Theoretical model for an MRI radio frequency resonator

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 47, NO. 4, APRIL 2000

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Theoretical Model for an MRI Radio Frequency Resonator Brian A. Baertlein*, Member, IEEE, Özlem Özbay, Tamer Ibrahim, Robert Lee, Member, IEEE, Ying Yu, Allahyar Kangarlu, and Pierre-Marie L. Robitaille

Abstract—A theoretical model is described for a magnetic resonance imaging (MRI) radio-frequency resonator (an MRI “coil”) that is useful at ultrahigh frequencies. The device is a “TEM resonator,” which is based on a concept originally proposed by Röschmann (1988). The coil comprises a circular cavity-like structure containing several coaxial transmission lines operating in a transverse electromagnetic (TEM) mode. The model developed herein treats the empty coil and is based on multiconductor transmission line theory. This work generalizes and extends similiar analyses of the device by Röschmann (1995) and Chingas and Zhang (1996). The model employs explicit calculation of per-unit-length parameters for TEM lines having arbitrary geometries. Calculations of the resonator’s frequency response are found to compare well with measurements. Fields produced by linear (single-point) and quadrature drive are also computed and compared to images of low-permittivity phantoms. Index Terms—Magnetic resonance imaging (MRI), radio frequency (RF) resonator.

I. INTRODUCTION

A

KEY element in a magnetic resonance imaging (MRI) system is the radio-frequency (RF) “coil,” a resonant device used for transmitting and receiving electromagnetic energy at the Larmor frequency of a given nucleus. The current generation of clinical MRI systems employ primary magnetic on the order of 1.5 T, for which the Larmor field intensities frequency of protons (the nucleus most commonly studied) is roughly 64 MHz. Desirable attributes of an RF coil are high Q and good field homogeneity across the region being imaged. Since the dimensions of an RF coil are roughly those of the anatomical region of interest (a human head or torso), typical coil dimensions are 0.3–0.6 m. It follows that the current generation of coils are electrically small. in MRI leads to improvments in Use of higher signal-to-noise ratio (SNR), and a number of so-called “high-field” MR systems operating at 3–4 T have been developed. Recently, The Ohio State University assembled an ultra-high-field MRI system, which operates at a magnetic field strength of 8 T. Since the Larmor frequency is proportional Manuscript received June 5, 1998; revised October 14, 1999. Asterisk indicates corresponding author. *B. A. Baertlein is with The Ohio State University, the ElectroScience Laboratory, Department of Electrical Engineering, 1320 Kinnear Road, Columbus, OH 43212 USA (e-mail: [email protected]). Ö. Özbay, T. Ibrahim, and R. Lee are with the ElectroScience Laboratory, Department of Electrical Engineering, Columbus, OH 43212 USA. Y. Yu, A. Kangarlu, and P.-M. L. Robitaille are with The Ohio State University, Center for Advanced Biological Imaging, Department of Radiology, Columbus, OH, 43210 USA. Publisher Item Identifier S 0018-9294(00)02645-8.

to coils operating at higher frequencies are of growing interest. At 8 T, the proton Larmor frequency is roughly 340 MHz. With the growth of high-field and ultra-high-field MR systems comes a need for greater attention to electromagnetic theory in coil design. At higher frequencies wave-like phenomena become more important, and naive use of low-frequency coil designs leads to poor field homogeneity. Recent innovations in coil design show promise for high-field applications. Prominent among the new designs is a concept that appears to have been proposed first in a 1988 patent by Röschmann [1]. In that work, transverse electromagnetic (TEM) transmission lines with adjustable capacitive end loads (open circuited coaxial transmission lines with variable length) were used in various configurations. A near-simultaneous 1988 patent by Bridges [2] describes a similar concept involving a circular array of TEM transmission lines surrounded by a complete cylindrical shield and terminated by fixed capacitive loads. The combination of adjustable loading and a circular geometry was later examined in a 1994 paper by Vaughan et al. [3]. Coils of this type have become known as “TEM resonators,” a name used in a 1987 paper by Röschmann [4] to describe coils that employ TEM fields. In this paper, an electromagnetic analysis of the TEM resonator is presented. The electromagnetic principles that govern this device (principally multiconductor transmission line theory) are well known and have been employed by others, as described below. A model based on this theory is presented here with a derivation of several important parameters. Experimental validation of the model is also presented. This analysis is restricted to empty coils. Understanding the operation of the empty coil is important, because its fields typically represent the intended operation of the coil. The insight gained from a study of the empty coil is also useful in understanding the field perturbations induced by tissue loads. In a parallel effort, the authors are using a numerical finitedifference time-domain (FDTD) model to study the device when loaded by human tissue [5]–[7]. That modeling, which is essential for studying such effects as tissue heating, is unattractive as a primary design tool because of the intensive calculations required. While the empty coil can be analyzed on a desktop computer in roughly 1 min using the model described below, several hours of supercomputer time are required for FDTD analysis of the loaded coil. Nonetheless, in most cases the behavior of the loaded coil differs markedly from that of the empty coil, and the empty coil models presented here are best used in exploratory studies.

