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Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI

coils: models and methods

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2004 Phys. Med. Biol. 49 1835

(http://iopscience.iop.org/0031-9155/49/10/001)

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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 49 (2004) 1835–1851 PII: S0031-9155(04)73798-7

Electromagnetic fields inside a lossy, multilayeredspherical head phantom excited by MRI coils: modelsand methods

Feng Liu and Stuart Crozier

The School of Information Technology and Electrical Engineering, The University ofQueensland, St Lucia, Brisbane, Queensland 4072, Australia

E-mail: Stuart@itee.uq.edu.au

Received 18 December 2003Published 29 April 2004Online at stacks.iop.org/PMB/49/1835DOI: 10.1088/0031-9155/49/10/001

AbstractThe precise evaluation of electromagnetic field (EMF) distributions insidebiological samples is becoming an increasingly important design requirementfor high field MRI systems. In evaluating the induced fields caused by magneticfield gradients and RF transmitter coils, a multilayered dielectric sphericalhead model is proposed to provide a better understanding of electromagneticinteractions when compared to a traditional homogeneous head phantom. Thispaper presents Debye potential (DP) and Dyadic Green’s function (DGF)-basedsolutions of the EMFs inside a head-sized, stratified sphere with similar radialconductivity and permittivity profiles as a human head. The DP approachis formulated for the symmetric case in which the source is a circular loopcarrying a harmonic-formed current over a wide frequency range. The DGFmethod is developed for generic cases in which the source may be any kindof RF coil whose current distribution can be evaluated using the method ofmoments. The calculated EMFs can then be used to deduce MRI imagingparameters. The proposed methods, while not representing the full complexityof a head model, offer advantages in rapid prototyping as the computation timesare much lower than a full finite difference time domain calculation using acomplex head model. Test examples demonstrate the capability of the proposedmodels/methods. It is anticipated that this model will be of particular valuefor high field MRI applications, especially the rapid evaluation of RF resonator(surface and volume coils) and high performance gradient set designs.

1. Introduction

In bioelectromagnetics, a problem that has been of significant interest is the interaction ofelectromagnetic fields (EMFs) with the human head. Examples include: electromagnetic

0031-9155/04/101835+17$30.00 © 2004 IOP Publishing Ltd Printed in the UK 1835

1836 F Liu and S Crozier

field patterns in human head models exposed to mobile phone radiation (MPR)(Wang and Fujiwarea 1999), transcranial magnetic stimulation (TMS) (Nadeem et al 2003,Miranda et al 2003), hyperthermia (HT) (Krairiksh et al 1995) and exposure to the gradientand RF fields in MRI (Keltner et al 1991, Strilka et al 1998, Jin 1999, Hoult 2000,Collins and Smith 2002, Yang et al 2002, Bencsik et al 2002). In this work we restrictourselves to MRI scenarios, but the formalisms are easily adapted to other situations. InMRI, a wide frequency range of induced fields is present. The rapid switching of magneticfield gradients in the low-frequency case induces eddy currents that may trigger peripheralnerve or cardiac stimulation (Bencsik et al 2002, Liu et al 2002, 2003a). For radiofrequencyexposure (in the FM band of the spectrum) both safety concerns and the field characteristicsneed to be investigated, especially for high-field systems (Keltner et al 1991, Strilka et al 1998,Jin 1999, Hoult 2000, Collins and Smith 2002, Yang et al 2002). At higher frequencies (say100 MHz and above), the shorter wavelength and larger conduction current losses introducedby EMFs, hitherto typically ignored or simplified at low field, are becoming dominant factorsin RF coil performance. It is hoped that the proposed methods and models presented herecan be adapted as a kernel in an optimization process for the design of novel coil structuresin MRI. Another motivation of the present work is to validate numerical codes such as newfinite difference variants which are being used to model both the gradient and the RF fields(Liu et al 2002, 2003a, 2003b) in body models.

A variety of head models and phantoms (including regular shaped and realistic structuredones) have been developed (Keltner et al 1991, Strilka et al 1998, Jin 1999, Hoult 2000,Collins and Smith 2002, Yang et al 2002, Bencsik et al 2002) to investigate field interactionsin MRI. Although medical imaging based anatomical head models more realistically representthe bioelectromagnetic structure of the human head, they require large computational resourcesfor field simulation and are usually not suitable for the iterative optimization of loaded coils.By contrast, regular-shaped phantoms have the feature of subject-independence, provide agross electrical approximation to the head, and can be described by a closed or semi-closedmathematical formulation. Recently, a series of work on layered head models have beenreported for biomedical applications (Eaton 1992, Srinivasan et al 1998, Reyhani and Glover2000, Moneda et al 2003, Yao 2000, Nikita et al 2000). Little work, however, has been carriedout in the MRI community on such models for MRI specific problems. At present, the head isbasically approximated by the homogenous sphere phantom, and corresponding solutionsof the electromagnetic fields (EMFs) induced in such model are available (Keltner et al1991, Strilka et al 1998, Jin 1999, Hoult 2000, Collins and Smith 2002, Yang et al 2002,Bencsik et al 2002). These are useful in a number of situations. However, since the humanhead is more like a stratified spherical structure, a multilayered spherical model might providea closer approximation to the real loading situation.

