Abstract
Objective
To experimentally characterize the effectiveness of a gradient nonlinearity correction method in removing ADC bias for different motioncompensated diffusion encoding waveforms.
Methods
The diffusion encoding waveforms used were the standard monopolar Stejskal–Tanner pulsed gradient spin echo (pgse) waveform, the symmetric bipolar velocitycompensated waveform (symvc), the asymmetric bipolar velocitycompensated waveform (asymvc) and the asymmetric bipolar partial velocitycompensated waveform (asympvc). The effectiveness of the gradient nonlinearity correction method using the spherical harmonic expansion of the gradient coil field was tested with the aforementioned waveforms in a phantom and in four healthy subjects.
Results
The gradient nonlinearity correction method reduced the ADC bias in the phantom experiments for all used waveforms. The range of the ADC values over a distance of ± 67.2 mm from isocenter reduced from 1.29 × 10^{–4} to 0.32 × 10^{–4} mm^{2}/s for pgse, 1.04 × 10^{–4} to 0.22 × 10^{–4} mm^{2}/s for symvc, 1.22 × 10^{–4} to 0.24 × 10^{–4} mm^{2}/s for asymvc and 1.07 × 10^{–4} to 0.11 × 10^{–4} mm^{2}/s for asympvc. The in vivo results showed that ADC overestimation due to motion or bright vessels can be increased even further by the gradient nonlinearity correction.
Conclusion
The investigated gradient nonlinearity correction method can be used effectively with various motioncompensated diffusion encoding waveforms. In coronal liver DWI, ADC errors caused by motion and residual vessel signal can be increased even further by the gradient nonlinearity correction.
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Introduction
Diffusionweighted imaging (DWI) remains a valuable tool in liver lesion detection and there is an ongoing interest in ADC mapping for tumor staging and therapy monitoring [1, 2]. However, cardiac and respiratory motion from the subject during the standard monopolar Stejskal–Tannerpulsed gradient spin echo (pgse) diffusion encoding gradients can cause intravoxel dephasing and therefore signal loss [3,4,5,6,7,8]. Since the apparent diffusion coefficient (ADC) is proportional to the diffusioninduced signal decay when diffusion encoding is applied, any undesired motioninduced signal loss will, therefore, cause an overestimation of the ADC. The close proximity of the left liver lobe to the heart makes the left liver lobe especially susceptible to cardiac motioninduced signal loss [7].
Respiratorytriggered scans are most frequently used to overcome issues caused by the breathing motion of the subject in liver DWI [3, 4, 9,10,11,12]. Cardiactriggered scans have also been proposed to overcome the issues caused by cardiac motion. However, cardiactriggered DWI scans are associated with low acquisition efficiency and require the optimisation of the trigger delay at each individual subject [13]. In addition, it has been recently suggested that a single trigger delay might not be adequate for removing artifacts induced by cardiac motion and vessel pulsation throughout the whole liver [14, 15]. Therefore, motioncompensated diffusion gradient encoding waveforms have recently been proposed as alternatives for motionrobust liver DWI [15,16,17,18,19,20,21,22,23,24].
The simplest motioncompensated diffusion encoding waveform that has been proposed is a symmetric bipolar waveform (symvc). An example of the symvc waveform design is given in Fig. 1b. In this waveform, the gradient first moment (\({m}_{1}=\gamma \int tG\left(t\right)dt\)) is zero, leading to no phase accumulation and hence no signal loss when spins move with constant velocities during diffusion encoding. When an echo planar imaging (EPI) readout is used in combination with the symmetric bipolar waveform, the unavoidable deadtime between the excitation and refocusing pulses means that this waveform is not optimal in terms of echo time. Various other diffusion encoding waveforms that use an asymmetric design to optimize the echo time have recently been proposed [15,16,17,18,19,20]. The ODGD [19] and CODE [16] formulations use constrained optimization algorithms to find arbitrarily shaped echo time optimized diffusion encoding waveforms.
When using asymmetric diffusion encoding waveforms (an example of one such waveform is given in Fig. 1c), spatially dependant concomitant gradient fields, which are characterized by Maxwell’s equations, can induce dephasing and therefore signal loss over the field of view in DWI sequences [25, 26]. These concomitant gradients must therefore be accounted for either in the pulse sequence design or with postprocessing. One issue with both the symmetric and asymmetric motioncompensated m_{1}nulled waveforms is that the vessel suppression capability is decreased and T_{2} shine through is increased [15, 20]. Therefore, vessels in liver DWI acquired with motioncompensated waveforms can appear bright, which can confound lesion detection and cause errors in ADC quantification. To address the bright vessel signal problem, adding m_{1} > 0 back to the motioncompensated waveforms (symvc or asymvc) has been proposed to find a balance between motion sensitivity and vessel signal suppression [15, 20, 27, 28]. An example of one such asymmetric partial velocitycompensated waveform is given in Fig. 1d.
