Rapid Dual-RF, Dual-Echo, Three-Dimensional Ultrashort Echo Time Craniofacial Imaging: A Feasibility Study

Abstract

Purpose:

To develop a dual-RF, dual-echo, three-dimensional ultrashort echo-time (UTE) pulse sequence and bone-selective image reconstruction for rapid high-resolution craniofacial MRI.

Methods:

The proposed pulse sequence builds on recently introduced dual-RF UTE imaging [Johnson et al, MRM, 2017]. While yielding enhanced bone specificity by exploiting high sensitivity of short T₂ signals to variable RF pulse widths, the parent technique exacts a two-fold scan time penalty relative to standard dual-echo UTE. In the proposed method, the parent sequence’s dual-RF scheme was incorporated into dual-echo acquisitions while radial view angles are varied every pulse-to-pulse repetition period. The resulting four echoes (two for each RF) were combined via view-sharing to construct two sets of k-space datasets, corresponding to short and long TEs, respectively, leading to a two-fold increase in imaging efficiency. Further, by exploiting the sparsity of bone signals in echo-difference images, acceleration was achieved by solving a bone-sparsity constrained image reconstruction problem. In vivo studies were performed to evaluate the effectiveness of the proposed acceleration approaches in comparison to the parent method.

Results:

The proposed technique achieves 1.1 millimeter isotropic skull imaging in 3 min without visual loss of image quality, compared to the parent technique (scan time = 12 min). Bone-specific images and corresponding 3D renderings of the skull were found to depict the expected craniofacial anatomy over the entire head.

Conclusion:

The proposed method is able to achieve high-resolution volumetric craniofacial images in a clinically practical imaging time, and thus may prove useful as a potential alternative to computed tomography.

1. INTRODUCTION

Recent developments in solid-state MRI via ultrashort echo-time (UTE) (1, 2) or zero TE (3, 4) methods have enabled detection and quantification of very short T₂ (ranging from tens of microseconds to a few milliseconds) species of the human body such as cortical bone water (5–7), myelin in white matter (8), lung tissue (1, 9), and phosphorus in bone mineral (10, 11). In particular, MR-based bone imaging, as a potential, noninvasive alternative to computed tomography (CT), has gained growing interest in many applications requiring information of bone structure and composition, e.g., the attenuation correction of positron emission tomography (PET) signals in MR/PET imaging (12) and MR-guided focused ultrasound (13).

In both UTE and ZTE imaging methods, signal reception begins immediately after transmit-receive switching (requiring a dead-time of tens of microseconds in modern clinical scanners). Typically, a center-out trajectory is employed for k-space sampling, thereby allowing capture of signals from sub-millisecond T₂ constituents, such as those from protons in water bound to the bone’s organic matrix. Nevertheless, long T₂ components (i.e., water in soft tissue or residing in the bone pore spaces), unless suppressed, also exhibit high signal intensities, which would obscure short T₂ contrast. To enhance short-T₂ conspicuity in reconstructed images, various approaches have previously been explored, generally falling into two categories: inversion recovery (IR)-based long T₂ signal attenuation (6, 14–16), and acquisition of two or more images at different TEs, with subsequent post-processing to filter out long T₂ components (12, 17–20). While the former, typically using adiabatic IR pulses, has been shown to be very effective for long T₂ suppression (21), it may result in impractically long imaging times, which may limit its clinical utility. On the other hand, there is some evidence to suggest the dual-echo subtraction method to yield higher SNR efficiency in short-T₂ imaging than IR-prepared UTE (21, 22).

It is well known that once the RF pulse duration approaches T₂, there is increasing damping of the resulting transverse magnetization (23). Recently, a method that exploits the sensitivity of bone proton magnetization to both, T₂ and RF pulse duration, has been reported (24). In that method, two hard RF pulses differing in duration and amplitude are applied alternately, and two sets of data are acquired, one from short RF at short TE and one from long RF at long TE. Compared to standard dual-echo based UTE, the technique enhances discrimination of bone signals between the two datasets, thus yielding a higher level of bone specificity in echo difference images. However, the two RF interleaves, obtained with identical k-space coverage, result in an inevitable two-fold increase in scan time.

