Three-dimensional numerical simulations of susceptibility ...pfeifer.phas.ubc.ca/refbase/files/Truong-MRI-2002-20-759.pdf · Three-dimensional numerical simulations of susceptibility-induced

Three-dimensional numerical simulations of susceptibility-inducedmagnetic field inhomogeneities in the human head

Trong-Kha Truonga, Bradley D. Clymerb, Donald W. Chakeresa, Petra Schmalbrocka,*aDepartment of Radiology, The Ohio State University, Columbus, Ohio 43210, USA

bDepartment of Electrical Engineering, The Ohio State University, Columbus, Ohio 43210, USA

Received 30 July 2002; accepted 16 September 2002

Abstract

Three-dimensional numerical simulations of the static magnetic field in the human head were carried out to assess the field inhomo-geneity due to magnetic susceptibility differences at tissue interfaces. We used a finite difference method and magnetic permeabilitydistributions obtained by segmentation of computed tomography images. Computations were carried out for four models, consisting of thehead and the neck; the head, neck, and shoulders; the head, neck, and thorax; and the head tilted backwards, including the neck and theshoulders. Considerable magnetic field inhomogeneities were observed in the inferior frontal lobes and inferior temporal lobes, particularlynear the sphenoid sinus and the temporal bones. Air/tissue interfaces at the shoulders were found to induce substantial magnetic fieldinhomogeneities in the occipital lobes and the cerebellum, whereas air/tissue interfaces in the lungs appeared to have less influence on themagnetic field in the brain. Tilting the head backwards could significantly reduce the field inhomogeneities superior to the planumsphenoidale as well as in the occipital lobes and the cerebellum. © 2002 Elsevier Science Inc. All rights reserved.

Keywords: Magnetic susceptibility; Susceptibility artifacts; High field MRI; Magnetic field map

1. Introduction

Magnetic susceptibility has proven to be a valuable con-trast mechanism for magnetic resonance imaging (MRI),especially for high resolution imaging at ultra-high field[1,2] and for functional MRI (fMRI) [3]. However, imagingmethods that are sensitive to intrinsic susceptibility contrastalso suffer from artifacts due to susceptibility differencesnear tissue interfaces, most prominently air/tissue inter-faces. Thus development and optimization of methods toreduce such artifacts yet maintain image contrast are ofparamount importance for ultra-high field MRI and fMRI.

Assessment of susceptibility artifact correction methodsrequires knowledge of the susceptibility-induced magneticfield inhomogeneities. While a variety of experimental tech-niques can be used to map the static magnetic field, numer-ical simulations of susceptibility-induced magnetic fields

are a promising alternative. Especially at ultra-high fieldstrength, experimental field mapping techniques may belimited because they are impeded by other sources of arti-facts such as radiofrequency (RF) coil B1 inhomogeneity,signal loss due to complete intravoxel dephasing, especiallyin regions of large magnetic field inhomogeneity, and/orsubject physiological motion. On the other hand, numericalsimulations are not affected by such problems and offermore flexibility in the selection of relevant parameters.

Susceptibility-induced magnetic fields have been numer-ically modeled by several investigators. Bhagwandien et al.[4–6] developed an iterative finite difference method forcomputing the static magnetic field for arbitrary two-dimen-sional (2D) or three-dimensional (3D) magnetic susceptibil-ity distributions, and used it to compute magnetic fielddistributions for two-component (air and water) models ofthe head, the chest, and the hand [6]. Li et al. used acommercial finite element algorithm to compute the 3Dmagnetic field distribution for two-component wire framemodels of a human head [7] and upper body [8] with a fairlycoarse description of the air spaces. More recently, Yang etal. [9] described preliminary work with a modified versionof Bhagwandien’s algorithm [5] using head models derived

1 Preliminary results of this work were presented at the Tenth ScientificMeeting of the International Society for Magnetic Resonance in Medicine,Honolulu, HI. 18-24 May 2002, p. 2323.

* Corresponding author. Tel.: �1-614-293-4139; fax: �1-614-293-6935.

E-mail address: [email protected] (Petra Schmalbrock).

Magnetic Resonance Imaging 20 (2002) 759–770

0730-725X/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved.S0730-725X(02)00601-X

from the Visible Human data and from MRI data. This workis very promising, however the Visible Human data canprovide geometric information for only one specific subject.On the other hand, MRI data can be acquired for differentsubjects, but does not allow differentiating air from corticalbone, thus limiting the models near the nasal cavity, sphe-noid sinus, and temporal bones. Finally, a new algorithmbased on a perturbation method was recently proposed byJenkinson et al. [10].

