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The finite-difference time-domain (FDTD) method is used to model a birdcage resonator. All the coil components, including the wires, lumped capacitors and the source, are geometrically modelled together. As such, the coupling effects within the

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I
NSTITUTE OF
P
HYSICS
P
UBLISHING
P
HYSICS IN
M
EDICINE AND
B
IOLOGY
Phys. Med. Biol.
46
(2001) 609–619 www.iop.org/Journals/pb PII: S0031-9155(01)14254-5
B
1
ﬁeld homogeneity and SAR calculations for thebirdcage coil
T S Ibrahim
1
,
2
, R Lee
1
, B A Baertlein
1
and P-M L Robitaille
2
,
3
1
Department of Electrical Engineering, The Ohio State University, 1320 Kinnear Road,Columbus, OH 43212, USA
2
MRI Facility, The Ohio State University, 1630 Upham Drive, Columbus, OH 43210, USAE-mail: robitaille.1@osu.edu
Received 26 May 2000, in ﬁnal form 30 October 2000
Abstract
The ﬁnite-difference time-domain (FDTD) method is used to model a birdcageresonator. All the coil components, including the wires, lumped capacitorsand the source, are geometrically modelled together. As such, the couplingeffects within the birdcage, including the interactions of coil, source andhuman head, are accurately computed. A study of the transverse magnetic(
B
1
) ﬁeld homogeneity and the speciﬁc absorption rate (SAR) is presented onan anatomically detailed human head model at 64 and 200 MHz representing1.5 and 4.7 T MRI systems respectively. Unlike that at 64 MHz, the
B
1
ﬁelddistribution is found to be inhomogeneous at 200 MHz. Also, high local SARvalues are observed in the tissue near the source due to the coupling betweenthe source and the head at 200 MHz.
1. Introduction
The radiofrequency (RF) coil is an essential element in magnetic resonance imaging (MRI)systems and therefore its correct design is important. Signiﬁcant effort has been devoted tomodellingtheelectricalcharacteristicsofRFcoils. Forinstance, boththebirdcagecoil(Hayes
et al
1985)andtheTEMresonator(Vaughan
et al
1994)havebeentheoreticallyanalysed. Theﬁniteelementmethod(FEM)hasbeenusedtomodeltheTEMresonatorloadedwithaphantomand human head model (Vaughan
et al
1994). A 2D FEM model has also been used to studythe
B
1
ﬁeld and the SAR in a birdcage coil loaded with a human head model (Jin and Chen1997). The FEM has also been utilized to calculate the SAR inside a human head model ina saddle shaped MRI head coil (Simunic
et al
1996). Since the wavelength was electricallylarge compared with the largest dimension of the saddle coil at 64 MHz, Simunic
et al
(1996)assumed quasistatic conditions in calculating the ﬁelds outside the human head. Jin
et al
(1996) employed the method of conjugate gradients with fast Fourier transform to evaluate theelectromagnetic ﬁelds inside a human head model placed within a birdcage coil. Recently, the
3
Author to whom correspondence should be addressed.0031-9155/01/020609+11$30.00 © 2001 IOP Publishing Ltd Printed in the UK 609
610 T S Ibrahim
et al
ﬁnite-difference time-domain (FDTD) method has been used to model a birdcage coil loadedwith a human head model (Collins
et al
1998, Chen
et al
1998).The aforementioned studies shared the common assumption that the RF coil functionsas an azimuthal transmission line at all the frequencies of interest. This was achieved bydeterminingthecurrentdistributioninthecoilwithouttheheadbeingpresentusingthemethodof moments (Chen
et al
1998), or by replacing the lumped capacitors with voltage sourceswhose magnitudes varied sinusoidally (Collins
et al
1998). Such assumptions, however, arenot valid when the human head is positioned within the coil, due to electromagnetic couplingbetween the coil and the head (Ibrahim
et al
2000b). Although the resulting inaccuracies maynot be too signiﬁcant at 64 MHz, the ﬁeld distribution calculations at higher frequencies aremuch more prone to being invalid. Even when the coil is empty, there are many cases wherethe ideal current distribution is not present (Ibrahim
et al
1999, 2000b). Chen
et al
(1998)stated that neglecting the effect of the head on the coil current distribution can be a majorsource of error and the most accurate simulation involves modelling the coil and the object tobe imaged as a single system. This, however, was considered to be a difﬁcult problem (Chen
et al
1998).The proper modelling of the RF coil has become of critical importance because of thedevelopment of ultra high-ﬁeld MRI systems (Robitaille
et al
1998), and it is anticipated thatthe optimization of RF coils for such a system will rely heavily upon numerical modelling(Ibrahim
et al
2000a, Vaughan
et al
2000, Collins
et al
2000). Assumptions about the coilcurrents can produce misleading results. The importance of RF homogeneity in a birdcageresonator at 4.0 T has been brought into question (Robitaille 1999). Since RF homogeneity isinherently linked to RF power requirements, this issue has also gained increased attention inthe light of recent reports of lower than expected RF power requirements at 8.0 T (Robitaille
et al
1998, Robitaille 1999, Abduljalil
et al
1999). As such, in order to further expand uponthis subject (Leroy-Willig 1999, Robitaille 1999), the electrical characteristics of the birdcagehead coil are now examined.In this work, in accordance with the suggestion of Chen
et al
(1998), both the RF coiland the human head are modelled as a single system using the FDTD method. Also, theRF coil is excited at one point for linear excitation and two points for quadrature excitationin exactly the same manner as the real system. The currents on the coil are then calculatedusing Maxwell’s equations. Unlike previous studies (Vaughan
et al
1994, Jin and Chen 1997,Simunic
et al
1996, Jin
et al
1996, Collins
et al
1998, Chen
et al
1998), these currents areobtained while considering the interaction between the human head and the resonator. Theimplementation of the current source as well as the inclusion of the interactions between thecoil and the head makes the simulation accurate in modelling the performance of the RF coil.Results for the distribution of the magnetic ﬁeld and the speciﬁc absorption rate (SAR) arepresented.