Another source of acquisitionrelated errors in ADC estimation is gradient nonlinearity, which is a static effect specific to the gradient coil design. With imaging gradients, gradient nonlinearity deviates the relationship between the spatial position and acquired data frequencies, leading to spatial image warping [29, 30]. With diffusion gradients, gradient nonlinearity causes the bvalue to be different than intended and can cause ADC quantification errors larger than 10% in clinically relevant field of views [31]. Specifically, since the calculation of the bvalue in DWI relies on a spatially uniform diffusionweighted gradient, any spatially nonuniform gradient leads to errors in the calculation of the bvalue and, therefore, to ADC bias [31].
Previous works have proposed methods to correct for gradient nonlinearity effects in diffusion MRI [31,32,33,34]. Bammer et al. formulated a general mathematical framework characterizing errors from gradient nonlinearity [32, 33]. The gradient field was approximated with a spherical harmonic expansion, which was then used to derive a gradient nonlinearity tensor, enabling the correction of both the magnitude and direction of the gradients. Although comprehensive, the above approach requires the acquisition of at least six diffusion directions. This is commonplace in diffusion tensor imaging (DTI) experiments, but in many body diffusion applications DWI protocols based on three orthogonal directions are typically acquired. Malyarenko et al. have proposed a simplified scheme to correct ADC bias due to gradient nonlinearity in DWI protocols employing three orthogonal directions [33]. The scope of the correction proposed by Malyarenko et al. was limited to DWI applications and was, therefore, not suitable for DTI applications [33]. The correction involved the rotation of the gradient nonlinearity tensor into the diffusion gradient frame, where the spatial bias of the bmatrix was approximated by the Euclidean norm. Although it has been theoretically predicted to work for arbitrary waveform designs, none of the previous works to correct ADC bias due to gradient nonlinearity in body DWI have been experimentally applied to any diffusion encoding gradient waveforms other than the conventional monopolar Stejskal–Tanner pgse diffusion encoding gradient waveform. Therefore, it is yet to be experimentally verified that the gradient nonlinearity correction method based on the work by Malyarenko et al. will be equally as effective for motioncompensated diffusion encoding waveforms as for the conventional monopolar diffusion encoding waveform.
The aim of the present work is to assess the effectiveness of the gradient nonlinearity correction for different motioncompensated diffusion encoding waveforms and specifically to characterize the effectiveness of the gradient nonlinearity correction in removing ADC bias due to gradient nonlinearity for different motioncompensated diffusion encoding waveforms in the presence of motion effects in the context of ADC mapping in the liver.
Methods
Motioncompensated diffusion encoding waveforms
The following diffusion encoding waveforms were presently investigated (Fig. 1).
Pulsed gradient spin echo waveform (pgse)
The standard monopolar Stejskal–Tanner pgse diffusion encoding waveform (Fig. 1a) was first used as it remains the most frequently used diffusion encoding waveform in body DWI.
Symmetric velocitycompensated waveform (symvc)
In the symmetric bipolar waveform (symvc), shown in Fig. 1b, the gradient first moment (\({m}_{1}=\gamma \int tG\left(t\right)dt\)) is zero.
Asymmetric velocitycompensated waveform (asymvc)
When an echo planar imaging (EPI) readout is used in combination with the symmetric bipolar waveform, the unavoidable deadtime between the excitation and refocusing pulses means that this waveform is not optimal in terms of echo time. An asymmetric velocitycompensated diffusion encoding waveform with optimized echo time was then used. The adoption of asymmetric diffusion encoding waveforms to minimize the echo time requires the additional consideration of concomitant gradient effects, which do not cancel out as in the case of symmetric diffusion encoding waveforms [25, 35, 36]. Different strategies have been previously performed to define velocitycompensated diffusion encoding with optimal TE and correction of concomitant field effects, including strategies that approximate the concomitant phase as a linear phase variation [16] and strategies that null the concomitant gradientinduced phase accrual [19]. Specifically, by optimizing the bvalue formulation for arbitrary gradient waveforms with appropriate constraints on system hardware, sequence timings, concomitant fields, and gradient moments, the optimized diffusionweighting gradient waveform design (ODGD) defined a TEoptimized concomitant field corrected motioncompensated waveform [19]. However, since the optimization for the ODGD waveform must typically be done offline (not at the scanner), a file that contains the optimized waveform has to be loaded onto the scanner, reducing in general the flexibility for protocol optimization.
