Abstract
Purpose Quantitative susceptibility mapping (QSM) is increasingly used for clinical research where oblique image acquisition is commonplace but its effects on QSM accuracy are not well understood.
Theory and Methods The QSM processing pipeline involves defining the unit magnetic dipole kernel, which requires knowledge of the direction of the main magnetic field with respect to the acquired image volume axes. The direction of is dependent upon the axis and angle of rotation in oblique acquisition. Using both a numerical brain phantom and in-vivo acquisitions in five healthy volunteers, we analysed the effects of oblique acquisition on magnetic susceptibility maps. We compared three tilt correction schemes at each step in the QSM pipeline: phase unwrapping, background field removal and susceptibility calculation, using the root-mean-squared error and QSM-tuned structural similarity index (XSIM).
Results Rotation of wrapped phase images gave severe artefacts. Background field removal with PDF gave the most accurate susceptibilities when the field map was first rotated into alignment with . LBV and V-SHARP background field removal methods gave accurate results without tilt correction. For susceptibility calculation, thresholded k-space division, iterative Tikhonov regularisation and weighted linear total variation regularisation all performed most accurately when local field maps were rotated into alignment with before susceptibility calculation.
Conclusion For accurate QSM, oblique acquisition must be taken into account. Rotation of images into alignment with should be carried out after phase unwrapping and before background field removal. We provide open-source tilt-correction code to incorporate easily into existing pipelines: https://github.com/o-snow/QSM_TiltCorrection.git.
Introduction
The acquisition of oblique image slices, or an oblique slab or volume in 3D MRI, is common in clinical practice to facilitate radiological viewing of brain MRI. For example, axial slices are often aligned along the subcallosal line for longitudinal studies that require consistent repositioning of acquired images1. Alternatively, slices may be aligned perpendicular to the principle axis of the hippocampus for accurate hippocampal volume measurements and sharper hippocampal boundary delineation2. Oblique slices are also acquired to reduce image artefacts from, for example, eye motion resulting in localised blurring around the eyes and ghosting along the phase encode direction3. Note that acquiring oblique slices does not require the subject to rotate their head, as only the acquisition volume is tilted.
Quantitative susceptibility mapping4–6 utilises the information in the (conventionally discarded) phase component, ϕ(r), of the complex MRI signal from a gradient echo (GRE) sequence to calculate the tissue magnetic susceptibility, χ. A typical QSM pipeline includes three key steps: 1) phase unwrapping of wraps present due to ϕ(r) being constrained to the [–π,π) interval, 2) background field removal separating the local field perturbations due to internal χ sources inside the volume of interest (e.g. the brain), ΔBint(r), from unwanted background field perturbations due to external sources, ΔBext(r), and 3) a local-field-to-χ(r) calculation to solve an ill-posed inverse problem: where is the Fourier transform, its inverse, B0 the magnetic field in Tesla, and dz(k) the z-component of the magnetic dipole in k-space (see Equation 5).
Calculation of dz(k) requires knowledge of the ‘z’ direction of the main magnetic field, B0, with respect to the image volume acquired. Therefore, oblique acquisition must be taken into account within the QSM pipeline otherwise incorrect χ estimates arise, as suggested by a preliminary study7 and our preliminary data8. With the increase in clinical applications of QSM9,10, accuracy in χ estimates for oblique acquisition, typical in clinical protocols, is of paramount importance in ensuring smooth translation of QSM into clinical practice. However, accurate QSM accounting for oblique acquisition is non-trivial and there are a number of techniques proposed to account for oblique acquisition in QSMn–14 including the most common methods of rotating the k-space dipole or the image volume into alignment with B0. The effect of these proposed tilt correction techniques on susceptibility values has not been evaluated and it is not known at which point in the QSM pipeline these techniques should be applied. Furthermore, it is not clear what is the optimal method for taking oblique acquisition into account in the QSM pipeline: simply defining the dipole at an angle (see DipK or DipIm below) has been shown to be non-optimal7. Therefore, the research presented here is the first quantitative and comparative evaluation of correction methods for oblique acquisition in QSM. We used a numerical phantom to carry out a comprehensive analysis of the effect of oblique acquisition on each step of the QSM pipeline, and propose three tilt correction schemes, analysing their effects on susceptibility values when applied at different points in the QSM pipeline. We also acquired several images, in five healthy volunteers, with volumes tilted at different angles and performed the same analysis of the effects of tilting and correction schemes in vivo. We provide open-source tilt-correction code at https://github.com/o-snow/QSM_TiltCorrection.git that uses the header information from NIfTI15 format images to correctly orient image volumes and account for tilted acquisition for accurate QSM.
