Evaluation of eddy current distortion and field inhomogeneity distortion corrections in MR diffusion imaging using log-demons DIR method

Diffusion-weighted (DW) magnetic resonance (MR) imaging provides image contrast that differs from conventional MR techniques by producing image contrast that is dependent on the molecular motion of water. Diffusion tensor imaging (DTI) is a powerful tool to characterize the magnitude, anisotropy, and orientation of the diffusion tensor. DTI is used to quantify three-dimensional (3D) spatial properties of water molecular diffusion processes, and as such probe tissue structure at a microscopic scale. Because DTI provides both anatomical and functional information it is a valuable tool to analyze the radiation-induced changes involved in radiation therapy treatments (Dong et al 2004, Chang et al 2014).

Clinical diffusion-weighted imaging (DWI) typically use single-shot echo-planar imaging (EPI) acquisitions that are sensitive to static magnetic field inhomogeneities, resulting in geometric distortions (Jezzard and Balaban 1995, Haselgrove and Moore 1996, Griffiths 1999, Doran et al 2005, Minjie et al 2008, Treiber et al 2016, Froeling et al 2017). 'Field inhomogeneity distortions' are spatial and intensity distortions due to B₀ field inhomogeneity induced by magnetic susceptibility variations. These field inhomogeneity distortions apply to all DWI and become more prominent with greater field strengths and are most pronounced along the phase encoding (PE) direction (Jezzard and Balaban 1995, Haselgrove and Moore 1996, Griffiths 1999, Doran et al 2005, Minjie et al 2008, Treiber et al 2016, Froeling et al 2017). In the past, several methods have been proposed to correct the field inhomogeneity distortions in diffusion MRI. In 1995, an unwarping method is proposed based on 'field map' (Jezzard and Balaban 1995) in which magnetic field inhomogeneity or field map is determined to correct field inhomogeneity distortion. A very different approach (Morgan et al 2004) is proposed to correct field inhomogeneity distortion using the reversed gradient method (Chang et al 2014) with additional acquisitions. Phase labeling for additional coordinate encoding (PLACE) is also proposed (Xiang and Ye 2007) which requires two EPI acquisitions. All these previously proposed correction methods for field inhomogeneity distortions require additional acquisitions, lengthening the scan time.

In addition to field inhomogeneity distortion, because DWI uses rapidly changing diffusion sensitized gradients, eddy currents are induced in either the scanner or patient which warps the image and creates two undesired occurrences: time-varying gradients and shifts in the main magnetic field. In diffusion-weighted EPI, eddy current-induced gradients and field shifts cause distinctive geometric distortions in the PE direction of the resulting images (Haselgrove and Moore 1996, David Doty 1998, Jezzard et al 1998, Thirion 1998, Hill et al 2001, Morgan et al 2004, Zhuang et al 2006, Xiang and Ye 2007, Vercauteren et al 2008, 2009, Spees et al 2011, Lombaert et al 2014, Okan Irfanoglu et al 2019). Several techniques have been proposed to deal with eddy current distortions such as active shielding gradients (David Doty 1998), pre-emphasis currents and post-processing techniques (Spees et al 2011).

Field inhomogeneity and Eddy current distortions alter anatomy, making accurate measurements more challenging. In order to correct these distortions, deformable image registrations (DIRs) have been used with image registration and/or field map-based registration (Minjie et al 2008, Wang et al 2017). Image registration-based correction methods use a deformable registration algorithm and score their registration using mutual information (MI). The field map-based method calculates the deformation field of the moving image and then 'unwarps' the image using this field. Another approach is the TOPUP method (Chang and Fitzpatrick 1992, Andersson et al 2003) which requires two acquisitions with reversed PE gradients of the same image. In theory the distortions of the image are identical but in opposite directions and the technique attempts to correct to the middle. Currently, the TOPUP method is widely adopted and used in neuroimaging, although the TOPUP method requires additional scans.

For this study, we will be investigating the feasibility of the log-demons DIR method to correct eddy current and field inhomogeneity distortions without the need for additional acquisitions, and quantitatively evaluate its performance on phantom and human data.

