Inverse Scattering MRI

474 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 2, FEBRUARY 2010

Imaging Electric Properties of BiologicalTissues by RF Field Mapping in MRIXiaotong Zhang, Student Member, IEEE, Shanan Zhu, and Bin He*, Fellow, IEEE

Abstract—The electric properties (EPs) of biological tissue, i.e.,the electric conductivity and permittivity, can provide importantinformation in the diagnosis of various diseases. The EPs also playan important role in specific absorption rate calculation, a majorconcern in high-field MRI, as well as in nonmedical areas such aswireless telecommunications. The high-field MRI system is accom-panied by significant wave propagation effects, and the RF radi-ation is dependent on the EPs of biological tissue. On the basis ofthe measurement of the active transverse magnetic component ofthe applied RF field (known as ��-mapping technique), we pro-pose a dual-excitation algorithm, which uses two sets of measured�� data to noninvasively reconstruct the EPs of biological tissues.The finite-element method was utilized in 3-D modeling and ��

field calculation. A series of computer simulations were conductedto evaluate the feasibility and performance of the proposed methodon a 3-D head model within a TEM coil and a birdcage coil. Usinga TEM coil, when noise free, the reconstructed EP distribution oftissues in the brain has relative errors of 12%–28% and correlatedcoefficients of greater than 0.91. Compared with other ��-map-ping-based reconstruction algorithms, our approach provides su-perior performance without the need for iterative computations.The present simulation results suggest that good reconstruction ofEPs from B1 mapping can be achieved.

Index Terms—��-mapping, electrical impedance imaging, elec-tric properties (EPs), magnetic resonance electric properties to-mography (MREPT).

I. INTRODUCTION

T HE electric properties (EPs) (conductivity and permit-tivity ) of biological tissues at radio and microwave

frequencies have been the subject of research for over fourdecades [1]. A comprehensive survey to date was conducted byGabriel et al. [2]–[4], which covered a large selection of healthytissue over a wide spectrum of frequencies (10 Hz–20 GHz).

Manuscript received October 02, 2009; revised November 06, 2009; acceptedNovember 06, 2009. Current version published February 03, 2010. This workwas supported in part by the National Institutes of Health (NIH) under GrantR21EB006070 and Grant R01EB007920, in part by the National Natural Sci-ence Foundation of China under Grant 50577055, by the National Science Foun-dation under Grant BES-0602957, and in part by the Supercomputing Instituteof the University of Minnesota. The work of X. T. Zhang was supported by theLu’s Postgraduate Education International Exchange Fund of Zhejiang Univer-sity. Asterisk indicates corresponding author.

X. Zhang is with the Department of Biomedical Engineering, University ofMinnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]).

S. Zhu is with the College of Electrical Engineering, Zhejiang University,Hangzhou 310027, China (e-mail: [email protected]).

*B. He is with the Department of Biomedical Engineering, University of Min-nesota, Minneapolis, MN 55455 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMI.2009.2036843

A few studies have also been conducted on EPs of a varietyof cancerous tissue types at radio and microwave frequencies[5]–[7]. These studies have consistently shown that the EPsof cancerous tissues are three or more times greater thanthose of healthy tissue, and so, it is anticipated that such EPinformation could be used in cancer detection. In addition,EPs are a determinant factor for the specific absorption rate(SAR), which is a major concern of safety in high-field MRIsystems and a critical problem in nonmedical areas such aswireless-telecommunications.

In the past two decades, many efforts have been madeto produce cross-sectional images of EPs in vivo by meansof electrical impedance tomography (EIT) [8]–[10] and itsvariants using magnetic induction tomography (MIT) [11].Both EIT and MIT are cost-effective and can provide dynamicinformation in regard to tissue EPs, but they are limited bylow spatial resolution due to the surface measurements andthe need to solve an ill-posed inverse problem. A techniquecalled “magnetic resonance electrical impedance tomography(MREIT),” which is based on the magnetic resonance cur-rent density imaging (MRCDI) technique [12] and measurescurrent injection inducted MR phase shifts, has been pursued[13]–[16]. While MREIT eliminates the need to solve theill-posed inverse problem and provides high spatial resolutionin imaging electrical conductivity in vivo, it requires currentinjection into the body within an MRI scanner, which limitsits applicability in medical applications. Another recentlydeveloped approach, magnetoacoustic tomography with mag-netic induction (MAT-MI), offers the promise of obtaininghigh-resolution tissue electrical conductivity profiles [17]–[20].In MAT-MI, an object is placed in a static magnetic field anda pulsed magnetic field is applied to induce acoustic signals,which are then detected to reconstruct the tissue conductivitydistribution. However, there have been no in vivo experimentsreported so far.