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Theoretical studies of the TEM resonator have appeared previously. The first analysis appears to be that of Vaughan et al. [3] who predicted the resonance frequencies of the fundamental (lowest order) mode and presented some field calculations. The resonance frequencies were calculated in terms of equivalent total lumped inductance and capacitance, which were in turn determined by approximating a circular array of coaxial lines by a continuous metal surface. Vaughan et al. also presented numerfield within a model of the human ical calculations of the head, but RF interaction between the head and the coil—a crucial part of any high-field model for a loaded coil—was not discussed. Röschmann [8] presented a brief analysis of the coil’s resonance frequencies. That analysis employs previously derived expressions for the characteristic impedance of an array of weakly coupled transmission lines. Those impedances were then used to predict the resonance frequency for the mode of interest. An accuracy of better than 1% was asserted for the frequency estimates obtained using this expression with an undisclosed second-order correction. In a subsequent more rigorous analysis, Chingas and Zhang [9] derived an expression for the resonance frequencies using transmission line theory and presented useful design curves. The work, however, is limited to cases in which the struts are 1) weakly coupled to each other, 2) symmetrically placed, and 3) equally loaded. A further limitation, albeit one currently of limited interest, is that all conductors must have circular cross sections. Finally, work by Tropp [10] has shown that the resonance frequencies can be calculated using lumped element models for the individual coaxial lines and another lumped circuit approximation for the mutual inductance between those lines. While such analysis is useful, it is difficult to incorporate the wave phenomena that occur at higher frequencies. The present analysis, like those of [8] and [9], is based on multiconductor transmission line theory. The model described here, however, is both more general and more complete than those presented previously. Transmission line voltages and currents are related to the applied RF sources. This relation leads to expressions for the coil modes, from which the results of [9] are obtained as a special case. The analysis can accommodate such characteristics as asymmetry in construction and loading. The latter occurs in dual-tuned coils [3]. By explicitly computing the per-unit-length transmission line parameters, the model is freed from the weak coupling approximations employed by others, and it is also amenable to arbitrary transmission line cross sections. The analysis presented here is also used to determine the coil’s internal fields. Calculation of those fields cannot be performed using the transmission line currents in a simple Biot Savart formulation, since to satisfy the boundary conditions at the conducting surfaces one must know the charge distribution around the periphery of the transmission lines and at the lines’ reference conductor (a cylindrical shield in this case). The paper comprises five major sections. In Section II, the TEM resonator and its operation are qualitatively described. In Section III, the theory of multiconductor transmission lines is reviewed and applied to the TEM resonator. In Section IV, the model parameters are derived. The fields and modal structure of

Fig. 1. Construction of a four-element TEM resonator.

Fig. 2.

Top view and cross section of a four-element TEM resonator.

the device are described in Section V. Experimental validation of the model, including its frequency response and field distributions, is presented in Section VI. Concluding remarks appear in Section VII. II. PHYSICAL AND FUNCTIONAL DESCRIPTIONS OF THE TEM RESONATOR The geometry of a typical TEM resonator is shown in Figs. 1 and 2. In practical applications, a large even number of elements (e.g., 16) are used, but only four elements are shown in the figure to simplify the illustration. Two types of coaxial TEM transmission lines play a role in its operation as described below. The outer surfaces of the coil comprise a cylindrical shell and end plates, which form a cavity. Apertures in the end plates permit the test object (nominally a human torso or head) to be the radius of the outer shell and by inserted. We denote by the cavity height. Within this incomplete cavity a number

BAERTLEIN et al.: THEORETICAL MODEL FOR AN MRI RADIO FREQUENCY RESONATOR

of axially oriented circular conducting tubes of radius are aris shown.) The center of each rayed in a circle. (The case tube is placed at a radius from the center of the cavity. These tubes are also separated from the cavity end plates by short gaps of height in the axial direction, which electrically isolate the tubes from the cavity walls. The coil is driven by sources connected across the gap at one or more tubes. The cylindrical shell and the tubes comprise eccentric coaxial lines. With an appropriate source, TEM waves will propagate in the axial direction on each of these lines. Furthermore, coupling between the lines produces modes that (as shown later) exhibit a sinusoidal field variation in the azimuthal direction. The interior of each conducting tube contains another conducting cylinder of radius which constitutes a concentric circular coax (a second type of coaxial line). The inner cylinder is supported by a dielectric sleeve of relative permittivity The center conductors of these smaller coaxial lines do not extend the full length of the tubes. Instead, the insertion depth , can be adjusted and, as of the inner conductors, denoted such, they comprise open-circuited transmission lines of variable length, which interact with the cavity fields through the following process: When a wave propagating in the large eccentric coax reaches the cavity end plate, it impresses a voltage across the gap between the conducting tube and the cavity end plate. The center conductors of the smaller coax lines make electrical contact with the end plates via a sliding contact and, as a result, the voltage present on the eccentric coax appears across the inner and outer conductors of the smaller coax. Waves are excited on the small coax, reflected by the open circuit termination, and then propagated back down the small coax lines, through the gaps and back into the large eccentric coax lines. The large and small coax lines comprise a finite transmission line that exhibits resonances. Because multiple transmission lines are involved, multiple TEM modes will arise in the cavity, one of which comprises the desired coil field. The process described above involves TEM fields only, and it will be assumed throughout this work that the fields in the coil are dominated by the TEM modes. The coil is explicitly designed to produce TEM fields in its interior, and this situation is supported by some simple analysis. First, it is observed that the outer shield of the TEM resonator comprises a finite section of a circular waveguide terminated in conducting plates. To this waveguide are added conductors (the tubes) terminated in adjustable loads. Ignoring the apertures in the end caps, the TEM resonator becomes a closed cavity bounded by perfectly conducting surfaces. The field in the cavity can be expressed as a sum of modes. Of these, only the TEM modes propagate at sufficiently low frequency. Any other mode will have a nonzero cutoff frequency Cutoff frequencies for the higher-order modes can be estimated for a given coil. As an example, consider a coil built for proton imaging of the head at 8 T with a radius of roughly 20 cm. A hollow circular waveguide (without tubes) supports only TE and TM modes. The waveguide mode with the lowest cutoff frequency is TE11 for which the cutoff frequency is where is the speed of light. The head coil radius leads to a TE11 cutoff of 440 MHz. The lowest-order mode of