Correction of eddy current distortions and field inhomogeneity distortions using the log-demons algorithm involved taking advantage of different MR acquisition sequences. Because eddy current distortions are dependent on the strength and direction of the diffusion gradient (Jezzard et al 1998, Zhuang et al 2006, Okan Irfanoglu et al 2019), we take advantage of the fact that the images should not have any eddy current distortions and only field inhomogeneity distortions. T1-weighted images do not have any field inhomogeneity distortions and are therefore used as an anatomical landmark when correcting these distortions. Registering the $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ and $b\,{\text{ = }}\,{\text{1}}\,{\text{000}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ to the T1-weighted image would correct for any field inhomogeneity and eddy current distortions present. For our study, we measure the best transformation based on the CC, and MI between two images. Two categories of transformations are available to register images, rigid body model and deformable registration model. In this study, DIR is applied to DWI. We study a few deformable registration techniques using the affine model, demons model and a log-demons model (Thirion 1998, Vercauteren et al 2008, 2009, Lombaert et al 2014), respectively.

Here we describe several techniques to test the efficacy of the log-demons registration technique for its ability to preserve anatomical and functional information.

Based on geometric models, registration techniques are broken down into two categories, rigid transformations and deformable registrations (non-rigid) (Hill et al 2001), in which a transformation, T, will be applied to the moving image to align with a static image. Rigid body transformations can be expanded to 9–12 parameters to allow for affine registration by adding three scaling parameters and three shear parameters (Maes et al 1997). Deformable models have a different optimization criterion that is locally defined and computed and the deformation is constrained by a regularization term (Antoine Maintz and Viergever 1998). For this study, we focus on the demons and log-demons deformable registration algorithms as described below.

The demons algorithm is based upon a thought experiment from James Clerk Maxwell in 1867 that suggested how the 2nd law of thermodynamics may be violated. Maxwell assumed a gas composed of two different particles and separated by a semi-permeable membrane that contains 'demons' that can differentiate the particles and allow them to diffuse in one direction. The 'demons' allow for unidirectional diffusion of the particles resulting in only blue particles in A and red particles in B, as shown in figure 1.

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Maxwell included the 'demons' into the membrane to generate a greater amount of entropy to avoid any contradiction with the 2nd law of thermodynamics (Thirion 1998). Thirion took this ideology and applied it to image registration to match a moving image to a target image by assuming the contour of an object in the moving image is the membrane, and 'demons' are scattered along that membrane (Thirion 1998). The relationship between the two images can be determined by the energy between them:

$$\begin{equation}\begin{array}{*{20}{c}} {E\left( {F,M,c,v} \right) = {{\left| {\left| {\left( {\frac{1}{{{\sigma _{\text{i}}}}}F - M \cdot c} \right)} \right|} \right|}^2} + \frac{1}{{\sigma _{\text{x}}^2}}{\text{dist}}\left( {v,c} \right) + \frac{1}{{\sigma _{\text{T}}^2}}{\text{Reg}}\left( v \right)}. \end{array}\end{equation} \tag{ 1 }$$

In which F is the fixed image, M is the moving image, ${\sigma _{\text{i}}}{\text{}}\,$ accounts for the noise on the image intensity, $\sigma _{\text{T}}^2{\text{}}\,$ controls the amount of regularization, $\sigma _{\text{x}}^2{\text{}}\,$accounts for the spatial uncertainty and c is the non-parametric spatial transformation (Thirion 1998, Vercauteren et al 2009). The regularization term, ${\text{Reg}}\left( v \right) = {\left| {\left| {\nabla v} \right|} \right|^2}$ and the ${\text{dist}}\left( {v,c} \right) = \left| {\left| {c - v} \right|} \right|$ are implemented to optimize the correlation between two images. The demons algorithm iteratively updates a displacement field, u, by minimizing the energy with respect to u (Thirion 1998, Vercauteren et al 2009) which is applied to equation (1):

$$\begin{equation}\begin{array}{*{20}{c}} {E_{\text{s}}^{{\text{corr}}} = {{\left| {\left| {F - M \cdot v \cdot \left( {Id + u} \right)} \right|} \right|}^2} + \frac{{{{\sigma }}_{\text{i}}^2}}{{{{\sigma }}_{\text{x}}^2}}\left| {\left| {{u^2}} \right|} \right|} \end{array}\end{equation} \tag{ 2 }$$

where s is the current transform and $s*\left( {Id + \boldsymbol{u}} \right)$ is a compositive adjustment.

