Nonlinear Gradient Field Mapping Using a Spherical Grid Phantom for 3 and 7 Tesla MR Imaging Systems Equipped with High-performance Gradient Coils

Abstract

Recent high-performance gradient coils are fabricated mainly at the expense of spatial linearity. In this study, we measured the spatial nonlinearity of the magnetic field generated by the gradient coils of two MRI systems with high-performance gradient coils. The nonlinearity of the gradient fields was measured using 3D gradient echo sequences and a spherical phantom with a built-in lattice structure. The spatial variation of the gradient field was approximated to the 3rd order polynomials. The coefficients of the polynomials were calculated using the steepest descent method. The geometric distortion of the acquired 3D MR images was corrected using the polynomials and compared with the 3D images corrected using the harmonic functions provided by the MRI venders. As a result, it was found that the nonlinearity correction formulae provided by the vendors were insufficient and needed to be verified or corrected using a geometric phantom such as used in this study.

Introduction

The gradient coil is one of the crucial components in the MRI system. The gradient coil is designed to meet many conflicting requirements.^1–4 These requirements include maximum gradient strength, efficiency of gradient strength per unit current, inductance, DC resistance, slew rate, and spatial linearity (the area of the linear region). Vibration, noise, and peripheral nerve stimulation when driving the gradient coil are also important factors in the gradient design.

These requirements are changing over time. In recent years, the gradient strength and slew rate have been particularly demanded, with gradient coils dedicated to the head having a maximum strength of 300–500 mT/m and a slew rate of 200–600 T/m/s.^5,6 Recent gradient coils for whole-body clinical MRI systems have a maximum strength of 70–100 mT/m and a slew rate of 200 T/m/s.^3–5 With this trend, the spatial nonlinearity of the magnetic field generated by the gradient coils is sacrificed and often becomes a problem for many users; as a result, many studies have been reported on this topic.^7–22

Nonlinearity of the magnetic field generated by the gradient coils leads to several problems. One is geometric distortion of MR images,^8–15 and the other is spatial variation of the measured diffusion coefficients.^7,16–22 It has also been pointed out that the nonlinearity of the gradient can cause image blurring in the image reconstruction of non-Caretesian sampling MR signal.²³ Therefore, it is very important to measure the nonlinearity of the gradient to correct image distortion or intensity of the diffusion coefficients.

The nonlinearity of the gradient magnetic field is, conversely, measured using the effects described above. In other words, the methods for measuring nonlinearity can be classified into two categories: methods that use phantoms with defined geometric positions to measure geometric distortions^8–15 and those that measure the diffusion coefficient distribution of phantoms made of homogeneous materials.^7,16–22

The former method, described in Refs. 8–15, has the advantage that the measurement principle is simple and conventional pulse sequences can be used. On the other hand, the hurdle for phantom fabrication is relatively high, and commercially available phantoms are expensive. The advantage of the latter method, described in Refs. 7 and 16–22, is that phantom fabrication is relatively easy and gradient field nonlinearity can be measured directly. On the other hand, the pulse sequences are relatively specific (e.g., diffusion weighted spin-echo echo planar imaging [EPI]) and sensitive to eddy currents, requiring complex image processing.

In this study, we aimed to measure the nonlinearity of gradients in the clinical and research MRI systems with high-end gradient coils, and to verify whether the nonlinearity is adequately corrected. For this purpose, a geometrically accurate phantom with negligible diamagnetic magnetic field even at high magnetic fields was developed by using a high-precision 2D grid and a spherical acrylic container. The advantage of this study over the past studies is the use of the spherical phantom in which the diamagnetic field caused by the presence of the phantom is uniform, which may reduce the imaging and analysis time and the measurement error.

Principle of gradient field measurement

There are two approaches to measure the magnetic field produced by the gradient coil using the MR images of a phantom containing geometrically defined control points.

One is to measure the coordinates of the control points along the phase-encoding direction in the acquired MR images of the phantom, and then calculate the magnetic field strength generated by the gradient coil. This technique takes advantage of the fact that the position in the phase-encoding direction determined by the gradient coil is independent of the static magnetic field inhomogeneity (ΔB₀) and the RF magnetic field inhomogeneity (ΔB₁). Therefore, to measure the gradient magnetic field in three directions (x, y, and z), at least two sets of 3D imaging sequences with different readout directions (x, y, or z) should be used, and the gradient field is measured from the coordinates of the control points along the phase encoding directions in the 3D MR images. In this study, this approach was used for the gradient field measurement on the 7T MRI system.