Theory
To accurately model the magnetic dipole kernel required for the field-to-χ calculation and, in some cases, for background field removal, it is necessary to know where the magnetic field B0 lies in acquired MRI images. Defining the two coordinate systems of interest7 as the acquired image frame and the scanner frame , the main magnetic field can be written as B0,im and B0sc in the image and scanner frames, respectively:
In the case of non-oblique acquisition, the coordinate systems are aligned and in the image frame (Figure 1, left).
For the local field, ΔBint(r), to χ(r) calculation (Equation 2), the magnetic dipole kernel must be calculated. Throughout this paper, references will be made to the dimensionless k-space dipole, dz(k) (Figure 1, middle row), and the dimensionless ‘image-space dipole’ defined in image space and Fourier transformed into k-space, dz,im(k) (Figure 1, bottom row), kernels defined as follows16,17: with the unit vector of k, is the unit vector of B0, V is the voxel volume, θ is the angle between and , the unit vector of r in image space, where , and is the Fourier transform. The periodicity of the Fourier transform constrains the boundaries of k-space, resulting in the dipole pattern becoming fixed along those boundaries. This causes a rotated image-space dipole to appear twisted, sheared or distorted (Figure 1, bottom row). It is possible to obtain the direction of relative to the (tilted) image axes from the image headers (e,g. DICOM or Nifti format) and, therefore, to correctly calculate the magnetic dipole kernel using either equation 5 or 6.
Methods
To determine the optimal method for taking oblique acquisition into account in the QSM pipeline we investigated three proposed tilt correction schemes, and, for comparison, an uncorrected analysis pipeline (Figure 2):
RotPrior: rotation of the oblique image into alignment with the scanner frame prior to phase unwrapping, background field removal or the susceptibility calculation method. In this method, the dipole is defined in k-space in the scanner frame (using equation 5)
DipK: the image is left unaligned to the scanner frame and the dipole used is defined in k- space in the oblique image frame (using equation 5). This is the default tilt correction method implemented in popular QSM toolboxes14,18,19. However, this method often requires the user to input the corrected B0 direction, which is optional in many of these toolboxes.
DipIm: the image is left unaligned to the scanner frame and the dipole used is defined in image-space in the oblique image frame (using equation 6).
NoRot: the oblique image is left unaligned to the scanner frame, and the k-space dipole is mistakenly defined in the scanner frame (using equation 4) and is thereby misaligned to the true magnetic field direction, B0. This is the uncorrected method which can easily result from users failing to input (the correct) B0 direction.
These schemes are the general names of the methods that we applied at different points in the pipeline (i.e. before phase unwrapping, background field removal, or susceptibility calculation) and for different methods or algorithms. For example, as no dipole kernel is necessary for phase unwrapping (and we substitute the dipole operations illustrated in Figure 2 with -orientation-independent unwrapping operations), we have called the only two schemes appropriate before unwrapping RotPrior and NoRot where the image volume is rotated prior to unwrapping and after, respectively.
All rotations were carried out about the x-axis (u-axis) to simulate single oblique acquisition, the y-axis (v-axis) for confirmation, and about the y = x axis (v = u axis) to simulate double oblique acquisition. Rotations were undertaken using FSL FLIRT20 with trilinear interpolation. To facilitate comparisons, all images left in the image-frame after correction (DipK, DipIm and NoRot) were rotated back into alignment with the scanner axes (see black arrow in Figure 2). Unless stated otherwise, all processing and analysis operations were carried out using MATLAB (MathWorks Inc., Natick, MA, USA).