The demons registration has been improved upon by Vercauteren in which the diffeomorphic transformation is related to the exponential map of the velocity field, v (Vercauteren et al 2008, Lombaert et al 2014). Diffeomorphic demons registration can be extended to represent a total spatial transform in the log-domain. The algorithm takes the ongoing transformations as an exponential of a velocity field v. The log-domain demons algorithm uses Lie group structures to relate diffeomorphic transformation ϕ to the exponential of the velocity field (Maes et al 1997, Antoine Maintz and Viergever 1998, Kristy et al 2017). The calculation of energy in equation (1) is modified to work in the log-domain by performing Gaussian smoothing in the log-domain making ${\text{dist}}\left( {s,c} \right) = \left| {\left| {{\text{log}}\left( {{s^{ - 1}} \cdot c} \right)} \right|} \right| {\text{and}} \, {\text{Reg}}\left( s \right) = {\left| {\left| {{{\Delta \text{log}}}\left( s \right)} \right|} \right|^2}$. Using this equation (1) then becomes

$$\begin{equation}\begin{array}{*{20}{c}} {E\left( {F,M,{\text{exp}}\left( c \right),{\text{exp}}\left( v \right)} \right) = \frac{1}{{2\sigma _{\text{i}}^2}}\left( {{\text{Sim}}\left( {F,M \cdot {\text{exp}}\left( {\text{c}} \right)} \right) + {\text{Sim}}\left( {F \cdot {\text{exp}}\left( { - c} \right),M} \right)} \right) + \frac{1}{{\sigma _{\text{x}}^2}}{\text{dist}}{{\left( {c,v} \right)}^2} + \frac{1}{{\sigma _{\text{T}}^2}}{\text{Reg}}\left( v \right)} \end{array}\end{equation} \tag{ 3 }$$

where $\sigma _{{\text{x,}}\,{\text{T}}}^2$ controls the step size and regularization and $\sigma _{\text{i}}^2{\text{}}\,$controls the deformations of the image geometry (Thirion 1998, Vercauteren et al 2009, Lombaert et al 2014).

3.1. Diffusion Standard Model 128

In order to study the efficacy of the log-demons registration algorithm, we took Diffusion Standard Model 128 phantom (figure 2) (Qalibre 2016) scans using a 3T Seimen's Skyra involving a T1-weighted scan (figure 3), one $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ scan and 20 diffusion-weighted datasets with $b\,{\text{ = }}\,{\text{1}}\,{\text{000}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$. The phantom's main components consist of 30 ml vials of polymer in aqueous solution.

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We rigidly register the T1-weighted image to the DWI, before applying any other registration technique to the images. For the DTI dataset, slices were selected based on geometric landmarks and each registration algorithm was performed on the slices. A log-demons deformable registration algorithm was applied to a 2D slice and compared to affine and demons registration algorithms. Because these distortions mainly occur in the PE direction, the transformations were only applied to the PE direction. The registered images were analyzed using MI for the algorithm's ability to correct eddy current and field inhomogeneity deformations. The phantom contains 13 vials of differing viscosity to study the differences in diffusion.

We tested the algorithms' ability to preserve information and image quality using the mean, and root mean square for 13 regions of interest (vials) in the apparent diffusion coefficient (ADC) and fractional anisotropy (FA) maps.

