Patient-to-Patient Variation of Susceptibility-Induced B₀ Field in Bilateral Breast MRI

Abstract

Purpose

To evaluate inter-subject variability of susceptibility-induced static field inhomogeneity in breast and to assess effectiveness of whole-body higher order shimming applied to bilateral breast.

Materials and Methods

A fast, computationally efficient method to calculate susceptibility-induced static field from anatomical images was developed. The method was validated against the conventional multi-echo B₀ mapping method and was used to generate data for linear and higher order shim simulation on thirteen volunteers.

Results

Most volunteers showed a significant anterior-posterior B₀ gradient. The majority of the subjects also exhibited a statistically significant left-right gradient. The second and third order shimming provided only minor (<5% each) improvements in B₀ homogeneity.

Conclusion

The shape of air-tissue boundary determines most of the observed B₀ distribution in bilateral breast. Despite significant variability among subjects, a common feature traceable to generic anatomy exists in the linear gradient. Nonlinear variation of susceptibility-induced B₀ field occurs over relatively short length scale and is likely best shimmed by slice-dependent or localized shimming. Keywords: B₀ mapping, susceptibility, breast shimming

INTRODUCTION

Static field (B₀) shimming is important to obtain high quality diagnostic imaging of breast at 3 T. Its impact on image quality is particularly strong in diffusion weighted imaging (DWI), which is more and more commonly added to standard breast protocols to improve specificity of breast cancer detection. The long echo-planar imaging (EPI) readout characteristic of this sequence can lead to significant image distortion; moreover, the quality of fat suppression can suffer in the presence of significant B₀ inhomogeneity. Both of these artifacts can undermine the clinical utility of diffusion weighted images.

Shimming is typically achieved by obtaining B₀ maps in vivo in a small number of slices, and performing an algorithmic search for the shim coefficients that minimize the standard deviation of B₀ in a predetermined region of interest (ROI). Several factors affect the quality of shimming, including B₀ mapping accuracy, conformity of B₀ variation and the available shim fields, selection of ROI, and efficiency of the search algorithm. Knowledge of common features in breast B₀ distribution and variability in population can help improve the quality of shimming by guiding shim strategies at both the hardware and the software level. For example, inclusion of higher order shim coils (1), breast-dedicated local shim coil design (2), and strategies for ROI selection and algorithmic search initialization can all benefit from reliable population studies on breast B₀ distribution.

A limited number of studies aimed at understanding the B₀ distribution in the breast region were previously published. In one of such studies (1), it was shown that the susceptibility-induced B₀ gradient in breast runs predominantly in the anterior-posterior (A-P) direction. The study provided quantitative analysis of the extent of such gradient in a single breast, while limited data were shown on the gradients in other directions (left-right, superior-inferior) in bilateral breast imaging. The authors also provided comparative analysis of shimming strategies in a single axial or sagittal slice in a limited number (seven) of subjects.

The previous work on breast B₀ distribution and shimming (1) is extended in this work. We first introduce and experimentally validate a method to rapidly calculate high resolution breast B₀ maps from 3D anatomical images of the upper torso. The calculation is based on summation of dipolar fields from a large number of susceptibility voxels surrounding the region of interest performed in the Fourier domain in a slice-by-slice manner. This method was chosen over experimentally acquired B₀ maps for subsequent shim simulation due to its acquisition speed, applicability to any breast composition (water, fat, silicone), and its ability to generate conclusions about inherent, body susceptibility-induced B₀ gradients that are independent of scanner configuration (table-top breast coil or in-table breast coil), magnet inhomogeneity, or motion artifacts.

Shim simulation performed on bilateral B₀ maps of thirteen volunteers confirmed strong A-P gradient due to the breast shape. In addition we found a statistically significant left-right (L-R) gradient likely traceable to asymmetric placement of heart and lungs in the upper torso. Predictions were then made about the improvements in 3D breast volume shimming achievable with higher order shims.