In this work, we propose a rapid bone MR imaging method by developing a three-dimensional (3D) DUal-RAdiofrequency aNd Dual-Echo (DURANDE) UTE pulse sequence along with bone-selective image reconstruction. Similar to the parent method, dual echoes are acquired following short and long RF pulses. However, in distinction to the method described in (24), encoding gradients are varied continuously along the entire pulse train, leading to a reduction of imaging time by a factor of two. During image reconstruction, two independent k-space datasets are generated by combining the four echoes via a view-sharing (VS) approach. Further, the sparsity of bone voxels in the corresponding difference images was exploited, thereby enabling substantially accelerated UTE bone imaging. In vivo studies were performed at 3 T to evaluate the feasibility of the proposed technique in achieving isotropic high-resolution volumetric craniofacial images within 3 min.

2. METHODS

2.1. Dual-RF, Dual-Echo UTE with View-Sharing (VS-DURANDE UTE)

Figure 1a shows a diagram of 3D VS-DURANDE UTE. Two hard RF pulses (RF₁, RF₂) differing in duration and amplitude for the same nominal flip angle are applied alternately in successive TR periods along the entire pulse train, while within each TR two echoes are collected at short TE (TE₁) and long TE (TE₂), from the beginning of gradient ramp-up. Thus, four echoes are acquired: ECHO₁₁, ECHO₁₂, ECHO₂₁, and ECHO₂₂. Here, the subscripts represent the corresponding RF and TE indices (Fig. 1a).

Figure 1.

a: Diagram of the VS-DURANDE UTE pulse sequence, in which RF₁ (short ~ 40μs) and RF₂ (long ~ 520μs) are alternately played out and four independent datasets are produced: ECHO₁₁, ECHO₁₂, ECHO₂₁, and ECHO₂₂. b: A schematic of k-space construction with VS between ECHO₁₁ and ECHO₂₁ (k₁) and between ECHO₁₂ and ECHO₂₂ (k₂). Note that varying gradients (radial view angles) on a TR basis (a) makes it possible to employ the VS approach (b), thus enabling scan time reduction by two-fold. Note also that the central portions of k₁ and k₂ are composed only of ECHO₁₁ and ECHO₂₂, respectively, to maximally differentiate bone signals between two corresponding images.

Bone proton magnetization, due to its very short T₂ relaxation time, exhibits a substantial level of signal decay during the relatively long duration of RF₂, while soft-tissue and pore water magnetizations remain nearly intact during nutation. Figure 2 provides numerical simulations for T₂ relaxation times (0.01 – 100 ms) versus transverse magnetization with varying RF durations (0.04, 0.25, 0.5, and 1 ms; flip angle fixed at 12°) under different off-resonance conditions (0 – ±300 Hz). With varying RF pulse durations, transverse magnetization of bone proton (T₂: 0.1 – 0.5 ms) is gradually saturated while that of soft-tissue (T₂: 50 – 100 ms) is nearly unaltered (Fig. 2a). However, in the presence of magnetic field inhomogeneities, longer RF₂ pulses result in undesirably elevated soft-tissue signals in the difference (Fig. 2b), which potentially leads to ambiguities in bone identification. Given the simulation results, a RF₂ duration of ~ 0.5 ms was chosen in this work.

Figure 2.

a: Excitation profile (transverse magnetization) for a range of T₂ species (0.01 – 100 ms) in response to various RF pulses differing in durations (0.04, 0.25, 0.5, and 1 ms) but employing the same flip angle (12°). b: (S₁-S₂)/S₁; (S₁, S₂: transverse magnetization produced by RF₁, RF₂) with varying the duration of RF₂ (0.1 – 2 ms) while keeping that of RF₁ fixed to 0.04 ms, under different off-resonance conditions (0 – ± 300 Hz). In b, T₂ values of 0.5, 100, and 100 ms were used for bone, fat, and soft-tissue, respectively. One could expect from the simulation a that in dual-RF based UTE with longer RF₂ pulse, short T₂ species are better discriminated from long T₂ tissues in the difference between S₁ and S₂. However, in the presence of off-resonance frequencies long RF₂ pulses result in elevated long T₂ signals in the difference, thus making it difficult to differentiate between bone and soft-tissues.

It has also been demonstrated in (24) that the dual-RF based technique with subtraction of ECHO₂₂ from ECHO_11, as opposed to merely subtracting ECHO₁₂ from ECHO₁₁, further enhances bone contrast. However, in the parent method all four echoes undergo identical spatial encoding over every two TR cycles, resulting in doubling of total imaging time compared to conventional single-RF, dual-echo UTE imaging. Furthermore, since ECHO₁₂ and ECHO₂₁ are not utilized in the subtraction process, acquisition of those two signals may be redundant.

