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Limitations of calculating field distributions and magnetic susceptibilities in MRI using a

Fourier based method

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2009 Phys. Med. Biol. 54 1169

(http://iopscience.iop.org/0031-9155/54/5/005)

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IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 54 (2009) 1169–1189 doi:10.1088/0031-9155/54/5/005

Limitations of calculating field distributions andmagnetic susceptibilities in MRI using a Fourier basedmethod

Yu-Chung N Cheng1,3, Jaladhar Neelavalli2 and E Mark Haacke1

1 Department of Radiology, Wayne State University, Detriot, MI, USA2 Department of Biomedical Engineering, Wayne State University, Detriot, MI, USA

E-mail: yxc16@wayne.edu

Received 23 October 2008, in final form 10 January 2009Published 30 January 2009Online at stacks.iop.org/PMB/54/1169

Abstract

A discrete Fourier based method for calculating field distributions and localmagnetic susceptibility in MRI is carefully studied. Simulations suggest thatthe method based on discrete Green’s functions in both 2D and 3D spaces hasless error than the method based on continuous Green’s functions. The 2D fieldcalculations require the correction of the ‘Lorentz disk’, which is similar to theLorentz sphere term in the 3D case. A standard least-squares fit is proposed forthe extraction of susceptibility for a single object from MR images. Simulationsand a phantom study confirm both the discrete method and the feasibility ofthe least-squares fit approach. Finding accurate susceptibility values of localstructures in the brain from MR images may be possible with this approach inthe future.

1. Introduction

The ability to measure the magnetic susceptibility from magnetic resonance imaging (MRI)data could prove very useful in a number of in vivo applications. The basal ganglia appear toincrease their iron content in diseases such as Parkinson’s, Huntington’s and multiple sclerosis(Haacke et al 2005, 2007). Knowing the magnetic susceptibility makes it possible to removeunwanted phase effects in gradient echo images in methods such as susceptibility weightedimaging (Haacke et al 2004). Further, using the geometry and the susceptibility makes itpossible to remove the phase (field) effects caused by the presence of air/tissue interfacesand even to predict and correct geometric distortion in echo planar images (Koch et al 2006).Recently a Fourier based method has been introduced to find local magnetic fields (Koch et al2006, Deville et al 1979, Salomir et al 2003, Marques and Bowtell 2005) and it is similar to

3 Address for correspondence: MRI Center/Concourse Research, Harper University Hospital, 3990 John R Street,Detroit, MI 48201, USA.

0031-9155/09/051169+21$30.00 © 2009 Institute of Physics and Engineering in Medicine Printed in the UK 1169

1170 Y-C N Cheng et al

the methods presented in Jenkinson et al (2004) and Yoder et al (2004). If sufficient accuracycan be achieved with this method, it can easily be implemented in the form of a software tooland is computationally fast.

Initial investigations (Koch et al 2006, Salomir et al 2003, Marques and Bowtell 2005)suggest that the method used in forward calculations of the magnetic field seems promisingwith tolerable errors if the field of view (FOV) is more than 50% of the object size (i.e., length,width or height). In this paper, we systematically study the source of the uncertainty from themethod itself. We present an improved method based on the use of discrete Green’s functionsthat can be applied on both 2D and 3D objects. We demonstrate the field distributions fromdifferent Green’s functions for a cylinder and a sphere. Through a least-squares fit approach,we quantify the susceptibilities from simulations of a spherical shell and a water phantom.In our approach, we use an iterative algorithm for the quantification of susceptibility in thepresence of noise and aliasing in MRI phase maps.

2. Theory and methods

2.1. 3D induced field distribution and its Green’s function

The induced magnetic field due to a magnetization distribution, �M(�r), is given by (seeequation (5.64) of Jackson (1999) or equation (3) of Marques and Bowtell (2005))

�B(�r) = μ0

4π

∫V ′

d3r ′{

3 �M(�r ′) · (�r − �r ′)|�r − �r ′|5 (�r − �r ′) −

�M(�r ′)|�r − �r ′|3

}. (1)

Equation (1) indicates that the induced field distribution can be expressed as a convolutionbetween the magnetization distribution and the Green’s function. As Deville et al (1979)pointed out, equation (1) may easily be calculated in the Fourier domain (k-space domain) andthen Fourier transformed back to the spatial domain. In addition, as discussed in Marques andBowtell (2005) or implied by equation (5.64) of Jackson (1999), the Lorentz sphere correctionhas been included in equation (1).

If the main field is along the z-axis, then only the z-component of the magnetic field isimportant in most MRI research. The z-component of the induced magnetic field Bz(�r) andits Fourier pair in k-space are

Bz(�r) = μ0

4π

∫V ′

d3r ′{

3Mz(�r ′)(z − z′)2

|�r − �r ′|5 − Mz(�r ′)|�r − �r ′|3

}≡ μ0

∫V ′

d3r ′ Mz(�r ′)Gz,3D(�r − �r ′), (2)

Bz(�k) ≡ F{Bz(�r)} = μ0Mz(�k)Gc,3D(�k), (3)

where Mz(�k) is the Fourier transformation of Mz(�r) and the Green’s function Gz,3D(�r) isdefined as

Gz,3D(�r) ≡ 1

4π· 3z2 − r2

r5≡ 1

4π· 3 cos2 θ3D − 1

r3, (4)

where θ3D is the azimuthal angle in the spherical coordinate system and r2 ≡ x2 + y2 + z2.The Fourier transformation of Gz,3D(�r) is

Gc,3D(�k) ≡ F{Gz,3D(�r)} = 1

3− k2

z

k2, (5)

where k2 ≡ k2x + k2

y + k2z and the notation Gc,3D denotes the continuous Fourier transformation

of the 3D Green’s function. Equation (5) is valid when k is not equal to zero or infinity. When

Limitations of calculating field distributions and magnetic susceptibilities 1171

k = 0, the derivation of Gc,3D(�k) shown in equation (A.2) indicates that Gc,3D(0) = 0 aslong as the integral is calculated over the entire space larger than a small sphere with radiusε. In fact, equation (A.2) at k = 0 is equivalent to the calculation of the Lorentz sphere term(Jackson 1999). Because the Lorentz sphere correction has been included in equation (1), itis consistent to assign zero to Gc,3D(0) for future numerical calculations (Koch et al 2006,Marques and Bowtell 2005).