We therefore introduce a generic spherical model which can be divided into an arbitrarynumber of concentric spherical layers of constant dielectric and conductive properties. In thismodel, the radii of the layers and the material properties can be chosen freely. Both analyticand numerical approaches have been developed to solve for the fields inside the nested sphere.Numerical methods, such as finite difference (FDTD) (Collins and Smith 2002, Jin 1999,Liu et al 2003a, Ibrahim et al 2001), finite element (FE) (Strilka et al 1998) and method ofmoments (MoM) (Jin 1999) could be used to investigate electromagnetic fields in realisticmodels. Among them, FDTD (Collins and Smith 2002, Collins et al 2002, Jin 1999, Liu et al2002, Ibrahim et al 2001) has possibly gained more widespread use.

Two full-wave, analytic algorithms are presented here based on Debye potential (DP)(Chew 1995) and dyadic Green’s function (DGF) (Chew 1995, Tai 1994) approaches. Forthe DP method, one or more current circular loops carrying ideal currents are placed in an

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils 1837

Table 1. Tissue layers, tissue densities (kg m−3) and the dielectric data ((σ, εr ): (conductivity(S m−1), relative permittivity) of the sphere model.

Frequency

1 KHz 64 MHz 170 MHz 300 MHz 470 MHzLayers (Gradient field) (1.5 T) (4 T) (7 T) (11 T)

White mattera (0.0626, 69813) (0.292, 67.8) (0.364, 48.8) (0.413, 43.8) (0.465, 41.3)Grey matterb (0.099, 164 060) (0.511, 97.4) (0.619, 67.7) (0.692, 60.0) (0.767, 56.3)CSFc (2.000, 109) (2.070, 97.3) (2.170, 79.1) (2.225, 72.7) (2.270, 70.3)Skulld (0.0202, 2702) (0.0596, 16.7) (0.0714, 14.2) (0.0827, 13.4) (0.0977, 13.0)Musclee (0.321, 434971) (0.688, 72.2) (0.734, 61.3) (0.7771, 58.2) (0.815, 56.6)Fatf (0.0224, 24105) (0.0353, 6.5) (0.0376, 5.8) (0.0396, 5.6) (0.0423, 5.6)Sking (0.0002, 1136) (0.436, 92.2) (0.560, 58.8) (0.641, 49.8) (0.717, 45.4)

Tissue densities: a, b: 1030; c: 1060; d: 1850; e: 1040; f: 1850; g:1100.

axisymmetric position over the sphere. For the DGF method, the coil can be positionedanywhere. The coil current is evaluated based on the well-known MoM algorithm. Thismethod also considers the coupling between the coils and head phantom, which is essentialfor near field analyses. The proposed methods cover all the operating frequencies in MRI, andwhile the DP method provides the most rapid solution, the DFG is a universal algorithm. Afterthe EMFs are calculated in the entire space, various imaging or antenna parameters such assignal-to-noise ratio (SNR), signal intensity (SI), eddy currents, specific absorption rate (SAR)and coil impedance can be found from the calculated values (Vesselle and Collin 1995).

The paper is organized as follows: in section 2, head model data are presented; insection 3, DP solutions for ideal currents are formulated; in section 4, DGF/MoM hybridsolutions for arbitrary coils are provided; in section 5, test examples are presented and, finally,conclusions are drawn in terms of the potential usage of the proposed model.

2. Spherical head phantom

The head model used herein is a multilayered, concentric spherical volume conductor, withsimilar radial conductivity and permittivity profiles as a human head. Each layer represents acorresponding group of structures within the head. The number of layers is somewhat arbitraryand the radii can be adjusted according to the physical structure under investigation.

The electromagnetic properties of the human head used in this work were obtained fromthe US Air Force Research Laboratory (http://www.brooks.af.mil/AFRL/HED/hedr/) whichrepresents a large male. The original spatial resolution of the model is 1 mm. For numericalcomputations, the model can be mapped onto a defined grid with volume-averaged conductiveproperties. There are a total of 40 segmented tissues in the whole body model and a tableof the electromagnetic data for the head phantom for a few different frequencies is providedin table 1.

From this table, it can be seen that tissues with higher water content such as muscle havelarger conductivity than low water content tissues such as bone. Tissue conductivity increaseswith increasing frequency, however the permittivity decreases with increasing frequency. Themagnetic permeability of tissues remains essentially constant with frequency. It should bepointed out that in the literature, there is some disagreement on the values of the materialproperties (Yao 2000); this uncertainty is possibly due to different measurement conditions.As a rough approximation, the head can be represented by a number of layers (white matter(WM), grey matter (GM), skull, muscle, fat, skin) from inner brain to the outer boundaries.

