Fast conjugate phase image reconstruction based on a Chebyshev approximation to correct for B0 field inhomogeneity and concomitant gradients

Abstract

Off-resonance effects can cause image blurring in spiral scanning and various forms of image degradation in other MRI methods. Off-resonance effects can be caused by both B0 inhomogeneity and concomitant gradient fields. Previously developed off-resonance correction methods focus on the correction of a single source of off-resonance. This work introduces a computationally efficient method of correcting for B0 inhomogeneity and concomitant gradients simultaneously. The method is a fast alternative to conjugate phase reconstruction, with the off-resonance phase term approximated by Chebyshev polynomials. The proposed algorithm is well suited for semiautomatic off-resonance correction, which works well even with an inaccurate or low-resolution field map. The proposed algorithm is demonstrated using phantom and in vivo data sets acquired by spiral scanning. Semiautomatic off-resonance correction alone is shown to provide a moderate amount of correction for concomitant gradient field effects, in addition to B0 imhomogeneity effects. However, better correction is provided by the proposed combined method. The best results were produced using the semiautomatic version of the proposed combined method.

INTRODUCTION

Off-resonance effects are common in magnetic resonance imaging (MRI). In contrast to Cartesian acquisitions, where off-resonance effects usually cause mainly geometrical distortion and intensity variations, off-resonance effects can cause unacceptable image degradation in non-Cartesian acquisitions. In particular, uncorrected off-resonance effects cause space-variant image blurring in spiral k-space scanning. This issue has been long recognized (1) and is one of the most challenging problems in non-Cartesian MRI‥

Off-resonance effects in MRI can arise from various sources. The off-resonance from chemical shift usually can be addressed by presaturation RF pulses or spectral-spatial excitation (2). Another source of off-resonance effects is B0 field inhomogeneity, including susceptibility-induced field inhomogeneity and main field inhomogeneity. Off-resonance effects can also come from concomitant gradient fields (3). A pure linear gradient does not satisfy Maxwell’s equations of electrodynamics. As a result, additional magnetic field terms known as concomitant gradients arise. When the off-resonance phase from B0 field inhomogeneity and concomitant gradients are both present, correction of only one of them is insufficient and sometimes can even increase blurring artifacts in some parts of the image (4).

Since both B0 field inhomogeneity and concomitant gradient fields vary slowly in space, we would expect that conjugate phase (CP) reconstruction (5–7) can afford effective simultaneous correction of both off-resonance effects. However, direct calculation of CP reconstruction is very time consuming. Fast alternatives to CP reconstruction with comparable accuracy have been developed, including time approximation methods (8–9), frequency approximation methods (10–12), and polynomial approximation methods (13–14). Each of these methods was originally developed for B0 field inhomogeneity correction. King et al (4) discussed how to extend fast CP methods for concomitant field correction in spiral imaging. No literature to date has discussed in detail simultaneous B0 inhomogeneity and concomitant field correction in non-Cartesian MRI.

In this paper, we introduce a new fast alternative to CP reconstruction, where the off-resonance phase term is approximated by its Chebyshev polynomials. Compared to similar ideas discussed in the past (11, 13), the proposed method is more computationally efficient and is well-suited for simultaneous B0 inhomogeneity and concomitant field correction. We apply the technique to deblurring images acquired using spiral scanning. We also demonstrate that the proposed method is efficient for combined semiautomatic inhomogeneity correction (17) and concomitant field correction. Semiautomatic correction has improved accuracy in off-resonance correction with an inaccurate or low-resolution field map.

THEORY

Conjugate phase reconstruction

Considering the presence of both B₀ field inhomogeneity and concomitant gradient fields and ignoring relaxation, the MR signal from an object with spin density m(r) is given by:

$$s\left(t\right)={\displaystyle {\int }_{\mathbf{r}}m\left(\mathbf{r}\right)}\text{exp}\left(-i2\pi \mathbf{k}\left(t\right)·\mathbf{r}-i\varphi \left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),t\right)\right)d\mathbf{r}$$ [1]

where k(t) is the k-space trajectory. The off-resonance phase term φ(Δω(r),Δω_c(r),t) can be expressed as:

$$\varphi \left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),t\right)=\mathrm{\Delta }\omega \left(\mathbf{r}\right)t+{\varphi }_c\left(\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),t\right),$$ [2]

where Δω(r) represents the angular off-resonance frequency of B₀ field inhomogeneity, and φ_c(Δω_c(r),t) represents the off-resonance phase due to concomitant fields.

