNR with Increase Size of Isotropic Voxels under Diffusion-Sensitizing Gradients of 100ms Duration. . 397. Fractional Anisotropy (FA), Eigenvalues (λ 1, λ 2, λ3) of Diffusion Tensor and MeanDiffusivity (MD) of all four TX-151 Phantoms Imaged over 10.5 mm X 10.5 mm X 10.5mm Voxels and Diffusion Gradients of 200ms Duration. . 418. Estimates to Measure Diffusion along V1 in Terms of the Largest Eigenvaluecompared to Diffusion along V2 and the Mean Diffusivity. . 419. Mean and Standard Error of the Principal Eigenvector in TX-151 Phantoms ofIncreasing Degree of Anisotropy. . 4210. Standard Deviation of Bz in TX-151 Phantoms Subjected to Horizontal CurrentInjection. . 4711. Local Spatial Averages of Laplacian of Conductivity in all Four Phantoms Subject to(A) Horizontal And (B) Diagonal Current Injection Pairs. 51vi
LIST OF FIGURESFigurePage1. Frequency Dependence of Dielectric Parameters (Relative Permittivity andConductivity) in Biological Tissues . . 22. (a) EIT using Boundary Measurements (b) MREIT using both Internal and BoundaryMeasurements . 83. Inverse Relationship between Electric Field, Gradients of Conductivity and Laplacianof Bz. . 124. (a) Definition of Domains and (b) Recessed Electrode Assembly . 145. Forward and Inverse Problems in MREIT . 166. Simple MR Pulse Sequence with Diffusion Weighting Added in one Direction. . 197. Inverse Relationship between Electric Field, Gradients of Conductivity and Laplacianof Bz. . 198. Schematic of the Diffusion Tensor Ellipsoid. . 209. TX-151 Gel Phantoms with (a) 1 (b) 3 (c) 27 (d) 47 Layers in Custom IdenticalSample Chambers used as Imaging Sample in MREIT Experiments. 2310. MR Signal Recorded in K-Space under Current Injection of Duration. . 2611. Structure of the New MREIT Current Source . 2712. Standard Spin Echo Pulse Sequence for MREIT . 2813. Conductivity of Phantom with Alternating High and Low Conductivity Gel LayersCalculated from the Impedance recorded by HP4192A. . 3014. Cross-section of COMSOL Models in the XY-plane for (a) 1 (b) 3 (c) 27 and (d) 47Gel Layers. . 31vii
FigurePage15. (a) Change in SNR with Increasing Length of Diffusion Gradients in IsotropicVoxels of Side 10.5 mm. (b) Change in SNR with Increasing Isotropic Voxel Size under100 ms Diffusion-Sensitizing Gradient. . 3816. 3D Plot of the Mean of Principal Eigenvector in all Four TX-151 Gel Phantoms. . 4317. SNR on Y-Axis and Square ROI of Sides in Pixels . 4418. 47 Layer TX-151 Phantom is Subjected to 10 mA Vertical (a,c,e) and Horizontal (b,d, f) AC Current. Wrapped Phase Images (a, b), Unwrapped Phase Images (c, d) and Bz(e, f) were Displayed for Vertical and Horizontal Current Injections Respectively. . 4519. Spatial Profiles of the (a) Z-Component of Internal Magnetic Flux Density (B) and(b) Standard Deviation of B in TX-151 Gel Phantoms Subjected to Horizontal CurrentInjection Pair. . 4720. Average and Standard Deviation of Bz in 3 Layer TX-151 Gel Phantom. . 4821. Voltage Distribution in 47 Layer TX - 151 Gel Phantom Arrangement Subjected toVertical and Horizontal Current Injections. . 4822. Laplacian of Sigma in (a) 1 (b)3 (c ) 27 and (d) 47 Layers TX-151 Phantoms Subjectto Horizontal and Vertical Current Injection Pair. . 5023. Laplacian of Sigma in (a) 1 (b)3 (c ) 27 and (d) 47 Layers TX-151 Phantoms Subjectto Diagonal Current Injection Pair. . 5124. Schematic Diagram to Measure the Impedance of TX-151 Gels.6725. Pictorial Representation of the Measurement of Impedance in High Conductivity TX151 Gel in a Sample Chamber (5 cm x 5 cm x 5 cm) using Four-Probe ElectrodeMethod.68viii
FigurePage26. Rectangular Sample Chamber with Current Injection and Voltage RecordingElectrodes. 6827. LabVIEW Code Designed to Communicate with Impedance Analyzer HP4192A andRecord Initial Resistance Values. . 68928. LabVIEW Code to Display the Time Course of Resistance Property in TX-151 GelPhantoms. . 6929. LabVIEW Code to Read the Resistance of TX-151 Phantoms at Time Intervals of 5Minutes Over a Total Duration of 4 Hours. . 7030. Conductivity of Phantom with Alternating High and Low Conductivity Gel LayersArranged Parallel to the Orientation of Electrodes Recorded by HP4192A.70ix
CHAPTER 1INTRODUCTIONThe interaction of an electromagnetic field with an object depends on the shapeand dielectric properties of the material composing the object. In particular, the complexrelative permittivity influences the relative amounts of electromagnetic radiationreflected, absorbed or transmitted from the object. Dielectric properties of a medium suchas relative permittivity and conductivity are obtained from the complex relativepermittivity as:Complex relative permittivity,where(1)is the relative permittivityis the out-of-phase loss factor ()σ is the total conductivityℰ0 is the permittivity of free spaceω is the angular frequency of the electromagnetic fieldAs biological molecules are polar, the complex relative permittivity is dependenton the frequency of applied alternating electromagnetic field. It follows that relativepermittivity decreases and conductivity increases with increasing frequency. Thisbehavior in biological tissues is shown in Figure 1. Some tissues such as muscle andwhite matter exhibit anisotropic conductivity at low frequency. However, a majority oftechniques assume isotropic or equivalent isotropic conductivity distribution [1] .1