3.2. MASSIVE dataset

The MASSIVE (multiple acquisitions for standardization of structural imaging validation and evaluation) dataset (Froeling et al 2017) is a brain dataset of a single healthy patient consisting of 8000 diffusion-weighted MR volumes. We aim to expand upon our simplistic analysis of the performance of the log-demons algorithm using the diffusion phantom by using the MASSIVE dataset. The MASSIVE dataset provides diffusion-weighted volumes divided into four sets with both positive (+) and negative (−) diffusion gradient directions and both anterior-to-posterior (AP) and posterior-to-anterior (PA) PE directions as seen in figure 4. To test the algorithms ability to correct eddy current distortions we register the AP+ and AP− $b\,{\text{ = }}\,{\text{1000 s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ images to the $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ image. A perfect registration/intensity correction algorithm would return identical images after registering the distorted images to the b0 volume. We also compared the difference maps of the AP+ and AP− images after registration by the affine, demons, and log-demons algorithm. A similar technique is used to correct field inhomogeneity distortions except the registrations were applied the AP+ and AP− $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ images to the T1-weighted image.

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To validate the registration systems, three experiments were conducted using various datasets. The tests were conducted using (MATLAB 2019a) on a Dell XPS 15 9560 PC with a Windows 10 OS. The registration techniques demonstrated in this chapter are the affine registration, demons deformable registration, and log-demons deformable registration.

4.1. Phantom study

4.1.1. Eddy current distortions

Correction for the eddy current distortions within the phantom involves registering the $b\,{\text{ = }}\,{\text{1000 s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ to the $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ and analyzed the images using the CC and MI to evaluate the similarity between the average of 20 2D slices.

4.1.2. Field inhomogeneity distortions

Correction for the eddy current distortions within the phantom involves registering the $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ to the T1-weighted image and analyzing using equations the CC and MI to evaluate the similarity between the two images.

4.1.3. Diffusion tensor analysis

The 20 $b\,{\text{ = }}\,{\text{1000 s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ slices were first registered to the $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ to remove eddy current distortions and then registered to the T1-weighted slice to correct any field inhomogeneity distortions. The ADC and FA were then calculated in MATLAB using the registered $b\,{\text{ = }}\,{\text{1000}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ slices and $b\,{\text{ = }}\,{\text{0}}\,{\text{s}}\,{\text{m}}{{\text{m}}^{ - {\text{2}}}}$ slice. Distortion correction and preservation of the images can be shown below in figure 5.

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The values were taken from each cylinder were used to determine similarity to the original ADC and FA image.

4.2. MASSIVE study

5.1. Phantom study

A phantom was initially used to test the efficacy of the log-demons algorithm on a phantom with known dimensions. Correction for eddy current and field inhomogeneity distortions are often corrected by registering to an atlas as no true ground truth would exist unless taking a brain slice.

The log-demons algorithm has comparable results with the affine registration method that is commonly used to correct for eddy current distortions in diffusion imaging. As for the field inhomogeneity distortion correction, the log-demons algorithm more easily corrects larger scale deformations as shown in table 2. The MI and CC were improved by, 2.15%, 0.89%, and 39.39% compared to no correction, affine, and demons algorithm respectively when correcting for eddy current distortions. MI and CC were improved by 8.89%, 9.33%, and 9.20% compared to no correction, and affine, and demons algorithm respectively when correcting for field inhomogeneity distortions. This difference in performance may be due to the algorithm's ability to contour the deformations. Because eddy current distortions are smaller in magnitude, the log-demons algorithm cannot accurately create the contour to show drastic improvement over the affine registration.

The log-demons algorithm outperformed both the affine and demons registration techniques in preserving diffusion tensor information as seen in tables 1 and 2 and figures 5–7. The performance of the log-demons algorithm can also be seen in figures 7 and 8, as the log-demons corrected images have the strongest correlation with the ADC and FA values from the 13 regions of interest in the original scan. The average percent difference of the ADC and FA for the log-demons registration technique was substantially better for corrections in the PE direction when compared to affine and demons registration techniques. Limiting the registration to the PE direction yields a more accurate result, which is expected as both distortions primarily occur in the PE.

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Table 1. Similarity measures for the eddy current distortion correction in the phase encoding direction.

