# An Antenna-Theory Method for Modeling High-Frequency  RF Coils: A Segmented Birdcage Example

Authors: Xin Chen; Victor  Taracila; Timothy Eagan; Hiroyuki Fujita; Xingxian Shou; Tanvir Baig; Robert Brown
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

We suggest that center-fed dipole antenna analytics can be employed in the optimized design of high-frequency MRI RF coil applications. The method is illustrated in the design of a single-segmented birdcage model and a short multisegmented birdcage model. As a byproduct, it is shown that for a long single-segmented birdcage model, the RF field within it is essentially a TEM mode and has excellent planar uniformity. For a short shielded multisegmented birdcage model, the RF field is optimized with a target-field approach with an average SAR functional. The planar homogeneity of the optimized RF field is significantly improved compared with that of a single-segmented birdcage model with the same geometry. The accuracy of the antenna formulae is also verified with numerical simulations performed via commercial software. The model discussed herein provides evidence for the effectiveness of antenna methods in future RF coil analysis.


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## Body

## 1. Introduction

Radiofrequency (RF) field inhomogeneity has been a major challenge in today’s high-field magnetic resonance imaging (MRI) mainly due to the shortened RF wavelength in human tissue at higher frequencies. It has been shown [1] that image quality can be significantly affected. Recently, both experimental and theoretical work have been presented to improve the RF field homogeneity, such as the TEM resonator [2, 3], closed-form analytical
solutions [4–8], and numerical simulations [9–11].  Techniques such as RF shimming [11] and parallel transmission [12] are applied to improve the RF field homogeneity with a focus on the transmit coil array. Although theoretical
progress has been made in analyzing the experimental results, and RF field
homogeneity can be improved with various techniques, a better understanding of
the relationship between the RF current sources and the RF fields is still
needed. A modeling control of the source-field relationship can be used to
optimize the RF performance, much as has been done, for instance, in [13], for
lower frequencies. Moreover, the specific absorption rate (SAR) is an important
RF safety concern as more RF energy tends to be deposited into human tissue in
high-field MRI experiments [14]. Although detailed numerical simulations have
been presented to analyze the SAR distribution in the human head within volume
and surface coils [15, 16], there is an absence of RF coil models that incorporate
both RF field and SAR calculation and optimization. In this paper, we show that
analytic formulae found in antenna theory can be applied with success in the
optimized design of RF coils. It is well known that a static surface current
distribution which is sinusoidal in the azimuthal direction on a surface of a
cylinder produces a homogeneous static magnetic field everywhere in the
cylinder [17]. We find that in a high frequency case, such sinusoidal current
distribution can produce magnetic field which is uniform in transverse planes
and oscillates in the direction perpendicular to the transverse plane; thus transverse
electromagnetic (TEM) mode can be supported. With our model we show that by
restricting the oscillation of the field required by Maxwell equations to the
longitudinal direction, homogeneity in transverse planes can be significantly
improved, as demonstrated in [5].