Although equation (5) may be derived from other methods, as pointed out by Koch et al(2006), derivations involving derivatives require special attention at the boundary of �M(�r).Therefore, we believe the derivation through Fourier transformation shown in Marques andBowtell (2005) is the most appropriate approach. Our derivation is also shown in appendix A.

2.2. 2D induced field distributions and its Green’s function

For an infinitely long object, with a uniform cross section along its length, its induced magneticfield can be calculated from a 2D Green’s function. Similar to the electric field distributiondue to an electric dipole presented in Jackson (1999), the magnetic field ( �H) on a 2D planedue to the scalar potential of a dipole �m may be written as (also see equation (13) in Sen andAxelrod (1999))

�H(�x) = − 1

2π�∇

( �m · �xρ2

)= 1

2π

(2�x( �m · �x)

ρ4− �m

ρ2

)(6)

whose spatial distribution agrees with that shown in Haacke et al (1999, p 751).Recall that the �B field is equal to μ0( �H + �M). After replacing the magnetic dipole moment

by its magnetization distribution, the induced magnetic field on a 2D plane becomes

�B(�x) = μ0

2π

∫d2x ′

{2 �M(�x ′) · (�x − �x ′)

|�x − �x ′|4 (�x − �x ′) −�M(�x ′)

|�x − �x ′|2}

+ μ0 �M(�x). (7)

It is obvious that �M(�x) in the last term can be replaced by∫

d2x ′ �M(�x ′)δ(2)(�x − �x ′) for alllater discussions.

However, neither equation (6) nor equation (7) has been properly defined at the origin,where the magnetic dipole moment is located. This term has been calculated in appendix B.Similar to equation (5.64) of Jackson (1999), the corrected magnetic field embedded in the 3Dspace is now the combination of equation (6) and equation (B.1),

�B(�x) � μ0

2π

∫d2x ′

{2 �M(�x ′) · (�x − �x ′)

|�x − �x ′|4 (�x − �x ′) −�M(�x ′)

|�x − �x ′|2}

+

(1 − 1

2sin2 θ

)μ0 �M(�x),

(8)

where θ is the angle between the main field direction and the axis of the infinitely long object.In the derivation of equation (8), we have automatically assumed that the integrand is zerowhen x = x ′. Furthermore, as we are interested in the magnetic field measured in the 3Dspace, the Lorentz sphere correction needs to be included in equation (8). The magnetic fieldnow becomes

�B(�x) � μ0

2π

∫d2x ′

{2 �M(�x ′) · (�x − �x ′)

|�x − �x ′|4 (�x − �x ′) −�M(�x ′)

|�x − �x ′|2}

+

(1

3− 1

2sin2 θ

)μ0 �M(�x).

(9)

1172 Y-C N Cheng et al

If the main field is along the ω-direction, only the ω-component of the induced magneticfield is important in our discussions here. Hence,

Bω(�x) = μ0 sin2 θ

2π

∫d2x ′

{2(z − z′)2Mω(�x ′)

|�x − �x ′|4 − Mω(�x ′)|�x − �x ′|2

}+

(1

3− 1

2sin2 θ

)μ0Mω(�x)

≡ μ0

∫d2x ′ Mω(�x ′)Gω,2D(�x − �x ′), (10)

where

Gω,2D(�x) ≡ sin2 θ

2π· z2 − y2

(z2 + y2)2+

3 cos2 θ − 1

6δ(2)(�x)

≡ sin2 θ

2π· 2 cos2 φ − 1

ρ2+

3 cos2 θ − 1

6δ(2)(�x), (11)

where ρ2 ≡ z2 +y2 and φ is the polar angle between z and y-axes. Here, the y-z plane is chosento be perpendicular to the axis of the infinitely long object and the plane contains the normalcross section of the object. We have properly inserted the factor sin2 θ in equation (10) asdiscussed in appendix B. Again, in the derivation of equation (11), the term, (2 cos2 φ−1)/ρ2,is already assumed to be zero at ρ = 0. The Fourier transformation of Gω,2D(�x), as derivedin appendix C, is

Gc,2D(�k) ≡ F{Gω,2D(�x)} = sin2 θ

2· k2

y − k2z

k2y + k2

z

+1

6(3 cos2 θ − 1) = 1

3− sin2 θ

k2z

k2y + k2

z

. (12)

Similar to the 3D case, equation (12) is valid only when k2y + k2

z is not equal to zero or

infinity. When �k = 0,Gc,2D(�k = 0) = (3 cos2 θ − 1)/6, as the integral of the first term inGω,2D(�x) is zero. This is because that term is (assumed) zero at ρ = 0 and the integral of(2 cos2 φ − 1)/ρ2 is also zero as long as the integral is calculated outside a small circle withradius ε. Furthermore, when the infinitely long object is parallel to the x-direction in the 3Dspace with the main field parallel to the z-direction, equation (12) is equal to equation (5) withkx = 0.

In the remainder of the paper, we will consider that the main field is parallel to thez-direction for the 3D case and ω-direction for the 2D case. We will label the induced fieldand magnetization with subscript z for any general discussions below. However, because ourmain interest here is the measured magnetic field from MR images, the consideration of thefield direction is not important at all.