MATERIALS AND METHODS

Theory

The method to rapidly convert a 3D susceptibility map into a 3D B₀ map was known for years (3,4). Such methods were successfully applied to calculation of susceptibility-induced B₀ inhomogeneity in brain (5) and in a single breast (6,7). One undesirable feature of the method when it is applied to high-resolution body B₀ mapping is its memory requirement for handling large matrices. Here we introduce a slice-by-slice Fourier domain method, which uses much less memory and still retains high computational speed (< 5 seconds per slice) adequate for our study.

Suppose that the main magnetic field is along the z direction. We will denote by B₀ the total position-dependent static magnetic field in a subject, and by B_d the part of the static field that is produced by induced magnetization (dipole) in the subject body due to finite tissue susceptibility. The z components of these magnetic fields are denoted as B₀ and B_d, respectively. Our interest is to calculate B_d on a two-dimensional lattice, or a target slice, contributed by another lattice of magnetized voxels, or a source slice. For simplicity let us consider axial slices only.

Suppose that the target and source slices are at axial positions z and z′, respectively, and h ≡ z − z′. The magnetic moment of a single voxel in the source slice is given by

$$p_z={vM}_z=v\chi H_z=v\chi \left(\frac{B_0}{{\mu }_0}-M_z\right)\approx v\chi \frac{B_0}{{\mu }_0}$$ [1]

where v is the volume of the voxel, M_z is the magnetization, χ is the susceptibility, and in the last approximation we used the fact that for a weakly magnetic material (χ ≪ 1) the applied magnetic field satisfies H_z ≈ B₀/μ₀ with an error χ.

At an arbitrary point (x,y,z) in the target slice, the dipolar magnetic field from a single source voxel at (x′,y′,z′) is

$$B_d(x,y,z;x^{\prime },y^{\prime },z^{\prime })={\mu }_0{p}_z(x^{\prime },y^{\prime },z^{\prime })·\left\{\frac{1}{4\pi }·\frac{2h^2-{(x-x^{\prime })}^2-{(y-y^{\prime })}^2}{{\left[h^2+{(x-x^{\prime })}^2+{(y-y^{\prime })}^2\right]}^{5/2}}\right\}.$$ [2]

The quantity in the braces, which we denote as K, has the dimension of 1/v and depends only on the relative coordinates of the two voxels. After substituting Eq. [1] to Eq. [2] and summing over the source voxels, we obtain the total off-resonance field contributed from one slice,

$$B_d(x,y,z;z^{\prime })=\sum _{x^{\prime },y^{\prime }}{B}_{0}vK(x-x^{\prime },y-y^{\prime },h)\chi (x^{\prime },y^{\prime },z^{\prime }).$$ [3]

A continuous representation of Eq. [3] is

$$B_d(x,y,z;z^{\prime })=B_0\mathrm{\Delta }{z}^{\prime }\iint {dx}^{\prime }{dy}^{\prime }K(x-x^{\prime },y-y^{\prime },h)\chi (x^{\prime },y^{\prime },z^{\prime })$$ [4]

where Δz′ is the thickness of the source slice and the double integral covers the entire transverse plane, with the understanding that χ is defined to be zero outside the source slice’s field-of-view. The convolution theorem applied to the first two (x, y) dimensions states that

$${\stackrel{\sim }{B}}_d(k_x,k_y,z;z^{\prime })=B_0\mathrm{\Delta }{z}^{\prime }\stackrel{\sim }{K}(k_x,k_y,h)\stackrel{\sim }{\chi }(k_x,k_y,z^{\prime })$$ [5]

where ~ indicates Fourier transformation applied to the arguments denoted with the letter k. The convolution core K in the k_x, k_y -space turns out to have a simple expression

$$\stackrel{\sim }{K}(k_x,k_y,h)=-\frac{2}{3}\delta (h)+\frac{1}{2}{ke}^{-k\mid h\mid },k\equiv \sqrt{k_x^2+k_y^2},$$ [6]

where δ(h) is the Dirac delta function with argument h. This can be verified by noting that the Fourier transform of Eq. [6] with respect to the third argument (h) equals, up to a constant scaling factor, the convolution core in the 3D Fourier method (see Eqs. 2,5,6 of (4)). The delta function term accounts for exclusion of the dipolar self-field (Lorentz sphere) in B_d calculation.