To improve imaging efficiency in the method of (24), in the present work a VS scheme is integrated into the DURANDE UTE scheme. To this end, radial view angles are varied every TR period (Fig. 1a) to make ECHO_11,12 and ECHO_21,22 traverse distinct k-space portions. Bone proton signal intensities in each echo follow the order: ECHO₁₁ > ECHO₂₁ >> ECHO₁₂ > ECHO₂₂. Thus, the first two and last two echoes are combined, respectively, to construct two independent k-space sets (k₁, k₂). Specifically, central regions of k₁ and k_2, in which intrinsic oversampling is performed, are composed only of ECHO₁₁ and ECHO₂₂ (Fig. 1b) to retain the highest and lowest bone signals, respectively, thereby maximizing bone signal specificity upon subtraction.

2.2. Bone-Selective Image Reconstruction

Since soft-tissue signals remain at a similar level over the four echoes while bone signals appear only at short TE echoes, subtraction of images pertaining to k₁ and k₂ highlights bone voxels. Given the sparse bone signals in the difference image, VS-DURANDE UTE scanning is further accelerated with fewer radial views by exploiting such sparsity during image reconstruction (25–27). The following sparse signal recovery problem can then be formulated:

$$\underset{I_1,I_2}{min}\frac{1}{2}{\sum }_{j=1}^{N_c}\left\{{\Vert k_{1,j}-F_{NU}\left({\mathbf{S}}_j{I}_1\right)\Vert }_2^2+{\Vert k_{2,j}-F_{NU}\left({\mathbf{S}}_j{I}_2\right)\Vert }_2^2\right\}+\lambda {\Vert I_1-I_2{e}^{-i\phi }\Vert }_1$$ [1]

where I₁ and I₂ are complex signals ($\in {\mathbb{C}}^{N\times 1};N=N_x{N}_y{N}_z$) to be solved for images (size: N_x × N_y × N_z) at TE₁ and TE_2, S_j is the receiver sensitivity matrix for the j-th coil ($\in {\mathbb{C}}^{N\times N}$), N_c is the number of receive coil elements, ${\mathcal{F}}_{NU}\left(\cdot \right):{\mathbb{C}}^{N\times 1}\to {\mathbb{C}}^{N_r{N}_p\times 1}$ is the non-uniform fast Fourier transform (NUFFT) operator that performs FFT on its argument and subsequent mapping onto prescribed radial coordinates with the size, N_r (number of readout points in each radial view) × N_p (number of projection lines), λ is the regularization parameter that balances data consistency with residual sparsity, φ is the phase accrual during ΔTE (=TE₂‒TE₁), and ‖∙‖₁ and ‖∙‖₂ are l₁- and l₂-norms. It is noted that as I₁ and I₂ are complex, phase correction with φ in the last term of Eq. [1] is essential, which otherwise potentially disrupts residual sparsity. Both S and φ are spatially smooth and thus can be estimated using over-sampled central low spatial frequency data. The solutions (I₁, I₂) can be found by employing an alternating minimization approach (28) that splits Eq. [1] into the following two sub-problems with respect to I₁ and I₂:

$$\{\begin{array}{l}\underset{I_1}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{2}{\sum }_{j=1}^{N_c}{\Vert k_{1,j}-{\mathcal{F}}_{NU}\left({\mathbf{S}}_j{I}_1\right)\Vert }_2^2+\lambda {\Vert I_1-I_2{e}^{-i\phi }\Vert }_1\\ \underset{I_2}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{2}{\sum }_{j=1}^{N_c}{\Vert k_{2,j}-{\mathcal{F}}_{NU}\left({\mathbf{S}}_j{I}_2\right)\Vert }_2^2+\lambda {\Vert I_2-I_1{e}^{i\phi }\Vert }_1\end{array}$$ Eq.[2a] Eq.[2b]

Each sub-problem is then solved by alternately applying the two operations: 1) sensitivity-encoding based projection to enforce data consistency (29, 30) and 2) soft-thresholding to control the sparsity of image difference (31). The two solutions are iteratively updated until convergence is reached. Implementation details of the above procedures are provided in the Appendix.