The presently employed asymmetric velocitycompensated (asymvc) waveform builds upon the ODGD formulation [19]. Without concomitant field correction, the optimal ODGD waveform is always a collection of trapezoids. Additionally, for a wide range of relevant sequence parameters in the employed bvalue range, the number of trapezoid lobes and polarities of an optimized waveform do not change. Therefore, the optimization of the waveform can be reduced to the optimization of the trapezoid lengths, which can be integrated into the pulse sequence environment and carried out efficiently online at the scanner. When concomitant field correction is added, the ODGD formulation yields optimal waveforms that are not trapezoidal. Alternatively, by making the slope of the trapezoidal waveform a variable and adding the concomitant field correction as a constraint, firstorder concomitant field correction is applied by adjusting the slopes of the trapezoids until the concomitant field term is nulled. This second concomitant correction approach is similar to Zhang et al. [15] and Szczepankiewicz et al. [23, 24] and is suitable for the waveform optimization at the scanner.
The design of the asymvc waveform is as follows. The asymvc waveform is assumed to have four lobes, two before and two after the refocusing pulse. Figure 1e shows a labelled diagram of the asymvc waveform, where \({\delta }_{1}, {\delta }_{2}, {\delta }_{3}\) and \({\delta }_{4}\) are the durations of the plateaus of each lobe, \({\tau }_{\mathrm{s}}\) is the duration of the nonadjustable slope, \({\tau }_{\mathrm{as}}\) is the duration of the adjustable slope and \({\tau }_{\mathrm{pre}180}\) and \({\tau }_{\mathrm{post}180}\) are the durations available before and after the refocusing pulse, respectively. The nonadjustable slopes are constant and are equal to the maximum slew rate. The other constraints for the timing of each lobe are the m_{0}, m_{1} and concomitant constraints. The variables to solve for, in terms of the amount of time before and after the refocusing pulse, are the duration of each plateau of each lobe and the slope in between the first and second lobes. The constraints are therefore
where the integrals of \({m}_{0}\) and \({m}_{1}\) are over the entire duration of the gradient waveforms, given by \(\tau\), and \(A1\) and \(A2\) are the time periods before and after the refocusing pulse, respectively. Equation 5 is the constraint imposed to minimize concomitant fields. With these five constraints, the five variables \({\delta }_{1}, {\delta }_{2}, {\delta }_{3}, {\delta }_{4}\) and \({\tau }_{\mathrm{as}}\) can be found.
To obtain the asymvc waveform, the bvalue equation is expressed in terms of the gradient lobe timings, which is then input into the scanner software environment and the echo time is optimized for a given bvalue. \({\tau }_{\mathrm{pre}180}\) and \({\tau }_{\mathrm{post}180}\) are found from each socalled “target echo time” in the optimization loop. As the asymvc waveform is restricted to trapezoidal waveforms and not arbitrarily shaped waveforms, the optimization procedure is not burdened by long computational times unsuitable for the direct scanner interface. Therefore, asymvc waveform can be designed onthefly on the scanner, instead of having to input a text file containing the offline optimized waveform whenever scan parameters are changed. The resulting waveform has an asymmetric bipolar design as shown in Fig. 1c.
A further description about the differences between the implementation of the asymvc, and representatives of existing optimized velocitycompensated waveforms such as the ODGD and the CODE [16] waveforms are included in the Supplementary Material.
Asymmetric partially velocitycompensated waveform (asympvc)
Compared to the pulse field gradient encoding (pgse) waveform, the symvc and asymvc waveforms, while decreasing sensitivity to motion, also decrease the vessel suppression capability and increase T_{2} shine through [15, 20]. Therefore, vessels in liver DWI acquired with symvc and asymvc can appear bright, which can confound lesion detection and cause errors in ADC quantification. The fact that the vessels can appear bright is caused in part by nulling the gradient first moment, m_{1}. By adding a small m_{1} > 0 back to the motioncompensated waveforms (symvc or asymvc) significant reduction of vessel signals can be achieved [15, 20, 27, 28]. The key idea of these methods is to find an appropriate m_{1} > 0 value that balances sensitivity to motion and vessel signal suppression. In the present work, small pgse gradients corresponding to the desired value of m_{1} were added to two of the gradient lobes of the asymvc waveform for all bvalues except for the maximum bvalue, similar to Zhang et al. [15]. At the maximum bvalue, in the present implementation, the lengths of each of the gradient lobes and the amplitude of the asymvc gradients were adjusted until the desired value of m_{1} was obtained without changing the bvalue. Although not fully TEoptimized, the above asymmetric partial motioncompensated waveform (asympvc) can be again designed onthefly at the scanner, facilitating protocol optimization. The asympvc waveform is shown in Fig. 1d. A detailed analysis of the employed asymmetric partially velocitycompensated waveform design is provided in the supplementary material, with a diagram shown in Fig. S1.