Numerical Phantom Investigations
Multi-echo (TE = 4,12, 20, 28 ms) magnitude and phase images, from a numerical phantom21, with (originally) no phase wraps or background fields present, were used to independently investigate the effect of the three tilt correction methods (described above), and no correction, on each step in the QSM pipeline.
We carried out these investigations with two image volumes: one unpadded with the original matrix size l64×205 ×205 and a second volume padded to 357 × 357 × 357. The padded matrix size was chosen as the long diagonal of the initial volume (padded to a cube: 205 × 205 × 205) and rounded up to the nearest odd integer. This was to ensure that none of the original frequency coefficients of the unit dipole field were cut off due to rotations about any of the three axes. An odd matrix size meant that there was a true centre of rotation correctly located within a single central voxel.
Numerical Phantom: Susceptibility Calculation
Local field maps, ΔBint(r), were calculated from a non-linear fit22 over all echo times (for the most accurate field estimates23) of the complex data set created by combining the magnitude and background field-free phase images. These local field maps, obtained from the supplied raw numerical phantom data, were free of any (synthetic) background fields or phase wraps, and therefore allowed investigation of the effect of oblique acquisition on χ calculation alone. To simulate oblique acquisition, local field maps were rotated between ±45° in 5° increments. All tilt correction methods described (and no correction) were compared for three χ calculation methods chosen to cover the two main approaches: direct non-iterative solutions (in k-space) and iterative solutions (in image-space).
The first method tested was direct, thresholded k-space division (TKD)24,25 (from open-source software) where a modified dipole kernel was generated in k-space with values below a threshold, δ = 2/3, replaced by the signed threshold value:
The dipole was originally defined according to DipK and DipIm and then always thresholded in k-space. Susceptibility underestimation was corrected by multiplication with a correction factor, cχ(δ), calculated according to26.
The second and third χ calculation methods aim to iteratively solve for χ through the minimisation of where M is a binary mask, W is a weighting term and R (χ) is the data regularisation term that reflects some prior information about χ. Iterative Tikhonov regularisation25,27 (open-source) was chosen as it has performed well in a variety of QSM applications including outside the brain28–30. It was applied with , a regularisation parameter α = 0.003 (chosen through an L-Curve analysis31), and W reflecting the spatially varying noise, and was also corrected for χ underestimation26. Weighted linear total variation regularisation (from the FANSI toolbox18,32) with R(χ) = α|∇χ|1, α = 6.31 × 10-5 (chosen through an L-Curve analysis) and W the magnitude of the complex data32 was also tested. This method was chosen as total variation based iterative approaches were shown to produce the most accurate susceptibility maps in the 2019 QSM challenge 2.033.
Mean χ values were calculated in five deep gray matter regions of interest (ROIs): the caudate nucleus, globus pallidus, putamen, thalamus and red nucleus. All susceptibility maps were compared using the root mean square error (RMSE) and QSM-tuned structural similarity index (XSIM)34 metrics relative to the supplied ground truth susceptibility map at 0°.
Numerical Phantom: Background Field Removal
For the background field removal step, local field maps from the numerical phantom required the addition of synthetic background fields, which were then removed following the three different tilt-correction methods (and no correction). After background field removal, the susceptibility maps were calculated from the resulting field maps using the χ calculation method found to be optimal in the above described assessment.
