Acceleration of Three-dimensional Diffusion Magnetic Resonance Imaging Using a Kernel Low-Rank Compressed Sensing Method

Abstract

Diffusion Magnetic Resonance Imaging (dMRI) has shown great potential in probing tissue microstructure and structural connectivity in the brain but is often limited by the lengthy scan time needed to sample the diffusion profile by acquiring multiple diffusion weighted images (DWIs). Although parallel imaging technique has improved the speed of dMRI acquisition, attaining high resolution three dimensional (3D) dMRI on preclinical MRI systems remained still time consuming. In this paper, kernel principal component analysis, a machine learning approach, was employed to estimate the correlation among DWIs. We demonstrated the feasibility of such correlation estimation from low-resolution training DWIs and used the correlation as a constraint to reconstruct high-resolution DWIs from highly under-sampled k-space data, which significantly reduced the scan time. Using full k-space 3D dMRI data of post-mortem mouse brains, we retrospectively compared the performance of the so-called kernel low rank (KLR) method with a conventional compressed sensing (CS) method in terms of image quality and ability to resolve complex fiber orientations and connectivity. The results demonstrated that the KLR-CS method outperformed the conventional CS method for acceleration factors up to 8 and was likely to enhance our ability to investigate brain microstructure and connectivity using high-resolution 3D dMRI.

1. Introduction

Diffusion magnetic resonance imaging (dMRI) is widely used in neuroscience research and clinics to examine the microstructural organization and integrity of white matter (WM) structures in the central nervous system (Le Bihan 2003). Over the last two decades, increasingly sophisticated dMRI acquisition methods and models have been developed to reveal important microstructural information but often required lengthy acquisitions. For example, diffusion tensor imaging (DTI) (Basser, Mattiello et al. 1994), which uses a tensor model to characterize water diffusion, requires at least six diffusion-weighted images (DWIs) and an unweighted image. Metrics derived from the estimated diffusion tensor data, such as mean diffusivity (MD) and fractional anisotropy (FA), have been widely used to examine white matter injuries (Horsfield and Jones 2002). In order to resolve complex microstructural organizations, such as crossing fibers, high angular resolution diffusion imaging (HARDI) (Frank 2001, Tuch, Reese et al. 2002) and diffusion spectrum imaging (DSI) (Wedeen, Hagmann et al. 2005) were introduced and both require acquiring at least thirty to sixty DWIs along different diffusion directions. While acquiring a large number of DWIs provides important anatomical information, e.g., reconstruction of complex fiber pathways (Mori and van Zijl, 2002, Jeurissen and Leemans, 2011, Berman and Chung, 2008, Yeh and Zaydan, 2019), it also prolongs the scan time, which in turn increases the motion sensititivy, increases the cost, reduces throughput, and limits the practicality. This is a major challenge for preclinical studies that use three-dimensional (3D) dMRI to examine mouse and rat brains at a much higher spatial resolution (~0.1 mm) than clinical studies (1–2 mm), which often takes several hours (Aggarwal, Mori et al. 2010, Calabrese, Badea et al. 2015).

Technical developments in the last decade, e.g., multiple channel head coil (Hutchinson and Raff, 1988, Navon and Einav, 1991, Carlson and Minemura, 1993, Ra and Rim, 1993, Sodickson and Manning, 1997), high performance gradients (Reeder and McVeigh, 1994, McNab and Edlow, 2013), and parallel imaging (Pruessmann et al., 1999, Griswold et al., 2002, Ying and Liang, 2010), have significantly shortened the scan time of dMRI. In recent years, applications of compressed sensing (CS) (i.e., constrained reconstruction from reduced acquisition) for reducing the acquisition time have also been explored (Lustig, Donoho et al. 2007). Among its applications to dMRI, most of them relied on under-sampling the q-space, either by reducing the number of diffusion-encoding directions or diffusion-weightings (Menzel, Tan et al. 2011, Michailovich, Rathi et al. 2011, Bilgic, Setsompop et al. 2012, Landman, Bogovic et al. 2012, Bilgic, Chatnuntawech et al. 2013, Merlet and Deriche 2013, Aranda, Ramirez-Manzanares et al. 2015, Auria, Daducci et al. 2015, Daducci, Canales-Rodriguez et al. 2015, Ning, Laun et al. 2015, Paquette, Merlet et al. 2015, Tobisch, Stirnberg et al. 2018). Specifically, important diffusion features, such as fiber orientation distribution (FOD) or ensemble average propagator (EAP), are estimated from reduced measurements using some sparsity constraints (Rathi, Gagoski et al. 2013). However, most of these studies only reported low acceleration factors (i.e., 2–3). On the other hand, a few groups attempted to reduce the acquisition time by under-sampling the k-space and reconstruct each DWI independently using the conventional CS technique. For example, Wang et al., used a conventional CS method to accelerate high-resolution 3D dMRI of post-mortem mouse brains (Wang, Anderson et al. 2018) and later used the technique to acquire extensive datasets to map neurite density in the mouse brain (Wang, Zhang et al. 2019). This approach benefits 3D dMRI using multi-shot acquisition but does not utilize potential correlation among DWIs for joint reconstruction, which can potentially achieve further acceleration. Previous studies on joint reconstruction of DWIs from under-sampled k-space data, however, depended on the models used to characterize the correlations among DWIs (e.g. reconstruction based on the diffusion tensor model) (Pu, Trouard et al. 2011, Welsh, Dibella et al. 2013, Wu, Zhu et al. 2014, Knoll, Raya et al. 2015, Zhu, Peng et al. 2017) and therefore inherited the limitations of the specific models they used. Some studies considered the inter-image correlation as one of the constraints in diffusion image reconstuction (Adluru, Hsu et al. 2007, Gao, Li et al. 2014, Shi, Ma et al. 2015) and denoising (Haldar, Wedeen et al., Lam, Babacan et al.). These methods, however, use linear models for the inter-image correlation with total variation and low rankness as constraint, while the actual correlations may be more complex and nonlinear.