Haacke et al. proposed a method of extracting EPs from MRIimages [21]. They found that in 1.5 T (and above) MRI sys-tems, the wavelength of the RF radiation was of the order of thebody size, leading to a distortion of the RF field inside the irra-diated object. By using an iterative sensitivity matrix algorithm,these authors suggested that the EPs can be estimated using MRIimages that reflect the disrupted RF profile. Wen later pointedout that the perturbation of the RF field in high-field MRI di-rectly related to the conductivity and permittivity distributionwithin the sample, and a modified Helmholtz equation-basednoniterative algorithm was proposed and tested in phantom ex-periments using 1.5 and 4.7 T MRI [22]. Recently, using an it-erative algorithm derived from Ampere’s Law, Katscher et al.

0278-0062/$26.00 © 2010 IEEE

ZHANG et al.: IMAGING ELECTRIC PROPERTIES OF BIOLOGICAL TISSUES BY RF FIELD MAPPING IN MRI 475

conducted an in vivo experiment using a 3-T MRI system toimage the EPs within a human head and leg [23], and namedthis approach “magnetic resonance electric properties tomog-raphy (MREPT).”

Signal intensity (SI) nonuniformity is distinct in high-fieldMRI systems. It can arise from a variety of factors, includingimperfections in the scanner hardware and interactions betweenthe magnetic field and biological tissues. Among these factors,electromagnetic (EM) interactions with the object have beenshown to be the primary cause of SI nonuniformity for volumecoils. It was reported that in 1.5 T MRI systems, loaded bird-cage (BC) coils had in-plane uniformity with 20% variationcompared with as little as 1% variation when unloaded [24].Through a series of experimental and simulation studies, it hasbeen shown that the magnetic field distribution in the transverseplane of the coil is strongly dependent on the EPs of the bio-logical tissues being studied [25]; moreover, the magnetic fielddistribution inside the RF coil at a high field can be studied andexplained using Maxwell equations [26], [27]. These findingshave provided us with the mathematical framework to link EPdistribution with MRI images.

The -mapping technique [28]–[30] was developed to mea-sure the rotating RF field components: the transmitted and re-ceived fields in RF coils. With the principle of reciprocity [31],the Cartesian transverse components of the RF magnetic field,

and , can be derived. MREPT utilizes these measure-ments to reconstruct the EP values within biological tissues,differing from other noninvasive imaging techniques in that noelectrode mounting is required and no external energy is intro-duced into the body during MRI scanning. It can be performedusing a standard MRI system with a regular volume coil, and itsspatial resolution is determined by MRI images and the qualityof the applied -mapping technique [23].

In this study, a new noniterative dual-excitation MREPT al-gorithm to image the tissue EPs is proposed, and the finite-el-ement method (FEM) [32] is utilized in RF coil modeling and

field calculation. The performance of the proposed MREPTalgorithm is evaluated using a five-tissue 3-D anatomically ac-curate head model, a shielded 12-rung BC, and a TEM coil.

II. DUAL-EXCITATION MREPT APPROACH

A. Computation in High-Frequency Time-Harmonic EMField—The Forward Problem

The RF field excited in MRI coils can be treated as a time-har-monic EM field [27]. Assuming the object of interest to beisotropic, and using the generalized Ampere circuit law andFaraday’s magnetic induction law in a time-harmonic EM field,Helmholtz equation can be derived as [33]

(1)

where is the electric field vector, the excitation currentdensity in the coil, the free-space permeability, the relativepermeability, the operating angular frequency, the vacuumwave number defined by with as the free-space permittivity, and the complex permittivity

with as the relative permittivity. Using the results ofthe electric field vector , the magnetic field strength can beobtained from [33]

(2)

The FEM is normally used to solve EM problems, and withits flexibility in grid size, FEM can deal with objects having ir-regular geometries with good accuracy. In this study, all EMfields were calculated through (1) and (2) by means of FEMusing ANSYS 11.0 (ANSYS, Inc., PA, USA) software with its“high-frequency harmonic EM analysis,” whereas the postpro-cessing of EM field data was completed using MATLAB 2008a(The Mathworks, Inc., Natick, MA).

The -mapping technique measures rotating RF field com-ponents: the transmitted and received field in RF coils.The principle of reciprocity [31] links magnetic field compo-nents in Cartesian and rotating frames via

(3)

where and are the - and -oriented RF magnetic fieldscreated by the coil. We use (2) to derive these transverse mag-netic components, and MREPT utilize these two measurementsto reconstruct the EP values of biological tissues.