2.3. Discrete Green’s functions in k-space

The continuous Green’s functions were derived based on an infinite field of view. However, allMR images have finite fields of view and are in a discrete form. In order to obtain consistentresults, both a discrete magnetization and a discrete Green’s function should be used in k-space.The discrete Green’s functions in k-space in the 3D and 2D cases can be numerically calculatedfrom the discrete Fourier transformation of the spatial Green’s functions in equations (4) and(11), respectively, with a given matrix size. The spatial Green’s functions are discretized andevaluated in units of the image resolutions. Nonetheless, the values at the origin of the Green’sfunctions require some discussion. In the 3D case, because the Lorentz sphere correction isincluded in equation (4), Gz,3D = 0 at �r = 0. In the 2D case, the 3D Lorentz sphere is correctedwith the presence of the 2D Lorentz disk. Therefore, at �x = 0,Gω,2D = (3 cos2 θ − 1)/6.

Limitations of calculating field distributions and magnetic susceptibilities 1173

2.4. Field calculations using the Green’s functions as a forward problem

For a non-ferromagnetic object, its magnetization can be expressed in terms of its magneticsusceptibility and an external field B0,

μ0 �M(�r) � χ(�r) �B0. (13)

If an object has a constant susceptibility χ , then the Fourier transformation of itsmagnetization along the main field direction, Mz(�k), is simply the Fourier transformationof the geometry of the object multiplied with χB0/μ0. As shown in equation (3), μ0Mz(�k)

should be multiplied with the Green’s function in k-space and the total result should be inverseFourier transformed back to the spatial domain. The induced magnetic field along the mainfield direction will then be given by

Bz(�r) = F−1{F{χ(�r)} · G(�k)}B0, (14)

where G(�k) represents either the 2D or 3D discrete Green’s function or the continuous Green’sfunctions. We evaluate the 3D and 2D Green’s functions and Mz(�r) at integer coordinates,i.e., x = y = z = 1 unit. Although one can evaluate these functions at a differentspacing between grid points, any choice of spacing has to be consistent between the evaluationof Green’s functions and geometry of the object (i.e., Mz(�r)).

When the induced field distribution of an infinitely long object is calculated from the 2DGreen’s functions, the geometry of the object should be taken as the cross section perpendicularto the object axis. The induced field should be calculated based on the coordinate systems onthe plane whose normal vector is parallel to the object axis. For example, if an infinitely longcylinder intersects at an angle θ with the main field direction, regardless of the value of θ , thegeometry used in calculating the magnetic field distribution should be a disk rather than anellipse.

For the forward calculations, we have simulated the magnetic fields of a sphere and acylinder with the continuous and discrete Green’s function. As the field decreases when avoxel is away from the object, the field difference between the calculated value (equation (14))and the theoretical value is quoted as error and is presented in percentage. However, whenthe field inside a sphere is calculated, its value is shown as in theory the magnetic field is zeroinside the sphere after the Lorentz sphere correction. In order to minimize the numerical errordue to the unsmoothed surface of the object (Koch et al 2006), the radius of any simulatedobject in this paper is at least 16 pixels with a fixed field of view of 256 pixels along eachdimension. The ratio of the object diameter to the field of view is used as an indication of howlarge an object is in the simulations.

2.5. Quantifying magnetic susceptibility from a least-squares fit

Because the Green’s functions in equations (5) and (12) contain null values, it is not appropriateto determine susceptibility through equation (3) by inverting the Green’s functions. Instead,for an object with a constant susceptibility χ , the following goodness-of-fit least-squaresfunction can be established based on equation (14) (Bevington and Robinson 1992). A similarconcept was briefly mentioned in Salomir et al (2003),

f =n∑

i=1

(Bi − χB0gi

δBi

)2

=n∑

i=1

(φi − γ TEχB0gi

δφi

)2

=n∑

i=1

SNR2i (φi − γ TEχB0gi)

2,

(15)

where n is the total number of voxels used in the fit, Bi is the measured magnetic field fromMR images at each voxel i, gi is the induced field per unit susceptibility and per unit main field

1174 Y-C N Cheng et al

calculated based on equation (14), δBi is the uncertainty of the measured field, φi ≡ −γ TEBi

is the phase value at each voxel, γ is the gyromagnetic ratio, TE is the echo time and SNRi isthe signal-to-noise ratio at each voxel from the magnitude image. Note that gi depends on thegeometry of the object, field of view, and the Green’s function.

By minimizing function f in equation (15) with respect to χ , i.e., ∂f/∂χ = 0, we obtain

χB0 =

n∑i=1

Bigi SNR2i

n∑i=1

g2i SNR2

i

. (16)

The uncertainty of χ can be found through the error propagation method (Bevington andRobinson 1992).

δχ = 1

γ TEB0

√n∑

i=1g2

i SNR2i

. (17)

Equations (16) and (17) show that the uncertainty of susceptibility measurement can be reducedwhen the SNR is increased.

It is important to note that equations (16) and (17) will fail if phase aliasing is not properlyunwrapped. The phase aliasing may be removed by complex dividing two phase imagesacquired at two different echo times with a short difference, TE , between the echo times.However, due to rapid field change within a voxel, usually phase aliasing would exist at voxelswith low SNRs. For this reason, we exclude those voxels from our analyses. Furthermore,voxels outside imaged objects containing no sufficient SNR in the magnitude images are alsoremoved. However, it can be seen from equations (16) and (17) that voxels with low SNRused in the fitting analysis will not significantly affect the susceptibility quantification or itsuncertainty, provided that enough voxels with sufficient SNR has been used in the analysis.

When acquiring images, a uniform phase φ0 can exist due to central frequency adjustmentby the scanner or rf excitation in the pulse sequence. Therefore, even for imaging one singleobject, practically, equation (15) should be modified to read

f =n∑

i=1

SNR2i (φi − φ0 − γ TEχB0gi)

2. (18)

Similar to the above derivation, both values of χ and φ0 can be determined by minimizingthe function f through the standard least-squares fit method (Bevington and Robinson 1992).The solutions and their associated standard deviations (i.e., uncertainties) derived through theerror propagation method are shown in Bevington and Robinson (1992).