In this paper, we studied the effectiveness of a kernel low rank (KLR) method to jointly reconstruct multiple DWIs from under-sampled k-space data. The KLR-CS method is based on machine learning and has been successfully used to accelerate dynamic MRI (Nakarmi, Wang et al. 2017). Specifically, the KLR-CS method was developed to learn the highly non-linear correlations among DWIs acquired in different diffusion encoding directions from low-resolution images, which were reconstructed from the fully sampled center region of the k-space and used as the training data. The correlations can then be used to reconstruct high-resolution images from extended, under-sampled k-space. In this way, the KLR-CS method can self-adapt to different dMRI acquisition schemes and tissue microstructure. In contrast to previous approaches (Pu, Trouard et al. 2011, Welsh, Dibella et al. 2013, Knoll, Raya et al. 2015, Zhu, Peng et al. 2017), which mostly exploited the tensor model in the reconstruction formulation, the KLR-CS method does not need any prior knowledge of DWI or a pre-defined model and is thereby more suitable to accommodate complex axonal orientations (e.g., crossing fibers), where the tensor model fails. On the other hand, different from the dictionary-learning-based approaches whose measurements in q-space are assumed to be linearly correlated (Bilgic, Setsompop et at., 2012, Bilgic, Chatnuntawech et al., 2013, Merlet, Caruyer et al. 2013), the KLR-CS method can accommodate nonlinear correlations between DWIs and therefore can be used with complex dMRI acquisition schemes without modification.

The main objective of this study is to evaluate the performance of the KLR-CS method comparing to a conventional CS method over a range of acceleration factors (AFs). High-resolution 3D dMRI data of post-mortem mouse brains were used as a testbed in this retrospective examination. Taking the fully sampled k-space data (FKS) as the ground truth, we compared the estimated MD, FA, FOD, and fiber tractography results with under-sampled DWI data reconstructed using KLR-CS and conventional CS.

2. Methods

2.1. Kernel low rank (KLR) image reconstruction

In conventional Fourier reconstruction, the k-space is fully-sampled, and the image is reconstructed by solving a linear equation. When the k-space is under-sampled, the linear equation becomes under-determined, which leads to infinite solutions. In conventional CS (Lustig, Donoho et al. 2007), known sparsity constraints (in the spatial direction) are added to the original linear equation for the reconstruction of each image, leading to an optimization problem that needs to be solved by nonlinear algorithms. In the KLR-CS method used in this study, a kernel low rank constraint in the q-space (defined as diffusion domain) was added to the set of original linear equations for joint reconstruction of all DWIs, whose mathematical representation for the constraint was learned from a number of low-resolution images obtained from the fully sampled central k-space data. Such a kernel low-rank constraint was represented as very few nonlinear diffusion bases, which were obtained using kernel principal component analysis (KPCA), a machine learning approach, from the training data.

Specifically, in the training step, the training data came from low-resolution DWIs reconstructed from the fully sampled center region of the k-space, excluding the undersampled data in the outer k-space during the same accelerated scan. Therefore no additional scans were needed for training. The validity of such self-training is demonstrated in the Results section. For each pixel, a training sample was extracted from the intensity values of all low-resolution DWIs at the location of the pixel, forming a 1D signal vector. Among all available training samples, 1,600 such signal vectors within the region-of-interest were randomly selected to reduce the training time. KPCA [Mika et al., Scholkopf, Smola, et al.] was then performed on the training data. Different from PCA, KPCA estimates nonlinear principal components from the training data and usually needs fewer principal components than PCA. During KPCA, a nonlinear function (a second-order polynomial was used here) maps the data from the original space to a high dimensional feature space, in which a linear PCA is performed. The resulting kernel principal components (i.e., eigenvectors after converted from the feature space back into the original space) are defined as diffusion bases. Specifically, the first diffusion basis came from the kernel principal component with the largest eigenvalue, and the later ones with the decreasing eigenvalues. Typically, the first few diffusion bases include most of the information on the variation in the diffusion domain, and higher-order bases capture more details. We used the first six diffusion bases for the KLR-CS reconstruction. The selection of the number of diffusion bases is discussed in Results.