Figure 1: Frequency dependence of dielectric parameters (relative permittivity andconductivity) in biological tissues [2].In biological tissues, electrical conductivity is highly dependent on the molecularcomposition, structure, concentration and mobility of ions, temperature, extra- and intracellular fluids and other factors. Conductivity is representative of the physiological andpathological state of a tissue and hence, provides useful diagnostic information [1]1.1PURPOSEThe purpose of the thesis is to identify incongruities in reconstructions of crosssectional conductivity distributions of electrically anisotropic phantoms. Stable andreproducible (accurate) gel phantoms with varying degrees of anisotropy were designedfor use as samples for imaging by Magnetic Resonance Electrical ImpedanceTomography and Diffusion Tensor Magnetic Resonance Imaging (DT-MRI).Thepresence of anisotropy in phantoms is demonstrated by Diffusion Tensor imaging and the2
effect of the measurement scale on DTI is demonstrated by changing the resolution. Theconductivity distributions of anisotropic phantoms were reconstructed using theHarmonic Bz algorithm, which assumes an isotropic conductivity distribution. Finiteelement models of the phantoms were solved numerically to calculate synthetic Bzdistributions. Conductivity distributions reconstructed using the Harmonic Bz algorithmfrom experimental and synthetic Bz were compared at different resolutions. Conductivitycontrast reconstruction resulting from the isotropic assumption were compared in termsof the laplacian of conductivity distributions.3
CHAPTER 2BACKGROUND2.1Previous Work2.1.1 Impedance imagingThe objective of Impedance Imaging is to map cross-sectional conductivitydistributions inside an electrically conducting subject. The subject is electricallyinterrogated by injecting current through a pair of surface electrodes and recordingresultant boundary voltages [3]. Internal current flow pathways establish internal currentdensity, internal magnetic flux density and voltage distributions. Internal current flowdepends on electrode configuration, conductivity distribution (σ) and geometry of thesubject. Under the assumption of fixed boundary geometry and electrode configuration,the internal current density is dictated by the conductivity distribution to be imaged [1]. Alocal change in the conductivity alters the internal current pathway, which is manifestedas a change in boundary voltage and internal magnetic flux density [4].2.1.2 Electrical Impedance TomographyElectrical Impedance Tomography (EIT) reconstructs conductivity images frommeasured boundary current-voltage data. However, spatial resolution and accuracy of thereconstructed conductivity distribution in EIT is poor due to the following reasons:1. The relationship between internal conductivity distribution and boundary currentvoltage data is highly non-linear. Additionally, boundary voltages are insensitive to local4
changes in conductivity. Owing to this non-linearity and sensitivity, the reconstruction ofconductivity images, based on boundary current-voltage measurement pairs, iscomplicated. This is formally described as, "The inverse problem of reconstructing theconductivity distribution is ill-posed in EIT".2. The inverse problem is sensitive to the boundary geometry and electrode positions.This information is inaccurately modeled thereby affecting the reconstruction by EIT.3. Current-voltage data is limited by a finite number of electrodes (usually 8 to 32) andthe data is contaminated by measurement artifacts and noise.Nevertheless, EIT is desirable in clinical applications for high temporal resolutionand portability. As of today, EIT is useful to track changes in conductivity over time orfrequency [1]. A number of different approaches were suggested to transform the inverseproblem in EIT into a well-posed one. One such proposal suggested integrating theresultant magnetic and electric fields induced in an electrically conducting subjectfollowing current injection through surface electrodes. This idea sparked interest in thescience community which was followed by extensive research on methods to measure theinternal magnetic field and utilize this newfound information in conductivity imagereconstruction [4].5