Correction method	No correction	Affine PE	Demons PE	Log-demons PE
CC	0.7515	0.7518	0.6745	0.7523
MI	1.9976	2.0255	1.4961	2.0148

Table 2. Similarity measures for field inhomogeneity distortion correction in both the phase and frequency encoding direction.

Correction method	No correction	Affine PE	Demons PE	Log-demons PE
CC	0.8445	0.8054	0.8398	0.8737
MI	1.266	1.2609	1.2624	1.3429

Table 3. The average, standard deviation, and root mean squared error of the percent difference of the original ADC value in the phase encoding direction of the registered image.

	Affine PE	Demons PE	Log-demons PE
Average	3.73%	16.34%	1.75%
STD dev	4.97%	27.37%	2.67%
RMS	4.83%	30.80%	2.58%

Table 4. The average, standard deviation, and root mean squared error of the percent difference of the original FA value in the phase encoding direction of the registered image.

	Original	Affine PE	Demons PE	Log-demons PE
Average	0.00%	30.66%	36.20%	11.67%
STD dev	0.00%	24.12%	30.82%	11.40%
RMS	0.00%	55.32%	60.17%	33.01%

Figure 5 provides a visualization of the performance of the algorithms. In the ADC images, the largest difference can be seen towards the edges of the cylinders and the phantom. As DWI becomes more prevalent in radiation oncology, distortion correction techniques will provide a better understanding of the anatomy and its function. The log-demons algorithm provides an image registration technique that can mitigate geometric and intensity distortions that affect the diffusion tensor. There is a noticeable difference in the affine and log-demons algorithms towards the bottom of the phantom. Preservation of the DTI is especially important when examining regions undergoing radiation treatment to ensure any changes in the region are due to radiation and not caused by geometric deformations. Because the phantom is filled with water, we expect the diffusion to be isotropic, which is why there is little signal from the FA maps. The FA maps of the affine and demons registered images are relatively brighter than the log-demons registered image. This visual analysis is a quick and simple method to ensure a deformation technique is preserving sensitive information. The diffusion phantom model 128 has provided a solid baseline in determining the efficacy of the log-demons algorithm and the rationale to apply these images to brain scans.

5.2. MASSIVE study

The massive dataset provides useful information for benchmarking new algorithms involved in correcting eddy current and field inhomogeneity distortions. The MASSIVE dataset offers another metric to measure distortion corrections and an image registration's effectiveness. Using MASSIVE brain scans we tested the algorithms ability to correct for these distortions without requiring additional atlas scans reducing scan time. When correcting for field inhomogeneity distortions, registering both the AP+ and AP− b = 1000 s mm⁻² images to a b = 0 s mm⁻² images the difference map of the two registered images provides a new metric to analyze distortion correction algorithms. An ideal correction would yield a zero-difference map. Surprisingly the algorithm that produced the highest MI did not yield the lowest average absolute difference.

When examining the field inhomogeneity distortion corrections in tables 5–8 the log-demons algorithm improves the distortions as seen by the increase in MI and a decrease in average absolute difference. The average absolute difference decreased by 0.39%, 8.03%, and 13.19% compared to no correction, and affine, and demons algorithm respectively when correcting for field inhomogeneity distortions. In figures 8 and 9, the difference maps of the registration algorithms, the log-demons algorithm appears to have the least sharp contrast at the edges of the images. Through an assessment of the MI similarity metric and taking advantage of the physical principles of MR distortions, we demonstrate the log-demons algorithm's ability to correct the eddy current distortions and field inhomogeneity distortions intrinsic to DWI. It should be noted that quantitative measures summarized in the tables are average values over large FOVs where errors would often occur at boundaries. Due to its limit, the corrected images should be carefully evaluated, especially at boundaries. Although the algorithm has been shown to work in diffusion tensor brain scans, the log-demons method could be extended for areas in which DWI is beneficial. It should also be noted that, due to the relative smoothness of the field variations in the phantom and brain data presented in this work, the potential errors at boundaries might be underestimated, especially for other anatomical sites with strong field variations.

