Automated Mapping of Residual Distortion Severity in Diffusion MRI

Abstract

Susceptibility-induced distortion is a common artifact in diffusion MRI (dMRI), which deforms the dMRI locally and poses significant challenges in connectivity analysis. While various methods were proposed to correct the distortion, residual distortions often persist at varying degrees across brain regions and subjects. Generating a voxel-level residual distortion severity map can thus be a valuable tool to better inform downstream connectivity analysis. To fill this current gap in dMRI analysis, we propose a supervised deep-learning network to predict a severity map of residual distortion. The training process is supervised using the structural similarity index measure (SSIM) of the fiber orientation distribution (FOD) in two opposite phase encoding (PE) directions. Only b0 images and related outputs from the distortion correction methods are needed as inputs in the testing process. The proposed method is applicable in large-scale datasets such as the UK Biobank, Adolescent Brain Cognitive Development (ABCD), and other emerging studies that only have complete dMRI data in one PE direction but acquires b0 images in both PEs. In our experiments, we trained the proposed model using the Lifespan Human Connectome Project Aging (HCP-Aging) dataset $(n=662)$ and apply the trained model to data $(n=1330)$ from UK Biobank. Our results show low training, validation, and test errors, and the severity map correlates excellently with an FOD integrity measure in both HCP-Aging and UK Biobank data. The proposed method is also highly efficient and can generate the severity map in around 1 second for each subject.

1. Introduction

Susceptibility-induced distortion is a common artifact in diffusion MRI (dMRI), which causes stretching or pressing of brain structures along the phase encoding (PE) direction, particularly in the brainstem, orbitofrontal and temporal cortical areas [1–3]. While various methods have been proposed for distortion correction, residual distortions often persist and pose a significant yet unaccounted for challenge in common tasks such as fiber tracking (Fig.1). To better inform connectivity analysis based on dMRI, there is thus a current gap in automatically predicting the severity of residual distortion across different brain regions, which would greatly enhance the rigor in connectome research by enabling the consideration of this uncertainty in downstream statistical modeling.

Fig. 1.

An illustration of the impact of residual distortion on connectivity analysis in the brainstem area of one HCP-Aging subject. (a) shows the first spherical harmonics component of the FOD image, which corresponds to the $L=0$ component in Eq. 1. (b) displays the residual distortion severity map generated from FOD-based features. (c) shows the severe residual distortion prevents the successful fiber tracking through the pons (dashed red ellipse).

Previous methods for distortion correction are mainly based on image registration techniques [4]. Based on the assumption that distortions have opposite directions and the same magnitude in the two opposing PEs, image registration using b0 images of opposite PEs have been proposed and incorporated into popular tools such as Topup in FSL [5, 6]. For data from Human Connectome Projects (HCP) [7–11] and related studies, complete dMRI were acquired in two opposite PEs. Recently, more informative features such as the fiber orientation distribution (FOD) [12] from both PEs were proposed to further remove the distortion artifacts in dMRI, but residual distortions still persist in varying degrees across subjects [13–15]. Nevertheless, with FOD-based features from opposite PEs, it is very straightforward to locate areas with significant residual distortions (Fig. 2) and hence generate a map of distortion severity. For many large-scale studies including UK-Biobank [16–18] and ABCD [19], and many other clinical studies, however, complete dMRI data are typically limited to one PE and only b0 images are collected from both PEs due to time constraints. This practically limits our ability to use FOD-based features for distortion correction and severity analysis. For these studies, it is desirable that we can generate the severity map of residual distortions based only on b0 images.

Fig. 2.

The presence of residual distortion in the pons area can be better observed from (b) the difference of the first component ($L=0$ component in Eq. 1) of FOD images between two PEs than (a) the differences of b0 images. The red ellipses mark the same position on brainstem that has high FOD differences.

However, current tools for dMRI quality control cannot yet address the residual distortion map generation. For example, Gaussian process (GP) based prediction is a widely accepted method used in the Eddy tool [20, 21] of FSL, which measures the error after the correction of eddy currents and head motion. Although the GP based prediction method performs well on the artifacts that have high variability between each volume of the dMRI data, such as eddy currents and head motion, it doesn’t work well for the residual susceptibility-induced distortion that affects all volumes of dMRI data to a similar degree. Fig. 3(b1) – (c4) show maps of the GP based prediction error for the data in Fig. 2. For AP data, the residual distortion at brainstem region cannot be extracted using the GP based prediction error maps. For PA data, the high-error regions at the brainstem have different shape, location and error value across different volumes of dMRI. Due to the inconsistent results in different volumes, the GP based prediction is not suitable for measuring the residual distortions that are constant across all volumes.