After obtaining the diffusion bases (i.e., the constraint is learned), an optimization problem was solved using an iterative approach satisfying two data consistency constrains: (a) each voxel of the reconstructed DWIs should be represented by the learned diffusion bases in the diffusion domain; (b) the k-space data of the reconstructed DWIs should be consistent with the acquired data at all locations. To enforce the first constraint, we first projected each testing sample onto the learned diffusion bases. Similar to the training samples, the testing sample was also extracted from the image intensity of each pixel along all diffusion directions, where the images were the aliased DWIs (zero-filled reconstruction directly from undersampled data). Such projection means the testing sample should be represented as a linear combination of diffusion bases. Similar to training, such a projection was performed in the feature space and then converted back to the original space. The diffusion weighted images in feature space can be represented by the multiplication of coefficients and diffusion bases. We then proceeded to enforce the second data consistency constraint. We replaced the reconstructed k-space data with the acquired k-space data at the sampled k-space locations after all the DWIs were converted from the feature space back to the original image domain. The conversion to the original image domain, know as pre-imaging, was performed by solving an optimization problem. The above two steps were repeated in an alternative manner.

More details on the KLR-CS method can be found in (Nakarmi, Wang et al. 2017).

2.2. Data Acquisition

All experimental procedures were approved by the Animal Use and Care Committee at the New York University School of Medicine. Five adult mice (C57BL/6, three-month-old, female, Jackson Laboratory, Bar Harbor, ME, USA) were used in this study. The mice were perfused transcardially with 4% paraformaldehyde in phosphate-buffered saline (PBS) and stored in 4% paraformaldehyde solution for 24 hours before returning to PBS. Before imaging, the mouse brains within the skull were placed in 15 mm diameter plastic tubes filled with Fomblin (Perfluoropolyether, Perkin Elmer LLC), which had a similar magnetic susceptibility to tissue but produced no signal in proton MRI.

Ex vivo MRI experiments were performed on a horizontal 7 Tesla MR scanner (Bruker Biospin, Billerica, MA, USA) with a triple-axis gradient system. Images were acquired using a quadrature volume excitation coil (72 mm diameter) and a 4-channel receive-only cryo-genic probe. While the excitation profile of the volume coil was relatively uniform for mouse brain specimens, the sensitivity profile of the cryogenic probe was not uniform, with higher sensitivity in the dorsal brain regions than the ventral brain regions. High-resolution dMRI data were acquired using a modified 3D diffusion-weighted gradient- and spin-echo (DW-GRASE) sequence (Aggarwal, Mori et al. 2010). The dMRI parameters were: echo time (TE)/repetition time (TR) = 33/500ms; two signal averages; field of view (FOV) = 12.8 mm x 10.4 mm x 16 mm, resolution = 0.1 mm x 0.1 mm x 0.1 mm; two non-diffusion weighted image (b0); 30 diffusion directions; and b = 2000 s/mm². The total scan time was approximately 8 hours.

2.3. Retrospective under-sampling and image reconstruction using conventional CS and KLR-CS

The acquired datasets were retrospectively under-sampled with different reduction factors ranging from 2 to 8. We adopted a 2D variable-density random sampling pattern (Wang and Arce 2010) along phase-encoding and slice-encoding directions as illustrated in Fig. 1B. The central k-space was fully sampled with a fixed radius, and the outer k-space was sparsely sampled using an exponential probability distribution function. Here we used a k-space center radius of 15 for all the 104 × 180 images, which corresponded to 697 data points, or 3.72% out of all k-space data points. The low-resolution DWIs from the fully sampled central k-space and zero-padded outer k-space were used for training. Both the KLR-CS and conventional CS methods used the same sampling patterns. Different DWIs had different sampling patterns that were randomly generated from the same distribution function. The conventional CS reconstruction was modeled as a convex optimization problem, in which wavelet transform was used as the sparsifying transform and total variation was used as a regularization term, and the optimization was solved by the conjugate gradient method (Lustig and Donoho, 2007, Wang and Anderson, 2018). Different from the KLR-CS method, correlations between different DWIs in the diffusion domain were not utilized in the conventional CS method, and each DWI was reconstructed independently. For KLR-CS, a total of six diffusion bases were obtained from the low-resolution training data and then used for the reconstruction of all DWIs simultaneously, as shown in Fig. 1C.

Fig. 1:

Under-sampled k-space data and diffusion-weighted images reconstructed using KLR-CS. (A) Representative DWIs and corresponding diffusion encoding directions (red dots on the unit sphere). Note the subtle differences in tissue contrasts between nearby DWIs. (B) Under-sampled k-space data were obtained by sampling the full k-space data retrospectively for acceleration factors (AFs) from 2 to 8. For the KLR-CS method used in this study, the under-sampling patterns in the ky-kz plane varied for different diffusion encoding directions. (C) Diffusion-weighted images were reconstructed from under-sampled k-space data.