For spiral scanning, King et al (4) gave an approximation of the off-resonance phase from the lowest order concomitant field for arbitrary scan plane orientation as follows:

$${\varphi }_c\left(\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),t\right)=\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right)t_c,$$ [3]

with

$$\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right)=\gamma \frac{g_m^2}{4B_0}\left(F_1{X}^2+F_2{Y}^2+F_3{Z}^2+F_{4}XZ+F_{5}YZ+F_{6}XY\right),\text{and}$$ [4]

$$t_c\left(t\right)=\frac{1}{g_m^2}{\displaystyle {\int }_0^t{g}_0^2(t^{\prime })d{t}^{\prime }}.$$ [5]

where B₀,g₀ (t) , and g_m are the static field, the amplitude of the readout gradient, and the maximum amplitude of the readout gradient, respectively. F₁ to F₆ are calculated from the rotation matrix used to rotate from the logical to the physical coordinate system. X and Y are the in-plane coordinates of the logical coordinate system, and Z is the through-plane coordinate of the logical coordinate system. The calculation of F₁ to F₆ is given in the appendix in (4).

When the off-resonance field varies smoothly in space, CP reconstruction is an effective method of off-resonance correction for various non-Cartesian acquisition methods. CP reconstruction corresponding to Eq. [1] can be expressed as:

$$m\left(\mathbf{r}\right)={\displaystyle {\int }_{t}s\left(t\right)}W(t)\text{exp}\left(i2\pi \mathbf{k}\left(t\right)·\mathbf{r}+i\varphi \left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),t\right)\right)dt$$ [6]

where W(t) is the density compensation function, which accounts for non-uniform sampling in k-space in non-Cartesian acquisition. W(t) can be calculated from geometric arguments, as the Jacobian of the coordinate transformation (18) or by numerical methods (19–20). Noll et al (21) discussed the implementation of a spatially-varying density compensation function in CP reconstruction.

Fast alternatives to CP reconstruction approximate the off-resonance phase term in different ways to reduce the computational cost, and consequently possess different features. Each of these methods was originally developed to correct for B0 field inhomogeneity. Fast CP reconstruction in the presence of both B0 field inhomogeneity and concomitant gradient fields has increased numerical complexity and has not been previously described to our knowledge. Extending the time approximation and frequency approximation methods to produce a combined correction would likely increase computation significantly. As we show below, it is possible to design a particular polynomial approximation that leads to fast and accurate correction for both B0 inhomogeneity and concomitant gradient fields.

Fast conjugate phase reconstruction based on Chebyshev approximation

Various polynomials can be used to approximate off-resonance phase term in CP reconstruction to reduce its computational cost. Among polynomials of the same order, Chebyshev approximation is nearly same as minimax polynomial approximation which has the smallest maximum deviation from actual function (22). The minimax polynomial is very difficult to find whereas Chebyshev approximation polynomial is very easy to calculate (22). Schomberg (13) has previously discussed Chebyshev approximation in CP reconstruction for the purpose of B₀ field inhomogeneity correction. In his algorithm, the off-resonance phase term is approximated by a Chebyshev polynomial that is a function of the product of B₀ field inhomogeneity and time. One drawback of Schomberg’s algorithm is that the calculation of Chebyshev coefficients becomes object dependent, since the field map is involved in the Chebyshev polynomial variable. Another problem associated with Schomberg’s algorithm is that it can not be applied for simultaneous B0 field inhomogeneity correction and concomitant gradient field correction.