The second approach uses 3D MR imaging to measure the gradient field from the coordinates of the control points on the MR images in the readout direction as well as in the phase-encoding direction. In this case, because the positional shift caused by ΔB₀ is superimposed in the readout direction, it is necessary to remove the effect of ΔB₀ by the following two methods. One method is to take two sets of 3D images with two pulse sequences, one with positive and the other with negative sign of the readout gradient, and to average or subtract (depending on the definition of the sign of the coordinate) the two measured coordinates on the images in the readout direction. This may be the gold standard method, but it requires twice the measurement and analysis time and has a relatively large error compared to the method described below. Another more efficient method is to make the signal bandwidth per pixel (PBW) much wider than the resonance frequency shift caused by ΔB₀. This method has the advantage of being able to measure the gradient magnetic field in the three directions by the single 3D MR image because the positional shift by ΔB₀ can be neglected. This method was used for the gradient field measurement on the 3T MRI system.

Materials and Methods

Spherical grid phantom

Figure 1(a) shows the spherical phantom developed in this study. The phantom consists of commercially available 2D polystyrene grids (grid spacing = 14.5 mm pitch, thickness = 10.8 mm) and a spherical acrylic container (outer diameter = 170 mm, inner diameter = 163 mm). Ten pieces of the grid were cut to efficiently fill the inside of the spherical container, stacked vertically with 2.4 mm thick silicone rubber O-rings between them, and fixed inside the spherical container. The plastic grids are mass-produced industrial products, and the reproducibility of the grid spacing and thickness is less than 0.1 mm. The grids were carefully stacked using plastic rods (12.0 mm diameter) and the O-rings, and the accuracy of the 3D coordinates of the grid control points is also less than 0.1 mm.

Fig. 1.

(a) The spherical water phantom developed in this study. Outer diameter = 170 mm, inner diameter = 163 mm. (b) Distribution of control points used for calculation of the nonlinearity of the magnetic field gradient. The total number of the control points is 1076 and the half of them (538 points) is plotted in this figure.

The spherical container was then sealed with adhesive and filled with a nickel chloride aqueous solution (approximately 8 mM, T₁ – 200 ms at 3T). In addition, 0.5 g of NaCl per liter was added to suppress dielectric resonance with high-frequency electromagnetic fields at 7T (resonance frequency – 300 MHz). This phantom contained 1764 control points in total. Figure 1(b) shows the distribution of the control points (1076 points) used in the measurements of the spatial coordinates. Figure 2 shows the whole proton spectrum of the spherical phantom measured using a clinical 3T MRI system (Vantage Titan 3T; Canon Medical systems, Otawara, Japan).²⁴ This spectrum clearly shows that the diamagnetic field within the spherical phantom was uniform, susceptibility difference between the grid material and the water was negligible, and the inhomogeneity of the magnetic field within the phantom was less than about 50 Hz, measured at about 3% of the peak height.

Fig. 2.

Proton spectrum of the whole spherical phantom measured using a clinical 3T system. The width of the spectrum near the bottom of the spectrum is about 50 Hz.

MRI systems and pulse sequences

The spherical phantom described in the previous subsection was imaged using two MRI systems. One was a clinical MRI system with a static field strength of 3T and a maximum gradient strength of 80 mT/m (SIGNA, Premier; GE Healthcare, Waukesha, WI, USA) and the other is a research MRI system with a static field strength of 7T and a maximum gradient strength of 70 mT/m (Magnetom 7T; Siemens Healthineers, Germany). The 3T system used a quadrature transmitter coil driven by a two-channel transmitter and a 48-channel head RF receiver coil, and the 7T MRI system used a single-channel transmitter and a 32-channel head RF receiver coil.

A 3D RF spoiled gradient echo sequence (TR = 4.915 ms, TE = 1.59 ms, flip angle = 30°, FOV = 220 mm × 220 mm × 230.4 mm, number of image matrix = 256³, voxel size = 0.86 mm × 0.86 mm × 0.9 mm, and pixel bandwidth = 976.562 Hz) was used for imaging at 3T. An MPRAGE sequence (TR = 1760 ms, TE = 3.2 ms, flip angle = 4°, FOV = (204 mm)³, number of image matrix = 256³, voxel size = (0.8 mm)³, and pixel bandwidth = 230 Hz) was used for imaging at 7T.