Given the two solutions, I₁ and I₂, a bone-specific image (I_Bone) is finally generated as in (24):

$$I_{\mathrm{B}\mathrm{o}\mathrm{n}\mathrm{e}}=\frac{\left|I_1\right|-\left|I_2\right|}{\left|I_1\right|+\left|I_2\right|}$$ [3]

The normalization in Eq. [3] removes proton-density weighting in the difference image while correcting for spatial variations of signal intensities due to transmit/receive field inhomogeneities.

2.3. Human Subject Studies

All human subject studies performed in this work were approved by the local Institutional Review Board. Informed written consent was obtained from all study participants (n = 5; 3 females, 2 males; ages = 32 ± 8 years) prior to imaging.

The VS-DURANDE UTE pulse sequence was implemented in the SequenceTree pulse programming environment (32). Unless otherwise stated, the following parameters were used: TR = 7 ms, TE₁/TE₂ = 0.06/2.36 ms, RF₁/RF₂ durations = 0.04/0.52 ms, flip angle = 12°, readout bandwidth = ±125 kHz, gradient ramp-up duration = 0.24 ms, N_r = 158, matrix size = 256 × 256 × 256, and field-of-view = 280 × 280 × 280 mm³. Additionally, a calibration scan was performed using the method in (33) to measure the gradient timing delay to correct for k-space trajectory mismatches.

To evaluate the effectiveness of the proposed VS approach and sparsity-constrained image reconstruction, a set of data was acquired in one subject at 3 T (TIM Trio; Siemens Medical Solutions, Erlangen, Germany) using the parent method (24) serving as reference. Here, 50,000 radial views were collected for each echo, resulting in 12 minutes of imaging time. A 12-channel head coil was used for signal reception. The acquired data was retrospectively undersampled by a factor of two by removing every other half projection lines, and subsequently processed with and without the VS scheme (Fig. 1b). Corresponding images were reconstructed directly by applying inverse NUFFT (iNUFFT) to each dataset. Furthermore, the reference data was decimated by a factor of four by retaining every fourth radial views only. Subsequently, images were reconstructed in three different ways for comparison: iNUFFT on k₁ and k₂, constructed with and without VS, respectively, and solving Eq. [1] for the view-shared data. In both cases (with and without VS), the k-space data contain 12,500 views each of ECHO₁₁ (k₁) and ECHO₂₂ (k₂), while in the former 12,500 views of ECHO₂₁ (k₁) and ECHO₁₂ (k₂) each were additionally utilized to fill peripheral k-space regions.

To investigate feasibility of the VS-DURANDE UTE with bone-selective image reconstruction method, four healthy subjects were scanned at 3 T (Prisma; Siemens Medical Solutions, Erlangen, Germany) for 3 minutes (N_p = 12,500 for each echo) using a 20-channel head/neck receiver coil. All I_Bone images obtained via Eqs. [1–3] were displayed in sagittal, coronal, and axial orientations using multi-planar reformatting. Furthermore, 3D renderings of the skull were produced in two study subjects, derived from the three image sets (I₁, I₂, and I_Bone). To this end, segmentation of bone voxels was performed using the ITK-SNAP software package (34, 35) in a semi-automatic fashion as follows: 1) manual labeling of representative voxels belonging to background, soft-tissue, and cortical bone, respectively (approximately 500 – 1000 voxels for each group), 2) training each group based on signal intensities of those voxels and their neighborhood, 3) generation of a classification-based probability map for cortical bone, and 4) automatic identification of bone voxels using an active contour segmentation pipeline.

3. RESULTS

Figures 3a-c compare three sets of images (|I₁|, |I₂|, and I_Bone) in one study subject obtained by the parent technique for reference (Fig. 3a; N_p = 50,000 each for ECHO₁₁ and ECHO₂₂, respectively, imaging time = 12 min), and its accelerated version with an undersampling factor of two without (Fig. 3b; N_p = 25,000 each for ECHO₁₁ and ECHO₂₂, 6 min scan time) and with (Fig. 3c; N_p = 25,000 each for all four echoes, 6 min scan time) the view-shared k-space ordering. A line profile of I_Bone in each method is plotted in Fig. 3d. Supporting Information Figure S1 shows I_Bone images of the whole-head in sagittal, coronal, and axial planes. When compared with the reference, the proposed VS scheme yields visually similar quality of images by effectively suppressing noise amplifications resulting from data undersampling.