To address these problems, we propose a fast CP reconstruction algorithm where the off-resonance phase term is approximated by a Chebyshev polynomial in time t . The proposed fast CP reconstruction algorithm can be expressed as follows:

$$m\left(\mathbf{r}\right)={\displaystyle \sum _{k=0}^{N-1}{w}_k\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)}{I}_k\left(\mathbf{r}\right)-\frac{1}{2}{I}_o\left(\mathbf{r}\right),$$ [7]

where τ is the readout length, I_k(r) are base images, and w_k)(Δω(r),Δω_c(),τ) are constant coefficients whose values depend on the B₀ field inhomogeneity, the concomitant gradient field, and the readout length. The base images, I_k(r) , are calculated in one of two ways from expressions of the following form:

$$q_l(\mathbf{r})={{\displaystyle {\int }_{t}s(t)\left(\frac{2t}{\tau }-1\right)}}^{l}W(t)\text{exp}(i(2\pi \mathbf{k}(t)·\mathbf{r})dt$$ [8]

The mathematical derivation of Eqs. [7] and [8] is given in Appendix A. Two forms of Eq. [7], including Eq. [A5] and Eq. [A10], are derived. Our experiments on spiral data sets indicate that these two formulas lead to equivalent off-resonance correction results. The proposed method can also be easily modified for B0 field inhomogeneity correction or concomitant field correction when only one off-resonance source is present.

The coefficients w_k(Δω (r),Δω_c(r),τ) can be calculated offline and saved as a table at a range of Δω(r)andΔω_c(r). The table is object independent and the same table can be used for any data set acquired with the same readout length and k-space trajectory. The table is three dimensional and the length of each dimension corresponds to the number of bins of Δω(r) , the number of bins of Δω_c(r) , and the number of base images, respectively.

Equation [8] involves reconstruction of the time signal s(t) multiplied by a modified sampling time followed by a fast reconstruction method such as gridding (15–16). To reduce the computational cost, we perform gridding on the time signal s(t) to calculate raw data on a Cartesian grid, and also interpolate the modified sampling time onto a Cartesian grid using Delaunay triangulation to form a Cartesian time mask. To compute the base images, we then simply multiply the Cartesian time signal by the time mask multiple times and then perform a Fourier transform. This strategy reduces the number of gridding operations performed during image reconstruction. Since the time mask is only a function of the k-space trajectory, it can also be calculated offline and used for data sets acquired with same k-space trajectory.

The number of base images needed in Eq. [7] is proportional to the range of B0 field inhomogeneity and concomitant gradient field. Center frequency correction can be incorporated into Eq. [8] to reduce the range of B0 field inhomogeneity. When the B0 field map is not highly non-linear, incorporation of linear off-resonance correction into Eq. [8] is desirable to further reduce the computational cost. To reduce the range of concomitant gradient field, we incorporate linear concomitant gradient correction (23) into Eq. [8]. For a given scan plane orientation, linear concomitant gradient correction reduces the frequency range of an off-center slice to that of a slice at isocenter. The derivation of this is given Appendix B.

Combination of semiautomatic off-resonance correction and concomitant gradient field correction

CP reconstruction requires knowledge of accurate off-resonance maps. The off-resonance map from concomitant gradient fields can be calculated from theory. For B0 inhomogeneity correction, a field map is usually acquired by two single shot spirals at different echo times. Such field map acquisitions, however, can be inaccurate in many applications and residual errors can appear in images. Recently, we proposed a method termed semiautomatic off-resonance correction (17) to address this problem. In semiautomatic off-resonance correction, the acquired map is used to provide a frequency constraint for following automatic off-resonance correction (24) rather than directly for CP reconstructions.

The computational strategy developed in semiautomatic correction requires calculating the value of individual pixels at an arbitrary constant frequency without reconstructing the whole image at that frequency. Multifrequency interpolation (11) was employed previously for this purpose. Note that the proposed fast CP reconstruction method also has this property. Therefore, the proposed CP reconstruction based on Chebyshev approximation is well suited for combined semiautomatic off-resonance correction and concomitant field correction. To perform combined semiautomatic correction and concomitant field correction, we reconstruct a series of images as follows:

$$m\left(\mathbf{r};\mathrm{\Delta }{\omega }_i\right)={\displaystyle \sum _{k=0}^{N-1}{w}_k}\left(\mathrm{\Delta }\tilde{\omega }\left(\mathbf{r}\right)+\mathrm{\Delta }{\omega }_i,\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)I_k\left(\mathbf{r}\right)-\frac{1}{2}{I}_o\left(\mathbf{r}\right),i=1,2\dots ,$$ [9]

where Δω̃(r) is the frequency constraint estimated from an acquired low resolution B0 field map and Δω_i(i = 1,2…) is a series of constant frequency shifts from ) Δω̃(r) (17). Note that concomitant field effects are corrected when reconstructing m(r;Δω_i) in Eq. [7]. After reconstructing m(r;Δω_i), semiautomatic correction is then performed for B0 field inhomogeneity correction (17).