## 2. Methods

A good
approximation of the oscillating current on a center-fed dipole antenna is a
sinusoidal standing wave current with nodes at both ends [18] (Figure 1):
(1)I(x′=0,y′=0,z′)={ezI0sin[k(l2−z′)],0≤z′≤l2,ezI0sin[k(l2+z′)],−l2≤z′≤0.Figure 1
Current distribution on a center-fed dipole antenna for different antenna’s lengths.λ is the wavelength of the oscillating current source.HereI0 is the
current amplitude, k is the wave
number (i.e., 2π/wavelength), and l is the length of the
antenna. It is assumed that the antenna has negligible cross section and is
aligned along the z direction with its center at the origin, as shown in Figure 2. The vector potential and the magnetic field
produced by this center-fed dipole antenna are therefore determined [18]:
(2)A(ρ,z)=μ04π∫−l/2l/2I(z′)e−ikρ2+(z−z′)2ρ2+(z−z′)2dz′,(3)B(ρ,z)=∇×A(ρ,z)=−μ0I04πi1ρ[e−ikρ2+(z−l/2)2+e−ikρ2+(z+l/2)2−2cos(kl2)e−ikρ2+z2]eφ.Figure 2
Schematic of an idealized center-fed dipole antenna with lengthl.In (2) and (3), the current
distribution is given in (1), (ρ,z) represents the cylindrical
coordinates of the observation point in space, and z′ determines
the location of the source point on the antenna. Given a sinusoidal standing wave
current distribution in (1), the magnetic field produced by this source can be calculated
analytically at any point in the space with (3).To model the
shorter wavelength effects over relevant dimensions and the average dielectric
properties of human head under an RF frequency of 400 MHz, we assume a conducting
dielectric with constant permittivity 49.75 and conductivity 0.59 S/m [19] fills
up all of space. Therefore, the wavelength for an RF field with frequency
400 MHz (corresponding to 9.4T main magnetic field in proton MRI) is
approximately 10.6 cm. We choose this frequency in our modeling as a
representative value of interest in today’s very high-field MRI research. With
the presence of a conducting dielectric and eiωt time dependence, the wave number k of the propagation in the dielectric
will be complex and satisfy
(4)k2=ω2μ0εε0−iωμ0σ.
Although
the magnetic field (3) is derived from a sinusoidal current distribution with a
real wave number k, we will still use
(3) with all the wave numbers k replaced
by the complex value given in (4) as an approximate representation of the RF
field in the conducting dielectric. With the presence of the conductivity,
Maxwell equation gives
(5)∇×B=μ0εε0∂E∂t+μ0σE=(iωμ0εε0+μ0σ)E.
The electric field can then be determined as
(6)E=1iωμ0εε0+μ0σ∇×B.
Once the electric field is known, the local SAR at a spatial
point r can be estimated as
(7)SAR(r)=σ2ρ|E(r)|2,
and the average SAR in a region of interest is
(8)SAR¯=1N∑i=1NSAR(ri),
where the density of the dielectric ρis 1030 kg/m3 [19] representing an average human head value and ri (i=1,2,3,…,N) are N spatial points
specified in a region of interest.A shielded single-segmented
birdcage model and a new shielded multisegmented birdcage model are constructed
with center-fed dipole antennas as building blocks respectively. The RF fields produced by the two models are separately calculated and compared, after
the multisegmented model is optimized with respect to the axial field
uniformity and the average SAR. The shielded single-segmented birdcage model (Figure3(a)) has 16 evenly spaced axial conductors forming a birdcage structure (a
model with 10 legs is shown in the picture for clarity) and a cylindrical
shield. The inner diameter is 25 cm and the outer diameter is 29 cm, attempting
to model a typical head RF coil. Each axial conductor is constructed as a
center-fed dipole antenna and there are no end rings. The azimuthal current
distribution among conductors is sinusoidal. Along the longitudinal (z)
direction, the current has a simple sinusoidal standing wave z-profile as given
in (1). It is noted
that when the length of the center-fed dipole antenna satisfies l=(n+1/2)λ (where λ is the wavelength of the
oscillating current source and n is
any even integer), the current distribution (1) becomes cos(kz). Thus, in
order to avoid more complicated charge and current distributions, the length of
the center-fed dipole antenna is always assumed to be (n+1/2)λ with n an even integer. Also, to
focus on the wavelength effect due to the shortened electromagnetic wavelength by