In the 3T imaging experiment, only one signal readout direction was sufficient due to the large pixel bandwidth (976.562 Hz). Because the raw image dataset (without distortion correction) was not available at 3T, the image dataset after image distortion correction provided by the vendor software was used for the gradient nonlinearity measurements at 3T. In the 7T imaging experiment, the signal readout direction was set to foot-to-head (FH) and right-to-left (RL) because of the relatively narrow pixel bandwidth (230 Hz) and the high static magnetic field strength. Two image datasets, one without image distortion correction and one with distortion correction, which could be selected by the user, were used for gradient nonlinearity measurements.

Measurements of the 3D coordinates of the control points

To improve the measurement accuracy of the 3D spatial coordinates of the control points, the acquired image datasets with 256³ voxels were interpolated using the zero-filling interpolation (ZIP) technique to obtain image datasets with 512³ voxels. This operation resulted in the voxel size of 0.43 mm × 0.43 mm × 0.45 mm for the image dataset acquired at 3T and (0.398 mm)³ for the image datasets acquired at 7T.

In the image dataset acquired at 3T, because the effect of the magnetic field inhomogeneity on the coordinates of the control points was negligible as shown Fig. 2, the 3D coordinates of the control points were manually measured using the single image dataset using an image display and processing program. In the image datasets acquired at 7T, because the effect of the magnetic field inhomogeneity on the coordinates of the control points was not negligible, the coordinates of the control points were measured using the coordinates in the phase encoding directions on the FH-read and RL-read 3D image datasets as described in the “Principle of gradient field measurement” section. The coordinates of the control points were measured for the image datasets acquired at 7T before and after the image distortion correction performed by the vendor software. The number of control points measured for each of the three image datasets was 1076. Reproducibility of the repeated manual measurements was very good.

Polynomial fitting of the 3D coordinates of the control points

The 3D coordinates $\left(u,v,w\right)$ on the MR images can generally be approximated by the following polynomials using the spatial coordinates $\left(x,y,z\right)$:

$$u\left(x,y,z\right)\approx \sum _{l=0}^L\sum _{m=0}^M\sum _{n=0}^N{a}_{lmn}{x}^l{y}^m{z}^n$$ (1)

$$v\left(x,y,z\right)\approx \sum _{l=0}^L\sum _{m=0}^M\sum _{n=0}^N{b}_{lmn}{x}^l{y}^m{z}^n$$ (2)

$$w\left(x,y,z\right)\approx \sum _{l=0}^L\sum _{m=0}^M\sum _{n=0}^N{c}_{lmn}{x}^l{y}^m{z}^n$$ (3)

where L, M, and N are zero or positive integers and $L+M+N$ is the order of the polynomials. In this study, third- and fifth-order polynomials were used. Furthermore, since $u$ and $x$, $v$ and $y$, and $z$ and $w$ are approximately equal, the following equation was used as the third-order polynomials (see Appendix for the fifth-order polynomials):

$$u\left(x,y,z\right)\approx a_{000}+a_{100}x+a_{200}{x}^2+a_{110}xy+a_{101}xz+a_{300}{x}^3+a_{210}{x}^{2}y+a_{201}{x}^{2}z+a_{120}x{y}^2+a_{111}xyz+a_{102}x{z}^2$$ (4)

$$v\left(x,y,z\right)\approx b_{000}+b_{010}y+b_{110}xy+b_{020}{y}^2+b_{011}yz+b_{030}{y}^3+b_{120}x{y}^2+b_{210}{x}^{2}y+b_{111}xyz+b_{021}{y}^{2}z+b_{012}y{z}^2$$ (5)

$$w\left(x,y,z\right)\approx c_{000}+c_{001}z+c_{101}xz+c_{011}yz+c_{002}{z}^2+c_{201}{x}^{2}z+c_{102}x{z}^2+c_{111}xyz+c_{021}{y}^{2}z+c_{012}y{z}^2+c_{003}{z}^3$$ (6)

The coefficients of these polynomials can be obtained by minimizing the sum of the squares of the errors $E_u$, $E_v$, and $E_w$ at the measured control points as follows:

$$E_u=\sum _{i=1}^P{\left(u_i-u\left(x_i,y_i,z_i\right)\right)}^2$$ (7)

$$E_v=\sum _{i=1}^P{\left(v_i-v\left(x_i,y_i,z_i\right)\right)}^2$$ (8)

$$E_w=\sum _{i=1}^P{\left(w_i-w\left(x_i,y_i,z_i\right)\right)}^2$$ (9)

Here, $\left(u_i,v_i,w_i\right)$ are the measured coordinates of the control points, $\left(x_i,y_i,z_i\right)$ are the designed (theoretical) coordinates of the control points, and P is the number of measured control points (1076). In the expression of the above polynomials, $x$, $y$, and $z$ coordinates may be written as $x-\mathrm{\Delta }x$, $y-\mathrm{\Delta }y$, and $z-\mathrm{\Delta }z$ to include the setting error of the phantom. However, for simplicity and clarity, we omitted these terms but these terms were used in the actual calculation.