1838 F Liu and S Crozier

d

a

b

r

φ

c

( 1)La a− =x

y

z θ

(0)a

(1)a

( )la

jwIe

Figure 1. A current loop is placed adjacent to a lossy, multilayered dielectric spherical phantom.In this figure, b is the radius of the current loop carry uniform current I exp(−jwt); a is the radiusof the sphere, which is centred and divided by L layers, each layer has its own constant dielectricvalue: (σ (l), ε

(l)r ), l = 0 ∼ L + 1. d is the distance between the loop and the sphere centre, and

c =√

b2 + d2.

Some layers can be regrouped or redefined freely for different purposes. In this model, we setup the basic radii of the above six layers to be 6, 8, 8.4, 8.8, 9.2, 9.8 and 10 cm, respectively.

3. Debye potential-based solution for ideal current loop

In this section, we describe an analysis method based on the DP to solve Maxwell’s equations,which is valid at both low (gradient field) and high (RF field) operating frequencies in MRI.

The coil/sphere structure is illustrated in figure 1, where a loop of radius b carryinguniform current I is placed adjacent to a multilayered, lossy dielectric sphere of radius a. Thesphere is centred and divided by L layers, each layer has its own constant dielectric values:(σ (l), ε(l)

r

). Free space is assumed for the region outside of the sphere and harmonic time

dependence exp(−jωt) is implicit throughout the analysis, but omitted from the equationsfor brevity. To calculate the induced electromagnetic field, a spherical coordinate systemis chosen as shown in this figure. Due to the axisymmetric structure, the EMFs have noφ-component; note that the electric field will have no radial component when the current hasonly a φ-component. Therefore, the EMF is a transverse electric (TE) field, which can beexpressed in terms of the Debye potential πm (Jin 1999, Chew 1995), and Maxwell’s equationscan be expressed by

(∇2 + k2)πm = 0 (1)

where k = ω√

µε. According to (Jin 1999), the solution of (1) has the general form

πm =∞∑

n=0

{jn(kr)

h(1)n (kr)

}Pn(cos θ) (2)

where the braces imply ‘linear superpositions of’, jn(kr) is the nth-order spherical Besselfunction of the first kind and h(1)

n (kr) is the nth-order spherical Hankel function of the firstkind; P 1

n (cos θ) is the associated Legendre polynomial.

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils 1839

Table 2. The parameters of the EMFs as expressed by Debye potential in all of the subregions(l = 0 ∼ L + 1).

Parameters

Expansion coefficients Wavenumber kSubregion l [C]n (k0 = ω

√µ0ε0)

r < a(0), l = 0 [C](0)n =

[C

(0)11 0

]n

k(0) = k0

√ε(0)r + iσ (0)/ωε0

a(l−1) < r < a(l), l = 1 ∼ L − 1, [C](l)n =[C

(l)11 C

(l)12

]n

k(l) = k0

√ε(l)r + iσ (l)/ωε0

(a(L−1) = a)

a < r < c, l = L [C](L)n =

[C

(L)11 C

(L)12

]n

k(L) = k0

r > c, l = L + 1 [C](L+1)n =

[0 C

(L+1)12

]n

k(L+1) = k0

Using the Debye potential πm, the electric field and magnetic field can be expressed as

E = ∇ × (rπm)

H = 1

iωµ∇ × ∇ × (rπm).

(3)

Substituting equation (2) into (3), after the vector implementation in the spherical coordinatesystem, the non-zero components of the EMFs are obtained as

E(l)φ = −∂πm

∂θ= −

∞∑n=0

[C](l)n [B](l)n [DP ]n

H(l)θ = 1

iωµ0

(∂2(rπm)

r∂r∂θ

)= 1

iωµ0r

∞∑n=0

[C](l)n [DB](l)n [DP ]n

H (l)r = 1

iωµ0

(k2 +

∂2

∂r2

)(rπm) = 1

iωµ0r

∞∑n=0

n(n + 1)[C](l)n [B](l)n [P ]n

(4)

with

[C](l)n = [C

(l)11 C

(l)12

]n

[B](l)n = [jn(k

(l)r) h(1)n (k(l)r)

]T

[DB](l)n = [∂(rjn(k

(l)r))/∂r ∂(rh(1)n (k(l)r))/∂r

]T

[DP ]n = ∂[P ]n/∂θ = ∂Pn(cos θ)/∂θ.

(5)

In the above, l denotes the layer number, [C](l)n is the expansion coefficients matrix;[B](l)n , [DB](l)n are the matrices of nth-order spherical Bessel function and its derivatives;[DP ]n is the derivate of the associated Legendre polynomial [P ]n; T denotes the transposeoperator.

All the other EMF components, i.e., Er,Eθ ,Hφ are zero. In general, the EMFs in theentire space can be expressed using equations (4) and (5), but fields can be simplified in somesubregions. For clarity, all of the expressions for the EMFs are given in table 2.

In this table, the subregion (r < a(0), l = 0), the element C012,n = 0 is due to the singularity

of the Hankel function at the origin. In the subregion (free space, r > c, l = L + 1), the EMFswill propagate to infinity, and so the spherical Bessel function is excluded and CL+2

11,n is zero.The task is then to determine the unknown elements of the coefficient matrix [C]n in each

subregion. For this, the boundary conditions at each spherical layer surfaces are considered.

