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Local susceptibility variations result in
*B*
_{0} field inhomogeneities, causing distortions and signal losses in MR imaging. Susceptibility variations become stronger with increasing
*B*
_{0} magnetic field strength. Active shimming is used to generate corrective magnetic fields, which can be used to improve
*B*
_{0} field homogeneity. FASTMAP is an effective shimming technique for computing optimal coil currents, which uses data from six projection directions (or columns): this technique is routinely used for shimming cubic volumes of interest (VOIs). In this paper, we propose several improvements to FASTMAP at 4T. For each shim coil, using a modified 3D gradient-echo pulse sequence, we compute
*B*
_{0} inhomogeneity maps and project them onto eight 1
^{st} and 2
^{nd} order spherical harmonic functions. This process is repeated for shim currents between -15,000 to 15,000 with increments of 5000 Digital to Analog Converter (DAC) units, and is used to compute the gradient between spherical harmonic coefficients and DAC values for all 8 shim coils—along with the R
^{2} values of linear fits. A method is proposed (based on R
^{2} values) to further refine optimal shim currents in respective coils. We present an analysis that is numerically robust and completely flexible in the selection of the VOIs for shimming. Performance analyses, phantom results, and
*in vivo* results of a human brain are presented, comparing our methods with the FASTMAP method.

Automatic shimming for optimizing magnetic field uniformities is highly desirable in MR spectroscopy. Objects are often heterogeneous and contain intrinsically unshimmable field variations due to rapid susceptibility changes, which can lead to distortions of the lineshape obtained from the volume [_{0} field homogeneity [

The use of linear shim coils is highly advantageous in MR imaging [_{0} field; specifically, whenever the origin of the VOI is offset from the isocenter [_{0} field inhomogeneity [_{0} field along six projection directions (or columns). FASTMAP, however, is restricted to selected cubic VOIs [

FASTMAP works well over reasonably homogeneous volumes with moderate field inhomogeneity [

The FASTMAP technique incorrectly assumes that shim coil fields can be fully characterized by a minimal set of spherical harmonics [

In this paper, we follow the same general principals outlined in FASTMAP but propose several improvements. In brief, we propose combining spherical harmonic functions and linear least squares fitting for estimating field inhomogeneity. This method entails the computation of 3D phase images and the determination of first and second-order spherical harmonic coefficients for specific shim currents, by changing the Digital to Analog Converter (DAC) settings, which control voltages across different shim coils. The spherical harmonic calibration constants are then determined by computing the gradients between spherical harmonic coefficients and the DAC values of each coil—followed by a first order correction [see Equation (5)]. Our analysis is numerically robust and completely flexible when selecting VOIs for shimming. A performance analysis comparing our technique with FASTMAP, on a phantom and a human brain, demonstrates how our proposed method outperforms the FASTMAP technique in terms of B_{0} homogeneity.

All experiments were performed on a 4T whole-body Varian INOVA (Palo Alto, CA) MRI scanner located in Cincinnati, Ohio. The system was equipped with the following resistive shim coils: X, Z, Y (n = 1, m = 1, 0, −1) and second-order [X^{2} - Y^{2}, ZX, Z^{2}, ZY, XY (n = 2, m = −2, −1, 0, 1, 2)]. A TEM volume head coil was used for RF transmission and reception.

The imaging protocol employed a modified 3D gradient-echo pulse sequence (see FigureS1 in supplementary material), which was used to obtain B_{0} field maps. Frequency distortion correction (along the read-out direction) was performed on B_{0} field maps. All acquisitions used a 256 × 256 ×256 mm field of view; a 128 × 64 × 64 acquisition matrix; a 10˚ pulse flip angle; a repetition time (TR) of 16 ms, and echo times of 5.25 ms and 7 ms. The data was acquired in the axial orientation, with a slab-selective pulse used for excitation.

After acquisition, inverse Fourier transformation was performed on the acquired 3D k-space data. Subsequently, 3D phase unwrapping was performed on the resultant phase images as necessary. Frequency maps were then computed from the difference of the two phase images (acquired at different echo times) with the following equation:

f ( x , z , y ) = γ ⋅ Δ B 0 ( x , y , z ) 2 π (1)

After calculation of the 3D frequency maps, voxels corresponding to the selected VOI were extracted. All image reconstruction steps were performed in Matlab (Mathworks, Natick, MA).