itself we neglect, for the moment, the conductivity of the dielectric for all
the calculations related to the RF field distribution of the single-segmented
birdcage model. The effect of
the shield can be simulated with the image method [20], which states that the
field produced by the currents induced on a shield is identical to the field
produced by the image of the original current with respect to the shield. In the
case of a cylindrical shield, the image of the current flowing in the axial
conductor is a current flowing with opposite direction and located coaxially at
a distance Rsh(Rsh−R0)/R0 from the shield, where Rsh and R0 are the outer and inner
radii of the birdcage model, respectively. The RF fields of the shielded birdcage model are then
determined with (3).(a) Schematics of a shielded single-segmented birdcage model and a representative current distribution on an axial conductor. A model with 10 conductors is shown. The lengthl satisfies l=(n+1/2)λ where n is an even integer and λ is the wavelength of the source current. A representative current distribution on a 2.5λ-long center-fed dipole antenna is also shown. (b) Schematics of a shielded multisegmented birdcage model and a representative current distribution on an axial conductor consisting with five half-wavelength-long center-fed dipole antennas.
(a)(b)For the shielded multisegmented birdcage model (Figure3(b)), the length is chosen to be 2.5λ (approximately 26.6 cm). The inner and the outer diameters of the model are 25 cm and
29 cm, respectively, the same as the single-segmented birdcage. There are eight
axial conductors and no end rings. Each axial conductor consists of five
half-wavelength-long center-fed dipole antennas. Three of them (the middle
segment and the two on one side of the middle segment) are independently fed and
the other two on the other side have current sources that are fixed by the
axial symmetry around the central transverse plane. The image method described
above is also applied here to incorporate the effect of the cylindrical
shielding. Based on (3) and (6), the magnetic and the electric fields produced
by each RF element (the half-wavelength-long center-fed dipole antenna) can be
calculated. Therefore, the magnetic and the electric fields of the model are
the superposition of the fields produced by all the individual RF elements and are
functions of current amplitudes and phases on these elements. In order to
obtain a uniform RF field profile with control over average SAR, a functional
is constructed as
(9)W=SAR¯+∑j=1Nrλj[B1,x(I1,ϕ1,I2,ϕ2,…,IN0,ϕN0;rj)−Bx(target)(rj)]+∑j=1Nrμj[B1,y(I1,ϕ1,I2,ϕ2,…,IN0,ϕN0;rj)−By(target)(rj)]=1NSAR∑i=1NSARσ2ρ|E(ri)|2+∑j=1Nrλj[B1,x(I1,ϕ1,I2,ϕ2,…,IN0,ϕN0;rj)−Bx(target)(rj)]+∑j=1Nrμj[B1,y(I1,ϕ1,I2,ϕ2,…,IN0,ϕN0;rj)−By(target)(rj)].
Here, Ii and ϕi are the unknown current amplitude and phase on the ith half-wavelength-long center-fed dipole antenna, N0 is the total number of the antennas (which is equal to 40 in the
present model), rj is the jth of the Nr total constraint points in
the central axial plane where we calculate the x and y components of the RF
field produced by the model (B1,xandB1,y) and impose
the desired target field with both uniform amplitude and phase (Bx(target)=0andBy(target)=constant),
and ri is the ith of the NSAR total spatial points in
a region of interest (ROI) where we determine the electric field E(ri)
and the local SAR. A cylindrical ROI with a diameter of 16 cm and a length of 8 cm
coaxially located within the multisegmented birdcage model, with their centers
coincide, is selected for an illustrative SAR calculation. λj and μj are the familiar
Lagrange multipliers. The functional W is minimized to obtain a uniform RF field using a functional approach to
control SAR. The minimization is achieved by differentiating with respect to Ii, ϕi, λj, and μj, and solving the
obtained 2N0+2Nr equations to find 2N0+2Nr unknowns Ii, ϕi, λj, and μj. Since the SAR term is a
quadratic function of unknown current amplitudes and phases, the 2N0+2Nr equations are all linear
and can be solved with matrix inversion. Optimal current amplitudes and phases
are found and can be employed as sources parameters to determine the
corresponding RF field and SAR profiles.