To investigate the effect of tilt correction schemes on the background field removal step, synthetic background fields, ΔBext(r) (Figure 3, bottom left), were added to the local field maps used in the Methods: Numerical Phantom: Susceptibility Calculation section. The background fields were calculated using the forward model, i.e. through a convolution, formulated as a multiplication in Fourier space, between the unit magnetic dipole field and a head-shaped susceptibility map16,17:
χhead is the head-shaped susceptibility map, with soft tissue (−9.4ppm) and bone (−11.4ppm)7,35 regions obtained by thresholding the magnitude (sum of squares over all echoes) and a pseudo-CT36–38 image, respectively (Figure 3). The magnitude and pseudo-CT images were padded from their original matrix size of 164 × 205 × 205 to 512 × 512 × 512 to ensure edge effects from the periodic Fourier transform were minimised around the volume of interest. These synthetic background fields were then cropped back to their original matrix size and added to the local field maps obtained previously simulating a total field map, ΔB(r) = ΔBint(r) + ΔBext(r). To simulate oblique acquisition, total field maps were rotated between ±45° in 5° increments.
To remove the synthetic background fields, ΔBext(r), from the tilted total field maps three different state-of-the-art background field removal methods39 were used, based on their widespread use, robustness and accuracy39,40. Projection onto dipole fields (PDF)41, from the MEDI Toolbox14, was used following tilt correction with all three correction schemes and no correction because PDF is orientation dependent, i.e. it uses the dipole field dz(k) (Equation 5). Laplacian boundary value (LBV)42, from the MEDI Toolbox14, and variable-kernel sophisticated harmonic artifact reduction for phase data (V-SHARP)43, from STI Suite19, were tested with RotPrior and NoRot only, as LBV and V-SHARP are orientation independent methods, i.e. they do not use the dipole field. Following rotation back into the reference frame (equivalent to RotPrior for the susceptibility calculation step), susceptibility maps were calculated from all local field maps using iterative Tikhonov regularisation (regularisation parameter α = 0.003) as this was found to be optimal. Susceptibility maps were compared using RMSE and XSIM34 relative to the ground truth susceptibility map at 0°.
Numerical Phantom: Phase Unwrapping
To investigate the effect of tilt correction on phase unwrapping, the synthetic background fields added in the previous section (Methods: Numerical Phantom: Background Field Removal) induced phase wraps when the phase was constrained to the [–π, π) interval, which were then unwrapped. Susceptibility maps were then calculated from these unwrapped field maps using background field removal and susceptibility calculation algorithms found to be optimal in the experiments described in the previous sections (Methods: Numerical Phantom: Susceptibility Calculation and Background Field Removal).
To investigate the effect of tilt correction on phase unwrapping, phase wraps were introduced into the wrap-free numerical phantom images via the additional synthetic background field described above. From each total field map at each angle, ΔB(r) = ΔBint(r) + ΔBext(r), multi-echo unwrapped phase images were simulated via scaling the tilted total field maps at each echo time according to ϕ(r, TE)=γ·TE · ΔB(r).
At every tilt angle, a complex data set (S) was made from the multi-echo magnitude images (M) and simulated phase images (ϕ) using , which constrained the phase to the range [—π, π), resulting in phase wraps. A wrapped total field map was calculated via a non-linear fit over all echo times22, which then underwent phase unwrapping using the commonly used Laplacian14,44, SEGUE45 (https://xip.uclb.com/product/SEGUE) and ROMEO46 (https://github.com/korbinian90/ROMEO/releases) techniques with the NoRot and RotPrior tilt correction methods. After rotating all the unwrapped images back into the reference frame (equivalent to RotPrior for the background field removal step and susceptibility calculation step), susceptibility maps were then calculated with PDF41 background field removal and susceptibility calculation using iterative Tikhonov regularisation (regularisation parameter α = 0.003) as we found these to provide optimal results.
Investigations In Vivo
In Vivo: MRI Acquisition
3D gradient-echo brain images of five healthy volunteers were acquired on a 3T Siemens Prisma-Fit MR system (National Hospital for Neurology and Neurosurgery, London, UK) using a 64-channel head coil across a range of image volume orientations. Note that the volunteers did not tilt their head but remained in the same position throughout the experiment. The image volume was tilted about the x-axis, as this is the most common in clinical practice, from −45° to +45° in 10° increments, with the reference image at 0° representing a non-oblique acquisition (subject 5 angles were between ±15°). Each image volume was acquired in 3 min 23 s with TR = 30ms; TEs = 4.92, 9.84, 14.76, 19.68, 24.60 ms; 1.23 mm isotropic voxels; FOV = 256 × 192 × 216.6 mm; Matrix Size = 208 × 156 × 176; bandwidth = 280 Hz/pixel; flip angle 15°; 6/8 partial Fourier along PE1 and PE2; on the scanner ASPIRE coil combination47; monopolar readout; and GRAPPAPE1 acceleration = 3 (FE direction: A>>P, PE1 direction: R>>L, PE2 direction F>>H).