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the cavity formed by this coil is TM010, the resonance frequency of which is The addition of conducting, eccentrically located tubes makes it possible for the cavity to support TEM modes. The tubes will also disturb the TE and TM modes of the equivalent waveguide. It has been shown [11] (also see [12]) that when the and located close to the walls of the tubes are thin near unity) they have a small effect on the cutoff structure frequencies of the waveguide. These conditions are satisfied by typical coils. Thus, when the operating frequency is somewhat below the cutoff frequency for the TE11 mode, all non-TEM waveguide and cavity modes will be cut off. It will be shown that using only the TEM modes it is possible to predict both the fields and the frequency dependence of the empty coil. That finding constitutes perhaps the best evidence that higher-order modes, if they exist, play a secondary role in the unloaded TEM resonator. The finding that only TEM modes are above cutoff has implications for the operation of the coil. The field components of these axially propagating waves are transverse. Vaughan et al. [3] assert that the coil supports azimuthally propagating waves. To produce an azimuthally directed Poynting vector, either or would require a component in the axial direction. Since the dominant fields are all transverse, we conclude that any azimuthally propagating energy is unrelated to the TEM fields. The nature of the fields within the TEM resonator is much different from those produced in a birdcage coil [13], even though a shielded birdcage is similar in appearance to a TEM resonator. In a birdcage coil the upper and lower rings form a closed two-wire transmission line that supports waves propagating in the azimuthal direction. Unlike the birdcage, the TEM resonator has no azimuthally oriented conductors that could support a TEM wave in that direction. III. MULTICONDUCTOR TRANSMISSION LINE THEORY The theory of multiconductor transmission lines is available in common textbooks [14] and in specialized works [15]–[17]. We present here those aspects of the theory necessary to establish our notation. A time harmonic excitation of the form is assumed and suppressed. MKS units are used throughout this work. A. General Theory transmission lines.1 Let the Consider an assemblage of and , revoltage and current on line be denoted spectively. These voltages and currents can be written as column and where vectors superscript denotes matrix transpose. These quantities satisfy vector transmission line equations (1) where and are the mutual impedance and admittance matrices of the lines. These matrices can, in turn, be expressed as 1A common return conductor is assumed for these lines and is not included in the lines.

N

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and which involve the line resisinductance conductance , and capacitance These matrices have several important properties, which are summarized in [16] and based in part on results in [18]. The properties exploited here are that and are symmetric with positive elements only on the diagonal. The matrices and are also symmetric, but they have only positive elements. It follows that and are symmetric. Straightforward substitution of the transmission line equations leads to Helmholtz equations for the line voltages and currents, which are solved via an eigen-decomposition of the matrix or Let be a matrix composed of the eigenproduct and let be its eigenvalues, vectors of which must satisfy tance

diag

(2)

The modal voltages at position , denoted2 and the transmission line voltages are related by ([16], [17]). The modal voltages can be expressed forward and backward propagating waves

matrix since it affects primarily the coil Q, and the Q of the unloaded coil is dominated by radiation losses through the apertures. (The Q of the loaded coil is dominated by losses in the tissue.) For a homogeneous medium the inductance and cawhere is the pacitance matrices are related by These relations can be exploited to identity matrix of order and be, determine the modal decomposition matrix. Let respectively, the matrix of column eigenvectors and the diagonal is orthogmatrix of eigenvalues for Since is symmetric, and onal. We can write These results imply that for a lossless coil or can be used as the modal decomposition matrix B. Solution for Modal Voltages are determined by applying The modal voltage vectors the boundary conditions at the ends of the line. The lines are asand by passive loads, sumed to be terminated at The RF genwhich are represented by impedance matrices erator(s) can be modeled by ideal current source(s) connected end, which leads to across tube gaps at the (7) (8)

(3) where the matrix

is given by diag

(4)

Using these relations and the transmission line equations the following results can be derived:

are the voltage wave and amplitudes to be determined. Using the transmission line equaare found to satisfy tions, the transmission line currents

(9) (10) where

are reflection coefficients at the ends

(5) where the current wave amplitudes have been expressed in terms of voltage waves and the characteristic admittance matrix (6) diag One can also define In this result a characteristic impedance which satisfies Although the preceeding analysis places no restrictions on these transmission lines, TEM resonators are typically built with symmetrically placed identical elements. In this case for both and , subsequent rows (columns) are found by barrel-shifting the preceeding row (column) right (down) by one position. The and have the same property. Matrices of products this kind are referred to as symmetric and circulant, and their properties have been known for some time [19]. The circulant matrix is a special case of the Toeplitz matrix, for which elements along each diagonal are constant. That symmetric circular resonators generate circulant matrices was noted previously by Joseph and Lu [20]. Some properties of the eigenvectors of these matrices were exploited by [21]–[23] and are used below. Additional simplification is possible. For an empty coil the conductance matrix vanishes. We can also set the resistance 2Here,

and in what follows, the superscript

m will indicate a modal quantity.