When the formalism represented by Eqs. [4–6] is applied to data on a discrete grid, the singularity at h=0 in Eq. [6] is removed by replacing Eq. [6] by its average over a finite thickness of the slice,

$$\stackrel{\sim }{K}(k_x,k_y,h=0)\equiv -\frac{2}{3\mathrm{\Delta }}+\frac{1}{\mathrm{\Delta }}\left(1-e^{-k\mathrm{\Delta }/2}\right)$$ [7]

where Δ is the thickness of the source (and target) slice at h=0.

To summarize, the procedure for slice-by-slice B_d calculation in the Fourier domain is the following.

• (Step 1)

  Identify a rectangular susceptibility matrix representing magnetized soft tissue. This can be done by tissue segmentation on a 2D anatomical MR image and assignment of a literature susceptibility value to each segment. Enlarge the lattice in both directions by zero susceptibility padding (buffer area) in order to prevent aliasing (4). The size of the buffer area is discussed later. In order to take advantage of the fast Fourier transform (FFT), select each linear dimension of the lattice as a power of 2.

• (Step 2)

  If the target slice is different from the source slice, use Eqs. [5–6] to calculate B_d on the target slice. This is done by first taking FFT of the susceptibility matrix, multiplying it by a convolution core matrix corresponding to Eq. [6], and taking an inverse FFT. The result is then multiplied by a constant factor accounting for the source slice thickness according to Eq. [5]. This produces a B_d map on a lattice of the same size as the source slice lattice.

• (Step 3)

  If the target slice is the same as the source slice, do the same as (Step 2) in which Eq. [7] replaces Eq. [6].

• (Step 4)

  Repeat (Step 1)–(Step 3) for all the source slices and add the resulting B_d to get the final B_d map on a given target slice. The source slices do not need to have the same thickness. Also, there is no need to add a buffer volume in the z direction, since dipolar fields are summed directly in that dimension and therefore there is no Fourier artifact.

The computation time to carry out this procedure is dominated by the Fourier transformation in Steps 2 and 3. If the susceptibility matrix has a dimension of N_x×N_y×N_z, and the number of target slices is N_t, the required time is proportional to N_xN_y ln(N_xN_y) ×N_tN_z. This compares with a full three-dimensional Fourier calculation (3,4) where the computation time scales as N_xN_yN_z ln(N_xN_yN_z), and direct summation of dipolar fields (3) for which the time scales as N_x²N_y²N_zN_t. We note that when N_t = 1, the 2D method is faster than the 3D method, which is independent of N_t. In practice, we found that the 2D method is comparable in speed to the 3D method up to N_t ≈3, and for a larger N_t it gets relatively slower roughly in proportion as N_t.

Figure 1(a) illustrates slice-by-slice B_d calculation described above with a representative tissue-segmented anatomical image (axial slice of an upper torso including lungs and both breasts). The procedure for B_d calculation was validated on a uniformly magnetized sphere. Figure 1(b) shows that, apart from a few pixels on the spherical boundary, the proposed method produces a B_d profile that is in excellent agreement with the analytical one.

Figure 1.

Illustration of slice-by-slice calculation of susceptibility-induced static field inhomogeneity. Shaded image on the right slice corresponds to the dipolar field map generated by the anatomy on the left slice with susceptibility segmentation (air, lung, tissue, in grey scale). (b) Susceptibility-induced static field profile in and around a uniform diamagnetic sphere, χ = −9 ppm, placed in a main magnetic field of 3 T. A 10 cm-diameter sphere was modeled on a 512 (x) × 512 (y) × 80 (z) grid with 1.5 mm isotropic resolution. Fourier-calculated (dots) and analytical (solid) B_d profiles are shown along a line containing a transverse diameter of the sphere. No adjustable parameter was used.