Fig. 3.

Maps of GP based prediction error for the data in Fig. 2. (a) shows the differences between the $L=0$ components of FODs in AP and PA directions. (b1)–(b4) and (c1)–(c4) show the loss maps for AP and PA dMRI data, respectively. The dMRI volumes of (b2) (c2), (b3) (c3) and (b4) (c4) have gradient directions close to (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. The high residual distortion region on brainstem is marked by the red ellipse in (a), and the same position is marked out in (b1) – (c4).

For the automated generation of residual distortion severity map from only b0 images of two PEs, we will develop in this work a novel supervised deep learning method. Using HCP style dMRI data that allows the calculation of FOD-based features in two PEs, we train a network that learns to predict a ground truth FOD-based severity map from only b0 related feature images, which can then be applied to more widely available data from studies such as UK-Biobank and ABCD. In our experiments, we use the HCP-Aging [9, 10] data to train the network and successfully test the method on both HCP-Aging and UK-Biobank data. We show that our method is highly efficient and generates reliable prediction of distortion severity that correlates excellently with a measure of FOD-based white matter integrity.

2. Method

In this study, a supervised deep-learning model was utilized to generate the residual distortion severity map for dMRI images. Initially, the model was trained on datasets containing dMRI images in two opposing PE directions. After that, the model was applied to and tested on datasets containing dMRI images in only one PE direction, and only b0 images in the opposite PE direction.

2.1. Ground Truth Calculation

The ground truth for the training of our model is the residual distortion severity map calculated by the FODs of dMRI data in two opposite PE directions. The residual distortion severity map is calculated by the structural similarity index measure (SSIM) map [22, 23] of the first 6 components of the FODs.

The FOD is calculated using dMRI images after pre-processing, including susceptibility-induced distortion correction and correction of artifacts of eddy currents and head motion. It reflects fiber orientation and provides detailed information derived from all gradient directions of dMRI images. The FOD is calculated using the method proposed in Ref. [24], where the authors defined a constrained minimization framework using an energy function that consists of a data fidelity term and a regularization term for the sparsity of the FOD, and then they developed a coordinate decent algorithm to solve the FOD reconstruction problem. The FOD is represented by spherical harmonics (SPHARM) up to the order $L$ [24], which is defined by Eq. 1:

$$f\left(p\right)=\sum _{l,m} s_l^m{\Phi }_l^m\left(p\right),\forall p\in \mathbb{S}$$ (1)

where $p$ is a fiber direction on the unit sphere $\mathbb{S},s_l^m$ represents the SPHARM coefficient at order $l=0,2,4,\dots ,L$, and ${\Phi }_l^m$ denotes the $m-th$ real SPHARM basis. We use $L=2$ in the generation of ground truth, since it is proven that $L=2$ can provide enough information for distortion detection [3]. The FOD contains $J=6$ components for $L=2$, where $J=(L+1)(L+2)/2$.

The SSIM map in the $j-th$ component of FOD (denoted as ${\mathit{\text{FOD}}}_{\mathit{\text{pos}}}\left(j\right)$ and ${\mathit{\text{FOD}}}_{\mathit{\text{neg}}}\left(j\right)$ for the FOD in the positive and negative PE directions, respectively) is calculated by Eq. 2 using voxels in each voxel’s $3\times 3\times 3$ neighborhood. It is a voxel-level measurement of the similarity of the $j-th$ FOD components in the positive and negative PE directions:

$$SSIM\left(j\right)=\frac{2{\mu }_{FO{D}_{pos}\left(j\right)}{\mu }_{FO{D}_{neg}\left(j\right)}+C_1}{{\mu }_{FO{D}_{pos}\left(j\right)}^2+{\mu }_{FO{D}_{neg}\left(j\right)}^2+C_1}\cdot \frac{2{\sigma }_{FO{D}_{pos}\left(j\right)}{\sigma }_{FO{D}_{neg}\left(j\right)}+C_2}{{\sigma }_{FO{D}_{pos}\left(j\right)}^2+{\sigma }_{FO{D}_{neg}\left(j\right)}^2+C_2}\cdot \frac{cov\left(FO{D}_{pos}\left(j\right),FO{D}_{neg}\left(j\right)\right)+C_3}{{\sigma }_{FO{D}_{pos}\left(j\right)}{\sigma }_{FO{D}_{neg}\left(j\right)}+C_3}$$ (2)

where $\mu$ and $\theta$ correspond to the voxel sample mean and variance of the value of the $j-\mathit{\text{th}}$ component of FOD coefficients of each voxel in ${\mathit{\text{FOD}}}_{\mathit{\text{pos}}}\left(j\right)$ and ${\mathit{\text{FOD}}}_{\mathit{\text{pos}}}\left(j\right)$ within its $3\times 3\times 3$ neighborhood window, respectively. The term $\mathit{\text{cov}}\left({\mathit{\text{FOD}}}_{\mathit{\text{pos}}},{\mathit{\text{FOD}}}_{\mathit{\text{neg}}}\right)$ represents the cross-covariance between voxels in each voxel’s $3\times 3\times 3$ neighborhood window in the $j-th$ component of ${\mathit{\text{FOD}}}_{\mathit{\text{pos}}}$ and ${\mathit{\text{FOD}}}_{\mathit{\text{neg}}}.C_1,C_2$ and $C_3$ are 3 small constants to avoid zero-division.

Compared with mean squared error (MSE) and local cross-correlation coefficients (LCC) parameters, SSIM can more comprehensively interpret the similarity of images and is more robust to intensity variations of FOD in different regions. As shown in Eq. 2, SSIM consists of three parts. The first part computes the difference in images’ pixel values; the second part computes the difference in image contrast, so that SSIM is sensitive to areas with low FOD values; the third part computes the similarity of image structure via LCC.

The residual distortion severity of each voxel in the severity map, $d(x,y,z)$, is calculated using the normalized $L2–norm$ value of the SSIM of each FOD component:

$$d(x,y,z)=\sqrt{\sum _{j=1}^6 (1-\mathit{\text{SSIM}}(j\left)\right(x,y,z){)}^2/6}$$ (3)

Since the range of SSIM is $[-1,1]$, where 1 means that two FODs have the same value, and $-1$ means the they have opposite signs and the same amplitude, the term $(1-\mathit{\text{SSIM}}(j\left)\right)$ is used to ensure the severity value is non-negative. Higher $d$ means more severe residual distortion at that voxel.

In Fig. 4, an example of a residual distortion severity map $d$ is shown. The yellow and red regions are areas where residual distortion is located. Higher value on the map indicates more severe residual distortion. The residual distortion at temporal lobe can also be detected, even that the FOD value at this region is much lower than the brainstem.

Fig. 4.

Residual distortion severity map. (a) is the FOD image $(L=0)$. (b) – (d) show the residual distortion maps in a sagittal, coronal and axial slice, respectively.

2.2. Network architecture

Fig. 5 shows the network architecture of our proposed method. The input of our method are distortion distribution map obtained from the distortion correction method, as well as only b0 images in two PE directions before and after being implemented the distortion correction and eddy currents and head motion correction methods. Therefore, the models trained in this way can be applied to datasets comprising dMRI data in only one PE direction.

Fig. 5.

Network architecture of the proposed method.

To provide the network with more information to facilitate learning, initially, we employ two convolution layers with a kernel size of $3\times 3\times 3$ to extract the correspondence between the b0 images in the two PE directions. Additionally, the distortion distribution map is included to provide the network with more insights into the distribution of distortion. It is noteworthy that residual distortion primarily occurs in areas with high distortion, and the distortion distribution map assists the network in identifying these regions.

B0 images appear dissimilar across different datasets. Therefore, we utilize a b0 image normalization to scale the b0 images such that the mean gray-level value of the voxels in the brain is set to 0.5 using Eq. 4:

$$b{0}_{\mathit{\text{norm}}}=r\times b0,\mathit{\text{where}}r=0.5/\underset{(x,y,z)\mathit{\text{in}}\mathit{\text{brain}}}{\mathit{\text{mean}}}b0(x,y,z)$$ (4)

Moreover, image up- and down-sampling methods are employed to make all data used in training and test have the same size and voxel spacing.