B. Dual-Excitation Algorithm—The Inverse Problem

Consider the magnetic permeability inside biological tissuesto be equal to that in a vacuum. Ampere’s Law reads

(4)

Using the complex permittivity , we have

(5)

Taking the curl of both sides of (5) and substituting with, we get

(6)

or its matrix form

(7)

According to Ampere’s Law, we replace the electric fieldcomponents with magnetic field components as follows:

(8)

For regular BC coils, the axial -component of theend-ring-generated magnetic field is nonproductivefor nuclear magnetic resonance (NMR) excitation, and can


become a source of ohmic loss to induced conduction cur-rents in the tissue load conductor. To solve this problem,the TEM resonator was developed [34], in which the coil’sreturn path follows the shield rather than the end rings, and arung-current-generated transverse magnetic field without thecounterproductive end ring currents is achieved. In this case,we can neglect the component in (8), and rewrite (7) as

(9)

In (9), and distributions correspond to -mappingresults, while , , , andare unknown. Upon acquiring another set of data, wecan form a set of linear equations with four equations and fourunknown variables, and thus, can be solved.

Two methods can be adopted to acquire two sets ofdata: 1) adjusting the RF coil excitation modes, such as

linear (vertical and/or horizontal polarization) and/or quadra-ture modes [35], to excite the imaging object twice, separately,but with the same subject orientation; or 2) rotating the subjectalong the -axis by an angle. In this study, we used the linearexcitation mode to generate two polarizations by switching thecurrent feed points to positions 90 apart from each other.

III. MATERIALS AND METHODS

A. FEM of the Head and Coil

T1-weighted MRI images ( , ) of ahuman head, which covered the entire brain, were acquired froma 3-T Siemens MRI system. The head images were then seg-mented into five tissues: scalp, skull, cerebrospinal fluid (CSF),white matter (WM), and grey matter (GM). The structural in-formation was imported into ANSYS software, and a five-tissueanatomically accurate head model was constructed using a hex-ahedral element with a mesh size of , which isequivalent to the voxel size of MRI images.

A 12-rung BC coil and a TEM coil with diameters of 28 cmand lengths of 28 cm were modeled. Either coil was enclosedby a cylindrical shield having a diameter of 32 cm and a lengthof 30 cm. Both the coils and the shield(s) were meshed with thetetrahedral element and assigned with copper material. Fig. 1 il-lustrates the FEMs of the head and shielded BC coil in ANSYS,and Fig. 2 shows the structural views on the axial, sagittal, andcoronal planes.

Since all RF coils are basically open structures, the field gen-erated by a coil radiates into a large space. However, a numericalsimulation can be performed only inside a finite space. There-fore, we must first truncate the simulation space by using an airvolume to enclose the solution area. In order to reduce the artifi-cial field reflection produced by the volume, we used a perfectlymatched layer (PML) [36] to surround the artificial surface ofthe air volume. For our shielded coils, the shield provided thetruncation at which the boundary condition shouldbe applied [35]. Both air and PML volumes were discretizedwith the tetrahedral element.

Fig. 1. Overview of the head FEM and a 12-rung-shielded BC coil FEM model(green: copper shield; cyan: rungs and end rings; magenta: head).

Fig. 2. (a) Axial, (b) sagittal, and (c) coronal views of the FEMs. The headmodel was meshed with hexahedral element while the coil with tetrahedralelement.

B. Coil Excitation

To perform a linear excitation, the equivalent circuit model[35] was utilized to calculate the electric current in each rung.All of the capacitors in the coil were replaced with currentsources at the required resonant frequency. For the BC coil, thecurrent flowing in the th rung is

(10)


TABLE ISPECIFICATIONS OF COIL EXCITATION SCHEMES

TABLE IIEP VALUES OF HEAD TISSUES

where denotes the maximum current. In our study, wasassumed to be unit current, so the current flowing in the endrings between the th and th rungs is

(11)

For the TEM coil, the current in each end ring was assignedto be zero. Similar current configurations can also be found in[37].

and offset was added on the phase of the currentflowing in the coil, separately, to simulate the current feed pointswitching. Three coil excitation schemes are listed in Table I.

C. Simulation Protocols

Simulations were conducted in 3 T (128 MHz) environment.The corresponding target EP values for different head tissueswere derived from the 4-Cole-Cole model [4] and are listed inTable II.

Using ANSYS software as the forward solver, we generatedthe simulated data of and by different coil excitationschemes, correspondingly. Simulations were conducted on boththe BC coil and TEM coil to evaluate the influence of neglectingthe component in (9).