To study the robustness of the KLR-CS method to noise, we added random Gaussian noise (both real and imaginary) to the under-sampled k-space data and then performed the KLR-CS reconstruction. The SNR was calculated in the image domain after Fourier transform and magnitude calculation as the ratio of the mean signal intensity of the region of interest (ROI) against the standard deviation of the noise. The results comparing the performance of KLR-CS at different SNR levels are shown later in Section 3.3.

2.4. Image analysis

Full k-space DWI data and DWI data reconstructed using conventional CS and KLR-CS approaches were processed in the same way. Quality of the reconstructed images was measured by computing the peak signal to noise ratio (PSNR) and the normalized mean squared error (NMSE). PSNR was defined as the ratio of maximum possible signal intensity to MSE. NMSE was defined as the average of the squares of the errors normalized by the average of the squares of the reference image intensity. From each DWI dataset, diffusion tensor was estimated using weighted linear least squares estimation as implemented in MRtrix (www.mrtrix.org), and FA and MD maps were calculated from the diffusion tensors. Mouse whole brain FOD maps were estimated by performing the constrained spherical deconvolution (CSD) with l_max set to 6 using MRtrix (Tournier, Calamante et al., 2004, 2007). Corpus callosum (CC), anterior commissure (AC), motor cortex (MO), and caudate putamen (CP) regions were manually segmented to examine FA, MD, and FOD values. We investigated whether the proposed KLR-CS method can resolve crossing fibers at various AFs and compared the accuracy of the estimated FODs using total FOD amplitude and orientation of the primary FOD peak as benchmarks.

Furthermore, we used fiber tractography approach for qualitative and quantitative assessment of the conventional CS and KLR-CS methods to map axonal trajectories in the thalamocortical pathway. Whole-brain probabilistic tractography was performed on the FOD image to generate 2.5 million streamlines using the second-order integration over FOD (iFOD2) algorithm (Tournier et al., 2010) with the following tractography parameters: FOD amplitude threshold for seeding and terminating tracks = 0.05, minimum fiber length = 3 mm, step size = 0.025 mm, angle between successive steps = 45°. Tractography reconstruction biases were tackled by applying the spherical deconvolution informed filtering of tractograms (SIFT) algorithm (Smith et al., 2013). Then we chose primary somatosensory barrel field area (SSp-bfd) as the seed and ventral posterior complex of the thalamus (VP) as the target region and extracted fibers connecting these two regions from the bias-corrected whole-brain tractograms. This approach was followed for all subjects individually, including the FKS data, which served as the ground truth, and data reconstructed with the conventional CS and KLR-CS.

All descriptive statistical analyses and visualizations were performed in R 3.2 (https://www.r-project.org) and GraphPad Prism version 7.05 for Windows (GraphPad Software, La Jolla California USA, www.graphpad.com).

3. Results

3.1. Validation of training using low-resolution images

We first tested whether training with the low-resolution images was sufficient for the proposed method. As shown in Table 1, the quantitative assessments (in terms of normalized mean squared error (NMSE) and Pearson’s coefficient (PC)) of the first six diffusion bases (corresponding to the six largest eigenvalues) learned from low-resolution images when compared to the full k-space data (ground truth). We assumed the reconstructed DWIs should be of high quality if the bases from low-resolution images were relatively close to the ground truth. The closeness was defined as small NMSEs and large PCs. Because each set of training data was randomly selected from 1600 different image pixels, we used the average results from 50 different sets of random selections. We heuristically assumed the learned diffusion bases to be acceptable if the NMSE was less than 3.5×10⁻⁴ and PC higher than 0.75. Accordingly, we can see that the first, second, and third diffusion bases (kernel principal components with the largest three eigenvalues of 62.88, 5.33, and 1.02, respectively) are acceptable using as little as 3.72% of the k-space data for training. For the 4^th diffusion basis (corresponding to an eigenvalue of 0.91), 6.65% of the k-space data are needed. The 5^th one (corresponding to an eigenvalue of 0.70) can be obtained using 15.01% of k-space data. We used 3.72% of the k-space data in this study to achieve high acceleration factors, while the first three bases can be estimated with acceptable accuracy. More central k-space data can be acquired for training (at the cost of longer acquisition time) to provide more accurate estimations for the diffusion bases. The reason that low-resolution images were sufficient in estimating the diffusion bases was that the bases represent the correlation in the diffusion domain and should be rather independent of the resolution in the spatial domain.

Table. 1:

Quantitative assessment of diffusion bases learned from low-resolution images. NMSE and PC were calculated for different percentages of the central k-space data (for low-resolution image) with the full k-space data (for high-resolution image) as the ground truth.