In Vivo: Phase Unwrapping
For all angles/volumes, a total field map and a noise map were obtained using a non-linear fit of the complex data22 from the MEDI toolbox14. A brain mask was created using the brain extraction tool (BET)48 with default settings applied to the final echo magnitude image of the reference 0° volume for a conservative brain mask estimate. This brain mask was then registered to all oblique acquired volumes to maintain consistency. As with the numerical phantom, both the RotPrior and NoRot correction schemes were applied. Residual phase wraps were then removed using Laplacian unwrapping44, SEGUE45 and ROMEO46. To investigate the effect of the correction schemes on this step in the pipeline, unwrapped total field maps were rotated back into the reference frame (equivalent to RotPrior for the background field removal step and susceptibility calculation step) and susceptibility maps were created using PDF background field removal and susceptibility calculation with iterative Tikhonov regularisation (α = 0.017 chosen through an L-Curve analysis).
As in the numerical phantom, and also due to very slight unavoidable changes in subject position between scans, the unwrapped field maps and susceptibility maps were registered into the reference image space to facilitate comparisons of results in vivo. To carry out this registration, the magnitude image (added in quadrature over all echoes) for each angle was rigidly registered to the 0° magnitude using NiftyReg49 resulting in a transformation matrix per angle/volume, which was applied to bring all angles/volumes into the same common reference space.
In Vivo: Background Field Removal
The ROMEO unwrapped field maps described above (in Methods: In Vivo: Phase Unwrapping) for volumes at all angles, and prior to any registrations or rotations, were used to investigate the effect of oblique acquisition on background field removal. ROMEO was chosen as it has been shown to outperform46 PRELUDE50 and BEST-PATH51. As for the numerical phantom, for each field map, at each angle, background fields were removed using PDF41 with all tilt correction schemes and no correction, and using LBV42 and V-SHARP43 with only RotPrior and NoRot. For all three background field removal methods, the brain mask was eroded by 4 outer voxels52. RotPrior was performed twice: with mask erosion either before or after the rotation to compare the effects of interpolation, particularly along the boundaries of the field map on PDF and V-SHARP, as it is known that boundary effects arise in these background field removal methods39.
For comparison purposes, after rotation and registration of the local field maps back into the reference frame (equivalent to RotPrior for the susceptibility calculation step), susceptibility maps were calculated from the local field maps using iterative Tikhonov regularisation (α = 0.017, chosen with an L-Curve). Local field maps and susceptibility maps were compared with RMSE and XSIM metrics (XSIM only for the susceptibility maps) averaged across all subjects relative to the 0° reference image.
In Vivo: Susceptibility Calculation
To investigate the effect of oblique acquisition on the χ calculation step in the pipeline, we used the local field maps following ROMEO unwrapping and LBV background field removal (described in the previous section Methods: In Vivo: Background Field Removal), prior to any registrations or rotations. LBV was chosen as it is orientation independent, thereby allowing our analysis to focus on the effect of oblique acquisition and the different correction schemes on the χ calculation step alone. The three tilt correction schemes (and no correction) were compared using the same three χ calculation methods as for the numerical phantom: TKD, iterative Tikhonov regularisation with a regularisation parameter α = 0.017 from an L-curve analysis, and weighted linear TV with a regularisation parameter α = 6.31 × 10-5 found also from an L-curve.