Human Body Model Simulation

A female human body model with 3 mm isotropic resolution was obtained by remeshing the “Visible Woman” human body model (based on National Library of Medicine’s Visible Human Project) (8) provided as part of a commercial electromagnetic simulation package, xFDTD (Remcom, USA, State College, PA). The model was used to demonstrate the proposed B_d calculation algorithm and investigate relative contributions to the breast B_d map from different regions of the body. Each voxel in the model was assigned a susceptibility value according to the following scheme: air (0 ppm), lung (−2.25 ppm), tissue (−9.0 ppm). This segmentation was also used for in vivo B_d calculation. Jordan et al (7) have shown that simple dichotomic segmentation of an anatomical image, namely into air/lung and tissue, produced B_d maps in a single breast in good agreement with experimentally determined maps. We verified that applying more complicated segmentation scheme (1) did not significantly change the conclusions of this paper. Especially, adding the effect of paramagnetic oxygen (0.4 ppm) in air and lung, and assigning larger diamagnetism to bone (−11.3 ppm) did not appear to significantly change the calculated B_d in the human body model study. The present segmentation choice was therefore made as a balance between simplicity and accuracy of the method.

The original body model fit to a matrix of size N_x×N_y×N_z = 183×100×618. From this a susceptibility matrix of size 512×512×618 was constructed by adding zero susceptibility voxels in the two lateral dimensions. In our method, buffer volume in the axial direction is not necessary.

The human body model was used to identify the axial range of the body making significant contribution to the breast B_d map. For this, the B_d map on an axial slice in the middle of the breast was calculated source-slice by source-slice, covering all 618 susceptibility slices. The maximum B_d contribution from each source slice was then plotted against the slice index. Due to the finite length of an MRI magnet, not all body parts are subject to the same static magnetic field, and tissue magnetization gets weaker the farther it is away from the magnet’s isocenter. This was taken into account. When calculating B_d, the contribution from slices away from the target breast slice was reduced according to the static magnetic field profile of a representative clinical 3 T whole-body magnet (provided by the manufacturer, Fig 2a inset).

Figure 2.

(a) Contribution to the breast B_d field from different slices in the human body model (Visible Woman). Inset is the main magnetic field profile used in the calculation. (b) B_d map at slice 448, contributed from the whole body (left), from the middle slab (center), and their difference (left – center, right). The middle slab is defined by the arrow span in (a). (c) Convergence of B_d calculation as matrix size increases.

In Vivo Anatomical Imaging And B₀ Field Mapping

Thirteen healthy female volunteers (height range: 157–188 cm, weight range: 54–86 kg), one of whom (volunteer 5) had silicone breast implants, were enrolled in this study. Informed consent was obtained in accordance with the guidelines of the Institutional Review Board of the authors’ institution.

Each volunteer underwent (1) a 3D anatomical scan covering the upper torso approximately between the neck and the navel, and (2) a 3D B₀ field map scan covering the breast area. A 3 T, whole-body scanner (Discovery MR750 3.0T, GE Healthcare, Waukesha, WI) was used for all scans, using a transmit/receive body RF coil. The two scan protocols were applied to each volunteer in a single exam without table movement. The parameters for the anatomical scan were: 3D spoiled gradient echo (SPGR), TR/TE/tip/bandwidth/locations/slice thickness/FOV = 1.43 ms/0.55 ms/10°/125 kHz/86/6 mm/40 cm. All volunteers completed the anatomical scan in a single breath-hold session lasting 17 seconds. The anatomical image was segmented into tissue, lung and air using a region-growing algorithm. Slice-by-slice Fourier calculation of B_d map was performed on nine axial slices covering both breasts.