Figure 3.

a-c: Three sets of images (|I₁|, |I₂|, I_Bone), acquired using the 3D DURANDE UTE pulse sequence and reconstruction based on the parent technique (using only ECHO₁₁ and ECHO₂₂) with 50,000 (a; 12 min) and 25,000 (b; 6 min) radial views for each echo, and the proposed technique with the VS scheme that utilizes all four echoes (Fig. 1b) with 25,000 views each (c; 6 min). d: Signal profiles of I_Bone, corresponding to the colored lines in a-c. Note that VS-DURANDE (c), when compared with the reference (a), suffers no appreciable loss in image quality with undersampling-induced noise amplifications (b and arrows in d) suppressed. Note further that in I_Bone the posteriorly positioned foam pad is also visible due to short proton T₂ of the polymeric material from which it is made.

Figure 4 shows three sets of images (|I₁|, |I₂|, and I_Bone) reconstructed from 12,500 radial half projections (simulating 3 min imaging time) of the reference data above using direct iNUFFT on k-space data constructed without (Fig. 4a) and with (Fig. 4b) the VS scheme, and the sparsity-constrained reconstruction (Fig. 4c), along with I_Bone signal intensity profiles for the three methods (Figs. 4d,e) in comparison to the reference (Fig. 3a). Corresponding I_Bone images of the whole-head in the three methods are provided in Supporting Figure S2. Amplified noise due to high undersampling in Fig. 3a impairs bone contrast, thus leading to ambiguity in identifying bone voxels (Fig. 4d). The VS method partially remedies the problem (Fig. 4b), while its combination with the sparsity constrained reconstruction yields significantly noise-reduced images (Fig. 4c) with the desired I_Bone signal profile retained (i.e., separation of inner and outer tables of cranial bone with diploe; Fig. 4e).

Figure 4.

a-c: I₁, I₂, and I_Bone images for retrospectively decimated data (12,500 views, 3 min), produced without (a) and with (b) the VS scheme, and by its combination with the bone-sparsity constrained reconstruction (c). d,e: Signal intensity profiles of I_Bone (horizontal colored lines in a-c) plotted together with the data from the reference scan (Fig. 3a). Note that image reconstruction using the VS + sparse prior (c) effectively suppresses amplified noise signals (a, arrow in d) resulting from data subsampling, thus leading to clear depiction of bone signals representing outer and inner table of the skull, comparable to those in the reference (e).

Figure 5 displays I_Bone images for the four additional subjects examined using the VS-DURANDE UTE with prospective undersampling (N_p = 12,500; imaging time = 3 min). All the bone-specific images in the sagittal, coronal, and axial orientations exhibit clear depiction of anatomic features of the skull over the entire head (e.g., mandible, sphenoid bone, zygomatic bone, and upper cervical spine), thus demonstrating feasibility of the proposed method in craniofacial imaging. Figure 6 displays 3D renderings of the skull in two subjects (Subjects 1 and 2 of Fig. 5), exhibiting the expected anatomic structures such as those mentioned above.

Figure 5.

Whole-skull bone images in four subjects, acquired and reconstructed using the proposed technique with prospective undersampling by a factor of four (imaging time = 3 min), displayed in sagittal, coronal, and axial planes. Note that diploe and separation of inner and outer layers of cranium are well visualized in all subjects. Note also clear demonstration of anatomic features such as mandible (yellow arrows), zygomatic bone (blue), sphenoid bone (red), and upper portion of cervical spine (green).

Figure 6.

3D renderings of the skull in two subjects (#1 and 2 in Fig. 4) in anterior, lateral, posterior, and superior views. Note the expected anatomic structures including mandible and upper portion of cervical spine. More subtle structures including zygomatic arch and mastoid process (blue and yellow arrows) are pointed out as well.

4. DISCUSSION AND CONCLUSIONS

VS-DURANDE UTE in combination with the bone-selective image reconstruction enables isotropic high-resolution (~ 1.1 mm) skull imaging of the whole head in a clinically practical scan time (~ 3 min). While building on the dual-RF based UTE bone imaging method (24) to enhance differentiation of cortical bone from long T₂ species, the proposed technique efficiently utilizes data redundancy both during signal acquisition (via VS) and image reconstruction (by solving the sparse recovery problem). The two acceleration strategies are synergistically combined, leading to four times higher imaging efficiency without visual loss of image quality when compared to the parent technique.