Gradient nonlinearity correction
The gradient nonlinearity correction (gnl) framework as proposed by Malyarenko et al. was adopted [33]. The method is briefly described here: in the case of linear gradients, the gradient strength is uniform across the whole magnet bore and does not depend on position. In the case of nonlinear gradients, the gradient strength is dependent on position within the magnet bore and spurious gradients orthogonal to the applied gradient will be produced. Firstorder spatial variations of the gradient can be described by a nonlinearity tensor \({\varvec{L}}({\varvec{r}})\), where
where \({{\varvec{g}}}_{0}\) is the gradient at isocenter, \({\varvec{r}}\) is the spatial position from isocenter and \({\varvec{g}}\left({\varvec{r}}\right)\) is the spatially varying gradient. \(\partial {B}_{z}^{{g}_{i}}\) is the z component of the magnetic field as a result of applying the gradient in the i direction and \({g}_{i0}\) is the i component of the gradient \({{\varvec{g}}}_{0}\). The spatially dependent magnetic field produced by each gradient coil can be described by a spherical harmonic expansion. Using the spherical harmonic coefficients provided by the manufacturer, the spatially varying magnetic field produced by each coil and, therefore, the gradient nonlinearity tensor \({\varvec{L}}\) can be derived. The spatially varying b matrix is given by \({{\varvec{b}}}^{^{\prime}}\left({\varvec{r}}\right)={\varvec{L}}\left({\varvec{r}}\right){{\varvec{b}}}_{0}{{\varvec{L}}}^{{\varvec{T}}}\left({\varvec{r}}\right)\), where \({{\varvec{b}}}_{0}\) is the nominal b matrix and includes b components due to the diffusion gradients, imaging gradients and imaging cross terms. The correction map is then obtained from (with further details given in [33])
for each diffusion direction k, where \(Tr\left\{.\right\}\) is the trace operator, \(F\) is the Frobenius norm and \({{\varvec{u}}}_{k}\) is a unit vector that defines the kth DW direction in the gradient coil coordinates. The corrected bvalue map is given by
where \({b}_{c}^{k}\left({\varvec{r}}\right)\) is the corrected spatially dependent bvalue map for each diffusion direction k, \({b}_{n}^{k}\) is the nominal bvalue and \({C}^{k}({\varvec{r}})\) is the correction map. Example correction maps are given in Fig. 2.
The code used to calculate the correction maps was provided by the vendor and was integrated into the reconstruction platform on the scanner. The individual bvalues and directions as well as the design parameters of the gradient coil and slice orientations were input to the reconstruction process, after which the correction maps were generated as a reconstruction output of the diffusion scans. The ADC maps were computed in MATLAB from the correction maps and the diffusionweighted data exported from the scanner.
MRI measurements
All MRI measurements were performed on a 3 T Ingenia Elition X scanner (Philips Healthcare, Best, Netherlands), which has a maximum gradient amplitude of 45 mT/m and a maximum slew rate of 200 T/m/s.
Phantom measurements
Phantom validation measurements were performed to assess the performance of the gradient nonlinearity correction with the pgse, symvc, asymvc and asympvc waveforms. A cylindrical phantom of length 200 mm and diameter 115 mm containing 31.5 mmol/L of NiCl_{2}6H_{2}O was placed inside an extremity coil at isocenter and axial slices were acquired with a ssEPI sequence in the S/I direction to investigate the gradient nonlinearity in the S/I direction. A separate scan in which sagittal slices in the L/R direction were acquired to investigate the gradient nonlinearity in the L/R direction was performed. More information is given in the supplementary material. Gradient nonlinearity was not measured in the A/P direction due to the unsuitability of the dimensions of the phantom for measuring in this direction.
For all waveforms in the axial slice experiment, seventeen slices were acquired with a FoV of 125 × 125 mm^{2}, a voxel size of 2 × 2 × 6 mm^{3} and a 2.4 mm slice gap. The parallel imaging acceleration factor R = 2, effective bandwidth over phase encode direction = 57 Hz/pixel. Bvalues of [0, 600] s/mm^{2} were used for the ADC calculation, with eight averages per bvalue and three orthogonal diffusion encoding directions along the readout (RO), phase encoding (PE) and slice selection (SS) directions. The echo times were 60/90/90/96 ms for pgse, asymvc, asympvc and symvc, respectively. TR = 8000 ms in all cases and no partial Fourier encoding was used for any of the waveforms. An m_{1} value of 0.1 s/mm was used for the asympvc waveform.
ADC maps were calculated for each individual diffusion direction, as well as for all directions combined. An ROI was drawn in the center of each slice and the mean ADC over the ROI was calculated.