2. Theory
For the electrically large geometries that are usually encountered in high-ﬁeld MRI, one canargue that it is better to use FDTD than FEM. This is due to the fact that the computation timeof the FDTD method is proportional to
N
1
.
33
where
N
is the number of unknowns, while thecomputationtimeoftheFEMmethodisproportionalto
N
>
1
.
5
. Inaddition, thereisalsoawidedisparity in terms of memory requirements. The memory needed to solve an FEM problemwith 2000000 unknowns can be used to solve an FDTD problem with 40000000 unknowns.The one disadvantage of FDTD relative to FEM is that it is less ﬂexible for modelling arbitrarygeometries, because FEM can be applied to an unstructured grid.
B
1
ﬁeld homogeneity and SAR calculations 611
TheFDTDmethodisbasedonaﬁnitedifferenceapproximationofMaxwell’stimedomainequations:
∇×
E
=−
∂
B∂t
−
σ
m
H
(1)
∇×
H
=
∂
D∂t
+
σ
e
E
(2)where
E
(V m
−
1
) and
H
(A m
−
1
) are the electric and the magnetic ﬁeld intensities and
D
(C m
−
2
) and
B
(Wb m
−
2
) are the electric and the magnetic ﬂux densities respectively.
σ
m
(
m
−
1
)
and
σ
e
(
−
1
m
−
1
)
are the magnetic and the electric conductivities respectively.The above two vector equations actually represent six scalar equations, one for each of the
x
,
y
and
z
components of the equation. In the ﬁnite difference method, time and space arediscretized. Timeisdividedintosmallincrements
t
(timestep). Spaceisdiscretizedalongallthree Cartesian coordinates, where the coordinates
x
,
y
,
z
are divided into increments
x
,
y
and
z
respectively. For this work, we choose
x
=
y
=
z
=
3 mm. The electric ﬁeld
E
can be broken up into its three Cartesian components:
E
x
,
E
y
and
E
z
. Similarly, the magneticﬁeld
H
is given by its three components,
H
x
,
H
y
and
H
z
. The notation used to represent anyﬁeld component, say for example
E
x
, is
E
nx
(i,j,k)
=
E
x
(ix,jy,kz,nt)
, where
i
,
j
,
k
are integer indices. All six components of the electric and magnetic ﬁelds can be solved interms of the neighbouring ﬁeld values. A central difference approximation is applied to all sixof the scalar equations which represent Maxwell’s equations. The resulting equations showthat the ﬁeld at a given time step is evaluated in terms of its value at a previous time step andof the ﬁelds from neighbouring cells at an earlier half time step (Ibrahim 1998).An automatic FDTD mesh generator was developed to produce the grid for the birdcagecoil. In order to operate, the mesh generator requires cell size, diameter and length of the coil,size of the transverse domain and the number of coil legs. To minimize the errors caused bystair stepping, the Yee cells were chosen to be small enough (3 mm) to fully characterize thestructure of the coil including the lumped capacitors and the excitation source. Based on thestability criterion (Ibrahim 1998), the time step was chosen to be 5.5 ps. A 16-leg high-passbirdcage resonator was used for the calculations in this work. The diameter and the length of the coil were set to be 28.8 cm and 40 cm respectively, representing the size of the actual 1.5 TGE Signa birdcage coil. The coil structure in this model was composed of perfect electricconductors and the size of each conductor was assumed to be negligible.The next step was the choice of the transverse domain size. With the given coil size, thedomain was chosen to have 148 cells in the
x
and
y
directions and 186 cells in the
z
direction(the number of cells in the grid is approximately 4000000 cells). The perfectly matched layer(PML) (Berenger 1994) was used for the outer boundary truncation of the grid. The grid sizeallows for 16 PML cells and a separation of at least 10 cells between the PML surface andclosest point on the coil geometry.A differentiated Gaussian pulse that has a suitable frequency spectrum between 50 and400 MHz was used to excite the coil. At the lumped capacitor locations, a lumped elementFDTD algorithm is used to model the tuning capacitors (Tsuei
et al
1993). Figure 1(
a
) showsa top view of the FDTD grid of an unloaded high-pass birdcage coil where the location of the lumped capacitors is observed on the upper coil ring. The exact deﬁnition of lumpedcapacitors is of great importance. The capacitance values (pF) give a realistic prediction of thelimitation of this speciﬁc coil design. For instance, the model had predicted that the resonancefrequency of mode 1 of this speciﬁc high-pass birdcage coil cannot surpass 270 MHz (0 pFcapacitance value). Also, as the frequency of operation rises, the modal ﬁeld distribution of the coil becomes more dependent on the values and the position of the lumped capacitors.