Metrics		Percentage of central k-space data used for a low-resolution image
		0.37%	1.63%	3.72%	6.65%	10.37%	15.01%
1st	NMSE (10−5)	4.42	2.08	1.36	1.02	0.82	0.69
	PC	0.96	0.98	0.99	0.99	0.99	0.99
2nd	NMSE (10−5)	33.40	16.24	10.72	8.03	6.39	5.28
	PC	0.75	0.86	0.90	0.93	0.94	0.95
3rd	NMSE (10−5)	93.94	61.49	34.52	21.96	15.31	11.10
	PC	0.45	0.59	0.75	0.84	0.88	0.91
4th	NMSE (10−5)	135.28	67.49	36.70	25.54	16.61	12.24
	PC	0.36	0.57	0.74	0.82	0.87	0.90
5th	NMSE (10−5)	152.81	88.42	62.51	46.44	35.62	28.36
	PC	0.33	0.49	0.59	0.67	0.73	0.78
6th	NMSE (10−5)	279.27	113.75	78.57	61.78	51.86	44.80
	PC	0.27	0.44	0.52	0.57	0.61	0.65

3.2. General image Quality Comparisons between KLR-CS and conventional CS

Fig. 2 compares average DWI, MD and FA maps generated from the KLR-CS and conventional CS results at different AFs. In the average DWI (Fig. 2A), the contrast between the corpus callosum (blue arrows) and surrounding tissues was better preserved in the KLR-CS result compared to conventional CS results at high AFs (e.g., 6 and 8). For example, a hypo-intense region in the hippocampus (red arrows) was consistently visible in the KLR-CS results for AFs from 2 to 8 but was indistinguishable from surrounding regions in the conventional CS results for AFs higher than 4. The MD maps (Fig. 2B) showed no apparent difference between the two CS methods for all AFs. The FA maps (Fig. 2C), on the other hand, showed a clear advantage of the KLR-CS method in preserving structural details. The contrasts between two hyper-intense regions in the hippocampus, which corresponded to the stratum radiatum and molecular layer of the dentate gyrus (indicated by green and yellow arrows, respectively), and the rests of the hippocampus, were well preserved in the KLR-CS generated FA maps, but gradually diminished in the conventional CS generated FA maps with increasing AFs (Fig. 2C).

Fig. 2:

Qualitative assessment of diffusion-weighted images (DWI) (A), maps of mean diffusivity (MD) (B) and fractional anisotropy (FA) (C) reconstructed using conventional CS and KLR-CS methods. In the DWI and FA maps, certain anatomical details are better preserved in the KLR-CS results than the conventional CS as indicated with arrows. D: The distributions of FA values in the KLR-CS results showed no apparent deviation from the full k-space result.

The violin plot in Fig. 2D shows no apparent change in the distribution of FA values over the whole brain at different AF levels of KLR-CS used in this study. Comparing with the full k-space data, the interquartile range (black rectangular bar) as well as the median (black mark) and mean values (blue diamonds) remained relatively unchanged for the images reconstructed at different AFs using KLR-CS. Similarly, the distribution of the full k-space MD map was also preserved (data not shown).

Two quantitative image quality measurements, PSNR and NMSE, also showed that the KLR-CS method better preserved the image quality for AF up to 8, as shown in Table 2. These results suggest that KLR-CS is better at preserving anatomical details than conventional CS.

Table. 2:

Image quality measurements of DWIs reconstructed using the KLR-CS and conventional CS methods. PSNR and NMSE were calculated for each DWI using the full k-space data as the ground truth.

AF		2	3	4	5	6	7	8
KLR-CS	PSNR (dB)	46.2±1.6	43.4±1.7	41.8±1.5	40.5±1.3	39.4±1.1	38.5±0.9	37.8±0.8
	NMSE (×10−12)	4.8±3.0	9.1±5.8	13.1±7.9	17.1±9.1	21.4±9.7	25.9±9.6	30.4±9.2
Conv. CS	PSNR(dB)	40.6±0.3	35.5±0.4	32.2±0.5	30.1±0.5	28.2±1.0	27.4±0.6	26.6±0.7
	NMSE (×10−12)	15.8±3.5	50.5±8.3	106.3±14.5	173.0±28.2	245.8±43.6	322.4±69.5	392.2±100.5

3.3. Regional analysis of the performance of KLR-CS

We further examined the performance of KLR-CS in several brain structures. The corpus callosum (CC) is an important WM structure and had relatively high SNRs in this study due to its dorsal location, which was within the high sensitivity region of the cryo-genic probe used for image acquisition. The differences in MD and FA values between full k-space and CS results increased with higher AF in the CC (Fig. 3A). The percentage differences in MD values based on KLR-CS remained below 5% even with an AF of 8. In comparison, the percentage differences in MD values based on conventional CS reached almost 20% with an AF of 8. The percentage differences in FA values were higher than MD values for both methods. With the AF of 8, the percentage differences from KLR-CS and conventional CS methods were 12% and 25%, respectively (Fig. 3A).

Fig. 3:

Performance of conventional CS and KLR-CS methods in terms of the estimated MD and FA values in the corpus callosum, anterior commissure, motor cortex, and caudate putamen. Differences in MD and FA values were calculated using the full k-space data as the ground truth. Data from five subjects (n=5) are included. * indicates a significant difference between the conventional CS and KLR-CS methods (two-sample t-test, p<0.05, FDR corrected).