2.6. Quantification of susceptibility from simulations

All simulations use MATLAB (The Mathworks Inc., Natick, MA) on a Windows XP platformand with a 1.54 GHz AMD TurionTM Mobile processor and with 2 GB RAM memory.Unless otherwise mentioned, in all simulations, the input main field is 1 tesla and the inputsusceptibility of an object is 1 ppm in SI units. The simulated objects are also assumed tobe surrounded by the vacuum. Equation (14) is used to simulate magnetic fields based ondiscrete and continuous Green’s functions as well as object geometries, which are spheres andcylinders at 14 different radii from 16 pixels to 120 pixels with an incremental size of 8 pixels.The field of view is always fixed at 256 pixels along each dimension.

With no white noise included in the simulations, equation (16) can be used to quantifythe susceptibility with a constant SNR in all pixels. The difference between the calculated

Limitations of calculating field distributions and magnetic susceptibilities 1175

and input susceptibility basically indicates the accuracy of the methods (or Green’s functions)themselves. Only pixels outside the objects are used in the quantification of susceptibilities,as the actual field inside a sphere is zero. One set of quantifications uses all pixels outside theobject. The other set uses pixels from a spherical shell or an annular ring, whose locations ofpixels satisfy R + 1 < r � R + 5 where R is the radius of the object.

In a simulation more close to practical imaging situations, we simulate magnitude andphase images with white noise added (Haacke et al 1999). An SNR of 5:1 is assigned andthe object is a spherical shell defined by two concentric spheres in this simulation. The innersphere has a radius of 16 pixels and the outer sphere has a radius of 96 pixels. The field ofview is again 256 pixels. The input susceptibility is 10 ppm in the shell and zero outside theshell and inside the inner sphere. The main field strength is 1 T and the echo time is 4 ms suchthat some phase aliasing occurs around the surface of the inner sphere. A constant backgroundphase value φ0 = 1 rad is also given as an input parameter in the simulation.

Only the discrete Green’s function is used in the quantification of susceptibility from thisshell simulation. Through equation (18), the least-squares fit approach allows us to determineboth the susceptibility and the constant background phase φ0. The actual phase value due tothe object at the ith voxel is φi −φ0, where φi is the simulated phase value. In order to removethe aliased phase points, the following criterion is used:

|φi − φ0 − γ TEχB0gi | � p/SNRi (19)

where SNRi is the signal-to-noise ratio of the ith voxel in the magnitude images and its inverseis the standard deviation of the phase in the same voxel. We choose p as an integer and it isusually one or two such that 68% or 95% of the non-aliased phase values are included in thesusceptibility quantification. Due to this criterion which helps to remove phase aliased voxels,an iterative procedure is used for the quantification as both χ and φ0 are unknowns. A flowchart of the iterative procedure is shown in figure 1. When the susceptibility value changesless than 0.1% (i.e., ε = 0.001 in figure 1), the iterative procedure will stop. In our program,we often choose the initial values of φ0 and χ as 0 rad and 7 ppm, respectively, compared tothe input values of 1 rad and 10 ppm, respectively.

2.7. Phantom study

One object with a somewhat complicated geometry was considered for our phantom study.As shown in figure 2, a CD case top made of polypropylene was filled with water and wasscanned on a 1.5 T Siemens Sonata MRI system. The CD case top had a wall thickness of1.3 mm, height of 180 mm and a diameter of 120 mm. It had a hollow cylindrical post inthe middle of one of its flat ends. The post had a varying diameter along its length, from12 mm at its base to 9 mm at its apex. Prior to the scans, a standard spherical phantom (witha diameter of 170 mm) containing the NiSO4 solution for the system quality control was usedfor shimming. The full width at half-maximum frequency was reduced to 6 Hz at the endof the shimming process. All the first- and second-order shim current values were noted andwere subsequently used for imaging the water phantom.

The water phantom was imaged transversely twice with echo times 6.58 ms and 9.58 msof a 3D gradient echo sequence and with the following parameters: TR 15 ms, flip angle 6◦,voxel size 0.78 mm × 0.78 mm × 0.78 mm, matrix size 256 × 256 × 192 and read bandwidth610 Hz per pixel. The images were later placed in a matrix of 256 × 256 × 256 with theadditional slices filled with zeros. The data acquired at the echo time 6.58 ms were complexdivided into data acquired at the echo time 9.58 ms. A set of images at an effective echo time3 ms was created and its phase values were used for the susceptibility quantification. TheSNR of the complex divided images was roughly 5:1. The complex images acquired from the

1176 Y-C N Cheng et al

Figure 1. A flow chart showing the iterative procedure of solving the susceptibility through theleast-squares fit method.

echo time 6.58 ms were used to obtain the geometry of the water phantom with a complexthresholding algorithm (Pandian et al 2008). The iterative procedure shown in figure 1 wasagain used for the susceptibility quantification. The initial values of φ0 and χ used in theiterative program were 0 rad and 6 ppm, respectively.

Limitations of calculating field distributions and magnetic susceptibilities 1177

(a) (b)

(c)

Figure 2. Images of the water phantom. (a) Transverse magnitude image at TE = 6.58 ms. (b) Itsassociated phase image. (c) The phase image after applying the complex division method. Voxelsinside the dashed box shown in (a) were used for susceptibility quantification.

3. Results

3.1. Spherical case: analytical solutions

We first assume a sphere with radius a and susceptibility χ embedded in vacuum. Usingequation (A.1), its magnetization in k-space is

μ0Mz(�k) =∫

d3r χB0 (a − r) e−i2π�k·�r

= 4πχB0

∫ a

0dr r2j0(2πkr)

= 2χB0a2

kj1(2πka), (20)

1178 Y-C N Cheng et al

where is the step function, and j0 and j1 are spherical Bessel functions. Based onequations (5) and (A.1), the induced magnetic field in the spatial domain is

Bz(�r) = F−1{μ0Mz(�k)Gc,3D(�k)}

= 8π

3χB0a

2(3 cos2 θ3D − 1)

∫ ∞

0dk kj1(2πka)j2(2πkr)

={

13χB0(3 cos2 θ3D − 1)

(ar

)3when r > a

0 when r < a.(21)

The last step requires the identity from Gradshteyn and Ryzhik (1994). The solution agreeswith the result shown in Haacke et al (1999, p 755).