1840 F Liu and S Crozier{Ea(l+)

φ = Ea(l−)

φ Ha(l+)

θ = Ha(l−)

θ (a(l−) ∈ l, a(l+) ∈ l + 1, l = 0, . . . , L − 1)

Ec+

φ = Ec−φ Hc+

θ = Hc−θ + Js (c− ∈ l = L, c+ ∈ l = L + 1)

(6)

where the source is defined as

Js(r, φ, θ) = I sin θ

cδ(r − c)δ(sin θ − b/c). (7)

Based on the orthogonal relation of the Legendre polynomial, from (6), one can obtain thefollowing matrix series:

[C

(1)11 C

(1)12

]n

[jn(k

(1)a(0)) k(1)j ′n(k

(1)a(0))

h(1)n (k(1)a(0)) k(1)h′(1)

n (k(1)a(0))

]

= [C

(0)11,n jn(k

(0)a(0)) C(0)11,n k(0)j ′

n(k(0)a(0))

]l = 0 ∼ 1 (8a)

[C

(l)11 C

(l)12

]n

[jn(k

(l)a(l)) k(l)j ′n(k

(l)a(l))

h(1)n (k(l)a(l)) k(l)h′(1)

n (k(l)a(l))

]

= [C

(l+1)11 C

(l+1)12

]n

[jn(k

(l+1)a(l)) k(l+1)j ′n(k

(l+1)a(l))

h(1)n (k(l+1)a(l)) k(l+1)h′(1)

n (k(l+1)a(l))

]l = 1 ∼ L − 1 (8b)

[C

(L)11 C

(L)12

]n

[jn(k0c) k0j

′n(k0c)

h(1)n (k0c) k0h

′(1)n (k0c)

]

= [C

(L+1)12,n h(1)

n (k0c) C(L+1)12,n k0h

′(1)n (k0c) + f (J (n))

]l = L ∼ L + 1 (8c)

and

f (J (n)) = iωµ0Ib

c2

2n + 1

2n(n + 1)∂P (cos θ)/∂θ |θ=θ0 . (9)

In the following sections, we present two methods to solve these equations.

3.1. Method 1

Consider equation (8c), after employing the Wronskian relations for the spherical Besselfunctions

jn(k0c)h′(1)n (k0c) − j ′

n(k0c)h(1)n (k0c) = i/(k0c)

2. (10)

The elements C(L)11,ncan be obtained as expressed by

C(L)11,n = −ωµ0Ik0b

2n + 1

2n(n + 1)h(1)

n (k0c)∂P (cos θ)

∂θ

∣∣∣∣ θ0. (11)

And also, in equation (8b), there exists a relationship:

[C](1)n = [C](L)

n Q(L−1)n Q(L−2)

n · · · Q(l)n · · · Q(2)

n Q(1)n (12)

where

Q(l)n =

[jn(k

(l+1)a(l)) k(l+1)j ′n(k

(l+1)a(l))

h(1)n (k(l+1)a(l)) k(l+1)h′(1)

n (k(l+1)a(l))

] [jn(k

(l)a(l)) k(l)j ′n(k

(l)a(l))

h(1)n (k(l)a(l)) k(l)h′(1)

n (k(l)a(l))

]−1

l = 1, . . . , L − 1. (13)

After substitution of equation (11) into (12), the combination of (12) and (8a) leads to a four-equation system with four unknowns (the expansion coefficients C

(0)11,n, C

(1)11,n, C

(1)12,n, C

(L)12,n).

These four parameters can therefore be easily found. Based on the recurrence relations, theexpansion coefficients of any layer can be obtained by solving equation (8b).

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils 1841

3.2. Method 2

Equation (8) can be rewritten as a matrix (14). All the expansion coefficients can be obtaineddirectly by solving this linear matrix system. Usually the number of layers of the sphere issmall and the sparse matrix A is not difficult to invert, hence it is computationally efficient tosolve in this manner. When the number of layers of the sphere is very large, method 1 is likelyto be more efficient.

A00 A0

1 A11

A10 A2

1 A31

A41 A5

1 A02 A1

2 0

A61 A7

1 A22 A3

2

. . .

A4l−1 A5

l−1 A0l A1

l

A6l−1 A7

l−1 A2l A3

l

. . .

A4L−1 A5

L−1 A0L A1

L

A6L−1 A7

L−1 A2L A3

L

A4L A5

L A0L+1

A6L A7

L A1L+1

×

C(0)11

C(1)11

C(1)12

...

C(l)11

C(l)12

...

C(L)11

C(L)12

C(L+1)12

n

=

0

0

0

...

0

0

...