We define relative error (RE) and correlation coefficient (CC)of the reconstructed EPs images as

(12)

Fig. 3. MRI T1-weighted image (a) of the slice of interest, segmented targetEP distributions (b1) &. (c1), reconstructed EP distributions (b2) &. (c2), andreconstructed EP distribution (b3) &. (c3) using � � � data when noisefree. Using the 28 cm-long TEM coil, coil excitation schemes S1 and S2 wereapplied. Two bright stripe-like artifacts along �- and �-axes can be observed in(b2) and (c2).

where and are the target EP values, and andare the estimated values in the th pixels within the specified

tissue.In order to test the noise tolerance of the proposed algorithm,

we added Gaussian white noise (GWN) to the simulated fielddata. From (3), we get the circularly polarized components ofthe field, then add GWN to the amplitude and phase ofand , respectively. The noise level is evaluated by ,which is defined as follows:

(13)

where is the amplitude of the noise-free field voxel signaland is the standard deviation (STD) of amplitude noise. Inthis study, we assumed , 150, and 100. Note thathere, is different from the commonly denoted SNR ofMRI systems . A related description can be found inSection V and the Appendix.

The Wiener adaptive low-pass filter was applied to preprocessthe “contaminated” maps, and (3) was reused to derive the

and components.

IV. SIMULATION RESULTS

We chose the central slice of the head model asthe slice of interest, and the corresponding MRI T1-weightedimage is depicted in Fig. 3(a). Fig. 3(b1) and (c1) draws thetarget EP distributions after image segmentation. Using the28-cm-long TEM coil, with and dataacquired from coil excitation schemes S1 and S2, the recon-structed EP distributions are shown in Fig. 3(b2) and (c2).


Fig. 4. �� distribution with the TEM coil when noise Free. Coil excitationschemes S1 and S2 were applied. (a) and (b) �� distribution for each excita-tion scheme. (c) and (d) �� distribution corresponding to our newly produced� field. Much lower intensity can be observed along �- and �-axes in (a) and(b), and in the central area in (c) and (d).

Two bright stripe-like artifacts along the - and -axes areobserved in earlier results, and this phenomenon resulted fromthe characteristics of the distribution inside the linearlypolarized RF coil. Fig. 4(a) and (b) depicts the modulus of the

distribution on the central axial slice within the head usingS1 and S2 excitations, in which the modulus varied rapidlyand dropped to zero in the area across the -axis in Fig. 4(a) andthe -axis in Fig. 4(b). We define ,and has the same modulus distribution with accordingto (8). Since acts as a denominator in our algorithm as(9) indicates, its “zero” area would introduce numerical errors,which are observed in the reconstruction results shown in Fig. 3as two bright stripe-like artifacts.

To solve the above problem, we produced two new sets ofdata by

(14)In a linear medium, the system described by Maxwell’s equa-

tions is a linear system; according to the superposition prin-ciple, and still hold for (9). Fig. 4(c) and (d) shows the

distribution corresponding to the new data,and Fig. 3(b3) and (c3) shows the reconstructed EP results using

data when noise free. The bright stripe-like artifactsdisappear in the reconstructed images; however, there are stilllittle artifacts in the reconstructed images, and they mainly re-sulted from much lower intensity of the distribution in thecentral area, as Fig. 4(c) and (d) depicts, which is about 1/100of the periphery intensity. Reconstruction results for CSF, WM,and GM tissues are listed in Table III, and their relative errors

TABLE IIIRELATIVE ERRORS AND CORRELATION COEFFICIENTS OF RECONSTRUCTED

ELECTRIC PROPERTIES WITHOUT AND WITH

HZ COMPONENT USING TEM AND BC COILS

Fig. 5. Reconstructed electric property distribution within a 28-cm-long anda 12-cm-long BC coil. (Top row) Conductivity. (Bottom row) Relative permit-tivity. (From left to right) Target EP distribution, reconstruction results withoutand with � component in the birdcage coils, respectively.

range within 12%–28% while correlated coefficients all above0.91. Similar data processing have been applied inthe following simulations.

In our algorithm, and are assumed to benegligible. Since BC coils have been widely used in clinicalapplications, it is necessary to evaluate the effects for reg-ular birdcage coils. We applied the dual-excitation algorithm onthe 28-cm-long BC, and we also took andinto (9) to complete the reconstruction using the same coil. Inaddition, we constructed a 12-cm-long birdcage coil (with thesame other geometrical parameters and configurations as the28-cm-long birdcage coil) to evaluate its performance. Coil ex-citation schemes S1 and S2 were applied. Reconstructed EPsdistribution and simulation results are shown and summarizedin Fig. 5 and Table III, respectively. It can be observed that,with the terms of -included component, we could surely im-prove EPs reconstruction results. However, while the experi-mental measurement of component is not feasible in MRI,it is reasonable to assume within a common BC coil,which could also provide well-differentiated structural informa-tion of biological tissues.