The resulting susceptibility maps were transformed into the reference space as described in the earlier section Methods: In Vivo: Phase Unwrapping. The same ROIs as in the numerical phantom were investigated and were obtained by registering the EVE53 magnitude image with the reference magnitude image (at the first echo time) and applying the resulting transformation to the EVE ROIs. Mean χ values were calculated in these ROIs for all tilt angles and all correction schemes. RMSE and XSIM measures averaged across all subjects were also used to compare the susceptibility maps.
Results
Numerical Phantom
All numerical phantom results shown here are for rotations of image volumes about the x-axis with unpadded matrices. Note that acquisitions tilted about the y-axis and y=x axis as well as images with padded matrices all gave similar results (see Supporting Figures S1–3). We chose to display these results as the padded matrix size leads to increased computation time, which is not recommended in a practical setting, and the x-axis is the most common axis of rotation for oblique acquisition.
When wrapped phase images are rotated prior to phase unwrapping with the correction scheme RotPrior, artefacts arise for both Laplacian and SEGUE unwrapping (Figure 4). SEGUE appears to fail with RotPrior as it incorrectly identified phase wraps and thereby removed a portion of the mask (Figure 4c).
When using PDF for background field removal, RotPrior is the most accurate method, and the largest errors arise from DipIm and NoRot (Figure 5). Striping artefacts are present in the local field map from the DipK method. LBV and V-SHARP are shown to be largely unaffected by oblique acquisition.
Figure 6 summarises the mean susceptibility in the Caudate Nucleus and Thalamus, alongside XSIM measurements across all angles for all three χ calculation methods (RMSE results are shown in Supporting Figure S4, and are similar). TKD and iterative Tikhonov methods are most accurate with RotPrior, and least accurate when the dipole is misaligned to the main magnetic field (NoRot). Weighted linear TV is relatively robust to oblique acquisition with RotPrior and DipK performing similarly. However, DipK shows more variability in χ at the ROI level than RotPrior. Weighted linear TV with DipIm fails at non-zero angles and NoRot results in the largest errors. Example susceptibility maps are shown in Figure 7, highlighting the widespread χ errors that arise when the magnetic dipole is defined incorrectly (NoRot).
In Vivo
In vivo, Laplacian and SEGUE phase unwrapping with RotPrior have the same image artefacts as in the numerical phantom (not shown here) compared with NoRot, with incorrect identification of phase wraps when wrapped field maps are rotated prior to phase unwrapping.
Figure 8 shows that PDF background field removal is most accurate with RotPrior and least accurate with DipIm followed by NoRot, confirming the results obtained in the numerical phantom (Figure 5). The average XSIM differences between tilt correction schemes in vivo (Figures 8 and 9) are smaller than in the numerical phantom (Figures 5 and 6), most likely due to issues inherent to in-vivo acquisition including motion and much greater noise. Striping artefacts are present in the DipK method for PDF in the local field maps prior to re-orientation for comparison purposes (Fig. 8c). Rotation interpolation obscures these artefacts in the in-vivo images. LBV and V-SHARP are shown to be largely orientation independent in the in vivo case as expected.
When RotPrior was performed with mask erosion before rotating the total field map, artefacts arose along the boundaries of the PDF local field map (See Supporting Figure S5). PDF performs more robustly if mask erosion is carried out after rotation, whereas V-SHARP appears to perform equally well in both scenarios.
Figure 9 shows the effect of all tilt correction schemes on susceptibility calculation in vivo and confirms that NoRot results in the largest susceptibility errors and that RotPrior is consistently the most robust tilt correction method compared to other methods. Both RotPrior and DipK perform better than DipIm, in agreement with results in the numerical phantom (Figure 6). Difference images (Figure 10) also confirm those obtained in the numerical phantom (Figure 7). Subtle effects found in several of the numerical phantom ROIs (Figure 6) were not apparent in vivo.
Discussion
We have shown that oblique acquisition must be accounted for in the QSM pipeline to ensure accurate susceptibility estimates throughout the brain. For all background field removal and susceptibility calculation methods tested, if the magnetic dipole kernel is left misaligned to the direction (NoRot), which can arise from user error in popular QSM toolboxes, then significant susceptibility errors result.