In the proposed method, there exists signal discontinuity around the boundary of the central portion of view-shared k-space. Although not prominent in the reconstructed images shown in this article, image artifacts and noise amplification can potentially result from the VS-related high-pass filter. Nevertheless, the level of such undesirable effects is minimized with the sparsity-constrained reconstruction, which is evident from the smoothly varying soft-tissue signals in the bone-specific image, obtained with the sparse prior (Fig. 4e). On the other hand, in order to completely avoid VS-induced artifacts, full readout samples of ECHO₂₁ and ECHO₁₂ could be employed to fill even the central portions of k-space. In this case, however, bone contrast in the difference image would be sacrificed because of signal averaging between ECHO₁₁ and ECHO₂₁, in which signal intensities of bone protons differ substantially (Fig. 2). Given the above considerations, the VS and sparse reconstruction approaches in this work are complementary to each other in reducing residual noise and artifacts while achieving optimal bone contrast.

The enhanced imaging speed should render the proposed technique useful as a potential alternative to CT. One possible clinical application is the evaluation and diagnosis of craniofacial abnormalities, including craniosynostosis, a relatively common disorder involving premature fusion of cranial sutures, in infants and young children (6, 36). In order to be a viable alternative to CT, the new MRI method would still need to be optimized for scan time, spatial resolution, minimizing number of views while preserving bone-specificity, and other image quality metrics. In particular, the receive bandwidth per pixel needs to be carefully chosen so as to exceed the spectral linewidth of the bone proton signal (~ 0.82 kHz at $T_2^{*}$ = 390 μs (6)), otherwise potentially resulting in loss of actual spatial resolution for bone voxels. Further investigation is also needed for quantitative inter-modality comparison.

Another potential application of the new method is attenuation correction in PET- MRI. It has been demonstrated that classification of bone, adipose tissue, and brain regions on UTE multi-echo images leads to accurate maps of ϒ-photon attenuation correction in PET imaging (12, 37). The flexibility of DURANDE UTE in adjusting TEs of ECHO₁₂ and ECHO₂₁ would enable water/fat separation based on the multi-point Dixon method (38). In this case, imaging efficiency should be maintained if the VS scheme were employed to construct an additional set of k-space data, in which low spatial frequency regions are occupied only by those echoes reflecting water/fat phase offset while high spatial frequency areas are filled with all four echo signals.

The normalized image via Eq. [3] provides adequate contrast between soft-tissues and bone voxels, facilitating skull segmentation. Nevertheless, a close look at the signal profiles of I_Bone in Figs. 3d and 4e suggests that soft-tissues are not completely suppressed because of $T_2^{*}$ weighting in I₂. A weighted subtraction of soft-tissue signals could be envisioned as an alternative with the following steps: 1) estimating a $T_2^{*}$ map using I₁ and I₂, 2) applying a $T_2^{*}$ threshold (e.g. 10 ms) to obtain rough segmentation of soft-tissue voxels, and 3) computing $I_1-I_2{e}^{\frac{\mathrm{\Delta }TE}{T_2^{*}}}$ for the nominator of Eq. [3]. While this $T_2^{*}$ correction would further enhance the bone contrast in normalized images, it may also result in locally amplified noise in the presence of errors in $T_2^{*}$ estimation.

Various solid-state MRI methods with short TR and small flip angles have recently demonstrated their potential for skull bone identification by means of ZTE (39, 40) and also UTE (41). In that work, bone segmentation was performed involving the following steps: 1) correction for intensity variations in proton-density weighted images, 2) inverse-log transformation, 3) search for optimal values of thresholding on Gaussian distribution models in a heuristic (39, 40) or analytic (41) manner, and 4) successive application of morphological refinements such as connected component analysis. In this process, ambiguities may arise from overlap of the histogram comprising either bone/air or bone/soft-tissue, resulting in falsely classified bone voxels. The method presented in this study produces self-normalized bone images with virtually complete elimination of soft-tissue signals via Eq. [3], allowing for simplification of post-processing. One limitation of DURANDE UTE, in combining data with different TEs, is that certain subregions are susceptible to errors caused by induced frequency shifts. Areas near air-tissue interfaces are particularly prone to errors, given the 9 ppm difference in magnetic susceptibility between air and tissue water.