Fig. 6 shows the reconstructed EPs distribution when noisewas added with , 150, and 100. The corre-sponding REs and CCs are summarized in Table IV. The28-cm-long TEM coil and S1 and S2 coil excitations wereused. We also applied S2 and S3, and S1 and S3—choosingtwo adjacent rungs as current feed points individually—toexcite the TEM coil with noise . A comparisonof the reconstruction results is depicted in Fig. 7. It can be


Fig. 6. Reconstructed EP distribution within a 28-cm-long TEM coil. (Top row)Conductivity. (Bottom row) Relative permittivity. (From left to right) Target EPdistribution, reconstruction results with decreasing �� .

TABLE IVRELATIVE ERRORS AND CORRELATION COEFFICIENTS OF RECONSTRUCTED

EP DISTRIBUTION WITH DECREASING SNRB1

Fig. 7. Reconstructed EP distribution with different coil excitation schemecombinations. (Top row) Conductivity. (Bottom row) Relative permittivity.(From left to right) Target EP distribution and reconstructed distribution withdifferent phase offset of the current flowing in the coil.

observed that, when added with noise, all the tissues could bedifferentiated in the reconstruction results; and similar tissuepatterns could also be produced while different coil excitationscheme combinations were utilized.

We also applied Wen’s method in [22] to the TEM model byS1 and S2 excitations to reconstruct the EPs distribution. Fig. 8shows the comparison of the reconstruction results when noisefree. It can be seen that we obtained better results around thetissue boundary regions, and this is because the term

in (6) was not taken into account in the modified Helmholtzequation in Wen’s method.

V. DISCUSSION

Since the conductivity and permittivity values of tissueschange with their physiological and pathological conditions,

Fig. 8. Reconstructed EP distribution with the TEM coil when noise free, andcoil excitation schemes S1 and S2 were applied (top row: � distribution, bottomrow: � distribution; from left to right: target EPs, using our algorithm, and usingWen’s method [22]).

EP maps could provide useful diagnostic information. MREPTpromises to reconstruct not only the conductivity distribution,as EIT, MIT, MREIT, and MAT-MI aim to do, but also thepermittivity distribution within biological tissues, thus en-riching our observations of their conditions. In addition, as it isbased on -mapping, which has been developed and pursuedfor decades, MREPT does not require electrode mountingor external energy deposition, which pose potential safetyconcerns. It could provide a high spatial resolution in imagingEPs with the aid of MRI. In this study, we have proposed anew dual-excitation MREPT algorithm, and as our simulationresults indicate, the dual-excitation algorithm furnishes uswith a practical approach to reconstruct the conductivity andpermittivity distributions within a human head while avoidingthe need to solve the ill-posed inverse problem present intraditional EIT. The desirable reconstruction results of tissueswith heavier implications in physiology and pathology (suchas CSF, WM, and GM) suggest that MREPT imaging deservesfurther investigation, can be extended to other organs of thehuman body, and may have wide clinical applications.

In our algorithm, the -component of the magnetic field in-tensity is neglected. Our simulation results suggest that thedual-excitation algorithm still works well, even using a typ-ical BC coils (see Fig. 5) in which the component promi-nently exists and is mainly generated by end rings. Using a TEMcoil, we can get better reconstruction EP images with higheraccuracy.

To eliminate the bright stripe-like artifacts shown in Fig. 2,we produced two new sets of according to the su-perposition principle. Alternatively, to generate a similardistribution as in Fig. 3(c) and (d), we may also excite the coilby quadrature polarization, which is commonly utilized in com-mercial RF coils. However, since our algorithm needs to excitethe coil twice, at least three current feed points will be requiredto perform the quadrature excitation twice. Therefore, our ap-proach of using data acquired from two linear excitations ap-pears to be more feasible and easy to implement.

In contrast to the previously proposed MREPT algorithmsin [21] and [23], our approach does not require iterative com-


putation that needs precise coil and subject modeling, and istime-consuming. Compared with the method presented in [22],ours is more capable to deal with regions where the EPs dis-tribute inhomogeneously and discontinuously. Fig. 8 illustratesthis point.