METHODS AND RESULTS

We tested our algorithm on both phantom and in vivo data sets collected by spiral scanning on a Siemens 1.5T Avanto scanner (Siemens Medical Solutions). We performed the following correction methods and compared the results: (1) concomitant field correction using the proposed method; (2) field-map-based B0 field inhomogeneity correction using fast CP reconstruction; (3) semiautomatic off-resonance correction; (4) combined field-map-based B0 field inhomogeneity correction and concomitant field correction (combined correction A); and (5) combined semiautomatic B0 field inhomogeneity correction and concomitant field correction (combined correction B). For semiautomatic correction and combined correction B, the following parameters are used for the semiautomatic portion of the image reconstruction: the size of summation window was 15×15; the power α of the objective function was 1; the searching range of the off-resonance frequency shift was from −50 Hz to 50 Hz; the increment of searching frequencies was 10 Hz; and the first 1.6 ms of the readout was used to reconstruct the low resolution images (after concomitant gradient field correction) whose phase was then removed from corresponding high resolution images before objective function calculation. The value of each of these parameters was the same as those specified in (17). To account for trajectory errors due to gradient delay and eddy current effects, modified k-space trajectories based on methods described by Tan et al (25) were used for image reconstruction.

A 3-D table of interpolation coefficients was precalculated for the proposed methods and used for each of the data set acquired with the same spiral trajectory. The 3-D table was calculated with B0 field inhomogeneity from −100 Hz to 100 Hz with a 1 Hz frequency increment, concomitant field off-resonance frequency $\left(=\frac{\mathrm{\Delta }{\omega }_c}{2\pi }\right)$ from −100 Hz to 100 Hz with a 1 Hz increment, and 12 base images. The specified range of concomitant field is sufficient in our application after incorporating linear concomitant field correction. For combined semiautomatic off-resonance correction and concomitant correction, an additional 3D table was precalculated for reconstruction of low resolution images used in phase calibration with the same range of B0 field inhomogeneity and concomitant field but with only 5 base images due to significantly reduced readout length.

We performed combined correction based on Eq. [A5] and Eq. [A10] on the data sets we acquired and found the results were equivalent. Figure 1 shows the results for a coronal scan of a resolution phantom. The parameters of the spiral sequence are as follows: 14 interleaves with 8192 samples and 2 microseconds ADC dwell time per interleaf, 5 mm slice thickness, and 512 by 512 reconstructed image matrix. A low resolution field map was acquired using two single shot spirals with 1 ms echo delay. The imaging slice was 6.4 cm off isocenter along the transverse direction, 2.3 cm off isocenter along the sagittal direction, and 1.4 cm off isocenter along the coronal direction. Figure 1a) is the image with no off-resonance correction. Figure 1b) is the result after concomitant gradient correction. Note that there is significant deblurring of concomitant field effects at the bottom of the image, but that the B0 blurring is still obvious at the top of the image. Figure 1c) is the result after B0 inhomogeneity correction using fast CP reconstruction. Note that now the B0 blurring artifacts are removed but the concomitant field blurring is obvious. Figure 1d) is the result after semiautomatic offresonance correction. Note that semiautomatic correction can provide both B0 inhomogeneity correction and moderate concomitant field correction, but residual concomitant field blurring is still obvious. Figures 1e) and 1f) are combined correction A and B, respectively. Both combined correction methods achieved simultaneous B0 inhomogeneity correction and concomitant field correction. Compared to Fig. 1e), Fig. 1f) shows better deblurring in regions indicated by the white arrows in Fig. 1e), indicating the inaccuracy of the acquired B0 field map in these regions.

Figure 1.