(11) and are modal deand compositions of the characteristic admittance and loads, respectively. As discussed earlier, each eccentric coaxial line in the TEM resonator is terminated by a small coaxial transmission line. Since there is essentially no potential for cross coupling among these lines (because of shielding by the tubes), we take to be diagonal. The elements of these matrices are simply the input impedances seen looking into the lines, namely where is the characteristic is its propimpedance of the small coax, is the insertion depth of the center agation constant and conductor in the small tube. For a single-tuned device we take Double-tuned coils, which involve inserting alternate stubs to a different depth [3], are modeled by alternating on the diagonal of values for C. Simplification for Identical Components For a single-tuned coil comprised of identical, symmetrically placed tubes a suitable transformation matrix is the discrete Fourier transform (DFT) operator [19]

(12)

BAERTLEIN et al.: THEORETICAL MODEL FOR AN MRI RADIO FREQUENCY RESONATOR

This orthogonal matrix satisfies where superscript indicates the Hermitian (conjugate transpose) matrix. For any circulant matrix the following relation holds: diag

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Because is an even function of its argument, written as a cosine series. We find

can be

(13)

Furthermore, it follows from the properties of the Fourier transform that if is symmetric, then even (14) distinct values for which implies that there are only Since diagonalizes circulant matrices, the modal admitand the modal load impedance matrices tance matrix are diagonal. Hence, are also diagonal, as is the factor

diag

odd (21) where was given For identical stub depths, of above. The th resonant frequency is then the root

(15) (22)

which leads to extensive simplification in the expressions for This result (with (9) and (10)) also shows that the condition for resonance is det (16) that satisfy this expresThe complex frequencies sion can be determined numerically. can be expressed analytically, Under special conditions This approach leads which simplifies the expressions for to the principal result of [9], which is now derived as a special case. For tubes well separated from each other and from the cavity shell, analytical expressions for the elements of and are available (cf. [14] and [16]). The inductance matrix for a weakly coupled symmetric device is where

(17) The modal characteristic impedance matrix of a lossless device is given by (18) is diagand, since is a circulant matrix, the quantity onal with values equal to the DFT of the mutual inductance diag

(19)

(20)

which is determined numerically. IV. CALCULATION OF MODEL PARAMETERS In this section, a numerical method for determing the transmission line parameters is presented. It is shown that the macan be calculated directly from the geometry. trices and One feature of the transmission lines, however, is not treated in this model. Coupling from the cavity into the small coaxial lines is accomplished via gaps of length in the ends of the tubes. Since the gaps couple the cavity waves to the small coax lines, this coupling is essential for coil operation. The junction between the cavity and the small coax line can be approximately described by a coaxial line discontinuity between the larger eccentric coax and the small coax. A detailed analysis of this discontinuity using both an integral equation and a mode-matching technique (similar to that described in [24]) appears in a separate work [25]. Therein, it is shown that the junction can be represented by a -network of capacitors, which account for static fields between: 1) the cavity wall and the inner conductor of the tube; 2) from the end of the tube to the outer shell; and 3) from the tube end to the tube inner conductor. For dimensions typically found in head coils, the net effect of this discontinuity can be represented by a shunt capacitance on the order of 1 pF. This small capacitance is equivalent to a slight increase in , and it has been omitted in this analysis. For lossless conductors in a homogeneous medium, it was noted that the inductance and capacitance matrices and are and related to each other and to via If coupling among the coaxial lines is weak, a closed-form approximation for is obtained. The assumption of weak coupling among the coaxial elements does not hold for all geometries of potential interest, and for this reason a numerical procedure to determine the per-unit-length parameters was developed. In any plane transverse to the direction of propagation, the behavior of a TEM field is that of a static field, which can be described by a suitable scalar potential. This fact can be used to evaluate the per-unit-length capacitance and inductance of the lines via two-dimensional solutions of the static field equations. In this work, it is assumed that all conductors have infinite

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conductivity and are therefore equipotential surfaces. Assume with rethat the th small coaxial line is held at potential spect to the outer cavity wall, and that all other lines are held at that satisfies zero potential. There exists a potential function in the plane and assumes the foregoing values on is the tubes. The boundary condition at the cavity wall To generate this potential a charge distribution, will exist The total charge on tube due to a on each tube over the pevoltage on tube is given by the integral of riphery of the tube. With the tube voltages as specified above, the per-unit-length mutual capacitance of tube driven by tube is for and is the self capacitance. When all coaxial tubes are identical and symmetrically located, the capacitance matrix is circulant with where is the self capacitance and is the mutual capacitance between an element and its th neighbor. Because of symmetry we also have The induced charge distribution is calculated using for an integral equation approach. The Green’s function The this problem is defined to satisfy at the cavity wall boundary condition on is The application of Green’s theorem in a plane and the foregoing boundary condition lead to the integral relation

Fig. 3. Geometry of the source point ; observation point  ; and image point  for the Green’s function G of a circular cylinder.

(23) Driving the point of observation to tube in terms of arc length leads to

and parameterizing

(24) which is the integral equation to be solved for the surface charge for excitation of tube This integral equation is solved using a standard technique referred to as the “method of moments” (MoM) [26]. The procedure is well known, and its implementation here poses no significant challenges. For the special case of a circular cavity, can be derived from image theory [18] using the well-known relation for the image of a line charge in a circular cylinder. The result is

Fig. 4. A comparison of per-unit-length capacitances calculated numerically using the MoM formulation and with the weak-coupling approximate form. (a) The case  cm,  : cm, cm. (b) Same geometry with  : cm.

! 38

= 29

= 12

1 = 25

(25) outside the where the image charge is located at cylinder along a line radially directed through the source point (see Fig. 3). The result of numerical calculations for a circular, symmetric array of identically placed elements are shown in Fig. 4. In these cm, a tube radius of results, a cavity diameter of cm, and a tube position of cm have been assumed. When the tubes are not close to each other or to the cavity wall [Fig. 4(a)], the value of derived from the weak-coupling approximation in (17) is within 5% of the numerical solution to

the exact equation. The error in is 11%. For this configuration, tubes more distant than the immediate nearest neighbor have negligible influence on the capacitance. When the tubes approach the cavity wall or each other, the error increases as shown cm was made, in Fig. 4(b). In Fig. 4(b), the change which causes the tube to approach the cavity wall, leading to an and 110% in For most cases the weak error of 12% in coupling approximation is reasonably accurate, but the authors as large as 40% for geomehave observed discrepancies in tries in which the tubes approach each other and the cavity wall simultaneously.