Uniqueness of solution is an important issue in inverse prob-lems. We applied three coil excitation schemes, and conductedthe reconstruction procedures by choosing two 30 apart (adja-cent), 60 apart, and 90 apart current feed points. It is shownin Fig. 7 that, under either excitation combination with noiseadded, our algorithm works well and produces overall similardistributions of the EPs. While these results suggest robustnessand potential uniqueness characteristic of our algorithm, it isalso indicated that when the switching current feed point is in-feasible, our algorithm is still effective by careful subject rota-tion over a small angle inside the coil. In the future, the theoret-ical study on the uniqueness of the MREPT problem remains tobe further performed.

In previous field simulation studies, the EP values werepreset according to experimentally measured data [38] or thepublished 4-Cole-Cole dispersion relation [3]. However, inmany instances, these two data are not in good agreement.Our algorithm will provide another effective way to accuratelypredict the EP values of biological tissues, and will also sub-stantially benefit future studies in the MRI domain as well asother areas that involve EP configuration in the range of RF.

For isotropic biological tissues, the SAR is defined by

(15)

where is the tissue density and is determined through exper-imental measurement. Once we get the distribution, the EPvalues can be obtained from the dual-excitation algorithm. Since

and are more dominant in a BC and TEM coil, theelectric field can be determined by Ampere’s Law withoutcomponent, and we can directly calculate the SAR distribution.Thus, the evaluation of heat effects becomes more convenient.

In our algorithm, the Laplacian operation amplifiesnoise effect. Thus, the present algorithm is relatively sensitiveto noise, as shown in Table IV, when SNR drops. We haveapplied Wiener filtering over images; in addition, otherfeasible ways can be used to reduce noise level in experiment,such as taking multi-MRI scans in experiments and averageimages. Furthermore, for common mapping techniques, thescanning parameters are usually chosen as ,and the MRI image SI depends on and via [21],[23], [39]

(16)

where is the equilibrium magnetization, the gyromagneticratio of , and the duration of rectangular RF pulse. For smallflip angle pulse, (16) can be written as

(17)

and image SI is propositional to . When we addnoise into and leveled by , it can be provedthat the corresponding will be lower than (seethe Appendix); in other words, is times higher than

. Moreover, note that we only consider noise in thefield in such a case; since there are other noise sources in

the MRI signal detection process, such as thermal fluctuationand electronics factors, is even higher than .Therefore, our assumption that equals to 100–200 isreasonable for regular MRI scanning.

A multiport quadrature coil can be driven at each one pointby well-controlled RF drive circuit [34]; thus, it is applicable toacquire different data to finish the reconstruction. Besides,our algorithm requires field measurements. Until now, many

-mapping techniques have been proposed, and the measure-ment of field intensity has been well established, while phasemeasurement has been studied by few groups [22], [23], [40].Recently, a novel phase mapping method was developed inlocal shimming studies for transceiver arrays [41], and it pro-vides us with the possibility for accurate phase mapping. Inthe future, more efforts should be focused on the pursuit ofcomplex image mapping in order to improve the efficiency andaccuracy of the MREPT technique.

APPENDIX

Noise in B1 field is leveled by in (13) via

(18)

where and are noise-free voxel intensity, andand are the STD of two uncorrelated GWN added intoand , respectively. We use and to represent thesetwo noise values. Then we have

(19)

with the variance as

(20)

and the corresponding can be derived from (17) and(20) by

(21)

ACKNOWLEDGMENT

The authors would like to thank Dr. Y. P. Du, Dr. Z. M. Liu,U. Katscher, X. Li, and Y. Liu for useful discussions, andW. H. Lee for technical assistance in the head finite-elementmodeling.


REFERENCES

[1] E. C. Fear, X. Li, S. C. Hagness, and M. A. Stuchly, “Confocal mi-crowave imaging for breast cancer detection: Localization of tumorsin three dimensions,” IEEE Trans. Biomed. Eng., vol. 49, no. 8, pp.812–822, Aug. 2002.

[2] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties ofbiological tissues. 1. Literature survey,” Phys. Med. Biol., vol. 41, no.11, pp. 2231–2249, Nov. 1996.

[3] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties ofbiological tissues: II. Measurements in the frequency range 10 Hzto 20 GHz,” Phys. Med. Biol., vol. 41, no. 11, pp. 2251–2269, Nov.1996.

[4] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties ofbiological tissues: III. Parametric models for the dielectric spectrumof tissues,” Phys. Med. Biol., vol. 41, no. 11, pp. 2271–2293, Nov.1996.

[5] S. S. Chaudhary, R. K. Mishra, A. Swarup, and J. M. Thomas, “Di-electric properties of normal & malignant human breast tissues at ra-diowave & microwave frequencies,” Indian J. Biochem. Biophys., vol.21, no. 1, pp. 76–79, Feb. 1984.