Images were obtained in both a phantom and in-vivo. The phantom was a water sphere with a diameter of 178 mm. In-vivo images of a human head were obtained from a single subject. Consent was obtained with an IRB protocol approved by the University of Cincinatti School of Medicine. The VOI for shimming was defined as the entire spherical phantom and the brain only, respectively (see supplementary material for details). B_{0} field maps were acquired both prior to, and after, the shimming procedure outlined below.

A one-time procedure was performed to construct shim calibration tables for active shimming. B_{0} field maps were acquired upon each of the system’s 8 shim coils at different shim current levels. Specifically, the shim current was varied from −15,000 to 15,000 by increments of 5000 per acquisition. Thus, 7 field maps were acquired per shim coil. A spherical phantom (d = 178 mm) was used as the reference object for this calibration procedure. After reconstruction of the 3D phase images for each shim coil, and shim current setting, frequency distribution maps were computed. The matrix representation of f ( x , y , z ) is given by:

f ( x , y , z ) = ∑ n = 0 ∞ ∑ m = 0 n F n , m ( x i , y j , z k ) ⋅ η n m (2)

where η_{nm} are the coefficients of spherical harmonics, and F_{n,m} is the Cartesian spherical harmonic spatial dependence function (see FigureS4). Using the linear least-squares method, the optimized spherical harmonic coefficients of the first- and second-order shim coils over the selected VOI can be estimated. The frequency distributions of all shim coils (at each DAC step) can be projected onto the spherical harmonics by using Equation (2). We assume that the η n m , g , l of each shim coil is linearly varying with the DAC values.

η n m , g , l = C n m , g ⋅ D A C l , g (3)

Here, C n m , g is the calibration constant for each spherical harmonic. These C n m , g values can be estimated using the following expression:

C n m , g = ( ( D A C l , g T ⋅ D A C l , g ) − 1 ⋅ D A C l , g ⋅ η n m , g , l T ) T (4)

The C n m , g values for all 8 shim coils are obtained from Figures S4-S6. Finally, the spherical harmonic calibration constants are computed by the gradient between spherical harmonic coefficients and the DAC values of each coil; this can be used to update the DAC settings. The R^{2} of this linear fit was also computed (see supplementary material for details).

Generally, first order coils should produce orthogonal fields that correspond to first order spherical harmonics. The second order coils could potentially produce fields that correspond to first and second-order spherical harmonics. Therefore, we propose the following correction when computing optimal DAC settings of first-order shims, in order to counter the contributions of second-order shims:

D A C 1 , m 1 | C o r r e c t e d = D A C 1 , m 1 − C 2 , m 2 × D A C 2 , m 2 C 1 , m 1 (5)

Here, D A C 1 , m 1 is the shim setting of the 1^{st} order m_{1}^{th} degree coil (X, Y, or Z), and D A C 1 , m 1 is the setting for the 2^{nd} order m_{2}^{th} degree shim coil for correct shimming of an object. C 1 , m 1 and C 2 , m 2 are the 1^{st} order m_{1}^{th} degree, and the 2^{nd} order m_{2}^{th} degree calibration coefficients of coils, respectively. The term C 2 , m 2 × D A C 2 , m 2 is the contribution of the second-order coil to the first-order spherical harmonics. Multiplying the term C 2 , m 2 × D A C 2 , m 2 by a proportionality

constant 1 C 1 , m 1 , then using Equation (4), we can compute an updated D A C 1 , m 1

setting. This new setting has effectively subtracted the contributions of the second order coil from the first order coil (or first order spherical harmonics).

Spherical harmonic calibration constants and corresponding R^{2} values of linear fits (for all shims) are tabulated in ^{2} values in spherical harmonic calibration constants for first-order shims (^{2} values that are ≥0.9 are highlighted in light blue in

Coefficient | Notation | Spherical Harmonic Calibration Constant (Cnm > g) | |||||||
---|---|---|---|---|---|---|---|---|---|

X-Coil | Z-Coil | Y-Coil | X^{2}Y^{2}-Coil | XZ-Coil | Z^{2}C-Coil | ZY-Coil | XY-Coil | ||