If two concentric spheres with radii Ri and Re (Re > Ri) and constant susceptibilities χi

and χe are embedded in vacuum, then the overall magnetization in the spatial domain can bewritten as

μ0Mz(�r) = (χi − χe)B0 (Ri − r) + χeB0 (Re − r). (22)

Because the Fourier transformation is a linear operation, based on the above derivations andresults from equation (21), the induced magnetic field due to the two concentric spheres caneasily be written down

Bz(�r) =

⎧⎪⎨⎪⎩

0 when r < Ri

13 (χi − χe)B0(3 cos2 θ3D − 1)

(Ri

r

)3when Ri < r < Re

13

[(χi − χe)

(Ri

r

)3+ χe

(Re

r

)3]B0(3 cos2 θ3D − 1) when Re < r.

(23)

This result agrees with solutions from solving the Laplace equation in the magnetostatics casewhen higher order terms of χi and χe are neglected.

Another example is to consider a sphere with radius Ri and constant susceptibility χi andan infinitely long cylinder with radius Re (Re > Ri) and constant susceptibility χe parallelto the main field along the z-direction. Outside the cylinder it is vacuum. The origin of thesphere is located on the axis of the cylinder. The magnetization of the entire system is

μ0Mz(�r) = (χi − χe)B0 (Ri − r) + χeB0 (Re −√

x2 + y2). (24)

Following the derivations of equation (21) and with the results shown in the followingsubsection, the induced magnetic field can readily be written down

Bz(�r) =

⎧⎪⎪⎨⎪⎪⎩

13χeB0 when r < Ri

13 (χi − χe)B0(3 cos2 θ3D − 1)

(Ri

r

)3+ 1

3χeB0 when Ri < r < Re

13 (χi − χe)B0(3 cos2 θ3D − 1)

(Ri

r

)3when Re < r.

(25)

Even if radius Re is increased to a size that is much larger than Ri in both equations (23)and (25), it is clear that a uniform induced field, χeB0/3, only exists in equation (25) within aregion less than radius Re. Thus, when calculating the magnetic field distribution, one needsto include the geometry of the largest object in the magnetic field. Outside the largest objectit should be vacuum or air with essentially only the main field. One cannot simply adjust thebackground field to be χeB0/3 in solving the field distributions, as incorrectly mentioned inMarques and Bowtell (2005).

Limitations of calculating field distributions and magnetic susceptibilities 1179

3.2. Infinitely long cylindrical case: analytical solution

In this subsection, we assume an infinitely long cylinder with radius a and susceptibility χ atan angle θ with the main field B0 which is taken to be parallel to the ω-direction. As discussedin section 2.4, the geometry of the object should be treated as a disk and the calculationsshould be performed in its own cross-sectional coordinate system,

μ0Mω(�x) = χB0 (a −√

z2 + y2). (26)

With the help of equation (C.1), its 2D Fourier transformation is

μ0Mω(�k) = F{μ0Mω(�x)} = 2πχB0

∫ a

0dρ ρJ0

(2π

√k2z + k2

yρ) = χB0a

J1(2π

√k2z + k2

ya)

√k2z + k2

y

.

(27)

In combination with the 2D continuous Green’s function shown in equation (12), the inducedmagnetic field in the spatial domain is

Bω(�x) = F−1{μ0Mω(�k)Gc,2D(�k)}= χB0a

sin2 θ

2· z2 − y2

z2 + y2

∫ ∞

0dk J1(ka)J2

(k√

z2 + y2)

+χB0

6(3 cos2 θ − 1)a

∫ ∞

0dk J1(ka)J0

(k√

z2 + y2)

=

⎧⎪⎪⎨⎪⎪⎩

χB0 sin2 θa2

2(z2 + y2)· z2 − y2

z2 + y2outside the cylinder

χB0

6(3 cos2 θ − 1) inside the cylinder.

(28)

The last step requires an identity from Gradshteyn and Ryzhik (1994). The coordinates y andz are defined by the plane perpendicular to the axis of the cylinder. This solution agrees withthe result from Haacke et al (1999, p 755). The solution of the θ = π/2 case was also derivedin Koch et al (2006).

3.3. Simulations

Figure 3 shows the magnetic field profiles for a sphere and a cylinder based on the theoreticalpredictions for the continuous Green’s functions (labeled as Gc) and the discrete Green’sfunctions (labeled as Gd ). When judging carefully, one can see the ringing effect of thefield profiles across the entire field of view calculated from the continuous Green’s functions.Actually this is the blurring effect due to the singularity (or discontinuity) of the continuousGreen’s functions at the origin (r = 0). This blurring effect is not observed with the use of thediscrete Green’s functions as every step involved in such a method is under the discrete Fouriertransformation. No truncation or finite sampling is introduced in k-space when the discreteGreen’s functions are applied. Compared to the continuous Green’s function, it is also clearthat the fields calculated from the discrete Green’s function agree well with the theoreticalvalues especially at distances further away from the object.

Another way to assess the accuracy due to different Green’s function is to compare theactual predictions with the theoretically known field values (table 1). In this comparison, theFOV is fixed but the diameter of either the spherical or the cylindrical object is changed from32 pixels by steps of 32 pixels. The pixels at the corners of the FOV whose locations are morethan 128 pixels from the center of the FOV are excluded in table 1. Only the numbers of pixels

1180 Y-C N Cheng et al

(a)

(c) (d)

(b)

Figure 3. Magnetic field profiles of (a) a sphere and (b) a cylinder along the main field direction.The cylinder is perpendicular to the main field. (c) The difference of fields in percentage betweenthe theoretical value from a sphere and the calculated values based on different Green’s functions.The horizontal axis represents the distance of a voxel measured from the surface of the object inthe percentage of the field of view. This is the same label used by Koch et al (2006). Panel (d) issimilar to (c) but the fields are calculated based on the cylinder. All objects have a radius of 16pixels in the simulations. The dashed curves represent the theoretical values. The black and graycurves show results from the continuous and discrete Green’s functions, respectively. Only oneside of field comparisons is shown in both (c) and (d). Each inset inside each panel is the magnifiedview of a portion of the plot in the panel.

outside the object that lie within a given uncertainty are listed. The results from table 1 clearlyshow that the field calculations from the discrete Green’s function are in better agreement withthe theoretical values than the calculations from the continuous Green’s function. Further, thetrue magnetic field should be zero inside the sphere (equation (21)). The root-mean-squaredfield inside a sphere as a function of its diameter is shown in figure 4. It is again seen that theresults based on the discrete Green’s function are better than the results from the continuousGreen’s function.