0

0

f J (n)

(2L+2)

(14)

where the matrix elements are defined as

{A0

0 = −jn(k(0)a(0)) A1

0 = −k(0)j ′n(k

(0)a(0)) l = 0

A0L+1 = hn(k0c) A1

L+1 = k0h′(1)n (k0c) l = L + 1{

A0l = jn(k

(l)a(l−1)) A1l = h(1)

n (k(l)a(l−1))

A2l = k(l)j ′

n(k(l)a(l−1)) A3

l = k(l)h′(1)n (k(l)a(l−1))

l = 1 ∼ L

{A4

l = −jn(k(l)a(l)) A5

l = −h(1)n (k(l)a(l))

A6l = −k(l)j ′

n(k(l)a(l)) A7

l = −k(l)h′(1)n (k(l)a(l)) l = 1 ∼ L.

(15)

1842 F Liu and S Crozier

Given the expansion coefficients, one can calculate the EMFs inside the sphere and also infree space.

For a single, homogeneous sphere, for example, i.e. L = 1, the linear system reduces to

A00 A0

1 A11 0

A10 A2

1 A31 0

0 A41 A5

1 A02

0 A61 A7

1 A12

n

C(0)11

C(1)11

C(1)12

C(2)12

n

=

0

0

0

f J (n)

. (16)

This single-layer formulation is the same as those derived by Jin (1999) and Keltner et al(1991).

This DP method/model can be used directly for the rapid evaluation of the fields associatedwith magnetic stimulation by z-gradient coils and also the RF behaviour of loaded surfacecoils.

4. Dyadic Green’s function/MoM based solutions for arbitrarily shapedand positioned coils

The above formulations are very efficient for the investigation of EMFs inside a head phantomin symmetric cases and where the effect of loading on the current in the coil may be ignored.These assumptions are reasonable enough for longitudinal gradients and low-field RF systems.However, recent advances in high-field MRI have reshaped coil-design methods and requirethe consideration of coil–tissue interactions. These are important for characterizing bothhigh-frequency RF resonator and phased array designs. To include the perturbation of the coilcurrents by the load, a hybrid technique is presented in this section. Again, a layered sphericalhead phantom is used but the RF coils are now arbitrary shaped or positioned. In this method,the total number of layers is L(one layer of free space and L − 1 layers in the phantom). TheDGF technique (Li et al 1994, Tai 1994) is extended to RF resonator analysis and combinedwith a MoM technique to completely account for changes in the coil current due to the load.

4.1. Dyadic Green’s function technique

The DGF is a dyad that relates a vector field to a vector current source (Chew 1995). TheDGFs enable the calculation of the EMFs in a compact fashion. It is constructed on sphericalwavefunctions that satisfy the vector wavefunctions in spherical coordinates, which are akinto Debye potentials (Chew 1995, Tai 1994).

It should be mentioned that all the DGFs involved in this section refer to the electric-type(Ge(r, r

′), for simplicity, ‘e’ is ignored in the context); for the magnetic-type DGF, the sameprocedure can be used, or the H-fields can be determined by invoking duality.

According to the principle of superposition, the EMFs inside and outside the layeredsphere can all be expressed by the DGF as

G(r, r ′) ={

−r̂ r̂/k2

0δ(r − r ′) + G0(r, r′) + G(L)

s (r, r ′) r > a

G(l)s (r, r ′) l = 0, . . . , L − 1 r � a

(17)

where −r̂ r̂/k2

0δ(r−r ′) is an explicit dyadic delta function term, which specifies the singularityof DGF at the source point. The source is at location r ′, the field point is at r;G0(r, r

′) is theGreen function (GFs) in free space outside the source region, and Gs(r, r

′) are the scatteringcomponents.

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils 1843

G0(r, r′) can be expressed by (Chew 1995, Tai 1994)

G0(r, r′) = ik0

4π

+∞∑n=1

n∑m=0

(2 − δm0)(2n + 1)

n2(n + 1)(n + m)

×{

M1mn(r, k0)M

3mn(r

′, k0) + N1mn(r, k0)N

3mn(r

′, k0) r < r ′

M3mn(r, k0)M

1mn(r

′, k0) + N3mn(r, k0)N

1mn(r

′, k0) r > r ′ (18)

where m, n denote the eigenvalue parameters, and δmn : (= 1 for m = n; and 0 for m �= n)

denotes the Kronecker symbol; and symbol Mmn,Nmn are the spherical vector wavefunctioncomponents in the spherical coordinate, i.e.,

Mmn(k)

Nmn(k)

Lmn(k)

=

0m

sin θzn(kr)P m

n T sc −zn(kr)P m′

n T cs

n(n + 1)

krzn(kr)P m

n T cs

1

kr(rzn(kr))′P m′

n T cs

∓1

kr(rzn(kr))′

m

sin θP m

n T sc

(zn(kr))′P mn T c

s

zn(kr)

rP m′

n T cs

∓mzn(kr)

r sin θP m

n T sc

×

r̂

θ̂

φ̂

(19)

where zn(kr)(r is the piloting radial vector) denotes the spherical Bessel function of thenth-order, and for simplicity, some abbreviations are used:

P mn = P m

n (cos θ) T cs = cos

sinmφ T s

c = sin

cosmφ

P m′n = ∂P m

n

/∂θ (rzn(kr))′ = ∂(rzn(kr))/∂r (zn(kr))′ = ∂(zn(kr))/∂r.