As an alternative to prior work, and to simplify thegeneration of models for the numerical simulations, wederived our models from computed tomography (CT) im-ages covering the head, neck, and thorax of a healthy vol-unteer, and then used a modified version of Bhagwandien’salgorithm [5] to compute the static magnetic field distribu-tions. We first tested the accuracy of our algorithm bycomparing numerical simulations with the analytical solu-tion for the magnetic field in a sphere. We then evaluatedhow the magnetic field in the head was affected by air/tissueinterfaces at the shoulders and in the lungs, and tested iftilting the head backwards could reduce some of the sus-ceptibility artifacts, as had been suggested by Herberlein etal. [11].

The long-term objective of this work is to use numericallycomputed magnetic field distributions for the evaluation ofsusceptibility artifact correction methods for high-resolutionultra-high field imaging, such as gradient compensation tech-niques [12,13], tailored RF pulses [14,15], post-processing[16–18], and local shimming [19–21].

2. Methods

2.1. Generation of the magnetic permeability distributions

Two sets of CT images were acquired on a healthy51-year-old male volunteer. The first set of images coveredthe head, neck, thorax, and abdomen, whereas the secondset covered the head, neck, and shoulders, with the headtilted backwards by 35°. The images were acquired with aGeneral Electric Highspeed CT scanner using kV 120, mA60, a field of view of 45.5 cm, a matrix size of 512 � 512,427 and 181 slices of 2 mm thickness respectively for thefirst and second data sets. The images were resized to matchthe slice resolution, thus resulting in 3D data sets with anisotropic resolution of (2 mm)3. Example images from thefirst data set are shown in Figs. 1a–e.

A threshold-based segmentation of these CT data setswas feasible and resulted in four masks for each data setcorresponding to air, fat, nonfat soft tissue, and bone voxels.Histograms and intensity plots were used to determine thefollowing threshold levels: �300 H.U. (Houndsfield Units)for air/fat, �50 H.U. for fat/nonfat soft tissue, and 250 H.U.for nonfat soft tissue/bone. Relative magnetic permeabilityvalues �r were determined from magnetic susceptibilityvalues � (�r � 1 � �) given in the literature (�air � 0.4 ·

10�6, �fat � �7.5 · 10�6, �nonfat soft tissue � �9.5 · 10�6, and�bone � �9.0 · 10�6 [6,7]), and these values were thenassigned to the corresponding masks (see Figs. 1f–j).

In order to study separately the influence of the air/tissueinterfaces at the shoulders and in the lungs on the magneticfield in the head, three models were created from the data setwith the head upright. They consisted of the permeabilitydistribution of the head and the neck (“head model”), thehead, neck, and shoulders (“head/shoulders model”), andthe head, neck, and whole thorax (“head/thorax model”). Inorder to study the influence of the head orientation on themagnetic field in the head, a fourth model was created fromthe data set with the tilted head, and consisted of the per-meability distribution of the head, neck, and shoulders(“ tilted head/shoulders model”).

2.2. Computation of the magnetic field

The computation of the static magnetic field from thepermeability distributions was carried out with a modifiedversion of the finite difference method developed by Bhag-wandien et al. [5,6]. Maxwell equations in a macroscopic

Fig. 1. Axial CT images of the head (a), thorax (b), and abdomen (c) of ahealthy volunteer, and coronal (d) and mid-sagittal (e) reconstructed CTimages of this data set. The solid arrow points to the planum sphenoidaleand the dashed arrow points to the upper clivus. The CT images weresegmented and assigned appropriate values of relative magnetic perme-ability. The resulting permeability distributions are shown in (f–j) with airin black, fat in light gray, nonfat soft tissue in dark gray, and bone in white.

760 T.-K. Truong et al. / Magnetic Resonance Imaging 20 (2002) 759–770

linear medium with no charges and no currents for thesteady-state lead to the Laplace equation:

�� r� x, y, z�� x, y, z� � 0, (1)

where �r(x, y, z) is the relative magnetic permeability and(x, y, z) [Wb/m] is the magnetic scalar potential. Since�r(x, y, z) is known, (x, y, z) can be determined. Then thez component of the molecular magnetic field Bz(x, y, z) [T]can be computed as follows:

Bz� x, y, z� � ��0��r� x, y, z� � 2

3� �� x, y, z�

� z,

(2)

where �0 � 4�10�7 H/m is the magnetic permeability invacuum and z is the direction of B0, the static magnetic fieldin vacuum in the absence of any sample. The Lorentzcorrection [22] has been taken into account in Eq. (2). Inorder to obtain a value independent of B0, the field isexpressed as the normalized magnetic field deviation fromthe applied field (the magnetic field in air in the absence ofany sample, B0�r,air):

�Bz� x, y, z� �Bz� x, y, z� � B0�r,air

B0�r,air. (3)