A separate scan was performed to test the performance of the concomitant gradient correction. The same phantom was placed inside the extremity coil and seventeen axial slices were acquired with a FoV of 128 × 128 mm^{2}, a voxel size of 2 × 2 × 6 mm^{3} and a 2.5 mm slice gap. This scan was performed without parallel imaging and also with a parallel imaging factor of R = 2. When a parallel imaging factor of R = 2 is used, the number of acquired kspace lines is reduced in comparison to when no parallel imaging is used. Therefore, the time interval available for diffusion encoding after the refocusing pulse is closer to the time interval before the refocusing pulse, meaning that the diffusion encoding waveforms before and after the refocusing pulse are more symmetrical. Since concomitant gradient effects are affected by the asymmetry of the waveforms, using a parallel imaging factor of R = 2 was expected to reduce the effect of the concomitant gradients. For both scans, TR = 5000 ms, bvalues of [0, 600] s/mm^{2} were used for the ADC calculation, with five averages per bvalue and three orthogonal diffusion encoding directions along the readout (RO), phase encoding (PE) and slice selection (SS) directions. For the scan without parallel imaging, effective bandwidth over phase encode direction = 27.2 Hz/pixel, the echo times were 98/103 ms for asymvc with and without concomitant gradient correction, respectively. For the scan with a parallel imaging factor of R = 2, effective bandwidth over phase encode direction = 54.5 Hz/pixel, the echo times were 89/90 ms for asymvc with and without concomitant gradient correction, respectively.
Using identical scan parameters apart from a TR of 6000 ms, an asympvc scan with an m_{1} value of 0.1 s/mm was performed with a parallel imaging factor of R = 2 and bvalues of [0, 600, 600] s/mm^{2}, with each of the b = 600 s/mm^{2} acquisitions using a different waveform design for inducing m_{1}. Waveform 1 is the design of the lower bvalues, with a separate pgse gradient added on top of the existing asymvc gradient to increase the m_{1}. Waveform 2 is the design of the higher bvalue, meaning that m_{1} is added by changing the relative lengths of the gradient lobes of the asymvc gradients. The above experiment was done to validate that both waveform designs for inducing m_{1} give the same ADC estimation.
In vivo measurements
In vivo measurements were carried out in 4 healthy volunteers (mean age, 34 ± 6 years) using the built in 12channel posterior and 16channel anterior coil. The study was approved by the local ethics commission and all volunteers have consented for their participation in the study.
Two slices of the liver were acquired in the coronal plane with a FoV of 240 × 312 mm^{2}, a voxel size of 3 × 3 × 6 mm^{3}, parallel imaging acceleration factor R = 3, effective bandwidth over phase encode direction = 53.3 Hz/pixel, number of packages = 1, bvalues of [200, 600] s/mm^{2} and corresponding averages of [3, 4]. The bvalue of 200 s/mm^{2} was chosen to minimize any perfusion signal contributions to the ADC estimation. To reduce echo time for better SNR, diffusion encoding directions [− 0.5 − 1 − 1], [1 0.5 − 1], [1 − 1 0.5] were used. These diffusion encoding directions (which shall be referred to from now on as the overplus diffusion directions) were used for all four subjects, giving echo times of 53/80/80/90 ms for pgse, asymvc, asympvc and symvc respectively. The TR was determined by the breathing and cardiac cycle of each volunteer, and was in the range of approximately 3000–6000 ms.
For three out of the four subjects, scans with diffusion encoding along the RO, PE and SS directions (magnet diffusion directions) were additionally performed, giving echo times of 60/88/88/94 ms for pgse, asymvc, asympvc and symvc, respectively. This was not done for the fourth subject due to time constraints and the fact that this subject had lower liver signal, meaning that the increased echo time would have reduced the image quality.
Scans with both cardiac and respiratory triggering (double triggering), as well as scans with only respiratory triggering were performed for both sets of diffusion directions (magnet and overplus). The imaging parameters were identical between the doubletriggered and respiratorytriggered scans.
ADC maps were generated, and subtraction maps were computed from subtracting the ADC maps without any gradient nonlinearity correction from the ADC maps with gradient nonlinearity correction. For the ROI analysis, ROIs were drawn on one slice of the pgse ADC map from the respiratorytriggered scans. These ROIs were then propagated to the subtraction maps for each diffusion encoding waveform. One of the ROIs was drawn in the left liver lobe in a region with an overestimated ADC due to motion corruption, taking care to avoid large vessels. This ROI was chosen to investigate interactive effects of both motion and gradient nonlinearity on the ADC estimation. The other ROI was drawn close to the right inferior side of the liver.