These three functions are the solutions of the vector Helmholtz equation, and Mmn(k),Nmn(k)

are two rotational modes whileLmn(k) is the irrotational type. The superscripts ‘1’, ‘3’ inequation (18) indicate the spherical Bessel functions of the first and third kinds (Hankelfunctions), respectively.

The scattering components(G(l)

s (r, r ′), l = 0, . . . , L)should have the following form:

G(l)s (r, r ′) =

+∞∑n=1

n∑m=0

[C](l)mn[MN ](l)mn l = 0, . . . , L (20)

with

[C](l)mn = [C

(l)11 δ0

l C(l)12 δ0

l C(l)13 δL

l C(l)14 δL

l

]mn

[MN ](l)mn = [M3

mn(r, k(l))M3

mn(r′, k0) N3

mn(r, k(l))N3

mn(r′, k0)

M1mn(r, k

(l))M3mn(r

′, k0) N1mn(r, k

(l))N3mn(r

′, k0)]T

.

(21)

In the above, l denotes the layer number, [C](l)mn is the unknown expansion coefficients matrix;[MN ](l)mn is the matrix of the spherical wavefunctions. It should be noted that the Hankelfunction is used in the free space but it should be avoided in the core layer due to the finitenessof the EMFs at the origin. Accordingly, delta functions are introduced for the expansioncoefficients in the inner layer and outside the sphere as

δLl =

{1 l �= L

0 l = Lδ0l =

{1 l �= 0

0 l = 0.(22)

The next task is to determine the expansion coefficients based on the continuity of the EMFcomponents at the layer interfaces. The boundary conditions are

1844 F Liu and S Crozier{r̂ × G(l)(r, r ′) = r̂ × G(l+1)(r, r ′)

r̂ × (∇ × G(l)(r, r ′)) = r̂ × (∇ × G(l+1)(r, r ′))r = a(l) l = 0, 1, . . . , L − 1. (23)

From (23), one can obtain the following matrix series (Li et al 1994):

[A](l)1 [C](l) = [A](l+1)2 [C](l+1) l = 0 ∼ L − 1 (24)

with

[A](l)1 =

h̄l,l 0 l,l 0

k(l)∂h̄l,l 0 k(l)∂l,l 0

0 ∂h̄l,l 0 ∂l,l

0 k(l)h̄l,l 0 k(l)l,l

[A](l)2 =

h̄l+1,l 0 l+1,l 0

k(l+1)∂h̄l+1,l 0 k(l+1)∂l+1,l 0

0 ∂h̄l+1,l 0 ∂l+1,l

0 k(l+1)h̄l+1,l 0 k(l+1)l+1,l

(25)

h̄st = h(1)n (ksat ) st = jn(ksat )

∂h̄st = 1

ρ

d(ρh̄st )

dρ

∣∣∣∣ρ=ksat

∂st = 1

ρ

d(ρst )

dρ

∣∣∣∣ρ=ksat

.(26)

It is easy to see that the coefficients can be solved by similar methods employed in the DPmethod: one is based on the recurrence formula and the other is involves introducing anintegrated linear equation. For convenience and simplicity, let us rewrite the linear equationof the coefficients in matrix form, resulting in the following equation:

[A](0)1 −[A](1)

2

[A](1)1 −[A](2)

2 [0]

· · ·[A](l)1 −[A](l+1)

2

[A](l+1)1 −[A](l+2)

2

· · ·[0] [A](L−2)

1 −[A](L−1)2

[A](L−1)1 −[A](L)

2

×

[C](0)

[C](1)

...

[C](l)

[C](l+1)

...

[C](L−1)

[C](L)

4L×1

= [0]4L×1. (27)

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils 1845

Again, all the expansion coefficients can be obtained directly by matrix inversion.At this stage, all the DGFs have been determined and the EMFs in the entire space are

readily found by integration:

E(r) = −jωµ0

∫ ∫ ∫V s

G(r, r ′) · Js(r′) dV

H(r) = −1

jωµ0∇ × E(r) =

∫ ∫ ∫V s

∇ × G(r, r ′) · Js(r′) dV

(28)

where Vs denotes the source zone.

4.2. MoM technique

The coil currents are determined by the MoM (Jin 1999, Chew 1995), a well-knownelectromagnetic technique to handle the antenna structure with a high accuracy. Withoutlosing generality, thin-wire conditions are only considered in this formulation.

For the present configuration, the distribution of the current in the coil depends on thebiological sample’s dielectric, conductive and relative positions of the coil/phantom andalso the frequency/polarization of the incident EMF. As the voltage source is applied at thefeeding position, it is equivalent to setting up an incident E-field, Ei . To satisfy a boundarycondition of zero tangential E-field on the coil, the scattered field, Es is generated from(Ei + Es)|tangential = 0 and can be expressed by the well-known integral equation (Jin 1999)with the unknowns of the induced currents on the coil, Icoil. To determine these currents, thewire is subdivided into N several segments and the well-known triangular basis functions fn

are employed for the linear current along wire elements.