2.1.3 Magnetic Resonance Current Density ImagingAn internal magnetic flux density B (Bx ,By ,Bz), current density J (Jx ,Jy ,Jz) andvoltage distribution is developed when a current I is injected into an electricallyconducting subject. A magnetic resonance imaging (MRI) scanner can measure thecomponent of B parallel to the main magnetic field B0. Assuming B0 is in the z-direction,the scanner can measure Bz. The other two components of B are measured similarlyfollowing two object rotations. The internal current density J is calculated usingAmpere's law. This technique, Magnetic Resonance Current Density Imaging (MRCDI),aims at non-invasively imaging and reconstruction of internal current density J fromAmpere's law (equation 2).Internal current density,0(2)where µ0 is the magnetic permeability of free space [4]2.1.4 Magnetic Resonance Electrical Impedance Tomography ImagingThe basic concept of MREIT was proposed by Zhang (1992), Woo et al (1994)and Ider and Birgul (1998) by combining EIT and MRCDI. The key idea of MREITemphasized the measurement of B using a current-injection MRI technique. Internalcurrent density J images from magnetic flux density B were constructed by Ampere's lawas in MRCDI. From B and/or J, it is possible to understand the internal current pathwaysdue to the conductivity distribution of the subject. In this way, Magnetic ResonanceElectrical Impedance Tomography (MREIT) was pioneered to overcome the technical6
difficulties in Electrical Impedance Tomography (EIT) and produce high-resolutionconductivity images [4].A serious problem in using equation 2 is the measurement of all three componentsof B. Currently available magnetic resonance scanners can only measure one componentof B that is parallel to the main magnetic field (B0). Despite this limitation, all threecomponents of B can be measured by rotating the subject. Theoretically, this seems like afeasible solution. However, it is discouraged because it misaligns pixels and isimpractical in a clinical setting [4]. Most recent MREIT techniques focus on investigatingthe relationship between the measured component of B and the current density orconductivity distribution to be imaged. Assuming B0 is in the z-direction, Oh (2003)invented a new method to extract conductivity information from Bz known as theHarmonic-Bz algorithm. Numerous non-biological and biological phantoms, postmortemanimal tissues, invivo animal and human experiments were conducted to validate and testthe new algorithm[1]. Potential clinical applications of MREIT include Functionalimaging, neuronal source localization and mapping, optimization of therapeutictreatments using electromagnetic energy.7
Figure 2: (a) EIT using boundary measurements (b) MREIT using both internal andboundary measurements [4]Comparing MREIT with EITMREITEITBetter spatial resolution and High temporal resolutionaccuracyInformation from MREITPortabilitycan be used as aprioriinformation in EITreconstructions for betterresults.DisadvantagesLong imaging timePoor spatial resolutionLack of portabilityInaccurateRequirement of anexpensive MR scannerTable 1: Comparison of the pros and cons of MREIT and EIT [4]Advantages8
2.2Theoretical considerations of MREIT2.2.1 Influence of current on the phase of MR signalsThe internal magnetic flux density induced during electrical interrogation iscrucial in determining the spatial resolution and accuracy of reconstructed conductivityimages in MREIT [4]. The current injected in MREIT experiments is in the form of pulseswith wide pulse-width similar to LF (low frequency) - MRCDIsource sequentially injects positiveand negative[4]. A constant currentcurrents through surfaceelectrodes in synchrony with an MR pulse sequence. Injected current induces a magneticflux density B (Bx,By,Bz) causing inhomogeneity in B0 changing B to (B B0). Thisleads to phase accumulation proportional to the z-component of B i.e. Bz. Positive andnegative currents with the same amplitude and width are injected sequentially to cancelout any systematic phase artifact of the MRI scanner and to increase the phase change bya factor of 2positive[1]. The MR spectrometer provides complex k-space data corresponding toand negativecurrents as:(3)(4)where M is the MR magnitude image representing the transverse magnetization,is any systematic phase error, 26.75 x 107 rad T-1 s-1 is the gyromagnetic ratio of hydrogenTc is the pulse width of the current in seconds.9
Two-dimensional discrete Fourier transformations ofimagesandandresult in complexrespectively as shown:(5)Incremental phase change is calculated by dividing the imaginary part of twocomplex images as:(6)where Arg(w) denotes the argument of a complex number w.The phase changezis wrapped in, and must be unwrapped usinga phase unwrapping algorithm such as Goldstein's branch cut algorithm.2.2.2 Phase UnwrappingGoldstein's branch cut algorithm is based on detecting inconsistencies whensumming wrapped phase gradients around every 2 x 2-sample path. The summationyields non-zero results at inconsistencies and are known as residues. Residues of oppositepolarities (i.e. signs) are balanced by connection with branch cuts. The cuts are generatedby a method to minimize the sum of cut lengths.A search of size 3 is placed around a residue and searched for another within thebox. If a residue of opposite polarity is found, a branch cut is placed between them andlabeled "uncharged". The search for another residue continues within the box. If a residueof same polarity was found, the box is moved to a new residue until an opposite charged10