In this study, the steepest descent method was used to find the coefficient sets $a_{lmn}$, $b_{lmn}$, and $c_{lmn}$ that minimize the sum of the squares of errors (7)–(9). In the steepest descent method, $E_u$, $E_v$, and $E_w$ were partially differentiated by the coefficients $a_{lmn}$, $b_{lmn}$, and $c_{lmn}$, respectively, and the calculation was repeated by changing each coefficient in the direction of decreasing error to obtain a coefficient set in which all partial differential coefficients were zero. The above operations were performed on the image dataset acquired at 3T and two image datasets acquired at 7T (datasets before and after the vendor-provided distortion correction software).

In the image datasets acquired at 7T before the distortion correction, $u\left(x,y,z\right)$, $v\left(x,y,z\right)$, and $w\left(x,y,z\right)$ are proportional to the magnetic field distribution in the static magnetic field direction (z direction) generated by each gradient coil. Therefore, the spatial distribution of the gradient magnetic fields in the 7T MRI system was obtained by partial derivative of these polynomials in x, y, and z, respectively.

Image distortion correction using the polynomial functions

Once $u\left(x,y,z\right)$, $v\left(x,y,z\right)$, and $w\left(x,y,z\right)$, which represent 3D geometric distortion, are obtained, a distortion-free image can be obtained by calculating the voxel values of the image whose coordinates are $\left(x,y,z\right)$ for each voxel. In practice, this calculation was performed on the ZIP-processed image datasets, and the voxel values were further calculated by linear interpolation to improve the accuracy of the voxel values. Since the voxel values of the acquired images varied from the original values due to the nonlinearity of the gradient magnetic field, the following Jacobian was used to correct the image intensity.

$$\frac{\mathrm{\partial }\left(u,v,w\right)}{\mathrm{\partial }\left(x,y,z\right)}=\left|\begin{array}{ccc}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}& \frac{\mathrm{\partial }u}{\mathrm{\partial }y}& \frac{\mathrm{\partial }u}{\mathrm{\partial }z}\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }x}& \frac{\mathrm{\partial }v}{\mathrm{\partial }y}& \frac{\mathrm{\partial }v}{\mathrm{\partial }z}\\ \frac{\mathrm{\partial }w}{\mathrm{\partial }x}& \frac{\mathrm{\partial }w}{\mathrm{\partial }y}& \frac{\mathrm{\partial }w}{\mathrm{\partial }z}\end{array}\right|$$ (10)

Results

Measurements of control points and polynomial fitting

Figure 3(a) shows a coronal cross-section selected from the 3D image dataset acquired at 3T. The yellow dots are the measured control points. Figure 3(b) and (c) show coronal cross-sections selected from the 3D image datasets acquired at 7T. Figure 3(b) shows the cross-section without distortion correction, and Fig. 3(c) shows that with distortion correction by the vendor software.

Fig. 3.

(a) A coronal cross-section selected from the 3D image dataset acquired at 3T. Distortion correction was performed using the vendor software. In-plane pixel resolution = 0.86 mm × 0.86 mm. The slice thickness = 0.9 mm. (b), (c) Coronal cross-sections selected from the 3D image datasets acquired at 7T. The voxel size = (0.8 mm)³ (nominal). No distortion correction was performed in (b). Distortion correction using the vender software was performed in (c). Image intensity variation caused by the inhomogeneous RF field was observed.

Table 1 shows the coefficients of the polynomiassls obtained from the coordinates of the control points of the image datasets acquired at 3T and 7T by the steepest descent method. The first-order coefficients of $u\left(x,y,z\right)$, $v\left(x,y,z\right)$, and $w\left(x,y,z\right)$ were all very close to 1.0, but the other terms were much smaller and shown in units of 10⁻⁶. In this table, x, y, and z are expressed in mm, and the range of the coordinates of the control points are –72.5 mm ≤ x ≤ 72.5 mm, –64.8 mm ≤ y ≤ 64.8 mm, and –72.5 mm ≤ z ≤ 72.5 mm. Second- and third-order coefficients that contribute significantly to the image distortion are indicated in red. It should be noted that image distortion remains in the image datasets acquired at 3T and 7T even after the distortion correction by the vendor software.