## 3. Results

The single-segmented birdcage model is assumed to be driven by a current that is
sinusoidal in the azimuthal direction around any transverse plane(10)Ij=I0ei(ωt+ϕj),
where Ij and ϕj are the current amplitude
and phase in the jth axial conductor,
and the phase ϕj coincides with the azimuthal angle of the axial
conductor. The RF fields produced by models with different lengths l=(n+1/2)λ are calculated with the
analytical formula (3). It is shown that with fixed diameter, the longer model has
increasingly better RF field uniformity. Figure 4 shows maps of the amplitude
of the clockwise component of the circularly polarized RF field (the component
contributing to the excitation of the nucleus spins) in the central transverse
plane for single-segmented birdcage models with lengths (a) 2.5λ (approximately 26.6 cm),
(b) 10.5λ
(approximately 1.12 m), (c) 20.5λ (approximately 2.18 m),
and (d) 100.5λ
(approximately 10.7 m). The normalized RF field amplitude across the central transverse plane
along the x-axis is also shown (Figure 4(e)). In terms of the maximum
percentage deviation from the field at the center, the model with a length of
2.5λ has a nonuniformity of 60%,
the one with a length of 10.5λ has
a value of 32%, the 20.5λ-long model has a value of 10%, and the 100.5λ-long
model only has a nonuniformity of 0.4%. The normalized amplitude of the RF
field along the longitudinal direction (z direction) is shown in Figure 5. As
the length of the model increases, the oscillation of the field in the z
direction approaches the absolute value of cos(kz), where k is the wave number of
propagation in the dielectric. In contrast to a low-frequency birdcage, the
length of the coil has to be very long before excellent transverse uniformity
is achieved.Amplitude of the clockwise component of the circularly polarized RF field in the central transverse planes for the single-segmented birdcage models with length (a) 2.5λ, (b) 10.5λ, (c) 20.5λ, and (d) 100.5λ. In (e), the normalized field variation across the central transverse plane along x-axis is shown.
(a)(b)(c)(d)(e)Figure 5
RF field variation along the longitudinal(z) direction for single-segmented birdcage models with different lengths. The results are all compared with the absolute value of the cosine function.The unknown current amplitudes and
phases of a multisegmented birdcage model are found through the aforementioned
optimization process. Table1 shows current amplitudes and phases among the
eight half-wavelength-long center-fed dipole antennas of three independent-fed
sections (i.e., the middle section, the section to the right (left) of the
middle section, and the rightmost (leftmost) section) for the B1 field strength
1.16 μT at the center of the
model. The symmetry of the current amplitudes implies that among the eight antenna segments of any section
only four of them are independent. The current phases, nevertheless, do not
exhibit any specific symmetries or relations with the azimuthal angles of the
RF elements. The RF field optimized
for uniformity in the central transverse plane is calculated with the solved
currents. Figure 6 shows the map of the amplitude of the clockwise component of the
circularly polarized RF field
and the variation of the field along the x-axis in the central transverse plane
of the optimized multisegmented birdcage model. It can bee seen that in the
central region, the RF field variation is below 20%, compared with a nearly 60%
variation of a single-segmented birdcage model (Figure 4). Figure 7 shows the
RF field variation along the longitudinal (z) direction for the optimized multisegmented
birdcage. The field pattern coincides with the absolute value of cos(αz) in the proximity of z=0, where αis the real part of
the complex wave number of the propagation in the dielectric. The average SAR
within the aforementioned cylindrical ROI of this optimized model is 3.7 W/kg.Table 1
Optimal current (a) amplitudes and (b) phases of the half-wavelength-long center-fed dipole antenna elements in the multisegmented birdcage model.
(a)
Current amplitudes/A
Azimuthal angle of RF elements/degree22.567.5112.5157.5202.5247.5292.5337.5The rightmost (leftmost) section10.957.453.4712.3810.957.453.4712.38The right (left) section6.354.232.657.436.354.232.657.43The middle section4.022.882.265.324.022.882.265.32(b)
Current phases/degree
Azimuthal angle of RF elements/degree22.567.5112.5157.5202.5247.5292.5337.5The rightmost (leftmost) section158.59216.59264.86317.69338.5936.5984.86137.69The right (left) section283.06335.2723.0590.43103.06155.27203.05270.43The middle section95.49145.82196.06268.50275.49325.8216.0688.5005Amplitude of the RF field in the central axial plane (a) and across the diameter along the horizontal(x) direction (b) for the optimized multisegmented birdcage model.Figure 7
RF field variation along the longitudinal(z) direction of the optimized multisegmented birdcage model.