A coronal scan of a resolution phantom acquired by a spiral sequence and reconstructed with different off-resonance correction methods. The imaging slice is 6.4 cm off isocenter along the transverse direction, 2.3 cm off isocenter along the sagittal direction, and 1.4 cm off isocenter along the coronal direction. Figure 1a) is the image without off-resonance correction. Figure 1b) is the image after concomitant gradient correction. Note significant deblurring of concomitant field effects at the bottom of the image, but B0 blurring is still obvious at the top of the image. Figure 1c) is the result after B0 inhomogeneity correction using fast conjugate phase reconstruction. Note that the B0 blurring artifacts are removed but the concomitant field blurring is obvious. Figure 1d) is the result after semiautomatic off-resonance correction. Note that semiautomatic correction can provide both B0 inhomogeneity correction and moderate concomitant field correction. Residual concomitant field blurring is still apparent in Figure 1d). Figure 1e) is the image after combined B0 field inhomogeneity correction based on conventional CP reconstruction and concomitant field correction and Fig. 1f) is the image after combined semiautomatic off-resonance correction and concomitant field correction. Both off-resonance methods achieve simultaneous B0 inhomogeneity correction and concomitant field correction. Compared to Fig. 1e), Fig. 1f) is sharper in regions indicated by the white arrows in Fig. 1e). The images show geometric distortion caused by gradient nonlinearity, which could be corrected using standard remapping methods.

Figure 2 is an example of an in vivo double-oblique head scan of a normal volunteer. The imaging slice was 8.4 cm off isocenter along the transverse direction, 2.6 cm off isocenter along the sagittal direction, and 4.8 cm off isocenter along the coronal direction. The imaging plane was tilted 30 degrees from transverse to coronal and then tilted 15 degrees from coronal toward sagittal. We used the method described by Wang et al (26) to create tagging lines to make off-resonance blurring more prominent. Twenty spiral interleaves with 8192 samples per interleaf and 2 microseconds ADC dwell time were used to acquire the data set. Other sequence parameters are the same as those used to acquire the data set in Fig. 1. Figure 2a) is the image without any off-resonance correction. Figure 2b) is the image after concomitant field correction using the proposed method. The deblurring is significant after concomitant correction, since the concomitant gradient field is severe in this scan plane. However, the residual blurring is still obvious in regions indicated by the white arrows due to the local B0 field inhomogeneity effects. Figure 2c) is the image after B0 inhomogeneity correction using fast CP reconstruction and Fig. 2d) is the image after semiautomatic correction. The blurring artifacts are significant in both images due to the strong off-resonance effects from concomitant gradient field. Figures 2e) and 2f) are the combined correction A and B, respectively. Both images are sharper than Fig. 2b). The tagging lines in Fig. 2f) is better defined than those in Fig. 2e) in the region indicated by the white arrow in Fig. 2e), indicating the inaccuracy of acquired the low-resolution B0 field map at this region.

Figure 2.

Head images of a normal volunteer acquired by a spiral sequence. Tagging lines were created to make the off-resonance effects more prominent. The imaging slice is 8.4 cm off isocenter along the transverse direction, 2.6 cm off isocenter along the sagittal direction, and 4.8 cm off isocenter along coronal direction. The imaging plane is tilted 30 degrees from transverse to coronal and then tilted 15 degrees from coronal toward sagittal. Figure 2a) is the image without off-resonance correction. Figure 2b) is the image after concomitant field correction. White arrows indicate the residual image blurring from B0 field inhomogeneity. Figure 2c) is the image after B0 inhomogeneity correction using fast CP reconstruction and Figure 2d) is the image after semiautomatic correction. The residual blurring is significant in both images due to the strong concomitant field effects. Figure 2e) is the image after combined B0 field inhomogeneity correction based on conventional CP reconstruction and concomitant field correction and Figure 2f) is the image after combined semiautomatic off-resonance correction and concomitant field correction. Both results provide a better deblurred image than that after concomitant field correction. The tagging lines are better defined in Fig. 2f) than Fig. 2e) in the region indicated by the white arrow in Fig. 2e).

We tested the algorithm with a variety of different parameters and in different anatomical areas. The algorithm continues to work well over larger fields of view. For example, when tested over a set of transverse slices covering the brain, the algorithm improved the image quality in each slice. In areas of the brain near the sinuses, the algorithm improved the image quality, although there was residual geometric distortion near the sinuses as expected. For slices farther from isocenter, the concomitant gradient correction becomes increasingly important. The algorithm also behaves well in cardiac and abdominal scans. Oblique scans of the coronary arteries show no flow-related degradation of the deblurring. All of the scans were fat suppressed; the algorithm leads to incomplete deblurring without fat suppression, as with most spiral deblurring methods.