BAERTLEIN et al.: THEORETICAL MODEL FOR AN MRI RADIO FREQUENCY RESONATOR

V. FIELD DISTRIBUTIONS The performance of a coil is determined by the magnetic field it generates. The relation between transmission line fields and equivalent voltages and currents is well known [27]. For a homogeneously filled waveguide the fields may be expanded in and terms of vector mode functions (26)

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5) Use to compute at interior points the modal magvia the gradient of (23) netic fields

(30) as in (28). The following expression for Normalize the gradient is appropriate for a circular shield: (31)

(27) Next, define a frequency of interest and are the modal amplitudes of forward- and where is the wavenumber of mode backward-propagating waves, , and evanescent fields have been ignored. Electric and magnetic mode functions are related via the modal characteristic For TEM modes and lossadmittance, viz: holds for all modes. For less conductors the relation these waves the electric mode functions are given by where is the transverse portion of the Laplais the potential distribution satisfying (23) when cian and the voltage on tube is the th component of the th modal in which voltage distribution is the th unit vector. The mode functions are normalized such that (28) Mode functions for distinct nondegenerate modes are orthonormal (29) is the Kronecker where is the interior of the cavity and for and 0, otherwise). delta Given a device geometry, the field distribution is computed as follows: First, perform the frequency-independent calculations listed below. for 1) Compute the tube charge density distribution individual excitation of the tubes as described in Section IV by solving (24) using the method of moments. over each tube and using Compute by integrating Compute using (Alternatively, for a circular shield and identical, weakly-coupled circular elements we can approximate from (17) and use 2) Determine the modal decomposition Use (12) for identical symmetric elements. 3) Form the modal characteristic admittance matrix using (6) and 4) For mode , let the voltage on tube be For this voltage distribution, by solving evaluate the modal charge distribution (24).

and proceed as follows.

. 1) Form the modal load matrices by solving (9) and 2) Determine the modal voltages (10) for a specific excitation . 3) Evaluate the cavity fields as the sum of modes defined in . (27). For the current amplitudes, use VI. NUMERICAL RESULTS AND VALIDATION In this section results computed from the foregoing model are presented. The model results are compared to data obtained experimentally and with an independent numerical model. A. Frequency Response element head coil The frequency response of an used in the OSU 8-T system was measured experimentally. For the purposes of this test, the device was tuned for operation of near 200 MHz. The coil has an outer shield diameter of 22 cm, a tube outer diameter of 34 cm, a height of 0.65 cm 1.59 cm (5/8 inches), a tuning stub diameter for the tuning (1/4 in), and a dielectric constant of stub line. The tubes were constructed of thin-wall copper pipe (0.08-cm approximate wall thickness) and were placed at a racm. The tuning coax characteristic impedance dius of was calculated using the tube’s thickness-corrected inner cm. The gap between the ends of the radius tubes and the end walls of the coil was 0.45 cm. measured at the input of this coil is shown The return loss modes are visible, in Fig. 5. It can be seen that although the positions of the higher order modes are unclear mode appears near as described below. The desired 177 MHz. There is obvious splitting of a degenerate mode at which has been attributed to imperfect symmetry. Some of the weaker nulls may also be due to asymmetry. The device is modeled as completely symmetric. The resonance frequencies for the same coil calculated using the above model are shown in Fig. 6. Table I presents the measured and calculated resonances. In general, good agreement exists in the location and structure of the model and experimental resonances. The model predicts that higher orders modes will bunch together, which makes them experimentally and numerically difficult to detect. In addition to the foregoing comparison to experimental data, the theoretical model has been compared to a detailed FDTD

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Fig. 5. Return loss measured for a 16-element coil.

Fig. 6. The logarithm of the inverse of (16) for the same coil studied in Fig. 5. The peaks in this figure indicate the modes of the device. TABLE I MEASURED AND PREDICTED RESONANCE FREQUENCIES FOR A 16-ELEMENT COIL

model of the coil being developed in the parallel effort mentioned previously. The FDTD model is a direct numerical implementation of Maxwell’s equations and, as such, it comprises an independent solution for the fields. The FDTD model includes all non-TEM modes and geometric features such as the end-plate apertures. The results are shown in Figs. 7 and 8 where the frequency response of an unloaded eight-element coil computed using FDTD and the present model are compared. Note that distinct quantities are being compared in these figures. In Fig. 7 the FDTD spectrum of the coil field is presented, while

Fig. 7. Spectral magnitude of magnetic fields present in an eight-element TEM resonator calculated via a FDTD model.

Fig. 8. The logarithm of the inverse of (16) for the same coil studied in Fig. 7. The peaks in this figure indicate the modes of the device.