## 4. Discussion

Due to the shortened RF field wavelength inside human tissue
for high-field MRI, it is more difficult to use conventional RF coils to
generate uniform RF field over the imaging region of interest. In terms of
theoretical considerations, the Biot-Savart law, which is widely applied to
determine the RF field in the low-frequency regime, is unable to give accurate
predictions for the RF field of high frequency. For high-field MRI, most of the
theoretical calculations and simulations are performed with commercial software
and iterative numerical algorithms. Whatever analytical solutions that have
been found previously available are mathematically complicated and have rarely
been used to simulate modern RF applications, such as RF shimming and parallel
transmission. In this paper, we have applied new analytic formulae from antenna
theory to calculate the RF fields for a representative class of RF coils, namely,
head coils with a birdcage structure. For the single-segmented birdcage model
(with single center-fed dipole antennas used for each axial conductor), at high
frequency, increasing the length leads to improved central planar uniformity
for the RF field. In fact, if the length of the model is much larger than the
fixed diameter, the oscillation of the RF field required by Maxwell equations
will be restricted to the longitudinal direction and excellent uniformity of
the RF field in any of the transverse slices can be achieved. (In the limit of infinite length, the
uniformity is perfect.)In
the long-coil limit, the electromagnetic fields inside approach a TEM mode. In
this limit, the current and charge distributions on the coils have the same
sinusoidal azimuthal dependence,sinφ as found
for the low-frequency birdcage, both yielding internal magnetic fields that are
spatially uniform and pointing along the x-axis. Let us see how this sinusoidal
azimuthal dependence arises from waveguide theory, where it is well known [17] that, for a TEM wave,
Maxwell equations become
(11)∇t2{EB}=(∇2−∂2∂z2){EB}=0.This means that the electric field is a solution
of a two-dimensional electrostatic-like problem in the transverse plane
perpendicular to the wave propagation direction. Although the TEM mode cannot
exist inside a single hollow (good) conductor, which generates an equipotential
surface, it can be supported with two or more conductor surfaces such as a
coaxial cable or transmission line, or a surface around which the potential is
not constrained to be constant. Suppose a cylinder is formed by such surfaces,
and a TEM electromagnetic wave (which can be either a standing wave or a
traveling wave) exists in this cylinder such that the electric and magnetic
fields in any transverse plane are uniform. For example, suppose a spatially
uniform magnetic field points along thex-axis and a spatially uniform electric
field points along the y-axis. At the surfaces, the electric and magnetic
fields must satisfy the familiar boundary discontinuity conditions
(12)(Eout−Ein)·n=σε0,n×(Bout−Bin)=μ0K,
where Ein
and Eout
(Bin and Bout)represent
the electric (magnetic) fields inside and outside the surface, respectively, σ(K) is the surface
electric charge (current) density, and the normal (to the surface) vector n
points from the inside of the cylinder toward the outside of the cylinder. In
the case where the electric and the magnetic fields outside the coil are zero,
and the electric (magnetic) field is uniform inside the cylinder and along the y-axis
(x-axis), we have
(13)(0−E0ey)·(cosφex+sinφey)=σε0,(cosφex+sinφey)×(0−B0ex)=μ0K,
where E0 (B0) is the magnitude of
the spatially uniform electric (magnetic) field, ex
and ey
are unit vectors pointing, respectively, along the x and y
directions, and φ is the azimuthal angle.
It can be seen that the azimuthal distribution of the electric charge and the
surface current density are both proportional to sinφ where φ is the azimuthal angle,
and the direction of the current flow points along the z direction. These
results are consistent with our calculations.Although
an infinitely long birdcage model with conventional sinusoidal current
distribution can support a perfectly uniform TEM mode, a short one with length
comparable to a normal RF head coil can not. Therefore, we have constructed a multisegmented
birdcage model with a large number of center-fed dipole antenna RF elements. Each
of the original axial conductors in the single-segmented birdcage model is now
replaced by five equal subleg segments in our example. With our analytical
tools the RF field can be calculated and optimized. The optimal RF field shows
excellent central planar uniformity. In other words, a kind of RF shimming is
achievable with our model and theory. Although excellent RF field planar
homogeneity is achieved with our optimization process in the central transverse
plane, the RF field uniformity degrades rapidly as the transverse plane moves
away from the center of the model. It can be seen in Figure7 that the oscillation
of the field coincides with the absolute value of the cosine function only
around z=0. It implies that perfect sinusoidal longitudinal (along z) oscillation
of the field is approximately restricted to the z=0 region, and, as a result,
the (transverse) field within the central transverse plane is approximately
uniform. Although in the present solution the planar uniformity of the RF field
degrades rapidly as the transverse plane moves away from the center, a similar
optimization can be performed over other transverse planes to improve the field
uniformity. Although the planar RF field homogeneity is improved, it is noted
that the field strength significantly increases toward the periphery of the
central axial plane. This is due to large current amplitudes in the antenna RF
elements (Table 1). In fact, in our present solution, the uniform field in the
central transverse plane is a result of partial cancellations among the fields
produced by individual antenna elements. Therefore, the current amplitudes on
the antenna elements tend to increase to produce a reasonable RF field strength
in the central region of the axial plane. The increased currents lead to
stronger both magnetic and electric fields at the periphery. As a result, the
average SAR is bigger than results presented in some other numerical
simulations such as [21]. It is difficult to reduce the SAR under the