As described in [5,6], the 3D problem in Eq. (1) is extendedto a four-dimensional one with an iteration variable �, andEq. (1) is defined as the equilibrium state of the followingiterative problem:

Cd� x, y, z; ��

d�

� �� r� x, y, z; �� x, y, z; ��,

(4)

where C [m�2] is a constant and d� � 1. This problem ismathematically analogous to a heat transfer problem where� is the time variable, whereas in our case � is an iterationvariable. This problem has a convergent solution with

lim�3�

� x, y, z; �� x, y, z�. (5)

Denoting a given voxel at position (x, y, z) by the index(0) and each of the six neighboring voxels (namely (x � dx,y, z), (x � dx, y, z), (x, y � dy, z), (x, y � dy, z), (x, y, z �dz), and (x, y, z � dz), where dx, dy, and dz represent thespatial resolution in the x, y, and z direction respectively) bythe indices (1) to (6), the magnetic scalar potential at thatvoxel can be iteratively computed as follows [5]:

�0; �� i�1

6

D�i��i; � � 1� � D�7��0; � � 2�,

(6)

where

D�i� �2B�i�

2 � B�7��i � 1, . . . , 6� (7)

D�7� �2 � B�7�

2 � B�7�(8)

B�i� �2�r�i��0

��r�i� � �0�C dx dy�i � 1, . . . , 6�

(9)

B�7� � ��i�1

6

B�i�. (10)

In other words, the potential at a given voxel and at agiven iteration step is computed as a linear combination ofthe potential at each of the six neighboring voxels from theprevious iteration step and the potential at that voxel fromtwo iteration steps before, where the coefficients of the linearcombination are functions of the permeability distribution.

This iterative algorithm requires suitable boundary con-ditions to be defined. Specifically, the permeability distri-bution of the sample must be surrounded by a buffer regionof uniform permeability (containing only air), which mustbe large enough so that the potential at its outside bound-aries is not influenced by susceptibility effects induced bythe sample. Bhagwandien et al. [5,6] used a buffer whosesize is a multiple of the size of the sample. For an elongatedsample (such as a human body), this results in a largerbuffer region along the long axis of the sample. To avoidthis effect, we generate the buffer by adding a buffer regionthat has the same thickness along each direction x, y,and z around the sample. This buffer is thus characterizedby a “buffer thickness,” defined as the smallest distancebetween the sample and the outside boundaries of the bufferalong any direction x, y, or z (see Table 1).

For the first step of the iteration, a magnetic scalarpotential uniform in x and y and linear in z is used as aninitial value:

� x, y, z; 0� � � x, y, z; 1� � �B0

�0z. (11)

The origin is not important since the magnetic scalar poten-tial is always defined with an arbitrary additive constant.The iteration stops when the steady-state is reached, asdefined by the following criterion:

max� x,y,z��sample

�� x, y, z; �� 10�10, (12)

where

� x, y, z; �� x, y, z; �� x, y, z; � � 1��

max� x,y,z��sample�� x, y, z; 0��

(13)

is the normalized absolute difference between the potentialat the current iteration step and the potential at the previous

761T.-K. Truong et al. / Magnetic Resonance Imaging 20 (2002) 759–770

iteration step. The maximum is taken only over voxelsinside the sample, because otherwise the largest differenceoccurs at the outside boundaries of the buffer and is there-fore dependent on the size of the buffer. This stop criterionthus controls the maximum absolute difference inside thesample, unlike the one used by Bhagwandien et al. [6],which is based on the sum of the minimal square differenceover the whole volume. The threshold value used in Eq. (12)was determined by comparing numerical simulations withthe analytical solution for the magnetic field in a sphere (seesection 2.3). This algorithm is unconditionally stable [5,6].The constant C determines the speed of convergence and itsoptimal value is found by trial and error.

The algorithm was written in Matlab (The MathWorksInc., Natick, MA) and executed on an SGi Origin 2000 witha memory limit of 1 GB and a computation time limit of10 h. The advantage of using Matlab is the ability to per-form very fast matrix operations, however at the expense oflarge memory requirements. Due to our computer memorylimitation, the computation for the head/thorax model,which is the largest one, could only be carried out at aresolution of (4 mm)3. Therefore, computations for all fourmodels were carried out at that resolution to allow directcomparisons between the different models.