[6] A. J. Surowiec, S. S. Stuchly, J. B. Barr, and A. Swarup, “Dielec-tric properties of breast carcinoma and the surrounding tissues,” IEEETrans. Biomed. Eng., vol. 35, no. 4, pp. 257–263, Apr. 1988.

[7] W. T. Joines, Y. Zhang, C. Li, and R. L. Jirtle, “The measured elec-trical properties of normal and malignant human tissues from 50 to 900MHz,” Med. Phys., vol. 21, no. 4, pp. 547–550, Apr. 1994.

[8] T. E. Kerner, K. D. Paulsen, A. Hartov, S. K. Soho, and S. P. Poplack,“Electrical impedance spectroscopy of the breast: Clinical imagingresults in 26 subjects,” IEEE Trans. Med. Imag., vol. 21, no. 6, pp.638–645, Jun. 2002.

[9] A. Malich, T. Bohm, M. Facius, M. Freessmeyer, M. Fleck, R. An-derson, and W. A. Kaiser, “Additional value of electrical impedancescanning: Experience of 240 histologically-proven breast lesions,” Eur.J. Cancer, vol. 37, no. 18, pp. 2324–2330, Dec. 2001.

[10] P. Metherall, D. C. Barber, R. H. Smallwood, and B. H. Brown, “Three-dimensional electrical impedance tomography,” Nature, vol. 380, no.6574, pp. 509–512, Apr. 1996.

[11] H. Griffiths, W. R. Stewart, and W. Gough, “Magnetic induction to-mography. A measuring system for biological tissues,” Ann. N. Y. Acad.Sci., vol. 873, pp. 335–345, Apr. 1999.

[12] M. Joy, G. Scott, and M. Henkelman, “In vivo detection of appliedelectric currents by magnetic resonance imaging,” Magn. Reson. Imag.,vol. 7, no. 1, pp. 89–94, 1989.

[13] N. Zhang, “Electrical Impedance Tomography Based on Current Den-sity Imaging,” M.S. thesis, Dept. Elect. Eng., Univ. Toronto, Toronto,ON, Canada, 1992.

[14] H. S. Khang, B. I. Lee, S. H. Oh, E. J. Woo, S. Y. Lee, M. Y. Cho, O.Kwon, J. R. Yoon, and J. K. Seo, “J-substitution algorithm in MagneticResonance Electrical Impedance Tomography (MREIT): Phantom ex-periments for static resistivity images,” IEEE Trans. Med. Imag., vol.21, no. 6, pp. 695–702, Jun. 2002.

[15] Y. Z. Ider and S. Onart, “Algebraic reconstruction for 3D magneticresonance-electrical impedance tomography (MREIT) using one com-ponent of magnetic flux density,” Physiol. Meas., vol. 25, no. 1, pp.281–294, Feb. 2004.

[16] X. T. Zhang, D. D. Yan, S. A. Zhu, and B. He, “Noninvasive imagingof head-brain conductivity profiles,” IEEE Eng. Med. Biol. Mag., vol.27, no. 5, pp. 78–83, Sep./Oct. 2008.

[17] Y. Xu and B. He, “Magnetoacoustic tomography with magnetic induc-tion (MAT-MI),” Phys. Med. Biol., vol. 50, pp. 5175–5187, 2005.

[18] X. Li, Y. Xu, and B. He, “Imaging electrical impedance from acousticmeasurements by means of magnetoacoustic tomography with mag-netic induction (MAT-MI),” IEEE Trans. Biomed. Eng., vol. 54, no. 2,pp. 323–330, Feb. 2007.

[19] R. M. Xia, X. Li, and B. He, “Magnetoacoustic tomographic imaging ofelectrical impedance with magnetic induction,” Appl. Phys. Lett., vol.91, no. 8, pp. 083903-1–083903-3, 2007.

[20] R. M. Xia, X. Li, and B. He, “Reconstruction of vectorial acousticsources in time-domain tomography,” IEEE Trans. Med. Imaging, vol.28, no. 5, pp. 669–675, May 2009.

[21] E. M. Haacke, L. S. Petropoulos, E. W. Nilges, and D. H. Wu, “Ex-traction of conductivity and permittivity using magnetic resonanceimaging,” Phys. Med. Biol., vol. 38, no. 6, pp. 723–734, 1991.

[22] H. Wen, “Noninvasive quantitative mapping of conductivity and dielec-tric distributions using RF wave propagation effects in high-field MRI,”in Proc. SPIE, 2003, vol. 5030, pp. 471–477.