Another WM structure, e.g., the anterior commissure (AC), had lower SNR than the CC due to its location in the ventral part of the brain, which is outside the high sensitivity region of the probe. The difference in the MD and FA values in the AC showed higher percentage differences than the CC but a similar overall trend (Fig. 3B). With the AF of 8, percentage differences in MD values reached 40% and 10% for the conventional CS and KLR-CS methods, respectively. Percentage differences in FA values reached more than 30% and 20% for the conventional CS and KLR-CS methods (Fig. 3C). Statistical analysis (two-sample t-test, p<0.05, FDR corrected) showed that the results from the KLR-CS method had significantly reduced percentage errors in these two WM regions compared to the conventional CS method for the AFs used here (except AF7 and AF8 for AC’s FA).

Similar advantage of the KLR-CS method over the conventional CS method in terms of precision of estimated MD and FA values was also observed in two gray matter (GM) structures: the motor cortex (MO) and caudate putamen (CP) (Fig. 3C–D). In the MO, which was closer to the coil and had higher SNR than the CP, differences in MD values were slightly above 10% for AF 2 to 8 using KLR-CS, whereas conventional CS showed a 15% difference at AF = 3 and 35% difference at AF = 8 (Fig. 3C). While comparing FA values, differences were in the range of 5% – 19% for AF up to 6 using KLR-CS and a maximum of 22% for AF = 7. In contrast, conventional CS technique showed a higher percentage of differences, reaching 22% at AF = 3 which is the maximum difference observed in the KLR-CS. Interestingly, MD difference in CP region was 7% for AF = 2 using conventional CS, which could be achieved with AF = 8 via KLR-CS. Regardless of the CS techniques, differences in FA values in the CP increased over AFs, however, conventional CS showed higher differences than KLR-CS. The KLR-CS method exhibited significantly lower percentage errors in MD values in the MO and CP up to AF = 8, compared with the conventional CS (Fig. 3 C–D). However, the KLR-CS method did not show a significant difference in FA values for AF = 2, 5, and 6 in the MO as well as AF = 4, 7 in the CP. Altogether, these findings by the KLR-CS method provided more precise MD and FA measurements than the conventional CS method at the same level of AF.

The performance of the KLR-CS method at different levels of SNR was also examined. As expected from the results shown in Figs. 3, percentage differences in MD and FA values for both WM and GM structures increased with the increment of AFs but decreased with higher SNR (Fig. 4). One-way ANOVA followed by Sidak’s multiple comparisons test (alpha = 0.05) revealed that higher SNR levels significantly increased the accuracy of MD and FA estimations via the KLR-CS method.

Fig. 4:

Performance of KLR-CS at different SNR levels. The percentage of differences in MD and FA values were calculated using the full k-space (FKS) data as the gold standard. Data from five subjects (n=5) were included. Low (10dB), medium (14dB), and high (20dB) SNR levels are indicated in light to dark blue colors and statistical significance levels (One-way ANOVA followed by Sidak’s multiple comparisons test) are indicated with different color bars, red: p<0.0001, green: p<0.0005, blue: p<0.005 and black: p<0.05.

3.4. Estimation of fiber orientation distribution (FOD) maps and resolving cortical and sub-cortical crossing fibers

We examined the performance of KLR-CS and conventional CS in preserving the capability of resolving multiple fiber orientations. As gray matter voxels often contain multiple fiber populations or crossing fibers, firstly we examined if the proposed KLR-CS was able to detect crossing fibers at different AFs up to 8.0 in the cortical and subcortical brain regions such as cingulate cortex, hippocampus, thalamus, and hypothalamus (Fig. 5). In all four regions, estimated FODs reflected the crossing fibers in these regions, and FODs based on both conventional CS and KLR-CS methods preserved the capability to resolve crossing fibers.

Fig. 5:

Estimated FODs in several GM regions (top to bottom: cingulate cortex, hippocampus, thalamus, and hypothalamus) based on conventional CS and KLR-CS methods.

As both CS methods preserved the capability to resolve crossing fibers, next, we exploited the variation in FOD amplitude in WM and GM regions. Fig. 6A represents the FOD amplitude difference maps from the FKS derived using the conventional CS and KLR-CS methods at different AFs. In the CC and AC, which mostly consist of axons running in the same direction, quantitative analyses revealed that the FODs from KLR-CS results had the amplitude closer to the full k-space result than conventional CS (Fig. 6B). However, only for AF = 2, 3 in the CC and AF = 2 in the AC, the difference between conventional CS and KLR-CS results became significant (two-sample t-test, p<0.05, False discovery rate corrected). Interestingly, we observed significant (two-sample t-test, p<0.05, FDR corrected) differences merely in the cortical FOD amplitude (Fig. 6B) as indicated with asterisks, but no significant difference in the sub-cortical regions, such as thalamus (Fig. 6B), hippocampus and hypothalamus (supplementary fig. 2) despite the FODs exhibited consistent shapes across various AFs using the KLR-CS method.