Furthermore, in order to reduce the computation time, amultiresolution approach was used [5,6]. A schematic dia-gram is shown in Fig. 2 and the parameters used are shownin Table 1. For the first step, the multiresolution implemen-tation starts with a low resolution version of the permeabil-ity distribution of the sample. This distribution is thensurrounded by an air filled buffer region with a given bufferthickness. The iterative algorithm is then run using thisextended permeability distribution, a linear estimate for theinitial magnetic scalar potential [Eq. (11)], and an appropri-

ate constant C. For the second step, the result of this itera-tion is interpolated to a higher resolution using linear inter-polation and the iterative algorithm is run again, now usingthis potential as the initial value and a higher resolutionversion of the permeability distribution. The constant C isadjusted as needed. This procedure is repeated until the finalresolution is reached. Finally, after the last step, the partialderivative of the magnetic scalar potential (x, y, z) withrespect to z is computed and multiplied by ��0[�r(x, y, z)� 2]/3 to yield the magnetic field Bz(x, y, z) [Eq. (2)].

Because of computer memory limitations, the multireso-lution approach used by Bhagwandien et al. [5,6] was mod-ified. As the resolution was increased, the buffer thicknesswas reduced in order to limit the total matrix size of thepermeability distribution (see Table 1).

2.3. Validation of the algorithm

The algorithm was validated with a sphere model, for whichan analytical solution for the magnetic field is available [5]:

Bzana� x, y, z� �

� B0�1 ��ext

3 � , r R

B0�1 ��ext

3�

��ext � �int� R3� x2 � y2 � 2z2�

3� x2 � y2 � z2�5/ 2 � ,

r � R (14)

where �int and �ext are the magnetic susceptibility valuesinside and outside the sphere respectively, r � (x2 � y2 �z2)1/2, and R is the radius of the sphere. The Lorentz cor-rection has been taken into account in Eq. (14). The follow-

Table 1Parameters used for the multiresolution implementation of the algorithm

stepvoxelsize

initialmatrix sizea

bufferthickness

totalmatrix sizea

memoryused iterations

computationtime

Head model1 (16 mm)3 16 � 15 � 16 384 mm 64 � 63 � 64 50 MB 132 3 min2 (8 mm)3 32 � 30 � 32 256 mm 96 � 94 � 96 170 MB 82 7 min3 (4 mm)3 64 � 60 � 64 128 mm 128 � 124 � 128 390 MB 99 22 min3' (4 mm)3 64 � 60 � 64 48 mm 88 � 84 � 88 130 MB 331 30 min4 (2 mm)3 128 � 120 � 128 48 mm 176 � 168 � 176 1 GB 752 6 h11Head/shoulders model1 (16 mm)3 28 � 18 � 22 384 mm 76 � 66 � 70 70 MB 206 5 min2 (8 mm)3 56 � 36 � 44 256 mm 120 � 100 � 108 250 MB 85 13 min3 (4 mm)3 112 � 72 � 88 128 mm 176 � 136 � 152 700 MB 86 38 minHead/thorax model1 (16 mm)3 28 � 18 � 32 384 mm 76 � 66 � 80 80 MB 316 8 min2 (8 mm)3 56 � 36 � 64 256 mm 120 � 100 � 128 300 MB 79 15 min3 (4 mm)3 112 � 72 � 128 128 mm 176 � 136 � 192 890 MB 357 3 h18Tilted head/shoulders model1 (16 mm)3 28 � 15 � 22 384 mm 76 � 63 � 70 70 MB 200 6 min2 (8 mm)3 56 � 30 � 44 256 mm 120 � 94 � 108 240 MB 85 16 min3 (4 mm)3 112 � 60 � 88 128 mm 176 � 124 � 152 640 MB 82 25 min

a Matrix sizes are given as lateral � anteroposterior � inferior-superior dimensions.


ing parameters were chosen for similarity with the headmodel: �int � �water � �9.1 · 10�6 [7], �ext � �air, R � 128mm, and a resolution of (4 mm)3.

The accuracy of the algorithm was evaluated by com-puting the normalized difference between the magnetic fieldobtained from the numerical simulation and from the ana-lytical solution:

�B� x, y, z� �Bz� x, y, z� � Bz

ana� x, y, z�

B0�r,air. (15)

Simulations using different threshold values for the stopcriterion [Eq. (12)] and different resolutions and buffer thick-nesses were carried out to determine the optimal parameters.

3. Results

3.1. Validation of the algorithm with the sphere model

Example images and plots of the error �B [Eq. (15)] forthe sphere model computed using different buffer thick-nesses are shown in Fig. 3. Figs. 3a,b clearly show that abuffer thickness of 112 mm does not give sufficiently ac-curate results. When the buffer thickness is increased to 128mm (Figs. 3c,d), the error �B inside the sphere decreases to�0.033 ppm. The error at the interface is substantiallylarger but it can be attributed to a lack of resolution (relativeto the diameter of the sphere). Simulations were carried out

at a higher resolution (or equivalently at the same resolutionbut with a sphere of larger diameter) and confirmed that thisis indeed the case, because the error at the interface becamesmaller. When the buffer thickness is further increased to144 mm (Figs. 3e,f), the error �B inside the sphere decreasesto �0.025 ppm. However, the error at the interface is notreduced since it is related to the resolution. Thus optimizingthe accuracy of the algorithm involves a trade-off betweenbuffer size and spatial resolution, because the total matrixsize of the permeability distributions is limited by the avail-able computer memory.