To extract information from the inputs, we utilize the U-Net architecture. U-Net [3, 25, 26] is known for its ability to efficiently extract information from various resolutions and frequency ranges. In our proposed method, U-Net uses convolutional layers with a kernel size of $3\times 3\times 3$ and a Leaky Rectified Linear Unit (ReLU) activation layer with a parameter of 0.2. In order to prevent overfitting, the $L2$ penalty is implemented in all convolution layers.

In the training process, we utilize a weighted mean squared error $e$ to measure the differences between each voxel in the ground truth $d$ and the predicted severity map $d_p$, shown in Eq. 5:

$$e(x,y,z)={\left(\left(d(x,y,z)-d_p(x,y,z)\right)\times (1+d(x,y,z\left)\right)\right)}^2$$ (5)

Notably, by the term $(1+d(x,y,z\left)\right)$, we use the value of the ground truth, $d(x,y,z)$, as the weight to enlarge the differences in regions with severe residual distortion. This is particularly useful in this study since such areas are relatively small compared to the overall volume size.

2.3. Datasets, Model Training and Validation

The proposed model is trained on the HCP-Aging dataset, and being tested on UK Biobank dataset. These datasets both collect dMRI images in anterior-posterior (AP) and posterior-anterior(PA) PE directions, and the subjects in these two datasets have a similar age range. Each scan in HCP-Aging has 99 gradient directions over 2 shells with b values of $1500$ and $3000s/{mm}^2$, while each scan in UK Biobank dataset has 100 gradient directions over two shells with b values of $1000$ and $2000s/{mm}^2$. Table 1 shows information of data used in this study. The size and spacing of the HCP-Aging data is interpolated and cropped to those of UK Biobank. The Topup [5] is employed to correct susceptibility-induced distortion, and Eddy in FSL is utilized to correct artifacts caused by eddy currents and head motion.

Table 1.

Information of HCP-Aging and UK Biobank datasets

Datasets	Set name	Number of subjects	Image size (voxel)	Resolution (mm3)
HCP-Aging	Training	500	140 × 140 × 92	1.5 × 1.5 × 1.5
HCP-Aging	Test	162	140 × 140 × 92	1.5 × 1.5 × 1.5
UK Biobank	Application	1330	104 × 104 × 72	2 × 2 × 2

Our proposed method was trained for a total of 1500 epochs with 25 steps per epoch by Anaconda3 using Python3, which requires GPU memory for 2 G bytes. Stochastic gradient descent was employed, where one training subject was randomly selected per step. Our optimization used the ADAM optimizer with a learning rate of 10⁻⁴, and an $L2$ penalty weight of 10⁻⁵.

We utilized 5-fold cross-validation [27, 28] during the training process to choose the value of hyperparameters including learning rate, $L2$ penalty rate and the number of epochs. The 662 HCP-Aging subjects were divided into two sets: 500 subjects for training and 162 subjects for independent testing. For the 500 subjects in the first set, we performed 5-fold cross-validation for training and hyperparameter tuning. More specifically, 500 subjects were randomly formed into 5 subsets, with 100 subjects in each subset. Then we trained 5 models for each set of hyperparameters. For each model, we chose 4 subsets to train the model and the rest was served as the validation set, and then each subset was used for validation in one model. The hyperparameters were selected to minimize the average loss in all five validation subsets to avoid overfitting. After the cross-validation, a model was defined with the best set of hyperparameters, trained with all 500 data and tested on the test set.

Compared with a random split of training and validation sets, 5-fold cross-validation can end in a more robust and reliable model, since it enables the model to make full use of all 500 data in the training set and generalizes the performances of the model on multiple validation subsets.