In conclusion, the results demonstrate the feasibility of the presented bone-selective MRI technique for whole-skull imaging at isotropic high spatial resolution in a clinically practical scan time. Further work is required to examine the accuracy of the method relative to CT imaging, and comparison to alternative MR approaches.

Supplementary Material

Figure S1.

Bone-specific images in sagittal, coronal, and axial planes, acquired using the 3D DURANDE UTE pulse sequence and reconstructed using 50,000 (a; scan time = 12 min) and 25,000 (b; scan time = 6 min) radial views for ECHO₁₁ and ECHO₂₂ each, and the proposed VS scheme that employs all four echoes with 25,000 views each (c; scan time = 6 min). Here, bone images of the whole-head are provided to supplement Fig. 3.

Figure S2.

Comparison of bone-specific images in sagittal, coronal, and axial planes, obtained from retrospective undersampling by a factor of 4 (12,500 radial views per each echo; scan time = 3 min) and reconstructed without (a) and with (b) the VS scheme, and its combination with the sparse prior on bone voxels (c). Here, bone images of the whole-head are provided to supplement Fig. 4.

APPENDIX

The following describes implementation steps to solve the sparsity-constrained linear inverse problems in Eq. [2] using the iterative soft-thresholding algorithm (31):

• Step 0: Iteration n = 0, $I_1^0={\sum }_{j=1}^{N_c}{{\mathbf{S}}_j^H\mathcal{F}}_{NU}^{-1}\left(\mathbf{D}{k}_{1,j}\right)$ and $I_2^0={\sum }_{j=1}^{N_c}{{\mathbf{S}}_j^H\mathcal{F}}_{NU}^{-1}\left(\mathbf{D}{k}_{2,j}\right)$

• Step 1: Enforce data consistency between measured k-space data and sensitivity-encoded images at current iteration by updating the solution as:

$$\widetilde{I_1^n}=I_1^n+\frac{1}{{\alpha }_n}{\sum }_{j=1}^{N_c}{{\mathbf{S}}_j^H\mathcal{F}}_{NU}^{-1}\left(\mathbf{D}\left(k_{1,j}-{\mathcal{F}}_{NU}\left({\mathbf{S}}_j{I}_1^n\right)\right)\right)$$

• Step 2: Control the sparsity of the difference image vector (I₁ − I₂e^−iφ) by solving:

$$\begin{gathered} I_1^{n+1}=\underset{I_1}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{2}{\Vert I_1-\widetilde{I_1^n}\Vert }_2^2+\frac{\lambda }{{\alpha }_n}{\Vert I_1-I_2{e}^{-i\phi }\Vert }_1 \\ =\mathcal{T}(\widetilde{I_1^n}-I_2{e}^{-i\phi },\frac{\lambda }{{\alpha }_n})+I_2{e}^{-i\phi } \end{gathered}$$

• Step 3: Repeat Steps 1 and 2 for Eq. [2b] with ${I_1=I}_1^{n+1}$.

• Step 4: Repeat Steps 1 ‒ 3 with ${I_2=I}_2^{n+1}$ and n = n + 1, if n ≤ N_iter.

Here, the ${\mathbf{S}}_j^H$ represents the Hermitian transpose of S_j, D is the sampling density compensation matrix, $\mathcal{T}$ is the soft-thresholding operator: $\mathcal{T}\left(z,\beta \right)=max\left(\left|z\right|-\beta \right)\cdot z/\left|z\right|$, and N_iter is the maximum number of iteration. In the present implementation, the step size α_n at each iteration was determined by ${\alpha }_n={\sum }_{j=1}^{N_c}{\Vert {\mathcal{F}}_{NU}\left({\mathbf{S}}_j{d}^n\right)\Vert }_2^2/{\Vert d^n\Vert }_2^2$ where dⁿ = Iⁿ − Iⁿ⁻¹ as suggested by (42). Convergence of the solutions was reached within N_iter = 20. Figure A1 provides a flow diagram of the above computation steps, implemented in Matlab (MathWorks Inc., Natick, MA) with the nfft software library (43) included for forward and inverse NUFFT operations.

Figure A1.

A flow diagram of the computation steps to solve Eqs. [2a] and [2b] in an iterative framework.