Correction maps
Figure 2 shows correction maps of the gradient nonlinearity correction for the RO, PE and SS diffusion directions. The correction maps calculated from the gradient field spherical harmonics for each diffusion direction were equal between the different diffusion encoding waveforms.
Results
Phantom results
Figure 3 shows the results from the phantom experiment in which slices were acquired in the S/I direction (gradient nonlinearity in the S/I direction). Without the gradient nonlinearity correction, along the S/I direction, the average ADCs exhibit the characteristic gradient nonlinearity curve, in which a local maximum or minimum is at the isocenter, and the mean ADC increases or decreases symmetrically away from the isocenter. After the gradient nonlinearity correction, the pgse waveform yields minimal variation of ADC for each diffusion direction. Before gradient nonlinearity correction, the standard deviation of the ADC (calculated from all directions combined) across all slices is 4.36 × 10^{–5} mm^{2}/s. After gradient nonlinearity correction, this is 1.02 × 10^{–5} mm^{2}/s. The range of the ADC values for pgse over a distance of ± 67.2 mm from isocenter is 1.29 × 10^{–4} mm^{2}/s before gnl correction, and 0.32 × 10^{–4} mm^{2}/s after gnl correction. Similar to the pgse waveform, the asymvc, asympvc and symvc waveforms also show the characteristic ADC variation when no gradient nonlinearity correction is performed. With the gradient nonlinearity correction, the curves were flattened for all waveforms. The asymvc waveform has a standard deviation of ADC (calculated from all directions combined) across all slices of 4.08 × 10^{–5} mm^{2}/s before gnl correction, and 0.75 × 10^{–5} mm^{2}/s after gnl correction. The range of the ADC values for asymvc over a distance of ± 67.2 mm from isocenter is 1.22 × 10^{–4} mm^{2}/s before gnl correction, and 0.24 × 10^{–4} mm^{2}/s after gnl correction. The symvc waveform has a standard deviation of ADC (calculated from all directions combined) across all slices of 3.16 × 10^{–5} mm^{2}/s before gnl correction, and 0.60 × 10^{–5} mm^{2}/s after gnl correction. The range of the ADC values for symvc over a distance of ± 67.2 mm from isocenter is 1.04 × 10^{–4} mm^{2}/s before gnl correction, and 0.22 × 10^{–4} mm^{2}/s after gnl correction. The asympvc waveform shows a slight overcompensation in the RO direction, as the curve looks inverted compared to the case without gradient nonlinearity correction; however, the variation in ADC is still reduced considerably. This waveform has a standard deviation of ADC (calculated from all directions combined) across all slices of 3.54 × 10^{–5} mm^{2}/s before gnl correction, and 0.26 × 10^{–5} mm^{2}/s after gnl correction. The range of the ADC values for asympvc over a distance of ± 67.2 mm from isocenter is 1.07 × 10^{–4} mm^{2}/s before gnl correction, and 0.11 × 10^{–4} mm^{2}/s after gnl correction. For all waveforms, with or without gradient nonlinearity correction, any observable dependence of the ADC variation on the diffusion encoding direction is small. The results from the phantom experiment in which slices were acquired in the R/L direction (gradient nonlinearity in the R/L direction) are shown in the supplementary material Fig. S2.
Concomitant gradient phantom experiment
A simulation of the effect of concomitant gradients on the ADC value in a phantom is given in the Supplementary Material. The difference in ADC for a range of distances from the isocenter is given in the Supplementary Material Fig. S4.
Figure 4 shows the phantom results for the asymvc waveform with and without concomitant gradient correction for two different parallel imaging factors. When no parallel imaging is used, the difference in ADC between a slice approximately 60 mm away from isocentre and a slice at isocentre was approximately 4.9 × 10^{–5} s/mm^{2}. When a parallel imaging factor of 2 was used, this ADC difference was 1.4 × 10^{–5} s/mm^{2}. When no parallel imaging was used, the standard deviation of the ADC of the asymvc with concomitant correction was 3.55 × 10^{–5} s/mm^{2} before gnl correction, and 0.30 × 10^{–5} s/mm^{2} after gnl correction. The standard deviation of the ADC of the asymvc without concomitant correction was 3.14 × 10^{–5} s/mm^{2} before gnl correction, and 2.14 × 10^{–5} s/mm^{2} after gnl correction, indicating that the CG correction is performing well. When a parallel imaging factor of 2 was used, meaning that the waveform was less asymmetric and therefore would not have as severe concomitant gradient effects, the standard deviation of the ADC of the asymvc without concomitant correction was 3.48 × 10^{–5} s/mm^{2} before gnl correction, and 0.93 × 10^{–5} s/mm^{2} after gnl correction. The standard deviation of the ADC of the asymvc with concomitant correction was 3.99 × 10^{–5} s/mm^{2} before gnl correction, and 0.47 × 10^{–5} s/mm^{2} after gnl correction, again remaining stable.
asympvc waveform comparison
Figure 5 shows the phantom results for the comparison between the two different waveforms used to make up asympvc. The maximum difference in ADC between the two waveforms at the same slice was 0.66 × 10^{–5} s/mm^{2} before gnl correction, and 0.65 × 10^{–5} s/mm^{2} after gnl correction.