[23] U. Katscher, T. Dorniok, C. Findeklee, P. Vernickel, and K. Nehrke,“In vivo determination of electric conductivity and permittivity usinga standard MR system,” in IFMBE Proc. H. Scharfetter and R. Merwa,Eds. Berlin, Germany, vol. 17, pp. 508–511, 2007, Springer.

[24] J. G. Sled and G. B. Pike, “Standing-wave and RF penetration arti-facts caused by elliptic geometry: An electrodynamic analysis of MRI,”IEEE Trans. Med. Imag., vol. 17, no. 4, pp. 653–662, Aug. 1998.

[25] M. Alecci, C. M. Collins, M. B. Smith, and P. Jezzard, “Radio fre-quency magnetic field mapping of a 3 Tesla birdcage coil: Experimentaland theoretical dependence on sample properties,” Magn. Reson. Med.,vol. 46, no. 2, pp. 379–385, 2001.

[26] Q. X. Yang, J. H. Wang, X. L. Zhang, C. M. Collins, M. B. Smith, H. Y.Liu, X. H. Zhu, J. T. Vaughan, K. Ugurbil, and W. Chen, “Analysis ofwave behavior in lossy dielectric samples at high field,” Magn. Reson.Med., vol. 47, no. 5, pp. 982–989, 2002.

[27] T. S. Ibrahim, C. Mitchell, R. Abraham, and P. Schmalbrock, “In-depthstudy of the electromagnetics of ultrahigh-field MRI,” NMR Biomed.,vol. 20, no. 1, pp. 58–68, Feb. 2007.

[28] S. Akoka, F. Franconi, F. Seguin, and A. Lepape, “Radiofrequency mapof an NMR coil by imaging,” Magn. Reson. Imag., vol. 11, no. 3, pp.437–441, 1993.

[29] E. K. Insko and L. Bolinger, “Mapping of the radiofrequency field,” J.Mag. Res. Ser. A, vol. 103, no. 1, pp. 82–85, Jun. 1993.

[30] J. H. Wang, M. L. Qiu, Q. X. Yang, M. B. Smith, and R. T. Con-stable, “Measurement and correction of transmitter and receiver in-duced nonuniformities in vivo,” Magn. Reson. Med., vol. 53, no. 2, pp.408–417, Feb. 2005.

[31] D. I. Hoult, “The principle of reciprocity in signal strength calcula-tions—A mathematical guide,” Concepts Mag. Res., vol. 12, no. 4, pp.173–187, 2000.

[32] P. P. Silvester and R. L. Ferrari, Finite Element for Electrical Engi-neers. Cambridge, U.K.: Cambridge Univ. Press, 1996.

[33] J. A. Kong, Electromagnetic Wave Theory. New York: Wiley, 1986.[34] J. T. Vaughan, G. Adriany, C. J. Snyder, J. Tian, T. Thiel, L. Bolinger,

H. Liu, L. DelaBarre, and K. Ugurbil, “Efficient high-frequency bodycoil for high-field MRI,” Magn. Reson. Med., vol. 52, no. 4, pp.851–859, Oct. 2004.

[35] J. M. Jin, Electromagnetic Analysis and Design in Magnetic ResonanceImaging. New York: CRC Press, 1999.

[36] J. P. Berenger, “A perfectly matched layer for the absorption of elec-tromagnetic-waves,” J. Comput. Phys., vol. 114, no. 2, pp. 185–200,Oct. 1994.

[37] S. Reza, S. Vijayakumar, M. Limkeman, F. Huang, and C. Saylor,“SAR simulation and the effect of mode coupling in a birdcage res-onator,” Concepts Mag. Res. B, vol. 31B, no. 3, pp. 133–139, 2007.

[38] H. P. Schwan, “Electrical properties of tissue and cell suspensions,”Adv. Biol. Med. Phys., vol. 5, pp. 147–209, 1957.

[39] E. M. Haacke, P. W. Brown, M. R. Thompson, and R. Venkatesan,Magnetic Resonance Imaging: Physical Principles and Sequence De-sign. New York: Wiley, 1999.

[40] U. Katscher, T. Voigt, and C. Findeklee, “Electrical conductivityimaging using magnetic resonance tomography,” in Proc. Conf. Proc.Annu. Int. Conf. IEEE Eng. Med. Biol. Soc., Minneapolis, MN, 2009,pp. 3162–3164.

[41] G. J. Metzger, C. Snyder, C. Akgun, T. Vaughan, K. Ugurbil, and P. F.Van de Moortele, “Local B-1(+) shimming for prostate imaging withtransceiver arrays at 7T based on subject-dependent transmit phasemeasurements,” Magn. Reson. Med., vol. 59, no. 2, pp. 396–409, Feb.2008.

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