residue is found or no residues can be found within the boxes. If no residues are found,the size of the box is increased by 2 and the algorithm repeats from the present startingresidue.2.2.3 Reconstruction of conductivity distributionBy sequentially injecting positive and negative currents, the systematic phaseartifactis rejected and the phase change is doubled. Bz is related to unwrapped phaseby a scaling factor and can be computed by:(7)Multi-slice magnetic resonance magnitude and phase images are reconstructedfrom k-space data. Magnitude images provide boundary geometry and electrode positionswhereas phase images provide Bz data.The spatial resolution of a reconstructed conductivity image is limited by thenoise measured in Bz data. The standard deviation of noise in Bz,signal-to-noise ratio (SNR) of the magnitude image,is related to theand total current injection timeTc as:(8)Incremental phase change (in equation 11) is the raw data in MREIT. This phasechange is proportional to the product Bz and Tc. Since Bz is proportional to I, theincremental phase change can be increased by optimizing MREIT pulse sequences to11
maximize the product of I and Tc.anddue to positiveand negativecurrentinjections were calculated as in equation 8. From the z-component of the curl of theAmpere's law 0, the following relationship is solved for theconductivity:Figure 3: Inverse relationship between electric field, gradients of conductivity andlaplacian of Bz.where u1 and u2 are voltages satisfying boundary-value condition due toand. This is iteratively solved in CoReHA software package which implements theHarmonic Bz algorithm [5].2.2.3.1Image ContrastImage contrast is an important parameter to overcome the disability of the humanvisual system to detect differences in absolute illuminance values.It is defined asdifferences in image intensity. Contrast depends on a multitude of factors such as spindensity, relaxation times and diffusion coefficients. This dependence is greatly influencedby the data acquisition protocol[10]. In this experiment, data acquisition parameters werechosen as described in Table 2 to enhance the T1 effect. Generally, enhancing the effectof either the spin density, T1 or T2 on image contrast is achieved by relatively varyingvalues of TR and TE. as shown in Table 2. The resultant image is said to carry a T1contrast because the image contrast is exponentially dependent on the T1 relaxation timeof the sample. MR imaging of normal soft tissues have significantly different T1 values12
thereby making it effective for good anatomical definition. Practically, TE and TR arelimited by system hardware performance and imaging time respectively[10].ContrastT1 - weightingT2 - opriateLongLongTable 2: Influence of echo and relaxation time (TE, TR) on image contrast.2.2.4. Forward ProblemA forward solver is extremely useful for algorithm development, experimentaldesign and verification. Image reconstruction in MREIT is inherently 3D, and therefore a3D forward solver is implemented. This model provides distributions of current density J,and voltage V within an electrically conducting domain (i.e. subject) following currentinjection using recessed electrodes.Consider Ω as an electrically conducting domain with isotropic conductivitydistribution σ and boundary Ω . Letcontainers (), electrodes (ℰ, ℰ andrepresent the area covered by plasticℰ ) and lead wires () respectively.Electrodes ℰ are recessed from the surface of the object Ω by plastic containers .Artifacts in Magnetic Resonance images occur due to the RF shielding effect ofconductive electrodes. To move these artifacts out of the domain Ω, recessed electrodesare preferred. Figure 4(b) displays the recessed electrode assembly. Use of recessedelectrodes ensures artifact-free MR images of the domain, including its boundary.13
Figure 4: (a) Definition of domains and (b) recessed electrode assemblyTo formulate the problem, considertwo plastic containers i.e.through ℰΩas the region comprising of the domain and. Assume a low-frequency currentℰ attached on , then the induced voltageinjectionsatisfies the followingboundary value problem with the Neumann boundary condition [4]:σ (10)σwhere n is the outward unit normal vector ong is a normal component of the current density ondue to Ir is a position vector in R3.g is zero on the portions of the boundary not in contact with the electrodes andover ℰ for j 1 or 2. To arrive at a unique solution for V in equation 10, areference voltage V(r0) 0 for r0is chosen. Having computed the voltagedistribution V, the current density J is given by:(11)whereis the electric field intensity.14
Considering the magnetic field produced by I, the induced magnetic flux densityB in Ω is :ΩwhereℰΩℰandare magnetic flux densities due to J in Ω,ℰ and I inrespectively.From the Biot-Savart
Neeta Ashok Kumar A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science Approved November 2015 by the Graduate Supervisory Committee: Rosalind Sadleir, Chair Vikram Kodibagkar Ji
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