In vivo results
Figure 6 shows the uncorrected ADC maps, corrected ADC maps, subtraction maps and correction maps for two different scans performed with the pgse waveform, one with double cardiac and respiratory triggering, and one with only respiratory triggering. It is clear that in the left liver lobe region when only respiratory triggering is used, the ADC is overestimated. In the subtraction maps, the left liver lobe region also has high values (with a mean ADC difference of 1.4 × 10^{–5} mm^{2}/s for the respiratorytriggered scans, and 1.0 × 10^{–5} mm^{2}/s for the doubletriggered scans), indicating that the gradient nonlinearity correction has increased the motioninduced overestimated ADC further. ADC maps and subtraction maps from two volunteers are shown in the supplementary material Fig. S5.
Figure 7 shows the uncorrected ADC maps, corrected ADC maps, subtraction maps and correction maps for all waveforms in which only respiratory triggering was used. The diffusion encoding directions were the overplus directions for the first volunteer, and RO, PE and SS for the second volunteer. The ADC in the left liver lobe when the pgse waveform was used is again overestimated. The subtraction map shows that the gradient nonlinearity correction has increased the motioninduced overestimated ADC even further. The motioncompensation capability of the asympvc, asymvc and symvc waveforms reduces the motioninduced overestimation of ADC at the cost of lower SNR, compared to when the pgse waveform was used. Subtraction maps show lower values in the left liver lobe for asympvc, asymvc and symvc waveforms compared to the pgse waveform. Since bright vessels will also have a high ADC, this is also further increased by the gradient nonlinearity correction in certain regions, depending on the correction map. This effect can also be seen in the subtraction maps. The effect of gnl correction on overestimated ADC is explained further in the supplementary material Fig. S3. ADC maps and subtraction maps from two volunteers are shown in the supplementary material Fig. S6.
Figure 8 shows the ROI analysis performed on the subtraction maps for all subjects for all waveforms. The plots show the mean value of the subtraction map over the ROI, and the error bars show the standard deviation over the ROI. In all subjects, the asympvc, asymvc and symvc waveforms had a lower mean subtraction map value in the left liver lobe region compared to the pgse waveform. In the right liver lobe region, the differences between all waveforms are typically smaller.
Discussion
The present work applies the gradient nonlinearity correction using the spherical harmonic expansion of the gradient coil field [33] to different motioncompensated diffusion encoding waveforms. Despite known disadvantages such as the prolonged echo time and reduced SNR, occurrence of bright vessel signal, concomitant effects if asymmetrical waveforms are used, motioncompensated diffusion encoding waveforms remain an alternative approach to more accurate ADC quantification. The gradient nonlinearity correction has, to the best of our knowledge, only been applied to the standard monopolar pgse diffusion encoding waveform and the present work, therefore, showed the feasibility of using the gradient nonlinearity correction with motioncompensated diffusion encoding waveforms.
Although not the main focus of the present study, the present work also proposes methods for online computation of nearTEoptimal motioncompensated and partial motioncompensated waveforms with concomitant field correction, allowing flexibility in protocol optimization [37, 38]. The presently employed methods utilize the knowledge from the offline TEoptimized gradient waveform design as proposed by PeñaNogales et al. [19]. However, the implementation of the gradient nonlinearity correction was found to be applicable to different motioncompensated waveform designs, and should also be applicable to any diffusion encoding waveform designed offline or onthefly as in [37, 38]. The asympvc achieved partial velocity compensation by adjusting the lengths of the lobes for the maximum bvalue, and by adding small pgse gradients to the asymvc waveform for the lower bvalues. The difference in design between the bvalues was done to mimic the design of Zhang et al. [15] and for ease of implementation.
Figure 2 shows the correction maps for the RO, PE and SS directions. Since the employed correction procedure holds for a general gradient waveform in which the polarity is reversed at TE/2 [33, 39] (to implicitly account for the effect of the spin echo RF pulse [40]), the correction maps were the same for all waveforms.
The phantom results show that the gradient nonlinearity correction can considerably reduce ADC bias for all waveforms. All waveforms showed some residual fluctuation after correction, which can be attributed to higher order gradient nonlinearity effects, or other factors such as eddy currents or residual concomitant gradient effects and is a subject for further investigation. The asympvc waveform showed a slight overcompensation in the RO direction, as the curve looks inverted compared to the case without gradient nonlinearity correction. This could be a result of residual concomitant gradient effects. When the current nonlinear gradient correction method is employed, similar levels of residual fluctuation of ADC values were observed across all waveforms, suggesting that the employed method can correct for gradient nonlinearity just as well in the case of the more complicated motioncompensated waveforms.