A_{11} | X | −2.3E−03 | −2.7E−07 | −8.0E−07 | −5.0E−05 | 1.3E−05 | 2.6E−05 | 2.1E−05 | −1.4E−05 |

A_{10} | Z | −6.2E−07 | 2.3E−03 | 2.3E−06 | 1.7E−04 | −2.2E−05 | 4.6E−05 | −7.0E−06 | −2.1E−04 |

A_{1-1} | Y | −1.1E−06 | −1.6E−06 | −2.3E−03 | 3.1E−06 | 2.2E−05 | −1.2E−05 | −2.0E−05 | 2.9E−05 |

A_{22} | X^{2} - Y^{2} | 3.9E−07 | −1.0E−07 | −1.8E−07 | −2.1E−04 | −1.6E−07 | −3.9E−07 | 1.7E−07 | 3.8E−06 |

A_{21} | ZX | −8.0E−07 | −3.6E−07 | −5.9E−08 | −6.3E−06 | −1.8E−04 | & 4E−07 | 2.4E−07 | 5.4E−06 |

A_{20} | Z^{2}C | −2.3E−07 | 9.3E−07 | −1.6E−07 | −7.3E−07 | −6.4E−07 | −4.5E−04 | −5.7E−07 | −1.8E−07 |

A_{2-1} | ZY | 1.6E−08 | 3.2E−08 | −7.6E−07 | 3.7E−06 | −1.3E−07 | & 3E−07 | 1.8E−04 | 9.7E−07 |

A_{2-2} | XY | 7.1E−09 | −1.3E−08 | 1.6E−07 | 3.0E−06 | −1.5E−07 | 3.3E−07 | −3.1E−07 | 2.1E−04 |

C | −1.0E−05 | 7.3E−06 | −2.4E−05 | 3.4E−04 | 2.2E−04 | −1.2E−06 | 2.0E−04 | −2.7E−04 |

Coefficient | Notation | Ra | |||||||
---|---|---|---|---|---|---|---|---|---|

X-Coil | Z-Coil | Y-Coil | X^{2}Y^{2}-Coil | XZ-Coil | Z^{2}C-Coil | ZY-Coil | XY-Coil | ||

A_{11} | X | 1.00E+00 | 7.27E−02 | 2.96E−01 | 9.96E−01 | 9.66E−01 | 9.94E−01 | 9.98E−0I | 9.95E−01 |

A_{10} | Z | 3.69E−02 | 1.00E+00 | 3.78E−01 | 9.99E−01 | 9.74E−01 | I.93E−01 | 7.98E−0I | 1.00E+00 |

A_{1-1} | Y | 2.82E−01 | 4.08E−01 | 1.00E+00 | 9.03E−01 | 9.98E−01 | 9.86E−01 | 9.88E−01 | 9.94E−01 |

A_{22} | X^{2} - Y^{2} | 2.23E−01 | 3.74E−01 | 4.17E−02 | 1.00E+00 | 1.01E−01 | 6.28E−01 | I.31E−01 | 9.99E−01 |

A_{21} | ZX | 7.99E−01 | 1.61E−01 | 5.03E−02 | 9.99E−01 | 9.99E−01 | 8.99E−01 | 6.25E−01 | 9.98E−01 |

A_{20} | Z^{2}C | 2.47E−01 | 3.69E−01 | 9.86E−02 | 7.06E−01 | 9.57E−01 | 1.00E+00 | 9.83E−01 | 1.07E−01 |

A_{2-1} | ZY | 1.11E−02 | 5.25E−03 | 6.73E−01 | 9.94E−01 | 4.75E−01 | 8.72E−01 | 9.99E−01 | 9.20E−01 |

A_{2-2} | XY | 6.45E−03 | 1.32E−02 | 1.87E−01 | 9.99E−01 | 6.53E−01 | 8.63E−01 | 8.55E−01 | 1.00E+00 |

C | 8.91E−02 | 4.02E−02 | 4.19E−01 | 9.92E−01 | 9.90E−01 | 1.07E−06 | 9.83E−01 | 9.91E−01 |

second-order harmonics (i.e., x^{2} - y^{2}, xz, and zy) in addition to first-order harmonics. On the other hand, both _{11}, A_{10}, A_{1-1}).

_{0} field distribution in the phantom before and after active shimming. The histograms of magnetic field distributions (over the entire phantom), before and after active shimming, are shown in _{0} homogeneity significantly within the phantom.

B_{0} maps following FASTMAP and active shimming methods are shown in _{0} homogeneity (

Performance analyses of phantom results and in vivo results of a human brain showed that our proposed method can significantly outperform FASTMAP. When field maps are derived using all data points within a VOI, B_{0} homogeneity can be improved by countering the contributions, or effects, of higher-order shims on first-order shims. First order shims play significant roles in B_{0} homogeneity within small VOIs. Accordingly, taking into account the contributions of higher order shims within small VOIs can be important for many MR spectroscopy applications. Specifically, our method highlights the advantage of using spherical harmonic expansion corrections for shimming spherical volumes.