As such, the computation for the head/thorax model,which is the largest one, could only be carried out with abuffer thickness of 128 mm. This value was therefore cho-sen for all four models to allow direct comparisons betweenthe different models. While simulations with the spheremodel showed that this buffer thickness gives overall accu-rate results, the lack of spatial resolution may introducesome errors, especially near tightly curved air/tissue inter-faces, but these errors appear to be limited to a thin layerfrom the interface (Figs. 3c,d). Nevertheless, increased spa-tial resolution is ultimately desirable.

3.2. Head model at (4 mm)3 vs. (2 mm)3

To further investigate to what extent the limited spatialresolution affects the accuracy of the simulations, we com-puted and compared the magnetic field for the head model

Fig. 2. Schematic diagram of the multiresolution implementation of the algorithm for the computation of the magnetic field. Ovals represent parameters andrectangles represent computation steps.


at resolutions of (4 mm)3 and (2 mm)3. Theses computationswere carried out using a buffer thickness of 48 mm at thefinal resolution in both cases (see Table 1, steps 1-2-3' and

steps 1-2-3-4 respectively). Although simulations with thesphere model showed that this buffer thickness is too smallto avoid errors, it is the maximum value achievable at aresolution of (2 mm)3 due to computer memory limitations.However, since the computations at both resolutions werecarried out using the same buffer thickness, we expect errorsrelated to the buffer to be identical, so that a comparisonbetween the two computations allows evaluation of theinfluence of spatial resolution near tightly curved air spacesin the sphenoid sinus and temporal bones. Example imagesof the normalized difference between the magnetic field atresolutions of (4 mm)3 and (2 mm)3 are shown in Fig. 4.

As expected, there are differences in regions near thesphenoid sinus (Figs. 4a,c) and the temporal bones (Figs.4b,c), as well as in the nasal cavity and the neck. In theseregions, the complex 3D geometry of the air spaces is betterdefined with a higher spatial resolution, thus resulting inmore detailed and more precise magnetic field distributionsfor the computation at (2 mm)3. However, it is important tonote that there are no significant differences in the brain.Specifically, the extended region of magnetic field inhomo-geneity observed just superior to the planum sphenoidale(see below, Figs. 5–8) is preserved at both resolutions.Although increased spatial resolution of the model is im-portant near tightly curved air spaces in the nasal cavity,sphenoid sinus, and temporal bones, it does not add signif-icantly to the assessment of the overall magnetic field dis-tribution in the brain.

3.3. Comparison between the head, head/shoulders, andhead/thorax models

Example 2D contour plots of the magnetic field deviation�Bz computed for the head, head/shoulders, head/thorax, andtilted head/shoulders models are shown in Fig. 5. Regions witha high density of contour lines correspond to large magneticfield inhomogeneities. As we are more interested in the inho-mogeneity of the magnetic field rather than its actual value, themagnetic field gradients along each direction �Bz/�x, �Bz/�y,and �Bz/�z were also computed. Example images of �Bz/�zcomputed for all four models are shown in Fig. 6, exampleimages of �Bz/�x and �Bz/�y computed for the head/shoulders

Fig. 3. Results for the sphere model showing images of the error �B [ppm]in the x-z plane passing through the center of the sphere (left) and plots ofthis error along a diameter parallel to the x and the z axes (right) computedusing a buffer thickness of 112 mm (a,b), 128 mm (c,d), and 144 mm (e,f)at the final resolution. The blue regions inside the sphere in (a) indicatelarge differences between the numerical simulation and the analyticalsolution. Note that the results have been cropped for ease of comparison,thus not showing the actual buffer used for the computations, and that onlythe error inside the sphere is shown. Also note that because of the sym-metry, results along the y axis are identical to those along the x axis.

Fig. 4. Mid-sagittal (a), coronal (b), and axial (c) images of the normalized difference between the magnetic field at a resolution of (4 mm)3 and (2 mm)3,i.e., (Bz

4 mm � Bz2 mm)/B0�r, air [ppm], for the head model computed using the same buffer thickness of 48 mm at the final resolution.


model are shown in Fig. 7, and 3D views of all three gradientscomputed for the head/shoulders and tilted head/shouldersmodels are shown in Fig. 8.