DISCUSSION

B0 field inhomogeneity and concomitant gradient fields can cause image degradation in non-Cartesian MRI. We propose an efficient algorithm for simultaneous B0 inhomogeneity and concomitant field correction in non-Cartesian MRI. Our algorithm is a fast alternative to CP reconstruction with the off-resonance phase term approximated by Chebyshev polynomials. Conventional B0 map acquisition can be inaccurate in some applications. We demonstrated that the proposed algorithms can be employed for combined semiautomatic correction and concomitant field correction to address this problem.

The proposed algorithms are computationally efficient. The interpolation coefficients are object-independent and can be precalculated and used for any sequence with a given readout length and k-space trajectory. If a precalculated table is not available, calculation of a table can be done in approximately 1 second, assuming a range of −200 to 200 Hz in Δω(r), −200 to 200 Hz in Δω_c(r), and 15 Chebyshev bins. This table calculation could be performed before or during data collection, because it requires only the scan prescription. Since the number of base images required is proportional to the off-resonance frequency range, linear concomitant field correction (23) is incorporated to reduce the computational cost. Incorporation of linear B0 inhomogeneity correction (27) is also desirable in many applications to reduce computational cost when the B0 inhomogeneity is not highly non-linear. To reduce the number of gridding operations, we also interchange the sequence of the gridding operation and multiplication of modified sampling time when calculating base images. Reconstruction on the scanner’s AMD Opteron-based computer requires approximately 1 second for a 256×256 image and approximately 4 seconds for a 512×512 image with 15 base images. The program deblurs data from individual channels in parallel using different processor cores. Additional speedups are likely with further code optimization.

During our experiments, we found that automatic off-resonance correction (24,28) and semiautomatic off-resonance correction (17) can sometimes achieve comparable results to combined B0 inhomogeneity and concomitant field correction when the concomitant gradient field is mild or moderate. The rationale behind this is that, in automatic or semiautomatic correction, the off-resonance phase ϕ(Δω(r),Δω_c(r),t) is approximated as a spatially-dependent angular off-resonance frequency times t , which is equivalent to approximating t_c in Eq. [3] as αt, where α is a constant.

This approximation is reasonable for the spiral gradients designed in our sequences, because the latter part of the spiral gradients has constant magnitude. Consequently, automatic correction and semiautomatic correction can achieve modest simultaneous B0 inhomogeneity correction and concomitant field correction even without explicitly including a concomitant field term. We constrained the searching range of off-resonance frequency within ±50Hz from the acquired B0 map when performing semiautomatic correction on acquired data sets. A least square fit of αt to t_c from our spiral gradient indicates that ±50Hz B0 off-resonance frequency corresponds to about ±100Hz searching range of concomitant field off-resonance frequency. This is sufficient for a moderate concomitant field, especially after incorporating linear concomitant field correction. A larger searching range can be employed in semiautomatic correction to accommodate a stronger concomitant gradient field. Automatic off-resonance correction (24, 28) does not have this issue since the searching range is not constrained by the B0 field map.

We focused on fast CP reconstruction in this paper. The idea of approximating a phase term by a Chebyshev polynomial can be applied to other image reconstruction methods where off-resonance phase is a concern, such as SPHERE (29) and off-resonance correction based on iterative reconstruction (9).

We only discussed the concomitant gradient field arising from spiral gradients in this paper. Since arbitrary time-varying phase terms can be approximated by Chebyshev polynomials, the proposed method should be able to address the concomitant gradient field from a more general readout gradient. The proposed method is also likely capable of addressing other time varying phase terms such as those induced by eddy current effects.

CONCLUSION

We developed a fast conjugate phase reconstruction method for simultaneous B0 field inhomogeneity and concomitant gradient field correction. Our method is based on Chebyshev approximation of the off-resonance phase term. When the acquired B0 field map is unreliable, the proposed method can be efficiently used for combined semiautomatic off-resonance correction and concomitant gradient field correction. The algorithm was demonstrated on data sets acquired by spiral scanning.