Fig. 6:

Representative difference maps derived using the conventional and KLR-CS for AF2, AF4, AF6 and AF8, showing differences in FOD amplitude (A & B) and primary FOD peak orientation (C & D). In B &D, means and standard deviations are shown here. * indicates a significant difference between the conventional CS and KLR-CS methods (two-sample t-test, p<0.05, FDR corrected).

We then examined how close the proposed KLR-CS method can estimate the fiber orientation along the primary FOD peak as the FKS and compared with the conventional CS. Fig. 6C shows the difference maps in prmary FOD peak orientation using the conventional and KLR-CS methods, where significant differences (two-sample t-test, p<0.05, False discovery rate corrected) were perceived in GM regions, such as, caudate putamen and thalamus at higher AFs (AF6 to AF8), but not in the WM regions like CC and AC (Fig. 6D).

3.5. Fiber tractography based on the KLR-CS results

Fig. 7 compares the tractography results of the thalamocortical pathway based on the KLR-CS and conventional CS results. Fibers connecting SSpbfd (seed) and VP (target) based on the full k-space data are shown in Fig. 7A. We found that the KLR-CS results preserve the overall trajectories of the connections better than the conventional CS results. For example, conventional CS at AF = 4 to AF = 8 generated a large number of spurious fibers running in the left-right direction (red) (Fig. 7B, indicated with white arrows), which were not presented in the full k-space results (Fig. 7A) or KLR-CS results (Fig. 7C). Moreover, in terms of total number of streamlines, the KLR-CS results remained largely unchanged at the level of full k-space data, whereas the conventional CS results showed an increased number of fibers with increasing AFs (Fig. 7D). Figure 7E displays the percentage of differences in the number of streamlines generated at different AFs using conventional CS in comparison with KLR-CS with the full k-space fiber trajectories as ground truth.

Fig. 7:

Reconstruction of thalamic cortical connections from DWIs generated using conventional and KLR-CS methods. A: Results from full k-space data. B-C: Results from conventional and KLR-CS results with different AFs. The streamlines were overlaid on the full k-space FA map. We used identical ROIs defined in the cortex and thalamus as seed regions for tractography. D-E: Comparisons of the number of streamlines from both results.

3.6. Motion robustness

Subject motion during in vivo 3D dMRI acquisition may affect the performance of KLR-CS. We tested the robustness of the KLR-CS method to motion using simulated shifts and rotations. Specifically, we randomly selected several diffusion-weighted images, and then shifted the selected images by one pixel and/or rotated them by 1 degree. Such random selection was repeated five times. The KLR-CS method was used to reconstruct DWIs with simulated motion. The mean and standard deviation of the PSNR and NMSE were calculated for five repetitions. Table 3 shows the quantitative results of reconstructed motion-induced diffusion-weighted images. In Table 3, (a) denotes randomly selected one image with 1 pixel shift, (b) randomly selected three images with 1 pixel shift, (c) randomly selected one image with 1 pixel shift and 1 degree rotation, and (d) randomly selected three images with 1 pixel shift and 1 degree rotation.

Table 3:

PSNR and NMSE for reconstructions with and without simulated motion.

PSNR (dB) (Mean±Std)	AF2	AF3	AF4	AF5	AF6	AF7	AF8
KLR-CS	46.9	44.2	42.7	41.5	40.3	39.2	38.2
(a)	46.5±0.1	43.7±0.1	42.0±0.1	40.6±0.3	39.4±0.4	38.5±0.3	37.6±0.3
(b)	46.4±0.2	43.5±0.2	41.8±0.1	40.5±0.2	39.3±0.2	38.3±0.3	37.5±0.2
(c)	46.5±0.2	43.7±0.1	42.0±0.1	40.5±0.3	39.4±0.3	38.4±0.3	37.5±0.3
(d)	46.2±0.2	43.4±0.3	41.7±0.1	40.3±0.1	39.2±0.2	38.2±0.2	37.4±0.2
NMSE (10−12) (Mean±Std)	AF2	AF3	AF4	AF5	AF6	AF7	AF8
KLR-CS	3.4	6.4	9.2	12.2	15.7	20.5	25.6
(a)	3.8±0.1	7.2±0.1	10.7±0.3	14.8±0.9	19.5±1.9	24.2±1.9	29.8±1.8
(b)	3.9±0.2	7.6±0.4	11.3±0.1	15.4±0.5	20.2±1.1	25.3±1.6	30.6±1.4
(c)	3.8±0.2	7.3±0.2	10.8±0.2	15.1±0.9	19.5±1.4	24.5±1.7	30.0±1.8
(d)	4.1±0.2	7.9±0.6	11.6±0.4	15.9±0.3	20.6±0.9	25.7±1.3	31.4±1.1

4. Discussion

In recent years, 3D dMRI has been increasingly used in preclinical studies to examine brain structures at high spatial resolutions. For example, many groups used high-resolution 3D dMRI to examine microstructural organizations in post-mortem rodent brains (Zhang, Richards et al. 2003, Aggarwal, Mori et al. 2010, Moldrich, Pannek et al. 2010, Richards, Calamante et al. 2014, Calabrese, Badea et al. 2015). In addition, in vivo 3D dMRI has also been demonstrated in rodent brains recently (Wu and Resisinger, 2014, Wu and Xu, 2013). 3D dMRI with more sophisticated dMRI acquisition or at even higher resolution can potentially extract additional structure information, but the associated lengthy acquisition remains a major obstacle. Among several approaches to overcome the obstacle, accelerated image acquisition tailored for 3D dMRI has great potential as it does not require a change in imaging hardware. As described in the introduction section, several acceleration strategies have been explored, and this study focused on the joint reconstruction of DWIs from under-sampled k-space data.