Figs. 5 to 8 show that in most regions of the brain,magnetic field changes are moderate (�1 ppm) and varyfairly slowly with location, decreasing from anterior toposterior and from superior to inferior. Noteworthy excep-tions are the regions near the frontal sinus, just superior tothe planum sphenoidale of the sphenoid sinus, just posteriorto the upper clivus, near the temporal bones and air filledmastoids, as well as in the inferior frontal lobes and inferiortemporal lobes. Considerable magnetic field inhomogene-ities are also observed in the nasal cavity, the mouth, and the

neck. Substantial magnetic field variations outside the headare of no consequence for image quality, but may presentproblems when fiduciary markers are placed in these re-gions, e.g., for surgical planning or for registration of mul-timodality data [6].

A comparison between the head model and the head/shoulders model shows that the magnetic field perturbationsnear the cut-off region of the head model (i.e., below theneck) are artifactual (Figs. 5 and 6). It is important to notethat inclusion of the shoulders for the numerical simulationsintroduces substantial magnetic field inhomogeneities in theoccipital lobes, the cerebellum, as well as the neck (Figs.5a–f and 6a–f). The resulting magnetic field gradient �Bz/�z

Fig. 5. Mid-sagittal (left), coronal (middle), and axial (right) coutour plots of the magnetic field deviation �Bz [ppm] for the head (a–c), head/shoulders (d–f),head/thorax (g–i), and tilted head/shoulders (j–l) models. The results for the tilted head/shoulders model have been rotated forwards for ease of comparison.The solid arrow points to the planum sphenoidale and the dashed arrow points to the upper clivus. The contour lines are 0.5 ppm apart.


is 0.21 ppm/cm at the occipito-temporal junction. This maybe attributed to the fact that the air/tissue interfaces at theshoulders are nearly orthogonal to B0. These results clearlyshow the necessity for including more distant anatomy,namely at least the neck and the shoulders, for accuratenumerical simulations of susceptibility-induced magneticfields in the head, and that if a trade-off between spatialresolution and inclusion of a larger anatomic region has to

be made because of computer memory limitations, thistrade-off should be made at the expense of resolution.

A comparison between the head/shoulders model and thehead/thorax model shows that the air/tissue interfaces in thelungs further increase the magnetic field inhomogeneities inthe occipital lobes, the cerebellum, as well as the neck (Figs.5d–i and 6d–i). The resulting magnetic field gradient �Bz/�zincreases from 0.21 to 0.23 ppm/cm at the occipito-temporal

Fig. 6. Mid-sagittal (left), coronal (middle), and axial (right) sections of the 3D magnetic field gradient �Bz/�z (z � inferior-superior direction) [ppm/cm] forthe head (a–c), head/shoulders (d–f), head/thorax (g–i), and tilted head/shoulders (j–l) models. The results for the head/shoulders, head/thorax, and tiltedhead/shoulders models have been cropped and those for the tilted head/shoulders model have been rotated forwards for ease of comparison. The planes shownare the same as those shown in Fig. 5. The solid arrow points to the planum sphenoidale and the dashed arrow points to the upper clivus. This gradient alongthe z direction is a major source of susceptibility artifacts in 2D axial slices.


junction. These effects appear to be substantially less pro-nounced than those due to the air/tissue interfaces at theshoulders. One reason for this may be that the air/tissueinterfaces in the lungs are curved, whereas those at the

shoulders are nearly orthogonal to B0. Another reason maybe that the lungs are further away from the brain than theshoulders. It should be noted that since the CT data used forour numerical simulations were acquired over multiple re-

Fig. 7. Mid-sagittal (left), coronal (middle), and axial (right) sections of the 3D magnetic field gradients �Bz/�x (a–c, x � lateral direction) and �Bz/�y (d–f,y � anteroposterior direction) [ppm/cm] for the head/shoulders model. The results have been cropped for ease of comparison. The planes shown are the sameas those shown in Figs. 5 and 6. Note that in the occipital lobes and cerebellum, these two gradients are smaller than the gradient along the z direction (Figs.6d–f), indicating the advantage of sagittal and coronal slice orientations over axial slices for 2D imaging.

Fig. 8. Three-dimensional views of the magnetic field gradients �Bz/�x (left, x � lateral direction), �Bz/�y (middle, y � anteroposterior direction), and �Bz/�z(right, z � inferior-superior direction) [ppm/cm] for the head/shoulders (a–c) and tilted head/shoulders (d–f) models. The results for both models have beencropped and those for the tilted head/shoulders model have been rotated forwards for ease of comparison. The cut planes shown are the same as those shownin Figs. 5 to 7. The solid arrow points to the planum sphenoidale and the dashed arrow points to the upper clivus.


spiratory cycles, and thus represent a static average of thechest position, our results do not yet allow drawing anyconclusions regarding respiratory effects in fMRI.