Table 1.

Coefficients of the third-order polynomials used for the image datasets acquired at 3T and 7T

(a) u(x, y, z)
Term	3T	7T (wo correction)	7T (w correction)
x	0.9967	0.9715	0.9731
x2(×10−6)	−4.0220	−47.199	−38.429
xy(×10−6)	3.6558	1.6124	−9.1177
xz(×10−6)	−0.8979	−16.168	0.4155
x3(×10−6)	0.4310	2.6311	1.0352
x2y(×10−6)	−0.5895	−2.9702	−3.0161
x2z(×10−6)	−0.0081	−1.4850	−0.0361
xy2(×10−6)	1.0073	3.9611	0.7369
xyz(×10−6)	0.0197	0.6058	0.0317
xz2(×10−6)	−0.00554	−6.4485	−0.2368
(b) v(x, y, z)
Term	3T	7T (wo correction)	7T (w correction)
y	0.9921	0.9640	0.9653
xy(×10−6)	−50.8640	−1.0369	3.4116
y2(×10−6)	183.5116	140.1424	194.571
yz(×10−6)	57.4647	−102.6217	−86.282
y3(×10−6)	3.3928	5.1600	3.2776
xy2(×10−6)	−1.9491	0.0481	−0.0140
x2y(×10−6)	1.7212	0.6922	0.1675
xyz(×10−6)	0.1923	0.0497	0.0470
y2z(×10−6)	2.3304	−0.9609	−1.0663
yz2(×10−6)	−7.0292	−7.4326	−0.5416
(c) w(x, y, z)
Term	3T	7T (wo correction)	7T (w correction)
z	0.9992	0.9950	0.9891
xz(×10−6)	−0.4177	1.1674	0.4537
yz(×10−6)	19.408	−22.395	37.4323
z2(×10−6)	−33.900	−49.715	−29.858
x2z(×10−6)	0.1685	−1.8791	0.03036
xz2(×10−6)	−0.01084	0.2914	−0.00569
xyz(×10−6)	0.00435	−0.1043	0.0112
y2z(×10−6)	0.58405	2.1230	1.0646
yz2(×10−6)	2.0706	−0.2781	−0.4214
z3(×10−6)	0.00658	−3.3395	0.0628

The fitting was performed for the image datasets with and without the vendor nonlinearity correction software at 7T. (a) Those for G_x. (b) Those for G_y. (c) Those for G_z. Major coefficients except the linear terms are shown in red.

Figure 4(a), (c), and (e) show histograms of the difference between the measured coordinates $\left(u_i,v_i,w_i\right)$ and the theoretical coordinates $\left(x_i,y_i,z_i\right)$ of the control points calculated for the uncorrected image datasets acquired at 7T. Figure 4(b), (d), and (f) show histograms of the difference between the measured coordinates $\left(u_i,v_i,w_i\right)$ and the coordinates $\left(u\left[x_i,y_i,z_i\right],v\left[x_i,y_i,z_i\right],w\left[x_i,y_i,z_i\right]\right)$ calculated from the third-order polynomials calculated for the uncorrected image datasets acquired at 7T. Theoretical coordinates $\left(x_i,y_i,z_i\right)$ of the control points are actually linear approximations of the measured coordinates to compensate for deviations of the origin of the phantom and the efficiency of the gradient coil. Thus, the difference between the left and right histograms represents the difference of the coordinates due to the second- and third-order terms of the respective polynomials. The standard deviation of the histogram changed from 0.536 to 0.281 mm for G_x, from 0.51 to 0.238 mm for G_y, and from 0.437 to 0.264 mm for G_z, respectively. When the fifth-order polynomials shown in Appendix were used for fitting, the standard deviation of the histogram further decreased from 0.281 to 0.250 mm for G_x, from 0.238 to 0.215 mm for G_y, and from 0.264 to 0.249 mm for G_z, respectively.

Fig. 4.

Histograms of the differences between the measured coordinates and the coordinates calculated using the polynomials. The left histograms show those calculated using the first-order polynomial (linear) fitting and the right histograms show those calculated using the third-order polynomial fitting. σshows the standard deviation of the histograms.

Table 2 shows the results of the same calculation using third-order polynomials for the image datasets acquired at 3T and 7T. For all image datasets, the standard deviations of the histograms were smaller when the third-order polynomial fitting was used than when the linear fitting was used. This result suggests that the vendor’s software was not sufficient to correct for image distortion at both 3T and 7T.