Table 5. Mutual information from the corrected AP+ or AP− b = 1000 s mm⁻² image and b = 0 s mm⁻² image after correction applied along PE direction in a single slice.

	Original	Affine PE	Demons PE	Log-demons PE
Mutual information	1.4010	1.4091	1.3829	1.4147

Table 6. The average absolute difference from corrected AP+ and AP− b = 1000 s mm⁻² image after correction applied along PE direction in a single slice.

	Original	Affine PE	Demons PE	Log-demons PE
Absolute difference	42.3306	41.4835	42.1386	41.2654

Table 7. Mutual information from the PE registered b = 0 s mm⁻² image and T1-weighted image in a single slice.

	Original	Affine PE	Demons PE	Log-demons PE
Mutual information	1.0244	1.0537	1.0060	1.0280

Table 8. The average absolute difference from corrected b = 0 s mm⁻² image and the T1-weighted image after correction applied along PE direction in a single slice.

	Original	Affine PE	Demons PE	Log-demons PE
Absolute difference	147.1033	229.7809	160.5785	146.6876

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Although the average change of MI of all slices increase with decreasing the average absolute difference for both eddy current and field inhomogeneity distortions, certain slices show opposite changes, which demands further improvement of the log-demons algorithm. Although still preliminary, our study suggests that the performance of the log-demons algorithm could potentially be further improved with optimization of σ_i, σ_T, and σ_x for each slice in the log-demons algorithm, at the cost of extra computational time. This is expected to be further investigated in future work.

The research was focused on the development of a deformable registration algorithm to aid in the improvements of eddy current and field inhomogeneity distortions without requiring additional scans which reduces the total scan time. Using the log-demons algorithm and the similarity metrics, CC and MI we examined the log-demons algorithm to correct the distortions intrinsic to DTI, but also the preserve functional information DTI has to offer.

The proposed method allows for the correction of eddy current distortions and field inhomogeneity distortions without requiring any pre-processing. By taking advantage of the log-demons ability to minimize the energy between two images and mitigate the distortions within the image. The algorithm outperformed the affine and demons registrations in correcting both eddy current and field inhomogeneity distortions in the PE direction and also preserving DTI information. When undergoing a study involving DTI analysis, measures should be taken to ensure that the information is being preserved if applying a transformation. The log-demons algorithm delivers an accurate correction for DTI that can provide a non-invasive measurement of treatment response. Although demonstrated with a DTI phantom study and the MASSIVE dataset, this method could be extended for areas in which DWI is beneficial.

This study may be limited in its structure as the MASSIVE dataset was taken with only one patient. Both the phantom study and MASSIVE dataset were acquired using a smaller field of view when compared to other clinical DWI scans. Scans with a larger field of view have greater geometric distortions due to the non-linearity of the gradient applied in the PE direction. To further strengthen the study, similar techniques may be applied to clinical brain scans to ensure the algorithm performs consistently across multiple patients. Further testing with scans involving a larger field of view may provide roe definitive evidence for the algorithm's performance.

This work was partially funded by the Duke Cancer Institute.

As previously mentioned, TOPUP method is commonly used in neuroimaging today. In this section, a comparison of the TOPUP method and the log-demons DIR method is provided to illustrate the differences in the implementation of the algorithms. To compare the algorithms' ability to correct for field inhomogeneity distortions, massive data shown in figure 9 are also used as below.

Figure A1 shows that the TOPUP method produces similar corrected images according to the difference map. The TOPUP program does seem to have a discrepancy in the right anterior frontal lobe, but overall produces similar images. The field map and deformation maps for the TOPUP method and log-demons DIR method are also calculated and shown in figure A2 below.

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According to figure A2 the TOPUP field map applies the deformation to the field in the upper quadrants of the peripheral brain, whereas the log-demons DIR method applies the transformations in the posterior and anterior directions. This might suggest that the log-demons method could potentially be further improved through optimization of algorithm.

Generally, the TOPUP method requires additional scans to perform distortion correction. In contrast, the methods presented in this work are to correct distortions without acquiring any additional scans. The TOPUP method and the methods presented in this work are fundamentally different and fall into different categories.