Experimental B₀ field map scan was performed as a means to verify the computed dipolar field map. The sequence, 3D IDEAL, was based on the three-point Dixon method which allowed B₀ map reconstruction in the presence of two different chemical species (fat, water) through iterative decomposition of fat, water, and the field map (9). The imaging parameters were: TR/TE/tip/bandwidth/locations/slice thickness/FOV = 4.32 ms/0.992, 1.784, 2.576 ms/10°/125 kHz/32/6 mm/40 cm. Out of the 32 slice B₀ maps acquired, nine slices that best matched (within 3 mm) the slice locations of the anatomy-based B_d maps were chosen for comparison. The imaging time for the field map scan was 53 seconds. All volunteers were scanned while breathing freely, with no respiratory gating. The scanner’s automatic linear shim settings were recorded and later subtracted from the acquired in-vivo B₀ maps. This procedure was verified to generate a uniform 3D B₀ map on a homogeneous spherical phantom with 30 cm diameter.

The outcome of IDEAL decomposition was found to be sensitive to the assignment of fat-water chemical shift frequency. We found that for many subjects manual adjustment of this parameter on the order of 20 Hz was necessary to produce B₀ maps varying smoothly across the fat-water boundary. For the volunteer with breast implants, the presence of three different chemical species makes unambiguous determination of a B₀ map from a three-point Dixon method impossible. In practice, we chose to manually adjust the chemical shift parameter in the reconstruction algorithm so as to minimize artificial B₀ discontinuity between water and silicone.

Shim Simulation

Nine axial, bilateral breast B_d maps were used for three-dimensional linear and higher-order shim simulations for each volunteer. Shimming was done over a rectangular volume of interest encompassing both breasts, but excluding the heart. The first order (linear) shim values were determined by subtracting a linear gradient field from the original B_d map until the standard deviation of the residual field is minimized. A simplex search algorithm (fminsearch) provided by Matlab (MathWorks, MA) was used for optimization. The second order shims were subsequently determined by fitting the linear-shimmed residual field map with a combination of eight (five second-order gradients in addition to linear gradients) field profiles. The third order shims were similarly determined by fitting the residual, second-order shimmed field map with a combination of all fifteen field maps accounting for field variation up to the third order (10).

Statistical Data Analysis

Minitab 12 (State College, PA) was used to perform all statistical data analysis included in this paper. This analysis included t-tests to validate the statistical significance of shim gradients in a population being significantly different from zero. The correlations between body type parameters and the gradients needed for shimming the breast area, as well as between gradients along different axes were examined. Body type parameters considered in the statistical analysis included height, weight and breast type (with three discrete levels assigned for mostly fatty tissue, mostly glandular, and equal mix of fatty and glandular tissue).

RESULTS

Human Body Model

The results of the human body model study are shown in Fig 2. Figure 2(a) shows the contributions to the breast B_d field from different slices. The tissue grey scale coding in the anatomical image in the lower panel is for an aid to the eye and does not represent the actual detail of susceptibility segmentation (only air, lung and soft tissue were considered in our model). The inset in the upper left corner indicates the main magnetic field profile used to weight the contributions from distant source slices.

The plot in Fig 2(a) shows that by far the largest contribution to the susceptibility-induced static field in breast comes from the immediate neighborhood of the target breast slice. It should be noted that due to the vector nature and the characteristic spatial distribution of the dipolar magnetic field, B_d contributions from distant slices tend to have the same sign and add up, whereas those from nearby slices can cancel each other; this is not obvious from the absolute B_d plot of Fig. 2(a). We found that when the slice-by-slice B_d field maps are added, contributions from the source slices located within about ±25 cm from the center of the breast in the axial direction accounts for more than 90% of the total B_d in breast. Such range is indicated as an arrow span in Fig. 2(a). Figure 2(b) compares the B_d map from the whole body (left) and that from the arrow span only (center). The difference map on the right shows that the residual B_d field from parts of the body outside the arrow span is less than 10% in the breast region.