constraints demanding uniform RF field. In order to address this problem, we
may perform the optimization process in a smaller region of interest to balance
the RF field homogeneity and the SAR. In our present model, we have assumed a current distribution that oscillates
with a dielectric wavelength to make an analytic solution possible. In fact,
with the more general endpoints for our sinusoidal current z-profile formula, by superposition treating different kinds of the current z-profiles, we can approximate, for
example, a more uniform current z-profile.In our modeling, we have assumed that the dielectric fills up
the whole of space instead of assuming a layer of air between the coil and a
cylindrical dielectric phantom [5–8] or using a heterogeneous head model [9–11].
We make this approximation mainly for simplifying our analytic calculations. To
estimate the error of this approximation, we performed numerical simulations
with HFSS (Ansoft
Corporation) to study the RF field produced by our building blocks (half-wave-length-long
center-fed dipole antennas) under several different conditions. Figure 8(a) shows
the amplitude of the complex magnetic field at z=0 as a function of the radial
distance from a half-wavelength-long center-fed dipole antenna aligned along
the z-axis with its center at the origin (as shown in Figure 2). The fields are
rescaled such that they are identical 0.8 cm away from the antenna. Fields under
the following conditions are plotted: (i) fields calculated with our analytic
antenna formula with a dielectric filling up the whole space, (ii) fields
calculated with numerical simulations with a dielectric filling up the whole
space, (iii) the antenna is located coaxially 12.5 cm from the central axis of a
dielectric cylinder with radius 11.5 cm (i.e., there is a 1 cm air gap between
the antenna and the surface of the cylinder), and (iv) the antenna is located
coaxially 12.5 cm from the central axis of a dielectric cylinder with radius
9.5 cm (i.e., there is a 3 cm air gap between the antenna and the surface of the
cylinder). It is shown that with the dielectric properties of an average human
head, there is an excellent agreement between the analytic antenna formula and
the numerical simulations without the presence of any air gap. The error
introduced by the approximation that ignores the air gaps is an overall scaling
difference of the field strength as long as the field is not estimated close to
the antenna (typically, at least 2 cm away from the antenna). The very
interesting point is that, by adjusting the conductivity to 3 S/m, the antenna
formula and the “dielectric space” assumption can provide an excellent fit to
the field distribution obtained with numerical simulation where there is an air
gap between the antenna and the dielectric cylinder, as shown in Figure 8(b). These
results suggest that the analytic antenna formulae are useful for modeling
practical RF coils. Although further calculations can be performed to address
the problem with different layers of dielectric, our present results illustrate
how RF field inhomogeneity is caused by the so-called wavelength effect and the
manner in which we can achieve a desired RF field pattern through RF shimming. In
a more realistic, heterogeneous case, improved RF field homogeneity can be
anticipated using the present antenna solution as input followed by numerical
algorithms in an iteration procedure, in order to converge to a better approximation.Complex magnetic field amplitude calculated with the analytic antenna formula and with the numerical simulation under different conditions. The field strength is rescaled for comparison.
(a)(b)Finally, the question arises about the construction of an RF
coil based on our model and theory. Although our present model has a total
number of 40 antenna RF elements, these elements are not independent. First of
all, the aforementioned symmetry with respect to the central axial plane of the
multisegmented birdcage model reduces the number of independent current sources
from5×8=40 to 3×8=24. Furthermore, the current amplitudes found
with the optimization (as shown in Table 1) have certain symmetry such that the
current amplitude of one antenna element is always identical to the current
amplitude of the antenna element diagonal to it. Although current phases do not
exhibit any specific symmetry, the optimal solution may lead to further
reduction of independent current sources. While the purpose of the present work is to present a
theoretical tool for an initial optimization, there are multiple solutions to
the practical problems. For example, with respect to decoupling in transmit
mode, we could start with overlapping the nearest neighbor coils and then add
combinations of capacitive or inductive elements for the decoupling of the
remaining (next nearest neighbor and beyond) interactions. In particular, the present
multisegmented birdcage model, in any case, can be implemented as a “parallel transmit
volume coil with independent control of currents on the array elements” [22].
The techniques presented in [22] can be applied to address the problems of
decoupling and feeding for constructing a practical coil based on the present
model.


## 5. Conclusion

Due to the shortened wavelength of the RF field, RF field
inhomogeneity has become a major challenge for high-field MRI. We present an
analytic tool to simulate the RF field in the appropriate high-frequency limit.
It is shown that an analytic calculation based on antenna theory and an
optimization on that modeling can be effectively carried out. RF shimming is
achievable to produce RF field with excellent planar uniformity. Our model and
analytical calculations can help understand the relationship between the
current sources and the RF fields. The major advantage of the present analytic
tools is efficiency; for instance, the calculations are much faster than
numerical iterative algorithms (by orders of magnitude in comparisons with
field calculations performed through numerical integrations). Important parameters
such as SAR can also be incorporated into the calculation and optimization. We believe
that the methods described here will be quite useful in future RF coil designs.



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*Source: 456019-2008-04-17.xml*