In the quantification of susceptibility, both figures 5 and 6 show that the methodusing the discrete Green’s function is better than the method using the continuous Green’sfunction. When all the pixels outside the object and the discrete Green’s function are used

Limitations of calculating field distributions and magnetic susceptibilities 1181

Figure 4. The root-mean-squared magnetic field divided by the main field inside a sphere as afunction of the object size. The horizontal axis is displayed as the ratio of the object diameter tothe field of view. The field of view is fixed at 256 pixels. Voxels used in the calculations are insidethe spheres and at least three voxels away from the boundary of each sphere, in order to minimizethe errors due to the Gibbs ringing effects. The notations Gc and Gd represent the continuous anddiscrete Green’s functions, respectively.

for susceptibility quantification, no significant error is produced by the method itself if thediameter of the sphere or cylinder studied is less than 40% of the field of view (see figure 5).If only certain pixels in spherical shells or annular rings are used, as shown in figure 6, themethod using the discrete Green’s functions produces no significant error for objects withdiameters up to almost 60% of the field of view. The results shown in table 1, figures 3, 5 and6 imply that voxels close to the object are more important for an accurate quantification withthe Fourier based method than voxels away from the object. This is consistent with the factthat fields in voxels close to the edge of the field of view are prone to errors (Marques andBowtell 2005).

In the simulation of the spherical shell with noise included, several results are observed.First, if we choose all voxels inside the shell for the susceptibility quantification, then weend up with a susceptibility that agrees with the result shown in figure 5. Thus, for a moreaccurate quantification, we only use the central 1283 voxels of the shell for the susceptibilityquantification. Second, when the initial choice of the susceptibility is between 7 and 10 ppm,the final answers from the iterative procedure do not seem to be affected. Third, whether oneor two standard deviations (i.e., p = 1 or 2) are used in equation (19) does not seem to affectthe final answers either. The results are shown in table 2(a). As the chi-square per number ofpoints in each calculation is close to one, each result is from a good fit. Although in general,the results in table 2(a) are in very good agreement with the input values, the differencesbetween the calculated values and the input values for p = 1 and 2 are due to the Fourierbased method itself (or the Green’s function).

As the number of voxels in the quantification is large, the statistical uncertainty is smallin the shell calculation. However, as the exact location or geometry of an object may notbe perfectly defined in MRI magnitude images due to the partial volume effect or the signal

1182 Y-C N Cheng et al

Table 1. This table lists the number of pixels outside a given object within a specified uncertaintyof the magnetic field distribution. Voxels that are more than 128 pixels away from the centerof the field of view are not counted in this table. The first column lists the ratio of the objectdiameter to the field of view, which are 256 pixels along each dimension. The second and thirdcolumns show the numbers of pixels with magnetic fields that are within 5% of the theoretical fieldvalues. Similarly, the fourth and fifth columns show the numbers of pixels with magnetic fieldsthat are within 20% of the theoretical values. The notations Gc and Gd represent the usages ofthe continuous and discrete Green’s functions in the simulations, respectively. (a) For spheres and(b) for cylinders.

Within 5% Within 20%

D/L Gc Gd Gc Gd

(a)1/8 2167 008 8720 240 5064 980 8762 2012/8 1942 810 8298 624 4870 682 8621 0223/8 1587 872 7421 496 4527 614 8183 4924/8 894 610 5906 220 3862 234 7345 6145/8 95 210 3533 400 2803 000 6011 0646/8 32 128 781 096 1193 108 4047 3807/8 21 248 43 696 131 472 1447 316

(b)1/8 15 384 49 756 31 312 50 2502/8 11 608 44 352 28 140 47 7763/8 7828 36 472 24 212 42 8524/8 1336 25 816 18 292 35 6485/8 280 10 884 10 428 26 2926/8 168 648 2312 15 0447/8 72 72 424 3128

Table 2. This table lists the results from (a) the simulated shell and (b) the phantom studies. Thefirst column lists the number of standard deviations used in the computer program. The secondcolumn lists the number of iterations. The third column lists the calculated susceptibilities and theirassociated statistical uncertainties from the fit. The fourth column lists the calculated backgroundphase values and their associated uncertainties. The last column lists the chi-square per number ofpoints used in the program. The input susceptibility and background phase were 10 ppm and 1 rad,respectively, for the simulated shell shown in (a). The theoretical susceptibility for the phantomstudy in (b) is −9.4 ppm.

p Iterations χ (ppm) φ0 (rad) χ2/N

(a)1 20 10.021 ± 0.0070 0.997 ± 0.000 17 0.292 8 10.025 ± 0.0058 0.999 ± 0.000 14 0.783 5 10.000 ± 0.0057 1.000 ± 0.000 14 1.01

(b)1 24 −9.199 ± 0.0044 0.748 ± 0.000 24 0.302 9 −9.186 ± 0.0035 0.750 ± 0.000 20 0.843 6 −9.131 ± 0.0034 0.749 ± 0.000 19 1.15

loss, the uncertainty due to the object geometry can be a concern. For that, we purposelycalculated the discrete Green’s function based on the radius of the inner sphere as 17 pixels.The calculated susceptibility through the least-squares fit deviated from the input value by

Limitations of calculating field distributions and magnetic susceptibilities 1183

(a)

(b)

Figure 5. Susceptibility error in percentage versus the ratio of the diameter of the object to thefield of view. The notations Gc and Gd represent the continuous and discrete Green’s functions,respectively. All pixels outside the objects are used in the quantifications, (a) for spheres and(b) for cylinders.

roughly 18%. This agrees with the volume increase of the inner sphere as the magnetic fielddistribution is proportional to the magnetic moment of the object, which is proportional to theproduct of the volume and susceptibility.