Table 2.

SD of the histograms of the differences between the measured coordinates and the calculated coordinates of the control points

(a) 3T with vendor software correction
Gradient component	SD (linear fitting) (mm)	SD (3rd order fitting) (mm)
Gx	0.1944	0.1887
Gy	0.5527	0.3737
Gz	0.2526	0.2236
(b) 7T without vendor software correction
Gradient component	SD (linear fitting) (mm)	SD (3rd order fitting) (mm)
Gx	0.5364	0.2814
Gy	0.5161	0.2383
Gz	0.4373	0.2640
(c) 7T with vendor software correction
Gradient component	SD (linear fitting) (mm)	SD (3rd order fitting) (mm)
Gx	0.2967	0.2461
Gy	0.2968	0.2390
Gz	0.2561	0.2059

The central column shows the SD for the linear fitting and the right column shows the SD for the third-order polynomial fitting. (a) For the image dataset acquired at 3T. (b) For the image dataset acquired at 7T without vendor-software distortion correction. (c) For the image dataset acquired at 7T with vendor-software distortion correction. The SD more than 0.3 mm is shown in red. SD, standard deviation.

Gradient nonlinearity map

Figure 5 shows the spatial distribution of the squared relative strength of the gradient magnetic fields obtained by partial derivatives of the third-order polynomials that approximate the gradient fields at 7T. As can be seen, a gradient field distribution similar to that reported in the previous study was obtained.⁷

Fig. 5.

(a) Relative intensity map of the square of G_x in the central sphere (diameter = 163 mm) shown in the axial planes calculated using the third-order polynomial for the gradient field. (b) Relative intensity map of the square of G_y in the central sphere shown in the axial planes calculated using the third-order polynomial for the gradient field. (c) Relative intensity map of the square of G_z in the central sphere shown in the axial planes calculated using the third-order polynomial for the gradient field.

Distortion correction

Figure 6(a) shows the central sagittal cross-section selected from the 3D image dataset acquired at 3T. Figure 6(b) shows the same cross-section corrected by the third-order polynomials. Figure 6(c) is the difference image between Fig. 6(a) and (b). As indicated by the yellow arrows, the curved rectangle in Fig. 6(a) became straight in Fig. 6(b).

Fig. 6.

Distortion correction using the fitted polynomials. (a) A sagittal cross-section selected from the 3D image dataset acquired at 3T. (b) The same cross-section after the distortion correction was performed using the third-order polynomials. (c) The difference image between (a) and (b). (d) A coronal cross-section selected from the 3D image dataset acquired at 7T and corrected using the vendor software. (e) The same cross-section after the distortion correction performed using the third-order polynomials. (f) Difference between (e) and (f). The same squares are drawn on the images in (d)–(f).

Figure 6(d) is a coronal cross-section selected from the 3D image dataset acquired at 7T and corrected for the image distortion using the vendor software, Fig. 6(e) is the same cross-section of the image dataset and corrected for the image distortion using the third-order polynomials, and Fig. 6(f) is the difference image between them. The yellow squares drawn on the images show that the vendor-provided image distortion correction software did not reproduce the correct shape (circular cross-section).

Figure 7 shows the normalized mean values of the absolute difference in voxel values between (1) the original image, (2) the image corrected by the vendor distortion correction software, (3) the image corrected by the third-order polynomials of the vender-software corrected image, and (4) the image corrected by the third-order polynomials of the original image, acquired at 7T. The mean absolute difference between the four images ([1]–[4]) were normalized by the mean voxel value of the original image. As can be seen from these values, the image distortion correction performed by the vendor software seemed insufficient.

Fig. 7.

The mean absolute value of the difference between 3D image datasets normalized by the mean absolute value of the raw 3D image dataset acquired at 7T. The difference between the images corrected by the third-order polynomials shown in red round rectangles is small (0.0276).

Discussion

The spherical grid phantom

Advantages of the phantom developed in this study are (1) fabrication with commercially available materials, (2) installation in head RF coils, (3) homogeneous demagnetizing magnetic field, and (4) ability to extract many control points. The use of the phantom also has the advantage that, unlike the relatively difficult measurement of the diffusion coefficient distribution, the nonlinearity of the gradient can be accurately measured by the widely used robust 3D gradient-echo sequences.