3.4. Susceptibility artifacts reduction with a head tilt

A comparison between the head/shoulders model and thetilted head/shoulders model (Figs. 5d–f, j–l, 6d–f, j–l, and 8)shows that tilting the head backwards significantly reducesthe magnetic field inhomogeneity in the region superior tothe planum sphenoidale. This may be explained by the factthat when the head is upright, the air/tissue interface at theplanum sphenoidale is almost orthogonal to B0 , thus result-ing in strong magnetic field gradients, whereas when thehead is tilted backwards (by 35° in our case), the anglebetween this air/tissue interface and B0 is reduced, resultingin weaker magnetic field gradients. Our numerical simula-tions thus confirm the experimental findings by Herberleinet al. [11]. In addition, it should be noted that when the headis tilted backwards, the magnetic field inhomogeneity in-creases in the region posterior to the upper clivus, slightlydecreases near the frontal sinus, and does not change aroundthe temporal bones. More importantly, tilting the head back-wards significantly reduces the magnetic field inhomogene-ities in the occipital lobes and the cerebellum. The resultingmagnetic field gradient �Bz/�z decreases from 0.21 to 0.09ppm/cm at the occipito-temporal junction. This improve-ment may be explained by the fact that the position of thetilted head is changed relative to the shoulders, such that theoccipital lobes and the cerebellum are less affected by thesusceptibility effects induced by the air/tissue interfaces atthe shoulders.

4. Discussion

Previous studies used finite difference [4–6,9], finiteelement [7,8], and perturbation methods [10] to compute 2Dand 3D susceptibility-induced magnetic field distributions.Several models were created to define the spatial distribu-tion of air and tissues, including two-component (air andwater) wire frame models coarsely approximating air spacesin the head [7] and upper body [8]; two- or multi-componentmodels generated by segmentation of MR images of thehead [6,9,10], the chest [6], and the hand [6]; and a multi-component model of the head and shoulders generated fromthe Visible Human data [9].

Unlike in previous studies, we used CT images to gen-erate multi-component models of the head and upper body,and a finite element algorithm to compute the 3D suscepti-bility-induced magnetic field distributions. CT images canbe easily segmented to generate magnetic permeability dis-tributions for air, fat, nonfat soft tissue, and bone. As such,they offer a major advantage over MR images, from whichair and bone cannot be differentiated. Our numerical simu-lations showed pronounced magnetic field inhomogeneities

in regions where air and bone are in close proximity andhave complex spatial distributions, such as near the frontaland sphenoid sinuses, the temporal bones and mastoids, aswell as in the nasal cavity and the neck. Since the mostpronounced susceptibility differences occur at air/tissue in-terfaces, unambiguous delineation of the air spaces is ofparamount importance for computations of susceptibility-induced magnetic fields in these regions.

On the other hand, CT images are less sensitive than MRimages for distinguishing various soft tissues. Susceptibilitydifferences between soft tissues are more subtle, and unlikesusceptibility differences at air/tissue interfaces, are notknown to lead to pronounced image artifacts. However, theyare very important for generating image contrast, and com-puting magnetic field changes due to susceptibility differ-ences between different tissues may be of interest for abetter understanding of contrast mechanisms. Using theVisible Human data as input for numerical simulations ofsusceptibility-induced magnetic fields has the advantagethat air and bone, as well as different soft tissues, can beclearly delineated. On the other hand, these data are uniqueand provide geometric information only for a single subject.While the numerical simulations presented in this paper areonly for a single subject also, it is feasible to use CT datathat are acquired for routine diagnostic examinations, aslong as a sufficiently large anatomic region is covered, sincethe resolution and CT image quality requirements are notparticularly stringent for this application. Such data couldbe used to compute susceptibility-induced magnetic fielddistributions for a range of head geometries, and serve as abasis for the development and testing of susceptibility arti-fact correction methods.

The work presented in this paper allows to draw conclu-sions about two approaches for reduction of susceptibilityartifacts, namely slice orientation and subject positioning.First, in 2D imaging, the slice thickness is usually largerthan the in-plane resolution, so that susceptibility artifactsare most pronounced in the slice select direction. Thusevaluation of the magnetic field gradients along all threedirections for a range of head geometries allows determin-ing an optimal acquisition plane for a given region of thebrain. For example, for the head/shoulders model, the gra-dient along the lateral direction in the region superior to theplanum sphenoidale (Figs. 7a and 8a) is much smaller thanthe gradients along the other two directions (Figs. 6d, 7d,and 8b,c). Therefore the optimal acquisition plane for thisregion of the brain is a sagittal plane. Similarly, the gradi-ents along the lateral and anteroposterior directions in theoccipital lobes and cerebellum (Fig. 7) are smaller than thegradient along the superior-inferior direction (Figs. 6d–f),making sagittal or coronal slices preferable to axial slices.