Fig. 8 presents the reciprocal of the determinant in (16). In both figures the location of the peaks correspond to the coil resonances. Excellent agreement in these results was obtained, which suggests that non-TEM modes and cavity apertures have a minimal effect on the resonance frequencies for the empty coil. The coil’s frequency response is an important specification, and (16) can be used to explore the dependence of this response on various design parameters. As an example, Fig. 9 illustrates how coil resonances are affected by tube position and tube radius. Fig. 9(a) is a baseline configuration. Moving the tubes toward the center of the cavity (i.e., decreasing ) tends to increase their coupling, causing the modal frequencies to spread apart [Fig. 9(b)]. Increasing the tube outer radius while mainincreases the modal impedance and shifts taining the ratio the resonances to higher frequency [Fig. 9(c)]. Decreasing the characteristic impedance of the stub tuning lines shifts the resonances down in frequency [Fig. 9(d)]. B. Field Distributions The model is capable of predicting field distributions for mode is of interest modes of all orders, but only the for MRI applications. Figs. 10 and 11 present, respectively, the vector sense and intensity of the transverse magnetic field at

BAERTLEIN et al.: THEORETICAL MODEL FOR AN MRI RADIO FREQUENCY RESONATOR

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(a)

(b)

(c)

(d)

Fig. 9. Variation of frequency response with changes in design parameters. (a) Nominal parameters. (b) Decrease tube location

 by 5% for fixed  = : (d) Decrease  = by 25% for fixed  :

1 by 5%. (c) Increase tube radius

Fig. 11. The B field magnetic energy density for mode 1 in a 16-element coil with linear drive.

Fig. 10. A vector plot of the B field for mode 1 in a 16-element TEM resonator with linear drive.

(the mid-plane) in the 16-element coil described in Section VI-A. A single source is used to excite the coil, resulting in linear polarization over the patient-accessible portions of the coil. It is evident that the majority of the magnetic field energy resides in inaccessible regions. This aspect of the coil’s operation is shared by all TEM resonator modes, by other TEM-based designs, and by a conventional birdcage design. Because circular polarization (CP) is most effective in exciting the MR signal, coils are commonly driven in a quadrature fashion to produce CP fields. When driven in this manner field is highly homogeneous over the central region. In the

Fig. 12 is shown the amplitude of the CP component over a centhrough the coil’s mid-plane. The extent of this tral cut cut is restricted to one tube diameter from the nearest tube, i.e., The figure shows the percent deviation in the magnetic field with respect to its value at the center of the coil. Very good homogeneity (better than 1%) is observed for points separated from the closest tube by more than two tube diameters. High uniformity is also observed experimentally, but making direct measurements of these fields is challenging. Evidence for field homogeneity is presented in Fig. 13 which shows a gradient recalled echo (GRE) image of a 18.5-cm diameter spherA slice taken ical mineral oil phantom through the horizontal mid-line of this image is shown in Fig. 14, variation across the phantom. Close which shows roughly

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Fig. 12. The difference in the amplitude of the B field (circular polarized component) for mode 1 in a 16-element coil. The field is sampled along a line and  <  : Here, cm and between two tubes with z  : cm.

2 = 19

=0

103

1 = 11

Fig. 14.

Fig. 13 Axial gradient recalled echo images from an 18.5-cm spherical mineral : , at 8.0 T [ , repetition time (TR) = oil phantom  0.5 s, echo time (TE) = 8 ms, field-of-view: = 22 cm, slice thickness (ST) = 10 mm, and bandwidth (BW) = 50 kHz].

( = 2 5 = 0)

256 2 256

agreement between the calculation in Fig. 12 and the measurements in Fig. 14 is not expected because the phantom couples to the source and distorts the fields in the coil. The asymmetry in the measured result has been attributed to source coupling and to errors in positioning the phantom at the center of the coil. The electric field is important for its contribution to tissue heating. A vector field plot of the electric field for a single source is shown in Fig. 15. An important feature of coils based on TEM waves is that the magnitudes of the transverse electric and magnetic fields are related by a constant factor—the mode’s characteristic impedance. For a small loop or a typical birdcage (particularly when the height exceeds the diameter), a minima of the electric field and a maxima of the magnetic field are coincident

A horizontal slice from the image of the mineral oil phantom.

Fig. 15. A vector plot of the electric field for mode 1 in a 16-element coil with linear drive.

with the center of the coil. It is not possible to construct a TEM resonator with this desirable property. A comparison of Figs. 15 and 10 suggests that the fields over the working volume of the empty coil approximate plane waves in that the electric and magnetic fields are orthogonal to each other and to the direction of propagation. Using the foregoing analysis, the characteristic impedance of this mode is found to be 237 , which indicates that the ratio of magnetic field to electric field for this wave is somewhat different (about 60% larger) than in a plane wave. The axial dependence of the fields is also of interest. A plot for the 16-element coil driven by a of the axial amplitude of single source is shown in Fig. 16. The figure shows symmetry in , which is required by equal tuning loads on both ends of the coil. The variation along the central axis is 8% with respect to the value at the coil center. Note that the end plate apertures are

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magnetic field. This fact may tend to produce radiation from the apertures, a phenomenon not included in the work presented here, but one which bears investigation in future works. The primary limitation of this analysis is the inability to treat the presence of the body within the coil. That limitation has been addressed in a separate numerical study by the authors [7]. The model presented here should be viewed as an exploratory design tool. Changes in field and resonance frequency resulting from a change in coil parameters can be quickly evaluated for an empty cavity with the present model. The numerical model permits attractive designs to be examined in more detail. ACKNOWLEDGMENT

Fig. 16. The amplitude of the B field for mode 1 in a 16-element coil with linear (single-point) drive. The figure shows an axial cut of the field in the central region.