Icoil =N∑

n=1

αnfn. (29)

After performing ‘testing’ for each basis function, the integral equations are transformed intoMoM matrix equations of the form

[Z][I ] = [V ] (30)

where [Z] is the impedance matrix, [I ] is the current to be determined, [V ] is the voltagevector involving the excitation source. Discrete lumped-circuit elements such as capacitorsare also considered. The impedance matrix is dense and hence equation (30) is usually solvedusing iterative techniques. Once the currents are obtained, the EMFs can be calculated basedon equation (28).

For a strip structure, in the MoM discretization, all the surfaces of the involved objects canbe divided into small triangular patches. The surface current on each patch is then decomposedinto a linear combination of basis functions.

Based on this hybrid approach, the perturbation of the coil currents by the phantom isincluded. It is hoped that the computational cost (memory and computational time) can bedramatically reduced in comparison with other kinds of techniques such as volume equivalenceprinciple. Moreover, the DGF/MOM framework is in a format that can be expanded toaccommodate many types of coil configurations.

5. Test examples

In this section, a number of examples were studied to test the proposed methods/models. Inall the simulations, a circular loop was used and the current waveform was harmonic and thegradient field was set to be 1 kHz, the cell size was 2 mm in the finite-difference calculations.

1846 F Liu and S Crozier

RecordPositions

Y

Z

X

Figure 2. Comparisons of the DP, QSFD and FDTD solutions for the magnitude of the inducededdy currents in the seven-layered sphere along the radial distance at y = 0. As in figure 1,a = 20, b = 26, d = 10 cm, respectively. The current frequency is 1 kHz. The radii of thespherical layers are 6, 8, 8.4, 8.8, 9.2, 9.8 and 10 cm, respectively. The dielectric data was obtainedfrom table 1.

Case 1: Eddy currents induced in a seven-layered sphere. The computational configurationis shown in figure 1, the eddy currents inside the sphere were calculated by DP, quasi-staticfinite difference (QSFD) and FDTD (Liu et al 2002) methods, respectively. In the QSFDcalculation, the vector magnetic potential (A) for each of the locations within the phantomwas evaluated using elliptic integrals (Liu et al 2003a). After the vector potential was obtained,the scalar potential φ was calculated using the iterative technique successive over relaxation(SOR) with a relaxation factor of w = 1.85. The simulation converged in an average of about2000 iterations. In the FDTD calculation, the cell size was 6.7 × 10−9 of the wavelengthof the problem and 12 layers of PML with a parabolic conductivity profile was constructedto surround the sphere–coil structure. There were about 10 yee-cells between the PML andthe sphere. Both of the low-frequency FDTD schemes were employed for the steady-fieldcalculations. The DP method was the fastest to compute taking only 10 s on a SUN E450, bycontrast, QSFD and FDTD took about 4 and 20 min, respectively.

The simulation results are shown in figure 2. From this figure, it can be seen all themethods generated similar values and the results reflected the effect of inclusion of the highlycontrasted conductivity profiles on the eddy current distribution. In this model, the currentdensity maximum was deep in the sample, even though the E-field maximum still occursat the surface. This type of information cannot be obtained from homogeneous isotropicmodels.

Case 2: The eddy currents induced in a seven-layered sphere excited by gradient z-coils. Inthis case, a typical z-gradient coil (figure 3(a)) (Crozier and Doddrell 1993, Liu et al 2003a)were used and the eddy currents induced in the seven-layered phantom calculated. Sincethe positioning of the subject plays a role in the severity of the response to stimulation, weinvestigated a series of phantom positions with respect to the gradient coil system (figure 3(a)):

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils 1847

i tIe ω i tIe ω−Layered Sphere

Gradient Z-coils

Z0 Z1 Z2

Registered |J|

(a)

(b)

Figure 3. The calculation of the eddy currents inside the seven-layered spherical phantom excitedby a typical z-gradient coil set. (a) Schematic arrangement of the phantom inside the z-gradientcoils. The z-coil windings consist of an anti-symmetric 20-loop-structure, each of radius r =345 mm, placed parallel and coaxial. To generate a slew rate of dB/dz/dt ≈ 100 T m−1 s−1, thecurrent flowing in z-coils is about 500.0 A and the frequency is 1 kHz. In this simulation, theinduced fields in the phantom were calculated as a function of the phantom positions with respectto the gradient coil system. The phantom is the same as that in figure 2. (b) Comparisons of the DPsolution and QSFD solution for the magnitude of the induced eddy currents in the layered spherealong the radial distance at y = 0. The field values were shown for three different positions: z = 0,0.3, 0.6 m, respectively.

from the coil centre to the end of the coil windings, where the largest magnetic fields exist.Since the calculation methods have already been verified against with each other in case 1; theinduced fields were calculated using the QSFD and the DP methods.