Appendix

Appendix A

The approximation to off-resonance phase by a Chebyshev polynomial in t can be expressed as (22):

$$\text{exp}\left(i\varphi \left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),t\right)\right)={\displaystyle \sum _{k=0}^{N-1}\left[c_k\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)T_k\left(t\right)\right]}-\frac{1}{2}{c}_0$$ [A1]

where τ is the readout length and T_k(t) are Chebyshev polynomials defined as

$$T_0=1;T_1\left(t\right)=\left(\frac{2t}{\tau }-1\right),\text{and}{T}_{n+1}\left(t\right)=2\left(\frac{2t}{\tau }-1\right)T_n\left(t\right)-T_{n-1}\left(t\right),n\ge 1,$$ [A2]

and c_k(Δω(r),Δω_c,(r),τ) are coefficients that depend on the strength of B0 field inhomogeneity Δω(r) , the strength of the concomitant gradient field Δω_c(r) , and the readout length τc_K(Δω(r),Δω_c(r),τ) can be calculated as follows (22):

$$c_k\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)=\frac{2}{N}{\displaystyle \sum _{n=1}^N\left[\text{cos}\left(\frac{\pi k\left(n-0.5\right)}{N}\right)\text{exp}\left(i\varphi \left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\frac{\tau }{2}\left(\text{cos}\left(\frac{\pi \left(n-0.5\right)}{N}\right)+1\right)\right)\right)\right]}$$ [A3]

Note the Chebyshev polynomial T_k(t) in Eq. [A2] can be expressed in polynomial form:

$$T_k\left(t\right)={\displaystyle \sum _{l=0}^k{b}_l^{\left(k\right)}}{\left(\frac{2t}{\tau }-1\right)}^l,$$ [A4]

where $b_l^{\left(k\right)}$ is the l^th coefficient term for T_k(t). Combining Eq. [A1], Eq. [A4] and Eq. [6], after mathematical rearrangement, we have:

$$m\left(\mathbf{r}\right)={\displaystyle \sum _{k=0}^{N-1}{c}_k}\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)p_k\left(\mathbf{r}\right)-\frac{1}{2}{c}_0{p}_0\left(\mathbf{r}\right)$$ [A5]

where

$$p_k\left(\mathbf{r}\right)={\displaystyle \sum _{l=0}^k{b}_l^{\left(k\right)}{q}_l\left(\mathbf{r}\right),}$$ [A6]

with

$$q_1\left(\mathbf{r}\right)={\displaystyle {\int }_{t}s\left(t\right)}{\left(\frac{2t}{\tau }-1\right)}^{l}W\left(t\right)\text{exp(i(2}\pi \mathbf{\text{k}}\left(t\right)\cdot \mathbf{r})dt$$ [A7]

Here q_l(r) denotes a series of images which can be reconstructed by replacing the signal s(t) with $s(t){\left(\frac{2t}{\tau }-1\right)}^l$ followed by gridding reconstruction or other fast reconstruction methods. In the theory section, we discussed exchanging the sequence of gridding and multiplication to reduce the number of gridding operations.

Equation [A5], Equation [A6], and Equation [A7] represent a direct implementation of Chebyshev approximation for fast CP reconstruction. A more computationally efficient implementation for fast CP reconstruction is to use the polynomial form of Chebyshev approximation. Note that by combining Eq. [A1] and Eq. [A4] and after mathematical simplification, we have:

$$\text{exp}\left(i\varphi \left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),t\right)\right)={\displaystyle \sum _{k=0}^{N-1}\left[h_k\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right){\left(\frac{2t}{\tau }-1\right)}^k\right]}-\frac{1}{2}{c}_0$$ [A8]

where

$$h_k\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)={\displaystyle \sum _{i=k}^{N-1}\left[b_k^{\left(i\right)}{c}_i\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)\right]}$$ [A9]

Substituting Eq. [A8] into Eq. [6], we have:

$$m\left(\mathbf{r}\right)={\displaystyle \sum _{k=0}^{N-1}{h}_k\left(\mathrm{\Delta }\omega \left(\mathbf{r}\right),\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right),\tau \right)q_k\left(\mathbf{r}\right)-\frac{1}{2}{c}_0{q}_0\left(\mathbf{r}\right)}$$ [A10]

where q_k(r) are same images as defined in Eq. [A7].