612 T S Ibrahim
et al
Figure 1.
Schematic representation of a high pass birdcage coil within the FDTD grid. A top viewof an unloaded coil (
a
) and axial (
b
), sagittal (
c
) and coronal (
d
) slices of a coil loaded with thehuman head model are displayed. The
B
1
ﬁeld and the SAR arrows (ﬁgure 1(
c
)) correspond tothe locations of the axial slices in which the FDTD results are presented in ﬁgures 3(
a
), 3(
c
), 3(
e
)and 3(
g
).
The simulation was run for 62500 steps. The time domain data were collected at arbitrarypoints inside the coil for different capacitor values. Figures 2(
a
) and 2(
b
) show the timedomain response at one point for the
H
y
ﬁeld component of a linearly excited coil. The ﬁeldsare calculated with different capacitor values inside an empty coil and a coil loaded with thehuman head model. Because of loss, the signal decays much faster for the loaded cases. It isapparent that with this number of time steps the signal has signiﬁcantly decayed (200 MHz:ﬁgure 2(
b
)) or reached steady state (64 MHz: ﬁgure 2(
a
)). Thus, it is clear that the frequencydomain data will not be signiﬁcantly contaminated by the late time signal.In reality, the goal was to obtain the ﬁeld distribution within the coil at the resonancefrequency where the lumped capacitors must be tuned to set the resonance at the Larmorfrequency. Finding the ﬁeld distribution is a two-step process. In the ﬁrst step, an initial guessis made for the capacitor values. A fast Fourier transform (FFT) is then applied to the FDTDsolution at a few points within the grid to obtain the frequency response of the coil. The actuallocation where the data are collected is not that important since the frequency response atany point within the coil should not differ signiﬁcantly. The only difference should be due tovariationsintheﬁelddistributionofthecoil. Iftheresonancefrequencyofthesolutionisnotatthedesiredlocation,thenthecapacitorvaluesarechangedandtheFDTDprogramisrerun. Thisstep is repeated until the desired resonance frequency is obtained. Figures 2(
c
) and 2(
d
), showtwo frequency responses of the birdcage coil, loaded and unloaded. The resonance frequencyof the mode of interest is obtained at 64 and 200 MHz which are the Larmor frequencies for1.5 and 4.7 T MRI systems respectively. To obtain the resonance frequency at 64 MHz, the
B
1
ﬁeld homogeneity and SAR calculations 613
Figure 2.
FDTD calculated time (
a
), (
b
) and frequency (
c
), (
d
) responses of one point in thecentre of the high-pass birdcage coil for an unloaded case (full curve) and a case where the coil isloaded with the human head model (broken curve). The
y
component of the
B
1
ﬁeld is plotted at64 MHz (
a
), (
c
) and 200 MHz (
b
), (
d
). The time and spatial steps were set to 5.5 ps and 3 mm,respectively. The FDTD simulations were run for 62500 time steps.
capacitor values are set to 48.6 pF for both the loaded and unloaded cases. However, capacitorvalues of 3 and 3.4 pF are needed to obtain the resonance frequency at 200 MHz for an emptycoil and a coil loaded with the human head model, respectively.In the second step, the FDTD simulation is run with the correct capacitor values, butinstead of applying an FFT at a few points in the grid, a discrete Fourier transform is appliedon-the-ﬂy at all the points in the grid at the resonant frequency. Thus, the time data do notneed to be stored.
2.1. The human head model
An anatomically detailed FDTD mesh of a male human head and shoulders, purchased in1997 from REMCOM (State College, PA (www.remcom.com)), was used in the birdcage coilsimulations. These data were obtained from MRI, CT and anatomical images and hence aresuitable for these types of calculations. Unlike many other meshes used in MRI coil studies,this model has shoulders which leads to more accurate results, especially in coil tuning andSAR calculations. The numerical instabilities (Jin
et al
1996) that occur when neglecting thehuman body below the neck are also avoided.The mesh consists of six tissue types: cartilage, muscle, eye, brain, dry skin and skullbone. The electrical constitutive parameters of these tissue are dispersive. Thus, in anyparticular simulation one must use the conductivity and the dielectric constant associatedwith the frequency of interest. This is done by tuning the coil to the frequency of interest and

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