The gnl correction can be affected by nongnl related ADC errors. Evidence of such interactions between gnl correction and nongnl related ADC errors was shown in Figs. 6 and 8 in the left liver lobe and regions with bright vessels.
The phantom experiment for the asymvc concomitant gradient correction in Fig. 4 showed that even when no parallel imaging factor was used, therefore making the waveform more asymmetric and increasing the impact of the concomitant fields, the CGcorrected waveform remained stable. The CGuncorrected waveform showed larger deviations in ADC away from isocentre when no parallel imaging factor was used. When a parallel imaging factor of 2 was used, the variation in ADC was lower; however, it was still larger than the CGcorrected waveform, indicating that the concomitant correction performed well.
The phantom experiment for the two different waveform designs for asympvc in Fig. 5 showed that the ADC calculated from both waveform designs was almost identical, justifying the usage of using different designs for the lower and higher bvalues in the asympvc waveform.
The subtraction map ROI analysis in Fig. 8 showed that for all subjects, the pgse waveform had a higher subtraction map value in the left liver lobe in a region which has overestimated ADC due to motion. The higher subtraction map value highlights the interaction between the gradient nonlinearity correction and ADC overestimation. In the right liver lobe region, the differences between the pgse waveform and the motioncompensated waveforms were smaller, as this is a region that is not as heavily affected by motion. The asympvc waveform also typically had a higher subtraction map value in the left liver lobe than the asymvc waveform, which could be due to the fact that the increase in m_{1} also increases sensitivity to motion. In the right liver lobe region, the motioncompensated waveforms also had a larger standard deviation over the ROI in the subtraction map than pgse, which could be explained by the lower SNR and bright vessel signal. Therefore, the presented results suggest that to get accurate ADC quantification, one has to take gradient nonlinearity, motion sensitivity, vessel signal suppression and the impact of these on each other into account.
The present work has some limitations. First, the phantom measurements were performed without any temperature control. In Malyarenko et al. [33], an ice water phantom was used to keep the temperature constant throughout the experiment. Second, only a distance of ± 67.2 mm from isocentre was measured with the presently used phantom. Despite these limitations, the pgse results without the gradient nonlinearity correction showed the characteristic curve of ADC bias, which was flattened by the gradient nonlinearity correction. Third, only four healthy volunteers were scanned in the present work. Further work would be required to fully characterize the interaction between gradient nonlinearityinduced and motioninduced bias in liver ADC quantification of subjects with different motion patterns and liver ADC variations.
Conclusion
The gradient nonlinearity correction based on the spherical harmonic expansion of the gradient coil field was used in conjunction with different motioncompensated waveforms, thereby showing the feasibility of the usage of the gradient nonlinearity correction with gradient waveforms other than the standard monopolar pgse diffusion encoding waveform. The phantom results showed a reduction in ADC bias in all waveforms after the gradient nonlinearity correction was applied. The in vivo results showed an interaction between the gradient nonlinearity correction, ADC overestimation in the left liver lobe due to motion, and bright vessel signal from the motioncompensated waveforms. This suggests that to get accurate ADC quantification, one has to take gradient nonlinearity, motion sensitivity, vessel signal suppression and the impact of these on each other into account.
Abbreviations
 DWI:

Diffusionweighted imaging
 ADC:

Apparent diffusion coefficient
 gnl:

Gradient nonlinearity
 EPI:

Echo planar imaging
 ODGD:

Optimized diffusionweighting gradient waveform design
 pgse:

Pulsed gradient spin echo
 symvc:

Symmetric velocity compensated
 asymvc:

Asymmetric velocity compensated
 asympvc:

Asymmetric partial velocity compensated
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Acknowledgements
The present work was supported by Philips Healthcare. The authors would like to thank Dr. Thomas Chenevert and Dr. Dariya Malyarenko for helpful discussions.
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by SM. The first draft of the manuscript was written by SM and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Johannes M. Peeters and Kilian Weiss are employees of Philips Healthcare. Dimitrios C. Karampinos receives grant support from Philips Healthcare.
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McTavish, S., Van, A.T., Peeters, J.M. et al. Gradient nonlinearity correction in liver DWI using motioncompensated diffusion encoding waveforms. Magn Reson Mater Phy 35, 827–841 (2022). https://doi.org/10.1007/s10334021009816
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DOI: https://doi.org/10.1007/s10334021009816