3.4. Phantom results

Most of the results of the phantom study are similar to the results of the shell simulation. Inorder to remove any significant error due to the method itself, we again restrict ourselves to

1184 Y-C N Cheng et al

(a)

(b)

Figure 6. Susceptibility error in percentage versus the ratio of the diameter of the object to thefield of view. This figure is similar to figure 5 but only certain pixels outside each object are usedin the quantifications, as described in the text. (a) For spheres and (b) for cylinders.

analyze the central 1283 voxels of the magnitude and complex divided images (see the dashedbox shown in figure 2(a)). The results are shown in table 2(b). The calculated susceptibilitiesare within 3% of the theoretical susceptibility of water relative to air, which is −9.4 ppm inSI units (Lide 2006-2007). The results indicate that the statistical uncertainties are smallerthan the actual uncertainties which are probably due to the non-prefect definition of the objectgeometry.

Limitations of calculating field distributions and magnetic susceptibilities 1185

In either the simulations or the phantom studies, at least two 2563 matrices are savedand renewed for data processing in the computer memory. As shown in figure 1, fast Fouriertransformation of the geometry or of the Green’s function is performed outside the iterationloop. Fast Fourier transformation of one 2563 matrix takes roughly 2 min. An iteration ofsusceptibility quantification also takes roughly 2 min. For example, the total time of the entireprocedure shown in figure 1 with 10 iterations is roughly 22 min.

4. Discussion and conclusions

As known in image reconstruction (see chapter 12 of Haacke et al (1999)), sampling in k-space domain will lead to a periodic reconstructed object in the spatial domain. As a result,the magnetic field distribution is also aliased (Marques and Bowtell 2005). This is the mainreason why all results based on the continuous Green’s function are worse than those from thediscrete Green’s function. Ideally, the magnetic field calculated based on a discrete Green’sfunction should not produce any error. However, as the discrete Green’s function is generatedby finite sampling the original Green’s function in the spatial domain, it is also inevitable thatthe discrete Green’s function used in this method are also aliased (albeit to a lesser degree)in k-space. Generally, the discrete Green’s function will start to fail when the object sizeincreases and becomes comparable to the field of view, as shown in table 1 and figures 4–6.The problem due to the finite sampling can be alleviated by enlarging the field of view relativeto the object size (Koch et al 2006, Marques and Bowtell 2005). This is consistent with theresults from all of our simulations. However, the immediate problem is that the computingtime will increase dramatically when a larger matrix size is used in calculations or practicallywhen collecting data with a pre-determined resolution. Despite these limitations, as long asthe discrete Green’s function is used, the method can be used accurately even when the objectsize (i.e., diameter) is as large as 60% of the field of view.

When the field of view is enlarged with a fixed image resolution, the resolution in k-spaceis increased. This obviously implies that enough points around the k = 0 point are neededfor the reconstruction of accurate magnetic field in the spatial domain. This is also due tothe discontinue value at k = 0. On the other hand, increasing the spatial resolution does notimprove the method. Increasing the spatial resolution at a fixed FOV is the same as increasingthe window function in k-space. A window function that consists of less than 10 points canlead to a significant blurring effect but this is often not the case in MRI. However, an improvedimage resolution with sufficient SNR will help in the definition of the object geometry and inthe measurement of local phase values near the edge of the object.

The numbers listed in table 1 may be used to explain the results shown in figure 6 and toestimate the size of a spherical shell used for the quantification of susceptibility, when a givenuncertainty is desired. For example, in the third column of table 1(a), when the radius of thespherical object is 96 pixels, a total of 781 096 voxels have uncertainties within 5% of the exactfield values. As shown in figure 6(a), if voxels in a shell are between radius 101 pixels and 97pixels and are used for the susceptibility quantification, then the calculated susceptibility isroughly 6% deviated from its true value. The number of voxels in the shell is roughly 490 000voxels, less than the above number of voxels. The difference (or excess) in the number ofvoxels indicates that more voxels can be included in the susceptibility quantification such thatthe error of the susceptibility will remain the same. The 781 096 voxels imply that the radiusof the larger sphere can be increased to 103 pixels rather than 101 pixels. The calculatedsusceptibility from the case of the larger shell (between 97 pixels and 103 pixels along theradius) is also roughly 6% deviated from the true susceptibility value. The method with

1186 Y-C N Cheng et al

the discrete Green’s function basically allows us to include more points for analysis withoutpenalty.

The background phase value, φ0, may be estimated from the phase value at the k = 0point and be used as an initial value in the iterative program. In theory, this is true if anobject has a uniform zero phase value or certain symmetry in its complex images. We haveconfirmed this conjecture from our simulated shell. In addition, the phase is roughly 0.54 radat the k = 0 point from the phantom images while our program found a φ0 of roughly 0.75 rad(table 2). Nonetheless, whether one starts φ0 with a zero estimate or the central k-spaceestimate seems to make little difference in our least-squares fit algorithm. In order to furtherobtain a better set of initial values, the value p can be assigned to three first and then reducedto either one or two in the program. The program seems to converge faster with a higher valueof p. However, the higher the value of p, the more the noisy data are included. Althoughthe 2D discrete Green’s function seems to produce slightly larger error than that from the 3DGreen’s functions (see figures 5 and 6), the 2D method only requires one slice for analysis.Therefore, one can afford computing time and enlarge the field of view. When the Fourierbased method is utilized, the overall geometry of the object is better to be included within thefield of view. Although it may be reasonable to neglect the field distribution due to a boundarythat is relatively far away from the region of interest (Koch et al 2006), one should not adjustthe background field by a constant without care as stated in results.