Through the analysis of the effect of tilted acquisition on a numerical phantom and five healthy volunteers in vivo, we have shown that any rotations that are applied to a wrapped field map prior to phase unwrapping will result in incorrect unwrapping, using Laplacian and SEGUE unwrapping techniques, and subsequent artefacts in the resulting QSMs. Results indicate that, for PDF background field removal, rotating the image into the scanner frame and using a k-space dipole defined in the scanner frame (RotPrior correction method) provides the most accurate susceptibility maps. If no image rotations are desired, due to unwanted interpolation effects, LBV or V-SHARP are recommended as they are largely unaffected by oblique acquisition. Both TKD and iterative Tikhonov susceptibility calculation methods provide the most accurate results when local field maps are rotated into alignment with the scanner axes and a k-space dipole, defined in the scanner frame, is used (RotPrior). The same conclusion holds for weighted linear TV, but susceptibility calculation can be carried out in the oblique image frame without any rotations provided the correct B0 direction is used in defining the k-space dipole (DipK correction method). We therefore recommend rotating the total field map into alignment with the scanner frame after phase unwrapping but prior to background field removal.
Both the numerical phantom and in vivo results indicate that when wrapped phase images are rotated prior to phase unwrapping (with the correction scheme RotPrior), artefacts arise for both Laplacian and SEGUE unwrapping methods. This is probably due to interpolation errors along phase wraps (Figure 4). Therefore, any phase unwrapping must be carried out in images left in the same orientation as acquired, with rotations only being applied afterwards to avoid artefacts.
When using PDF for background field removal, numerical phantom and in vivo results show that RotPrior consistently provides the most accurate susceptibility maps while NoRot performs the worst (Figures 5 and 8). Striping artefacts arise in local field maps in both the numerical phantom and in vivo when using PDF with the DipK method. First identified by Dixon, E.7, these striping artefacts are present due to the violations in circular continuity when defining the tilted dipole in k-space and using the inverse discrete Fourier transform to transform the susceptibility maps into image-space (which enforces periodicity, see Supporting Figure S7). Striping artefacts arise from regions of high susceptibility changes, such as on the brain boundaries (Figures 5c and 8c). DipIm also resulted in poor background field removal, most likely due to fitting to the incorrect twisted or sheared unit dipole field (bottom row of Figure 1). To avoid artefacts and robustly achieve background field removal with PDF, total field maps must be rotated into alignment with the scanner frame prior to PDF background field removal as it is then possible to use the non-oblique dipole, which doesn’t violate circular continuity (Supporting Figure S8). We showed that LBV and V-SHARP were mostly unaffected by oblique acquisition, with the differences between zero and non-zero tilt angles arising solely from rotation interpolation effects.
Given that RotPrior is the most accurate method for PDF, the typically necessary mask erosion must be carried out after rotation into the reference space. Artefacts that arise along the boundaries of the local field map if erosion is carried out prior to rotation (Supporting Figure S5a and S5b) probably arise from distortion of the dipolar background fields due to interpolation at the edges. In contrast, V-SHARP does not show substantial differences with mask erosion before v. after rotation which suggests that the interpolation may not substantially affect the harmonic nature of the background fields on which this method relies.
We found TKD and iterative Tikhonov regularisation to be affected by oblique image orientation and most accurate with RotPrior. We showed weighted linear TV to be relatively robust to oblique acquisition, however RotPrior is still maintained to be the most consistently robust method (Figure 9). For all susceptibility calculation methods tested, a unit dipole field misaligned to the main magnetic field (NoRot) leads to artefacts and substantial errors in susceptibility maps. The subtle differences between correction methods found in the numerical phantom ROIs (Figure 6) were not apparent in vivo probably due to noise, motion and the expected variability in susceptibility maps over repeated acquisitions27,54.