The discrete Fourier-based B_d calculation is subject to an aliasing error when not enough zero susceptibility buffer zone is added to the source susceptibility matrix (4). This is illustrated in Fig. 2(c). Here the B_d maps such as shown in Fig. 2(b) (left) were calculated with a different in-plane susceptibility matrix size, as indicated in the table. A larger matrix size means correspondingly larger buffer zone around the same tissue susceptibility map; the physical pixel resolution was unchanged. The resulting B_d maps were then compared in terms of their maximum departure from the largest matrix case (case 5). The plot shows that B_d becomes relatively insensitive to the matrix size for a matrix larger than 512 × 512. In all the B_d maps shown in this paper, we have used an in-plane matrix size of 512 × 512.

In Vivo Validation

The middle column of Fig. 3(a) shows an example of a multi-slice B_d map of bilateral breast calculated from in-vivo anatomical images. The three slices (s1 to s3) correspond to successive axial displacement by 54 mm in the inferior to superior direction. The images in the left column are the IDEAL-measured B₀ field maps on the same subject in corresponding slices. The difference map on the right and the linear profiles in Fig. 3(b) indicate that there is a very good agreement between the B₀ and B_d maps in the breast region. The difference between the two maps can be attributed to several factors: respiration, cardiac motion, susceptibility assignment error, chemical shift assignment error, and susceptibility of distant body parts and clothing.

Figure 3.

(a) Comparison of measured (left column) and dipolar-field-calculated (middle column) static field maps of a volunteer on three (labeled s1, s2, s3) slices. The rightmost column displays the difference (left – middle). (b) Field profiles along the dotted lines shown in (a). (c) B₀ (IDEAL) and B_d (dipole) maps from a volunteer with silicone implants in both breasts. Strong artificial B₀ boundary around fat is visible (arrows). (d) The 80% distribution range of the B₀ (IDEAL), B_d (dipole), and the difference maps in all volunteers studied. Also shown for each volunteer is the mean absolute difference (labeled mean error) between B₀ and B_d.

Potential use of the anatomy-based B_d calculation as a chemical shift-insensitive method to predict B₀ inhomogeneity in breast is highlighted in Fig. 3(c). On the left is shown a three-echo IDEAL-based B₀ map on a volunteer with breast implants, where the chemical shift parameter was adjusted to match the difference between water and silicone. This produces a strong artifact in the B₀ of fat. The dipolar field map on the right, on the other hand, captures the overall, slowly varying field variation without an artifact. A four-echo IDEAL could in principle resolve water, fat, silicone, and the B₀ inhomogeneity (9). Such measurement, however, would be more time consuming and more susceptible to motion and respiration artifacts.

Figure 3(d) summarizes the conformity between the measured (B₀) and calculated (B_d) field maps in all volunteers in terms of the 80% range in the histogram of the field maps. Here a histogram was constructed from field map pixels in nine slices representing three-dimensional bilateral breast. Heart was not included. The 80% range in the histogram was then obtained for the measured (B₀), calculated (B_d), and the difference (B₀−B_d) field maps. For each volunteer, the mean absolute difference between the B₀ and B_d maps was also calculated and is displayed in the bar graph as the fourth bar in each volunteer. As expected, B₀−B_d conformity was the lowest for volunteer 5, with implants. For the rest of the subjects subtracting B_d from B₀ did remove a large share of the field variation. For these subjects the mean error in B₀−B_d was also small and was comparable to the earlier results (7). As in (7) we found somewhat larger spread of field values in the measured (B₀) field map compared to the calculated (B_d) map. This could be attributed to respiratory variation in the acquired B₀ maps (4,11,12) as well as contributions from sources other than tissue susceptibility.

Shim Simulation

In order to quantify the first order spatial variation of susceptibility-induced field in breast, we fitted the nine-slice field maps (B_d) of each volunteer to a three-dimensional linear gradient field. The data shown in Figure 4 indicate that across the volunteer pool, there is a relatively strong negative (decreasing from nipple to chest wall) A-P gradient in breast. The mean and standard deviation of this gradient were −29 Hz/cm and 6.5 Hz/cm respectively. The mean and standard deviation for the other gradients were the following: G_x, −1.0 Hz/cm (mean), 1.1 Hz/cm (stdev); G_z, −1.4 Hz/cm (mean), 2.6 Hz/cm (stdev).