Second, our study confirmed the experimental finding byHerberlein et al. [11] that tilting the head backwards cansignificantly reduce the magnetic field inhomogeneity in theregion superior to the planum sphenoidale (Figs. 5j, 6j, and8f). However, the observed improvement may be highly


dependent on the subject’s air space anatomy. For examplein this study, since the air/tissue interface extended to themost posterior recess of the sphenoid sinus, improvement inthe region superior to the planum sphenoidale appeared tobe associated with larger magnetic field inhomogeneitiesadjacent to the upper clivus. In addition, our study alsoshowed that tilting the head backwards can significantlyreduce the magnetic field inhomogeneities in the occipitallobes and the cerebellum (Figs. 5j,k and 6j,k).

The long-term objective of our work is to use numeri-cally computed magnetic field distributions as a tool for thefundamental assessment and optimization of various sus-ceptibility artifact correction methods. The two examplesdescribed above show that numerical simulations are usefulin this context, because they are not affected by othersources of artifacts such as RF coil inhomogeneity prevalentat ultra-high field strength, signal loss due to completeintravoxel dephasing in regions with very high magneticfield inhomogeneity, or subject physiological motion, andthus allow susceptibility effects to be studied in isolation. Inthe future, we plan to use our numerical simulations toassess active [23] and passive shimming [19–21], as well asthe design of tailored RF pulses [14,15] for susceptibilityartifact reduction. Numerical simulations may also be usefulfor the assessment of post-processing [16–18] and gradientcompensation techniques [12,13], in particular a priori de-termination of an optimized k-space stepping scheme formost time efficient gradient compensation techniques[24,25].

We expect that numerical simulations will be especiallyuseful for the assessment of active and passive shimming,because they allow shim coil configurations and currents, aswell as passive ferroshim configurations, to be arbitrarilyselected and optimized. Similarly, numerical simulationsallow easy evaluation of a variety of shapes and materialsfor extending the passive diamagnetic shimming methodproposed by Wilson et al. [21]. Direct experimental mea-surement of the static magnetic field would be compara-tively laborious and uncomfortable for subjects. We expectthat the current accuracy of our numerical simulations issufficient, despite the restrictions in spatial resolution andbuffer size imposed by computer memory limitations. An-other limitation with numerical simulations is the lack ofperfect match between the model generated from one sub-ject and the anatomy of another subject. This limitation maybe overcome by carrying out numerical simulations basedon CT data obtained from routine diagnostic examinationsin order to create a database of parameters suitable for arange of head geometries. While some susceptibility artifactcorrection methods, especially post-processing techniques,will likely require direct acquisition of static magnetic fieldmaps for the subject to be imaged, numerical simulationsare nevertheless useful, because they offer more flexibilityin the assignment of tissue properties and are not contami-nated by other image artifacts.

Last but not least, numerical simulations based on more

realistic anatomic models than the simple geometric approx-imations used in prior studies [26,27] may allow gaining abetter understanding of respiratory artifacts in fMRI. Sev-eral authors have suggested that such artifacts are caused bybulk susceptibility changes during respiration due to move-ment of the chest and the diaphragm and/or variations in theoxygen concentration. While there is clear evidence thatfMRI studies are contaminated by respiratory artifacts, thereare still some deviations between experimental findings [28]and simple models [26]. Numerical simulations based onmore realistic tissue distributions during inhalation and ex-halation would allow a more detailed assessment, and sincechanging the susceptibility value for different oxygenationlevels in the lungs is trivial with numerical simulations, itwould be possible to better distinguish effects due tochanges in shape due to movement of the chest and dia-phragm from effects due to changes in oxygenation. How-ever, it should be noted that respiratory-induced magneticfield fluctuations measured experimentally are on the orderof 0.01 ppm [28], whereas the accuracy of our computationsis currently only about 0.03 ppm due to computer memorylimitations. Thus an improved implementation of the algo-rithm, allowing use of a larger buffer size, increased spatialresolution, and inclusion of yet larger anatomic regions, ismandatory for this application. We are currently evaluatingthe possibility to parallelize the algorithm in order to avoidthe memory limitations with the current implementation.

Acknowledgments

We would like to thank Mr. S. Bair for his help withMatlab questions and Mr. T. S. Ibrahim for helpful com-ments on the manuscript. This work was supported by theDepartment of Radiology of The Ohio State UniversityMedical Center. The computations were carried out at theOhio Supercomputer Center.

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