Accuracy of polynomial fitting

Since the voxel size is anisotropic on the 3T MRI system, we discuss the accuracy of the polynomial fitting on the 7T MRI system, where the voxel size is isotropic (0.8 mm cube, nominal). For the measurements of the coordinates of the control points, image datasets with 0.4 mm cubic voxels were obtained by the ZIP processing. Thus, the average error of the coordinates in the measurements can be 0.2 mm (half of the interpolated voxel size). Of course, as pointed out by Wang et al., it was reported that measurement accuracy of about 0.1 mm could be achieved with a similar phantom by using a 3D Prewitt operation.⁸ In this study, this operation was not used because the calculation became unstable due to large changes in voxel values around the surface of the sphere caused by the dielectric resonance at 7T.

In the ideal case, if the number of control points is infinite and the order of the approximation polynomial is infinitely large, the average difference between the coordinates of the control points given by the measurements and those of the approximation is supposed to be 0.2 mm, the half size of the voxel. In other words, the ideal measurement error is expected to be distributed between –0.2 and +0.2 mm.

For the image data set at 7T without the vendor-provided distortion correction, the standard deviations of the histograms were 0.281 and 0.250 mm for G_x, and 0.238 and 0.215 mm for G_y, and 0.264 and 0.249 mm for G_z, respectively, when using the third- and fifth-order polynomials. Thus, compared to the ideal case, the actual error distribution (± standard deviation: about 68% for Gaussian) was approaching the ideal case. The result that the standard deviation of the histograms ranged from 0.215 to 0.281 is a reasonable result considering the manual measurements of the coordinates of the control points, the limited number of the control points (1076 points), and the finite order (fifth-order) of the polynomials. Therefore, we concluded that the control points were almost correctly measured, and the polynomial approximation of the measured coordinates was also correct.

Gradient nonlinearity in the 3T clinical MRI system

The 3T clinical MRI system was a product of GE company, and it was reported in the literature that GE corrects the nonlinearity of the gradient magnetic field with five solid harmonic functions.¹² However, when the gradient magnetic field measured for the “corrected” image dataset was approximated by the third-order polynomials, there were several nonlinear terms that could not be ignored, as shown in Table 1. This result suggested that the representation of the nonlinear magnetic field by the gradient coil set with five solid harmonic functions was insufficient. In fact, as shown in Fig. 6(a)–(c), significant image distortion in the yz-plane (sagittal plane) was observed, which was corrected using the calculated third-order polynomials.

Another possibility is that the coefficients of the solid harmonic functions for image distortion correction used for the MRI system were measured at a factory and might differ from those in the gradient system installed in the individual MRI system. In such cases, the image distortion correction method using such phantoms used in this study can solve this problem.

The nonlinearity of the gradient field perpendicular to the slice plane does not directly affect the image distortion in the slice plane, but it does affect the position and thickness of the slice plane, which, in turn, affects the pixel intensity of the image. However, correction of the pixel intensity is not possible by correcting the gradient nonlinearity alone, and is difficult to perform within the scope of this study.

Gradient nonlinearity in the 7T research-oriented MRI system

The 7T MRI system used in this study allowed the vendor-provided image distortion correction to be turned on and off during the image reconstruction process. However, as shown in Table 1, the coefficients of the nonlinear terms in the third-order polynomials indicated that the vendor-provided image distortion correction was not satisfactory. The normalized mean absolute difference in the voxel values between images with and without image distortion correction shown in Fig. 7 also suggested that the vendor’s image distortion correction was not sufficient. Siemens, the developer of this system, was said to have used 11 solid harmonic functions to correct for the nonlinearity of the gradient magnetic field.¹² However, as with the 3T clinical MRI, these coefficients were likely to be from a similar gradient system in a factory, rather than measured for the installed individual MRI system. Therefore, it is desirable to evaluate the nonlinearity of the gradient field by phantoms such as those used in this study for such research MRI systems as well as clinical MRI systems.

Future extension of this study

In this study, the FOV was limited because only the head coils were used. However, even with this limited FOV, image distortion was clearly detectable in the images. In particular, the presence of the large third-order term was remarkable. From this result, it is expected that in the body region, the image distortion due to the nonlinearity of the gradient field would be much larger. However, since extending the polynomial obtained for the head FOV to the body region will result in a larger error in image distortion, it is desirable to measure the nonlinearity using a larger phantom with the same method used in this study.

The accurate evaluation of the nonlinearity of the gradient field performed in this study is expected to be useful in a wide range of fields in MRI. For example, it will be extremely useful for measuring the volume and shape of tissues and organs, for surgical planning, and for superimposing or matching with images acquired with other modalities.