not included in this model and, hence, the degree of uniformity predicted here is optimistic. VII. CONCLUDING REMARKS A transmission line model has been developed for the TEM resonator, an MRI coil design that has been used at high field strength. The model, which addresses only the empty coil, can replicate both the frequency dependence of the device and the structure of its fields. Non-TEM waves are neglected in this analysis. The agreement obtained in our experiments shows that these higher-order modes have a limited role in the frequency response of the empty coil. The model provides insight regarding the behavior of the device and the relation between its performance and various design parameters. Some of these relations were demonstrated in Fig. 9. When driven by two sources in quadrature, the fields of the empty TEM resonator are highly uniform over much of the patient-accessible region. Field calculations in the transverse plane show less than 1% deviation over the region at least two tube diMeasureameters from the nearest tube (i.e., for ments with a low-permittivity phantom (mineral oil, show about variation, which can be attributed in part to coupling between the source and phantom. The magnetic field has a maxima at the cavity center and decreases slowly as one moves axially away from the center. A feature of the fields produced by TEM-based coils is that the electric and magnetic fields achieve their maxima at the same points. This basic property of TEM fields has implications for the coil’s achievable SNR. Although the internal magnetic fields of the TEM resonator and those of a birdcage appear similar, the electromagnetic behavior of these devices is quite different. For example, a birdcage involves wave propagation in the azimuthal direction, and it produces an electric field minima coincident with the magnetic field maxima at the center of the coil. The axial nature of propagation in the TEM resonator leads to near-planar wave fronts over the working region of the empty coil. The characteristic impedance of those waves is about 240 , which indicates moderate enhancement in the

The authors would like to thank the anonymous reviewers for their helpful comments. REFERENCES [1] P. Röschmann, “High-frequency coil system for a magnetic resonance imaging apparatus,” U.S. Patent 4 746 866, May 24, 1988. [2] J. F. Bridges, “Cavity resonator with improved magnetic field uniformity for high frequency ooperation and reduced dielectric heating in NMR imaging devices,” U.S. Patent 4 751 464, June 14, 1988. [3] J. T. Vaughan, H. P. Hetherington, J. O. Otu, J. W. Pan, and G. M. Pohost, “High frequency volume coils for clinical NMR imaging and spectroscopy,” Magn. Reson. Med., vol. 32, pp. 206–218, 1994. [4] P. Röschmann, “Radiofrequency penetration and absorption in the human body: Limitations to high-field whole-body nuclear magnetic resonance imaging,” Med. Phys., vol. 14, no. 6, pp. 922–932, Nov./Dec. 1987. [5] T. S. Ibrahim, R. Lee, B. A. Baertlein, A. Kangarlu, and P.-M. L. Robitaille, “Three-dimensional full wave analysis for MRI RF coils,” presented at the 7th Annual Meeting of International Society of Magnetic Resonance in Medicine, Philadelphia, PA, May 1999. , “SAR and the B field homogeneity study at high-field MRI: [6] 3T-9T,” presented at the 7th Annual Meeting of International Society of Magnetic Resonance in Medicine, Philadelphia, PA, May 1999. [7] , “Modeling the TEM resonator in the presence of a human head for high field MRI,” presented at the 8th Annu. Meeting Int. Soc. Magnetic Resonance in Medicine, Denver, CO, Apr. 2000. [8] P. Röschmann, “Analysis of mode spectra in cylindrical N-conductor transmission line resonators with expansion to low-, high- and bandpass birdcage structures,” in Proc. 3rd Annu. Meeting Int. Soc. Magnetic Resonance in Medicine, Nice, France, 1995, p. 1000. [9] G. C. Chingas and N. Zhang, “Design strategy for TEM high field resonators,” in Proc. 4th Annu. Meeting Int. Soc. Magnetic Resonance in Medicine, New York, Apr. 1996, p. 1426. [10] J. Tropp, “Mutual inductance in the bird cage resonator,” J. Magn. Reson., vol. 126, pp. 9–17, 1997. [11] G. I. Veselov and S. G. Semenov, “Theory of circular waveguide with eccentrically placed metallic conductor,” Radio Eng. Electron. Phys., vol. 15, no. 4, pp. 687–690, 1970. [12] M. Davidovitz and Y. T. Lo, “Cutoff wavenumbers and modes for annular-cross-section waveguide with eccentric inner conductor of small radius,” IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp. 510–515, May 1987. [13] C. E. Hayes, W. A. Edelstein, J. F. Schenck, O. M. Mueller, and M. Eash, “An efficient highly homogeneous radio frequency coil for whole-body NMR imaging,” J. Magn. Reson., vol. 63, pp. 622–628, 1985. [14] C. R. Paul, Introduction to Electromagnetic Compatibility. New York: Wiley, 1992. [15] N. Fache, F. Olyslager, and D. De Zutter, Electromagnetic and Circuit Modeling of Multiconductor Transmission Lines. Oxford, U.K.: Clarendon, 1993. [16] K. S. H. Lee, EMP Interaction, Principles, Techniques, and Reference Data. New York: Hemisphere, 1986. [17] F. M. Tesche, M. V. Ianoz, and T. Karlsson, EMC Analysis Methods and Computational Models, New York: Wiley, 1997. [18] W. R. Smythe, Static and Dynamic Electricity, 3rd ed. New York: Hemisphere, 1989.