Considering full k-space data as the ground truth, we have investigated whether the KLR-CS method can provide better data reconstruction than conventional CS. We examined several commonly used DTI indices, such as FA and MD, as well as FOD maps and tractography results. Overall, our results suggest that the KLR-CS method out-performed conventional CS with the same acceleration factor. While comparisons based on FA and MD values were straightfoward, comparisons based on FODs, which are not scalar, need additional considerations. Even though there are multiple ways to project FODs into scalar indices for comparison, we chose FOD amplitude and peak orientation as benchmarks due to their simplicity and their impacts on tractograpy. Both CS methods preserved the ability to resolve crossing fibers (Fig. 5). In the two white matter regions studied here, we observed limited improvement with KLR-CS based on FOD amplitude and peak orientation measurements (Fig. 6), most likely due to the coherent organization of axons in large white matter tracts. In gray matter regions, which may contain crossing fibers, comparions based on peak orientation (Fig. 6D) showed that the KLR-CS method outperformed conventional CS in gray matter regions at high AFs. It is necessary to note that small errors in estimated peark orientation can accumulate in tractography, which may explain the apparent differences in tractography results as shown in Fig. 7 between conventional and KLR-CS.

Altogether, our results demonstrated that the proposed KLR-CS method preserved more anatomical details than the conventional CS method used here. The reason lies in the fact that the former incorporates the constraint in the diffusion domain in addition to the spatial direction, while the latter only in the spatial direction. Because in conventional CS reconstruction, the constraint in the spatial direction typically makes the image look smoother spatially (i.e., spatial filtering). On the other hand, the KLR-CS method basically spreads the filtering to both spatial and diffusion domains (i.e., both spatial and diffusion filtering) and therefore can preserve more anatomical details.

The constraint in the diffusion domain is enforced by projecting each voxel of the under-sampled DWIs at all diffusion directions onto the learned diffusion bases, which can be interpreted as nonlinear principal components. Specifically, for DWIs with N pixels, there are at most N diffusion bases. Because many pixels may have highly similar variations in the diffusion domain (e.g., pixels with the same fiber orientation), the diffusion bases for these pixels are highly redundant. Therefore, the actual number of diffusion bases, learned through nonlinear kernel principal component analysis, is much smaller than N. The projection onto these diffusion bases serves as filtering of the under-sampled DWIs in the diffusion domain. As a result, the KLR-CS method preserves the diffusion information better than the conventional CS, where only the spatial constraint is applied. When large phased-array coils are available, the KLR-CS can be generalized to combine with SENSE-based parallel imaging methods for further accelerations.

Although linear principal component analysis has been successfully used for accelerating cardiac imaging, it is not applicable in diffusion imaging. In (Nakarmi and wang, 2017), it is shown that when there are very few time frames (< 50) in cardiac imaging, nonlinear principal component analysis has a significant advantage over the linear one. In dMRI, the number of directions is typically smaller (30–60) than the number of time frames in cardiac imaging. In such a case, linear principal components analysis requires much more diffusion bases, thus reducing the effect of filtering.

One unique feature of the KLR-based method is that the method is self-trained, that is, the diffusion bases can be learned from the low-resolution DWIs reconstructed only from the center of k-space of the desired high-resolution DWIs, and therefore no additional training data are needed. The advantage of such self-training is that the learned basis can represent the specific correlations among DWIs for the target dataset. The assumption here is that the bases learned from the low-resolution DWIs are close to those from high-resolution DWIs. Such an assumption may be violated when the diffusion bases present significant spatial variations, for example, when the fiber orientation changes abruptly between adjacent voxels.

Both k-space and q-space under-samplings have shown success in accelerating diffusion imaging. For the KLR-CS method, a sufficient number of diffusion directions (≥ 30) are needed for training the diffusion bases. Therefore, only the under-sampling of k-space is studied here. For the case with a large number of diffusion directions, it worthwhile to study the benefit of under-sampling in both k-space and q-space for the KLR-CS method.

5. Conclusion

In this paper, we demonstrated the feasibility of accelerating 3D dMRI using the KLR-CS method. The KLR-CS method outperformed a conventional CS method in terms of both image quality and quantitative metrics. Tractography results based on the data reconstructed using KLR-CS showed good agreement with the ground truth from full k-space data at the acceleration factors up to 8. In summary, the KLR-CS method can be used to dramatically shorten the acquisition time of 3D dMRI.