Eq. [A10] is another approach for fast CP reconstruction based on Chebyshev approximation. Compared to Eq. [A5], Eq. [A10] is more computationally efficient, since Eq. [A5] involves an additional combination of images. We found both methods achieved equivalent results on the spiral data sets we acquired.

Appendix B

The concomitant gradient field, as given in [4] is:

$$\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right)=\gamma \frac{g_m^2}{4B_0}\left(F_1{X}^2+F_2{Y}^2+F_3{Z}^2+F_{4}XZ+F_{5}YZ+F_{6}XY\right)$$ [B1]

Assuming a fixed scan plan orientation, F₁ through F₆ and Z are constant throughout the rest of this derivation. Adding an offset of A, B, C to the scan plane in the X, Y, and Z axes leads to [B2]:

$$\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right)=\gamma \frac{g_m^2}{4B_0}\left(\begin{array}{c}{F}_1{\left(X+A\right)}^2+F_2{\left(Y+B\right)}^2+F_3{\left(Z+C\right)}^2+F_4\left(X+A\right)\left(Z+C\right)+\hfill \\ F_5\left(Y+B\right)\left(Z+C\right)+F_6\left(X+A\right)\left(Y+B\right)\end{array}\right)$$ [B2]

Expansion of the squared terms in [B2] leads to [B3]

$$\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right)=\gamma \frac{g_m^2}{4B_0}\left(\begin{array}{c}{F}_1\left(X^2+2XA+A^2\right)+F_2\left(Y^2+2YB+B^2\right)+F_3\left(Z^2+2ZC+C^2\right)\hfill \\ +F_4\left(XZ+XC+AZ+AC\right)+F_5\left(YZ+YC+BZ+BC\right)\\ +F_6\left(XY+XB+AY+AB\right)\end{array}\right)$$ [B3]

Collecting all the terms in [B3] into 3 categories (constant, linear, and squared) leads to [B4]

$$\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right)=\gamma \frac{g_m^2}{4B_0}\left(\begin{array}{c}\left[F_1{X}^2+F_2{Y}^2+F_3{Z}^2+F_{4}XZ+F_{5}YZ+F_{6}XY\right]+\hfill \\ \left[F_{1}2XA+F_{2}2YB+F_{4}XC+F_{5}YC+F_{6}XB+F_{6}AY\right]+\\ [F_1{A}^2+F_2{B}^2+F_3{C}^2+F_{3}2ZC+F_{4}AC+F_{4}AZ+\\ F_{5}BC+F_{5}BZ+F_{6}AB]\end{array}\right)$$ [B4]

Concomitant linear correction applies a linear fit to the concomitant field, with a spatially constant f₀ term and gradients in the X and Y directions. The constant term is used to demodulate the raw data, and the gradients to warp the k-space trajectory. Mathematically:

$$\mathrm{\Delta }{\tilde{\omega }}_c(\mathbf{r})=f_0+\alpha X+\beta Y$$ [B5]

$$s^{\prime }\left(t\right)=s\left(t\right)\text{exp}\left[i2\pi f_0{t}_c\right],k_x^{\prime }\left(t\right)=k_x\left(t\right)+\alpha t_c,k_y^{\prime }\left(t\right)=k_y\left(t\right)+\beta t_c$$ [B6]

The linear fit will completely remove off-resonance phase due to the 2^nd and 3^rd square brackets in [B4], as those terms are either constant or linear in X or Y. The remaining terms in [B4], given in [B7], are identical to [B1]. These terms in [B7], which are all square and cross terms, will be partially corrected by the linear fit, principally by the center frequency term.

$$\mathrm{\Delta }{\omega }_c\left(\mathbf{r}\right)=\gamma \frac{g_m^2}{4B_0}\left(F_1{X}^2+F_2{Y}^2+F_3{Z}^2+F_{4}XZ+F_{5}YZ+F_{6}XY\right)$$ [B7]

For a fixed scan plane orientation, the linear concomitant correction completely removes any blurring due to phase resulting from off-center terms ( the 2^nd and 3^rd square brackets in [B4]). The residual blurring pattern is the same as that of an isocenter scan that has been linear corrected.