Figure 4.

Linear gradient fits to the calculated breast field maps for all volunteers. The inset defines the direction of the gradients as well as the approximate location of the region of interest where gradient fitting was done.

Figure 5 shows the result of higher order shimming. The plot in Fig. 5(a) indicates the degree to which the field homogeneity was improved by higher order shimming for each volunteer (solid line) and on the average (dotted line). It can be seen that the second and third order shimming provide only minor improvements in the field homogeneity for most of the volunteers. On the average, the reduction in the standard deviation provided by the first order shim was 39 %, whereas the incremental improvements with the second and third order shims were 4.5% and 3.0 %, respectively. The relative ineffectiveness of higher order shimming indicates that static field inhomogeneity in bilateral breast is significantly localized in nature (Fig. 5(c)), and conventional whole body shim coils may do little to remove it when applied to the entire breast volume.

Figure 5.

(a) Static field inhomogeneity as a function of the shim order. Solid lines represent individual data from thirteen volunteers. Dotted line is the average over the volunteers. (b,c) Unshimmed (first row), and residual static field maps after linear (2^nd row), second-order (3^rd row), and third-order (4^th row) shimming. Numbers on the axes indicate pixel count (1 pixel = 3.2 mm). Field maps from two volunteers are shown for whom higher-order shimming made significant difference (b), and no significant difference (c).

Statistical Data Analysis

The t-test for null hypothesis yielded p < 0.01 for both G_x and G_y, indicating that the average gradients needed to shim a population of subjects is significantly different from zero on both these axes. While the strong G_y gradient needed to shim each subject was previously reported (1), we also found that a small, but statistically significant G_x gradient is required to shim most of the subjects. The left-right gradient was such that the left breast (closer to heart) has higher off-resonance field than the right breast. This fact is consistent with a diamagnetic heart contributing positive off-resonance field preferentially to the left breast. Significant scatter in the S-I gradient data made determination of a statistically meaningful anatomy-induced axial gradient in breast inconclusive (p = 0.071).

Correlations between body type parameters and the gradients required to best shim the breast volume were mostly inconclusive. No correlations were found between subject height, weight or breast type and any of the shim gradients (p > 0.05). The single statistically significant correlation found was between G_x and G_y; a Pearson correlation coefficient of 0.8 (p=0.002) indicated a tendency for stronger G_y gradients to be associated with stronger G_x gradients.

DISCUSSION

Whereas actual off-resonance field in body MRI is a combination of fields from a number of sources, analysis of body susceptibility-induced B₀ inhomogeneity alone is important to develop a generic shimming strategy that is valid across population and different scanner environments. Such study also provides data to test effectiveness of high-order body shim coils applied to particular anatomy of interest.

To this end we have developed a computationally efficient method to calculate susceptibility-induced B₀ maps in bilateral breast from segmented anatomical images of the upper torso. The anatomical scan necessary to produce reliable B₀ maps takes only a single breath-hold session, which makes the method insensitive to respiratory artifacts. In our work the image segmentation was done with two-dimensional region growing algorithm which was relatively slow. A number of three-dimensional image segmentation algorithms are available, however, which will substantially expedite computation. It is also worthwhile to note that body image segmentation is currently considered for SAR calculation in ultra-high field MRI (13). If detailed image segmentation needed for such purpose is offered and routinely performed in a clinical scanner, anatomy-based B₀ calculation could be done with little additional computational load to the scanner. Once the image is segmented and assigned susceptibility values, B₀ calculation in our method takes less than five seconds per slice on a standard laptop computer (Intel Core i5 CPU with 2 GB RAM). This can be made even faster by exploiting the fact that in our method the susceptibility data do not have to have a uniform spatial resolution across the volume, and slice thickness for susceptibility slices away from the immediate neighborhood of the target slice can be made larger without significantly impacting the accuracy of the result. Initial numerical study indicates that calculation speed can be doubled with negligible error by choosing coarser slice division away from the breast region and thereby reducing the number of source-target pairs by a factor of two.