1848 F Liu and S Crozier

Frequency [Hz]

S1

1[d

B]

50 55 60 65 70 75 800

10

20

30

40

50

With load &after tuning

No loadWith load

Reflection Coefficient vs Frequency

ResonanceFrequency Shift

Figure 4. Reflection coefficient (S11) versus frequency. In the coil loop, there are four capacitorsdistributed equally and a voltage source (amplitude: 1 V, phase: 0◦) was used as the RF input.When the resonance frequency shifted after loading, the capacitors were ‘tuned’ to make the coilresonant at desired frequency, e.g., 64 MHz for this case. In the figure, a = 10, b = 6, d = 12 cm,respectively. The radii of the spherical layers are 8, 8.4, 9.8 and 10 cm, respectively. The dielectricdata were obtained from table 1.

Radial Distance (m)

|Hr|

(arb

.)

0 0.025 0.05 0.075 0.1

0.4

0.5

0.6

0.7

0.8

0.9

1DPDGF/MoM

Sphere center Border

Figure 5. Comparison of the DP and the DFG/MoM solutions for the magnetic fields in thelayered sphere along the radial distance at y = 0. The configuration is the same as figure 4. Forthe DP calculation, the current of the coil was ‘predefined’ from the MoM evaluation, where theloading effect was considered.

Figure 3(b) illustrates that the induced eddy currents were larger near the winding endand lower at the centre, and also the conductivity profile influenced the local-field values,indicating the importance of considering conductivity profiles.

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils 1849

Figure 6. Comparison of the DFG/MoM solution for the magnetic fields at different resonancefrequencies (64, 170, 300, 470 MHz) in the layered sphere in the x = 0.002 m plane. Theconfiguration is given in figures 1 and 4.

Case 3: The RF field inside a four-layered sphere excited by surface coil. For the RF studies,the DFG/MoM hybrid method was employed. In the configuration of figure 1, the radius ofthe coil was b = 6 cm, the distance between the coil centre and the sphere centre was d =8 cm, the radius of the sphere was a = 8 cm. The RF source was an input voltage (amplitude:1 V, phase: 0◦), and only the linear polarization was analyzed. Firstly, we set up the resonantcircular loop by introducing four capacitors distributed equally along the coil. The value ofeach capacitor was initially determined by considering the resonance of the LC circuits (L:self-inductance of the loop and C: four capacitors), and then verified by MoM calculations.When the spherical phantom was ‘loaded,’ the resonance frequency was shifted and hencethe capacitor values were ‘tuned’ to make the coil resonant at desired frequencies (64 MHz,170 MHz, 300 MHz and 470 MHz) (see figure 4). In this calculation, the Green functions for

1850 F Liu and S Crozier

a dielectric sphere consisted of a core and three dielectric layers, at the origin of the coordinatesystem. Thus the head phantom was a four-concentric sphere model, representing brain, CSF,skull and scalp. Since the GFs are expressed using series expansions, a convergence criteria(1 × 10−3) for the summation of the rows of GFs had to be chosen before calculation. TheMoM calculations were performed using the electromagnetic module of the MoM packageFEKO (http://www.feko.co.za). A comparison between the DFG/MoM solutions and the DPresults was also undertaken. In DP calculations, a predefined current distribution along thesurface coil was derived from the MoM evaluations, where the effect of the head model wasalso included.

Figure 5 shows the comparison of the DP and the DFG/MoM solutions for the magneticfields at 64 MHz in the layered sphere along the radial distance at y = 0. The same results areobtained from both approaches for the symmetric case. When the loads are complicated andconsequently the current distribution changes, the DFG/MoM method has to be employed.Figure 6 shows the magnetic fields at different resonance frequencies (64, 170, 300,470 MHz) in the layered sphere in the x = 0.002 m plane. The results indicate that themagnetic fields were disturbed less by the phantom at low frequencies such as 64 MHz;however, as the frequency increases, it tends to influence the field profile strongly, particularlyat locations with large contrast in electrical properties. The DFG method computation timefor the sphere-loop case (figure 5) was 1.3 min for a convergence criterion of 1 × 10−2 and0.6 min for a 1 × 10−1 criterion.

6. Conclusion

The spatial distribution of the EMFs inside biological samples and the mechanism of interactionbetween the coils and tissues is an important consideration in high-field MRI, for both patientsafety and system design. Suitable phantom-based analysis is essential for MRI coil designand evaluation. This paper presents rapid analysis methods for MRI coils in the presence of amultilayered lossy sphere, loosely approximating the human head. The proposed approaches,DP and DGF/MoM, enable one to quickly assess the loading effects upon the operatingcharacteristics of the coils and also the field behaviour inside the objects excited by boththe gradients and RF resonators. The methods have been validated for multilayer problems.While generic shapes and structures are considered here, the procedure is versatile and canbe easily extended to other configurations such as multilayered cylinders or spheroids. Thesemethods are intended for use in the rapid evaluation of new coil designs and the induced fieldsin high-field scenarios.

Acknowledgments

Financial support for this project from the Australian Research Council (Australia) and TheNational Institute of Health (USA) P41 RR16105 is gratefully acknowledged.

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