Conclusion

A spherical grid phantom was used to measure the nonlinearity of the magnetic field generated from the gradient coil sets of clinical 3T and research 7T MRI systems. The nonlinearity of the magnetic field was approximated by third- or fifth-order polynomials whose coefficients were determined by the steepest descent method. The spatial distribution of the nonlinearity was consistent with the results of previous studies. However, considerable nonlinearity of the gradient magnetic field was observed even after the image datasets acquired on the 3T and 7T MRI systems were processed using the vendor-supplied distortion correction software. The remaining image distortions could be corrected using the polynomials obtained from the phantom measurements. Therefore, we concluded that the correction of the image distortion due to the nonlinearity of the gradient by the vendor software was not always sufficient and that it was important to measure the nonlinearity of the gradient magnetic field using geometrically precise phantoms such as those used in this study.

Appendix

In the main text, the third-order polynomial used for fitting the 3D coordinates of the control points was shown. The following are the fourth- and fifth-order polynomials additionally used in this study.

$$\begin{array}{rl}& u{\left(x,y,z\right)}_{4,5}=a_{400}{x}^4+a_{310}{x}^{3}y+a_{301}{x}^{3}z+a_{220}{x}^2{y}^2\\ & +a_{211}{x}^{2}yz+a_{202}{x}^2{z}^2+a_{130}x{y}^3+a_{121}x{y}^{2}z\\ & +a_{112}xy{z}^2+a_{103}x{z}^3+a_{500}{x}^5+a_{410}{x}^{4}y\\ & +a_{401}{x}^{4}z+a_{320}{x}^3{y}^2+a_{311}{x}^{3}yz+a_{302}{x}^3{z}^2\\ & +a_{230}{x}^2{y}^3+a_{221}{x}^2{y}^{2}z+a_{212}{x}^{2}y{z}^2+a_{203}{x}^2{z}^3\\ & +a_{140}x{y}^4+a_{131}x{y}^{3}z+a_{122}x{y}^2{z}^2+a_{113}xy{z}^3\\ & +a_{104}x{z}^4\end{array}$$ (A.1)

$$\begin{array}{rl}& v{\left(x,y,z\right)}_{4,5}=b_{040}{y}^4+b_{130}x{y}^3+b_{031}{y}^{3}z+b_{220}{x}^2{y}^2+b_{121}x{y}^{2}z\\ & +b_{022}{y}^2{z}^2+b_{310}{x}^{3}y+b_{013}y{z}^3+b_{112}xy{z}^2\\ & +b_{211}{x}^{2}yz+b_{050}{y}^5+b_{140}x{y}^4+b_{041}{y}^{4}z\\ & +b_{230}{x}^2{y}^3+b_{131}x{y}^{3}z+b_{032}{y}^3{z}^2+b_{320}{x}^3{y}^2\\ & +b_{221}{x}^2{y}^{2}z+b_{122}x{y}^2{z}^2+b_{023}{y}^2{z}^3+b_{410}{x}^{4}y\\ & +b_{311}{x}^{3}yz+b_{212}{x}^{2}y{z}^2+b_{113}xy{z}^3+b_{014}y{z}^4\end{array}$$ (A.2)

$$\begin{array}{rl}& w{\left(x,y,z\right)}_{4,5}=c_{004}{z}^4+c_{103}x{z}^3+c_{013}y{z}^3+c_{202}{x}^2{z}^2+c_{022}{y}^2{z}^2\\ & +c_{112}xy{z}^2+c_{211}{x}^{2}yz+c_{121}x{y}^{2}z+c_{301}{x}^{3}z\\ & +c_{031}{y}^{3}z+c_{005}{z}^5+c_{104}x{z}^4+c_{014}y{z}^4\\ & +c_{203}{x}^2{z}^3+c_{113}xy{z}^3+c_{023}{y}^2{z}^2+c_{302}{x}^3{z}^2\\ & +c_{212}{x}^{2}y{z}^2+c_{122}x{y}^2{z}^2+c_{032}{y}^3{z}^2+c_{401}{x}^{4}z\\ & +c_{311}{x}^{3}yz+c_{221}{x}^2{y}^{2}z+c_{131}x{y}^{3}z+c_{041}{y}^{4}z\end{array}$$ (A.3)

The above polynomials were added to the third-order polynomials ([4]–[6]), and the sets of coefficients that gave the smallest error were determined by the steepest descent method. The number of iterations was about 1 million, and the computation time was a few minutes using a conventional PC.