3. Results

3.1. Results on HCP-Aging Dataset

The root mean squared error (RMSE) in the training, validation, and test sets on HCP-Aging dataset of the model with the selected hyperparameters is shown in Fig. 6. In Fig. 6(a), the maximum RMSE for the training, validation and test sets is less than 0.04, and the mean value of the RMSE is less than 0.02. Student’s t-test result shows no significant differences between the training set and test set. Similarly, the RMSE shows that all validation sets have no significant difference with the test set. This indicates that the model is robust and not over-fitted to the training set. In Fig. 6(b) , we visualized one data from the test set in Fig. 6(a) . Compared with the ground truth, the calculated residual distortion severity map has similar high-distortion regions and severity values.

Fig. 6.

The training, validation, and test loss of the HCP-Aging dataset (a) and a case in the test set that presents the ground truth and the calculated map (b). In figure (a), “ns” indicates that the difference is not statistically significant with $p>0.05$.

3.2. Results on UK Biobank Dataset

We used the model trained on the HCP-Aging dataset to predict the severity of residual distortion in the UK Biobank dataset. To quantitatively evaluate our method’s performance, we calculated the mean residual distortion severity and mean FOD value within a region of interest in the brainstem, which is well-known for severe residual distortions, shown in Fig. 7(a). Because the FODs are computed with compartment-based models, the mean FOD equals to the $L=0$ component in Eq. 1, which reflects the intra-axonal volume fraction of the compartment model and hence the integrity of the FOD model [24]. A reduction of mean FOD could be due to white matter degeneration or distortion artifacts. Given that only healthy subjects were included in our experiments, we use the mean FOD as an independent surrogate measure of distortion artifacts. Fig. 7 shows scatter plots of the mean FOD versus the mean severity of the two datasets. To locate the region of interest of each subject, we registered the Topup and Eddy results of b0 images in the AP direction to the FMRIB58 FA common space [29] using Elastix [30], and resampled the mean FOD to that space.

Fig. 7.

The scatter map of the mean FOD and mean residual distortion severity in the brainstem. Figure (a) illustrates the brainstem region, while figures (b) and (c) show the scatter plots for data in the HCP-Aging and UK Biobank datasets, respectively.

The negative correlation between the mean severity and mean FOD of HCP-Aging dataset is shown in Fig. 7(b), and the Pearson correlation coefficient is −0.665. The similar correlation in the UK Biobank dataset is shown in Fig. 7(c), and the Pearson correlation coefficient is −0.646. Similar distribution and correlation in the two datasets prove that our method is reliable to predict residual distortion’s severity. Fig. 8 displays four examples, whose residual distortion severity gradually increase and their mean FOD values gradually decrease. The FODs are more organized under low residual distortion severity, as shown in Fig. 8(a). For Figs 8(b) to (d), as the residual distortion severity increases, the FODs are more and more disorganized.

Fig. 8.

The FOD and residual distortion severity maps of the brainstem for four cases in UK Biobank dataset. The FODs in the high distortion regions marked by the ellipse get more and more disorganized as residual distortion severity increases. The left subfigures of (a) – (d) show FODs, and the right subfigures are the residual distortion severity maps of each case.

4. Discussion and Conclusion

This study draws attention to an understudied but important question and proposes a supervised deep learning solution, which aimed at predicting the voxel-level residual distortion severity map of dMRI distortion correction results. The calculation time of our method is around one second and it avoids FOD calculation in testing, which makes it applicable to large-scale data analysis. Result shows that our method can reliably and efficiently generate the localized residual distortion map for both HCP-Aging and UK Biobank datasets.

The proposed method can efficiently and reliably map the voxel-level localized severity of the residual distortion for the preprocessed dMRI data. With the severity map, researchers can locate the voxels or brain regions that have high residual distortions which may change brain connectivity. They can strategically process or exclude those distorted regions, which are similar with pathological regions but caused by residual distortions, so as to mitigate negative impact on subsequent downstream connectivity and statistical analysis.

One limitation of our method is the limited number of datasets that has dMRI data in 2 PE directions, and these datasets are mandatory for training. Another limitation is that our method is not applicable to datasets with b0 images in only one phase encoding direction, such as the Alzheimer’s Disease Neuroimaging Initiative (ADNI) [31].

In the future, we will further test the proposed method on additional datasets, such as the Lifespan Human Connectome Project Development (HCP-D) [32] and the ABCD Study. We will also focus on enhancing the accuracy of predicted residual distortion map and extending the application of predicting the severity of artifacts like off-resonance effects and subject movement [33, 34].