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[19] L. A. Pipes, “Circulant matrices and the theory of symmetrical components,” Matrix Tensor Quart., pp. 35–50, Dec. 1966. [20] P. M. Joseph and D. Lu, “A technique for double resonant operation of birdcage imaging coils,” IEEE Trans. Med. Imag., vol. 8, no. 3, pp. 286–294, Sept. 1989. [21] J. Tropp, “The theory of the bird-cage resonator,” J. Magn. Reson., vol. 82, pp. 51–62, 1989. [22] P. Mansfield, M. McJury, and P. Glover, “High frequency cavity resonator designs for NMR,” Meas. Sci. Technol., vol. 1, pp. 1052–1059, Oct. 1990. [23] M. C. Leifer, “Resonator modes of the birdcage coil,” J. Magn. Res., vol. 124, pp. 51–60, 1997. [24] J. R. Whinnery, H. W. Jamieson, and T. E. Robbins, “Coaxial-line discontinuities,” in Proc. I.R.E., Nov. 1944, pp. 695–709. [25] B. A. Baertlein, Ö. Özbay, T. Ibrahim, R. Lee, Y. Yu, A. Kangarlu, and P.-M. L. Robitaille, “Scattering analysis of re-entrant coaxial lines,” IEEE Trans. Microwave Theory Tech., in preparation. [26] R. F. Harrington, Field Computation by Moment Methods. Malabar, FL: Robert Krieger, 1968. [27] R. E. Collin, Field Theory of Guided Waves, 2nd ed. Piscataway, NJ: IEEE Press, 1991.

Brian A. Baertlein (S’88–M’88) received the Ph.D. degree in electrical engineering from the University of Arizona, Tucson, in 1988. His professional career includes 20 years of experience in electrical engineering and applied physics comprising analyses of scattering and propagation phenomena, antennas, electromagnetic compatibility, sensor systems of various types, sensor fusion, and signal processing. He is currently a Research Scientist and Adjunct Associate Professor of Electrical Engineering at the Ohio State University (OSU) ElectroScience Laboratory (ESL), Columbus. Before joining OSU he was a Senior Scientist with several small businesses doing work for the US DoD and DoE.

Özlem Özbay received the B.Sc. and M.Sc. degrees in electrical engineering from Istanbul Technical University, Istanbul, Turkey in 1993, and 1995, respectively. She is currently a Ph.D. degree student in the Electrical Engineering Department at The Ohio State University, Columbus. Her research interests include propagation in complex environments, numerical techniques in electromagnetics, antenna design, and measurement techniques.

Tamer Ibrahim was born in Alexandria, Egypt, on November 24, 1972. He received the B.S.E.E. degree with distinction (Honors thesis in ground penetration radar) and the M.S.E.E. degree from The Ohio State University, Columbus, in 1996 and 1998, respectively. He is working towards the Ph.D. degree in Electrical Engineering at the same university. In June 1995, he joined The ElectroScience Laboratory (ESL) at The Ohio State University where currently he is a Graduate Research Associate. His areas of interest include numerical techniques in electromagnetics, interaction between electromagnetic fields and biological tissues, and design and analysis of magnetic resonance imaging radiofrequency coils. Mr. Ibrahim is a member of Phi Kappa Phi, Tau Beta Pi, and Eta Kappa Nu.

Robert Lee (S’82–M’83) received the B.S.E.E. in 1983 from Lehigh University, Bethlehem, PA, and the M.S.E.E. and Ph.D. degrees in 1988 and 1990, respectively, from the University of Arizona, Tucson. From 1983 to 1984, he worked for Microwave Semiconductor Corporation in Somerset, NJ as a Microwave Engineer. From 1984 to 1986, he was a Member of the Technical Staff at Hughes Aircraft Company in Tucson, Arizona. From 1986 to 1990, he was a Research Assistant at the University of Arizona. In addition, during the summers of 1987 to 1989, he worked at Sandia National Laboratories, Albuquerque, NM. Since 1990, he has been at The Ohio State University where he is currently an Associate Professor. His major research interests are in the analysis and development of finite methods for electromagnetics. Dr. Lee is a member of the International Union of Radio Science (URSI) and was a recipient of the URSI Young Scientist Award in 1996.

Ying Yu received the B.S. in biomedical engineering at Zhejiang University, Hangzhou, China, in 1992. She is currently a Ph.D degree candidate in biomedical engineering at The Ohio State University, Columbus. She is a member of International Society of Magnetic Resonance in Medicine (ISMRM). Her research interests include: high field imaging, high-resolution imaging, and RF coil design.

Allahyar Kangarlu received the undergraduate degree education in Physics at the Arya Mehr University, Tehran, Iran, and completed the Ph.D. degree in physics at the University of Missouri, Colmbia, in 1987. He is with The Ohio State University, Columbus, where he is currently a Member of the Radiology Department. He has been involved with the 8-T human magnetic resonance imaging (MRI) scanner since its inception. He has conducted theoretical work on radio-frequency (RF) pulses and RF penetration for MRI applications. He has also developed computational tools for solving the Bloch equation, analysis of adiabatic pulses, and pulse power minimization. He has worked in the understanding of dielectric resonances, fast imaging, and the study of the interaction of the electromagnetic fields with human tissues while dealing with the safety aspects of the operation of the 8 T magnet. Presently, he is developing diffusion/perfusion imaging at high fields for application to multiple sclerosis, cancer, and stroke.

Pierre-Marie L. Robitaille was born in North Bay, ON, Canada, on July 12, 1960. He received the M.S. degree in biochemistry from Iowa State University, Ames, in 1984. He received the Ph.D. degree In 1986, from the same institution, completing a double major in Inorganic chemistry and zoology. Prior to becoming the Director of Magnetic Resonance Research at The Ohio State University in 1989, he undertook 3 years of postdoctoral training in biophysics at the University of Minnesota, Minneapolis. He was responsible for the conception and implementation of the 8-T MRI project at OSU guiding both experimental and theoretical aspects of the project. He currently a Professor and holds academic appointments in biophysics, chemical physics, biomedical engineering, medical biochemistry, and radiology. Dr. Robitaille is a member of the International Society of Magnetic Resonance in Medicine and a life member of the American Physical Society.