At non-zero tilt angles, XSIM (and RMSE in Supporting Figure S4) values have a respectively high (or low) baseline level arising from rotation (no matter how small the angle) and registration interpolations, and imperfections inherent to in vivo acquisition. Additional discrepancies in similarity measures between rotated and unrotated QSM may also have occurred due to slight differences in repeated acquisitions as evidenced by the slightly larger (0° to 5°) XSIM discrepancy in vivo than in the numerical phantom (Figures 8 and 9 compared with Figures 5 and 6). The effect of these discrepancies has been minimised by averaging across all five healthy volunteers.
Our results also indicate that at larger tilt angles in the numerical phantom, DipK is less accurate than RotPrior, therefore, for certain imaging applications including cardiac imaging and pelvic imaging where large tilt angles of up to 45° are often required, tilt correction is likely to be essential for accurate susceptibility mapping.
Therefore, we recommend accounting for oblique acquisition by using the RotPrior tilt-correction method before background field removal since this method gave the most accurate susceptibility maps in both the numerical phantom and in vivo. If desired, the susceptibility map can be rotated back into the original orientation after susceptibility calculation to facilitate comparison with other (processed) images. In the future, it would be interesting to carry out a similar investigation of tilt correction methods for total field inversion (TFI)55,56, in which background field removal and susceptibility calculation are combined into a single step. Due to the reliance of TFI on the correct definition of the magnetic dipole, we would expect to see similar results, with RotPrior being more robust to oblique acquisition. However, this would need to be confirmed with further work. It is possible to build an alternative rotation-free pipeline of methods relatively unaffected by oblique acquisition (such as LBV and weighted linear TV) but those methods must be checked to ensure true independence of image orientation. However, such an approach limits the choice of methods for the steps in the QSM pipeline, which could lead to suboptimal susceptibility maps. For example, LBV’s highly specific boundary approximations can be easily violated making it easier to simply rotate the field maps in some cases. These aspects must be considered carefully when designing a QSM pipeline.
Conclusions
Oblique acquisition must be accounted for in the QSM pipeline to avoid artefacts and erroneous susceptibility estimates. We recommend rotating the total field map into alignment with the scanner frame after phase unwrapping but before background field removal (and then rotating the final susceptibility map back into the original orientation). Alternatively, a QSM pipeline relatively robust to oblique acquisition can be built from a more limited number of image-orientation-independent methods (e.g. LBV or V-SHARP for background field removal and weighted linear TV for susceptibility calculation). However, care must be taken in weighing up the minimal effects of image interpolation (from tilt correction rotations) versus choosing from a smaller range of methods that are orientation independent and may not be as robust oblique acquisition, as they may not be as accurate nor optimal for a given data set. It would also be vital to ensure a chosen method is independent of slice orientation, which may require further investigation. Our recommended correction scheme ensures that all methods developed for each stage of the QSM pipeline can be used and optimised.
We provide an open-source MATLAB function that can be easily incorporated into existing MATLAB-based QSM pipelines and used both to align oblique 3D volumes with the scanner frame and to rotate image volumes (e.g., susceptibility maps) back into the original oblique image orientation. This function is freely available at: https://github.com/o-snow/QSM_TiltCorrection.git. Our results allow accurate susceptibility maps to be obtained from oblique image acquisitions. This is an important step in translating QSM into clinical practice.
Supporting Information Figures
Acknowledgments
Oliver Kiersnowski’s work was supported by the EPSRC-funded UCL Centre for Doctoral Training in Intelligent, Integrated Imaging in Healthcare (i4health) (EP/S021930/1). John Thornton received support from the National Institute for Health Research University College London Hospitals Biomedical Research Centre. Karin Shmueli and Anita Karsa were supported by European Research Council Consolidator Grant DiSCo MRI SFN 770939. We would like to thank Carlos Milovic for his help with the FANSI toolbox, and the healthy volunteers for their time.
Footnotes
New in vivo data has been acquired in four more healthy volunteers, which has confirmed numerical phantom results in the original version. Further clarifications have been made for ease of understanding throughout the manuscript.