Voxels containing low SNR in the magnitude images will lead to high uncertainty in thequantification of object susceptibility. In order to remove phase aliasing, phase unwrappingor complex division method may be applied in combination with the Fourier based method.Alternatively, our iterative procedure can automatically exclude the phase aliasing voxels. Thisis better than to select non-aliasing voxels manually as we want to keep voxels close to theobject which have high phase-to-noise ratios. Induced eddy currents or shimming performedon the scanner can change the phase values in images. The latter problem is minimized byshimming a spherical phantom for quality control prior to scans.

Despite the fact that the statistical uncertainty of each susceptibility value shown intable 2(a) is very small, the statistical uncertainty is not large enough to explain the differencebetween the quantified susceptibility and the true susceptibility. This means that the uncertaintyof the method itself (see, for example, figures 5 and 6) rather than the statistical uncertaintyfrom white noise is a major source of the uncertainty. Fundamentally, the Fourier basedmethod will completely fail when the object size is equal to the image resolution (Sepulvedaet al 1994, Tan et al 1996). Practically, the geometry of small objects and their induced fielddistributions cannot be accurately prescribed without sufficient resolution. The problem is thatthe magnetic field will decrease quickly so that there will not be enough voxels to accuratelyestimate the susceptibility. In addition, due to the partial volume effect or the signal loss,the definition of the object geometry can also significantly affect the answer. As a result, theuncertainty of susceptibility quantification will increase. These factors require further carefulstudies of this method.

In summary, we have shown that discrete Green’s functions used in the Fourier basedmethod will lead to much better results in estimating local magnetic fields from a given objectof interest and that a least-squares fit approach based on this method can be applied to quantifysusceptibility accurately.

Acknowledgments

This work was supported by the Department of Radiology at Wayne State University and byNIH HL062983-04A2.

Limitations of calculating field distributions and magnetic susceptibilities 1187

Appendix A. Derivation of the continuous 3D Green’s function in k-space

Equation (5) can easily be derived with the following identity provided in Jackson (1999):

ei�k·�r ≡ 4π

∞∑�=0

i�j�(kr)

�∑m=−�

Y ∗�m(θ, φ)Y�m(θk, φk), (A.1)

where θk and φk are the angles used in the spherical coordinate system.The Fourier transformation of the Green’s function, Gz,3D(�r), is

Gc,3D(�k) ≡ F{Gz,3D(�r)}≡

∫d3r Gz,3D(�r) e−i2π�k·�r

= (1 − 3 cos2 θk)

∫ ∞

0dr

1

rj2(2πkr)

= 1

3− cos2 θk

≡ 1

3− k2

z

k2, (A.2)

where cos θk ≡ kz/k. This derivation is valid only when k is not equal to zero or infinity.Throughout this paper, we choose exp{−i2π�k ·�r} as the Fourier transformation. Although

it is well understood that such a choice is not unique, it is acceptable as long as its inverseFourier transformation is consistently defined.

Appendix B. Calculation of the 2D Lorentz disk

Equation (6) was established on a 2D plane. If the direction of a dipole is always assumed tobe parallel to the main field and if the dipole is embedded in 3D space, due to the inner productin equation (6), only the transverse component of the dipole on the 2D plane will generate anon-zero magnetic field. If only the field produced by equation (6) parallel to the main field isof interest, only the projected component along the main field direction is needed. For thesetwo reasons, an additional factor sin2 θ is added in front of equation (10) and is also seen inthe first term of equation (B.1).

Similar to discussions under section 5.6 in Jackson (1999) and equation (6), the magneticfield at the dipole with a magnitude of m can be calculated from averaging the field within aninfinitesimal disk with a radius R∫

ρ<R

m · �B(�x) dS = − μ0

2π

∫ρ<R

dS m · �∇( �m · �x

ρ2

)+

∫ρ<R

dS μ0m · �mδ(2)(�x)

= −μ0m

2π

∮(m · n)2

R(R dφ′) + μ0m

= −μ0m sin2 θ

2π

∫ 2π

0dφ′ cos2(φ′ − φm) + μ0m

= −μ0m sin2 θ

2+ μ0m

=(

1 − sin2 θ

2

)μ0m. (B.1)

Note that in this derivation we have defined the unit vector n ≡ �x/R = cos φ′x +sin φ′y wherethe x and y-axes define the plane of the disk and m ≡ sin θ cos φmx + sin θ sin φmy + cos θ z.

1188 Y-C N Cheng et al

The component perpendicular to the plane of the disk (i.e., the component parallel to theaxis of the infinitely long object) has been neglected for all discussions relevant to 2Dcalculations. The result of equation (B.1) can be considered as the Lorentz disk and it isthe induced magnetic field inside an infinitely long cylinder (see equation (25.35) of Haackeet al (1999)). The Lorentz sphere in the 3D case can also be calculated with a similar derivationof equation (B.1).

Appendix C. Derivation of the continuous 2D Green’s function in k-space

The continuous Green’s function Gc,2D(�k) in k-space can be derived with the help of thefollowing definition of the Bessel functions:

Jm(x) = 1

2π im

∫ 2π

0dφ′ eix cos φ′−imφ′

. (C.1)

Therefore,

Gc,2D(�k) ≡ sin2 θ

2π

∫d2x e−i2π�k·�x 2 cos2 φ − 1

ρ2+

1

6(3 cos2 θ − 1)

= −sin2 θ cos 2θk,2D

∫ ∞

0dρ

1

ρJ2(2πkρ) +

1

6(3 cos2 θ − 1)

= sin2 θ

2· k2

y − k2z

k2y + k2

z

+1

6(3 cos2 θ − 1), (C.2)

where k ≡√

k2z + k2

y in this derivation and cos 2θk,2D ≡ (k2z − k2

y

)/(k2z + k2

y

). This result is

valid only when k is not equal to zero or infinity.

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