Our method, however, could not improve the magnetic field homogeneity near regions of the nasal sinus to a satisfying degree: these regions are known for significant susceptibility variations. Future research, focusing on combining active and passive shimming, must be pursued in order to further improve field homogeneity in the frontal brain [

Magnetic field gradient pulses can produce eddy-currents in conductive brain regions [

There may be instances where simultaneous shimming of arbitrary volumes (with differing levels of field uniformity) becomes necessary. For example: to establish a shim over a particular organ, with a tight B_{0} range, while maintaining a coarser uniformity over the entire abdominal slice to prevent frequency-based fat-suppression techniques from failing. Thus, our method provides greater flexibility and can be advantageous for shimming arbitrary volumes over FASTMAP.

Here, we followed the method of projecting shim maps onto spherical harmonics: an a priori basis set to represent field maps. Due to some arguments suggesting that the use of spherical harmonics may be sub-optimal [_{0} fields over large volumes [

The authors declare no conflicts of interest regarding the publication of this paper.

Jayatilake, M., Sica, C.T., Elyan, R. and Karunanayaka, P. (2020) Comparison of FASTMAP and B_{0} Field Map Shimming at 4T: Magnetic Field Mapping Using a Gradient-Echo Pulse Sequence. Journal of Electromagnetic Analysis and Applications, 12, 115-130. https://doi.org/10.4236/jemaa.2020.128010

Inhomogeneous magnetic fields in the MRI scanner can be corrected by adjusting shim coils to produce additional magnetic fields. These shim coils generate unique magnetic field distributions which are modelled using orthogonal spherical harmonic functions [

Below, we present the theory and methods to: 1) numerically estimate inhomogeneous magnetic fields by varying shim settings; 2) derive calibration tables, and (3) determine appropriate shim currents for the first and second-order shim coils.

S.1. Modeling the B_{0}Static Magnetic Field

Assuming a current density of zero ( J ¯ = 0 ), the static inhomogeneous magnetic field Δ B 0 in a region of interest is given by Laplace’s equation (S1).

∇ 2 ( Δ B 0 ) = 0 (S1)

The solution to this equation Δ B 0 can be expressed as a sum of spherical harmonics [

Δ B 0 ( r , θ , ϕ ) = ∑ n = 0 ∞ ∑ m = 0 n A n m r n ⋅ P n , m ( cos θ ) ⋅ e j m ϕ (S2)

Here, r , θ and ϕ are the spherical coordinates. n and m are integers satisfying the conditions n ≥ m ≥ 0 ; n is the order and m is the degree of a given spherical harmonic. A n m are the coefficients of spherical harmonic functions. The P n , m ( cos θ ) is Ferrer’s associated Legendre polynomial [

Δ B 0 ( x , y , z ) = c + α 11 x + α 10 z + α 1 − 1 y + α 22 ( x 2 − y 2 ) + α 21 z x + α 20 ( z 2 − 1 / 2 ( x 2 + y 2 ) ) + α 2 − 1 z y + α 2 − 2 x y + ⋯ (S3)

n | m | Short-hand notation | Coefficient (α_{nm}) | Spatial dependence function | |
---|---|---|---|---|---|

Spherical | Cartesian | ||||

1 | 1 | X | α_{11} | r ⋅ sin θ ⋅ cos ϕ | x |

1 | 0 | Z | α_{10} | r ⋅ cos θ | z |

1 | −1 | Y | α_{1-1} | r ⋅ sin θ ⋅ sin ϕ | y |

2 | 2 | X^{2}-Y^{2} | α_{22} | r 2 ⋅ sin 2 θ ⋅ cos 2 ϕ | x 2 − y 2 |

2 | 1 | ZX | α_{21} | r 2 ⋅ sin θ ⋅ cos θ ⋅ cos ϕ | xz |

2 | 0 | Z2C | α_{20} | r 2 ⋅ ( 3 cos 2 θ − 1 ) / 2 | z 2 − ( x 2 + y 2 ) / 2 |

2 | −1 | ZY | α_{2-1} | r 2 ⋅ sin θ ⋅ cos θ ⋅ sin ϕ | yz |

2 | −2 | XY | α_{2-2} | r 2 ⋅ sin 2 θ ⋅ cos ϕ ⋅ sin ϕ | xz |

We performed a phantom study using the pulse sequence shown in FigureS1 to compute the B_{0} field maps. Δ B 0 ( x , y , z ) was computed by comparing two phase images with different echo times.