The proposed anatomy-based B₀ calculation was verified through comparison with in-vivo B₀ maps on thirteen volunteers. The relatively small variation of the difference field (Fig. 3) confirmed that the main contributor to the observed B₀ distribution in breast is the susceptibility and the shape of organs in and around the breast region. The residual field variation, on the order of a few tens of Hz, was not thoroughly analyzed in the present study. Two potential sources of discrepancy, however, could be considered in more detail: respiration and remote dipolar fields. For the former, Bolan et al (11) have reported respiration-induced breast B₀ variation on the order of 0.2 ppm, or 26 Hz at 3 T (between normal and maximally inhaled or exhaled state). Whereas this value is not inconsistent with the residual field observed here, our own pilot study on a single slice B₀ map has found a much weaker effect, about 5 Hz at 3 T. We suspect that respiration effect has significant subject dependent behavior. Furthermore, it is reasonable to expect that dynamic B₀ variation synchronized with respiratory cycle would be substantially averaged out during a relatively long (53 s) B₀ mapping sequence. On these grounds, we believe that respiration alone is not likely to materially alter the outcome of our investigation, which was based on average behavior of breast B₀ in multiple volunteers.

Another potential source of error is the remote dipolar field from susceptibility voxels outside the 50 cm range of the anatomical image. Although this field was estimated to be an order of magnitude smaller than the torso-contributed field (Fig 2), it could affect higher order shim analysis if there existed strong higher order harmonic contents. We analyzed the harmonic content of the remote dipolar field in the Visible Woman and found that such field varied much more smoothly than the torso-induced field. Specifically, the standard deviation of the remote field decreased from 7.2 Hz to 5.6, 0.7, 0.4 Hz after the 1st, 2nd, and 3rd order shim was applied, respectively. If this additional variation is added to the torso-induced field variation (dotted line in Fig. 5a) in quadrature, the total standard deviation at any shim order increases by less than 1.5 %. Therefore, it is unlikely that omission of the remote field would have qualitatively changed the result of our shim analysis.

Our work suggests that, as in the single slice case reported in (1), A-P gradient is the strongest linear component in a 3D breast B₀ map. The average value of this gradient, about 30 Hz/cm and increasing toward the nipple, could be used as a starting point for automatic shim searches in clinical scans. We found experimentally that such strategy improves the chance of arriving at more accurate shim values in in-vivo breast scans. Although certain volunteers also displayed substantial L-R gradient, the small average value (−1 Hz/cm) of this gradient does not seem to warrant a non-zero starting value for automatic shimming in general.

We found that, after linear shim fields were subtracted, the remaining nonlinear breast B₀ field could not be easily shimmed away using the second or the third order harmonic functions. As a simple explanation, consider bilateral breast shimming in which both breasts exhibit a half-sphere like field profile. Then, an axial B₀ map across each breast would show a parabola-like profile along the L-R direction. Such case was explicitly demonstrated in (1). Shimming both breasts in this case would require a fourth order harmonic function which is not commonly available in clinical scanners. We hypothesize that breast-specific local shim coils providing second order correction centered around each breast could be more efficient than whole body third order shims for bilateral breast shimming.

In conclusion, we have analyzed susceptibility-induced static field variation in bilateral breast at 3 T, and found statistically significant linear field gradients that can be traced to human anatomy. Higher-order shimming over three-dimensional bilateral breast was found to have limited effectiveness, which suggests need for slice-by-slice shimming or local coil approach to address localized inhomogeneity. The proposed method of static field calculation based on anatomical images can be useful for patient-specific B₀ map generation for real-time shimming unaffected by chemical composition of breast. Given high degree of inter-patient variation and localized nature of susceptibility-induced B₀ distribution, slice- or breast-dependent shimming approach in breast based on accurate, patient-specific B₀ maps could significantly enhance quality of shimming and performance of imaging protocols such as diffusion weighted imaging.