At each voxel, the relationship between Δ B 0 ( x , y , z ) , phase evolution Δ ϕ ( x , y , z ) , and echo time (DTE) is given by Equation (S4):

Δ B 0 ( x , y , z ) = Δ ϕ ( x , y , z ) γ ⋅ Δ T E . (S4)

Here γ is the gyromagnetic ratio in radian/s/T for proton γ ( H 1 ) = 2.675 × 10 8 rad ⋅ s − 1 ⋅ T − 1 . Since the phase can only have magnitudes between − 2 π < ϕ < 2 π , phase unwrapping must be performed on an as needed basis. At each voxel, the distribution of the precessional frequency f ( x , y , z ) is related to Δ B 0 ( x , y , z ) by Equation (S5):

f ( x , z , y ) = γ ⋅ Δ B 0 ( x , y , z ) 2 π . (S5)

These f ( x , y , z ) maps were computed for DAC values: (A) −15,000, (B) −10,000, (C) −5000, (D) 0, (E) 5000, (F) 10,000, and (G) 15,000.

S.3. Phantom StudyThis procedure was repeated on a water phantom to compute frequency distribution maps. FigureS2 and FigureS3 show f ( x , y , z ) maps for the water phantom, for all 8 shim coils.

S.3.1. Computing Calibration TablesBy combining Equation (S3) and Equation (S5) we obtain the following expression

f ( x , y , z ) = c ′ + η 11 x + η 10 z + η 1 − 1 y + ⋯ + η 22 ( x 2 − y 2 ) + η 21 z x + η 20 ( z 2 − 1 / 2 ( x 2 + y 2 ) ) + η 2 − 1 z y + η 2 − 2 x y + ⋯ (S6)

Here, c ′ and η represent the 0^{th} and higher-order coefficients of spherical harmonics. The matrix representation of f ( x , y , z ) is given by:

f ( x , y , z ) = ∑ n = 0 ∞ ∑ m = 0 n F n , m ( x i , y j , z k ) ⋅ η n m . (S7)

where η_{nm} are the coefficients of spherical harmonics. Using the linear least-squares method, the optimized spherical harmonic coefficientsof the first and second-ordershim coils can be estimated. The frequency distributions of all shim coils (at each DAC step) can be projected onto spherical harmonics by using Equation (S7). We assume that the η n m , g , l of each shim coil is linearly varying with the DAC values, i.e.,

η n m , g , l = C n m , g ⋅ D A C l , g . (S9)

Here C n m , g is the calibration constant for each spherical harmonic. These C n m , g values can be estimated using the following expression:

C n m , g = ( ( D A C l , g T ⋅ D A C l , g ) − 1 ⋅ D A C l , g ⋅ η n m , g , l T ) T . (S10)

The values for all 8 shim coils are computed using tables and Figures S4-S6 shown below. Finally, the spherical harmonic calibration constants are computed by the gradients between spherical harmonic coefficients and the DAC values of each coil. This can be used to update the DAC settings.

The calibration constants and their corresponding R^{2} are obtained from FigureS4 & FigureS5 and used in

B_{0}: Static main magnetic field

FASTMAP: Fast automatic shimming technique, by mapping along projections

DAC: Digital to Analog Converter

n: Order of a spherical harmonic

m: Degree of a spherical harmonic

η_{nm}: Coefficients of spherical harmonics

P n , m ( x , y , z ) : Cartesian spherical harmonic spatial dependence function

C n m , g : Calibration constant.

R^{2}: Linear fit

J ¯ : Current density

Δ B 0 : Static inhomogeneous magnetic field

P n , m ( cos θ ) : Ferrer’s associated Legendre polynomial

Δ ϕ ( x , y , z ) : Phase evolution

ΔTE: Echo time

γ : Gyromagnetic ratio in radian/s/T.

f ( x , y , z ) : Precessional frequency distribution at each voxel

c ′ : 0^{th} coefficients of spherical harmonic

η_{nm}: Coefficients of spherical harmonics.

C n m , g : Calibration constant for each spherical harmonic.

VOI: Volume of Interest