Model-Based Image Reconstruction with Wavelet Sparsity Regularization for Through-Plane Resolution Restoration in T2-Weighted Spin-Echo Prostate MRI

Abstract

Purpose

The purpose is to develop a model-based image reconstruction method using wavelet sparsity regularization for maintaining restoration of through-plane resolution but with improved retention of signal-to-noise ratio (SNR) vs. linear reconstruction using Tikhonov (TK) regularization in high through-plane resolution (1mm) T2-weighted spin-echo (T2SE) images of the prostate.

Methods

A wavelet sparsity-regularized (WS) image reconstruction was developed which takes as input a set of ≈80 overlapped 3mm-thick slices acquired using a T2SE multislice scan and typically 30 coil elements. After testing in contrast and resolution phantoms and calibration in six subjects, the WS reconstruction was evaluated in 16 consecutive prostate T2SE MRI exams. Results reconstructed with nominal 1mm thickness were compared with those from the TK reconstruction with the same raw data. Results were evaluated radiologically. The ratio of magnitude of prostate signal to peri-prostatic muscle signal was used to assess presence of noise reduction. Technical performance was also compared with a commercial 3D T2SE sequence.

Results

The new WS reconstruction was assessed as superior statistically to TK for overall SNR, contrast, and multiple evaluation criteria related to sharpness while retaining the high (1mm) through-plane resolution. WS tended to provide improved overall diagnostic quality vs. TK but not significantly so. In all 16 studies the prostate-to-muscle signal ratio increased.

Conclusion

Model-based wavelet sparsity-regularized reconstruction consistently provides improved SNR in high (1mm) through-plane resolution images of prostate T2SE MRI vs. linear reconstruction using Tikhonov regularization.

INTRODUCTION

T2-weighted spin-echo (T2SE) imaging is one of the most commonly used pulse sequences in MRI, performed in nearly 100% of all clinical exams. This is generally done with a two-dimensional (2D) multislice acquisition [1] using a fast-spin-echo readout [2]. Although actual spatial resolution is dependent on the anatomic region under study, the slice thickness is generally no smaller than 2 mm and is typically 3 - 6× coarser than the inplane resolution. Multiple approaches have been studied to obtain improved through-plane resolution such as smaller slice thicknesses, e.g. [3], three-dimensional (3D) T2SE [4–6], T2-prepared steady-state free precession (SSFP) [7, 8] with centric encoding [9], and more recently a 3D FSE approach allowing flexible echo time selection [10]. These 3D FSE methods have been studied in, e.g. musculoskeletal [11, 12], prostate [13, 14], and neuro [15] applications. However, conventional 2D T2SE multislice is still commonly used. As a specific example, for prostate MRI the current (v2.1) recommendation of the Prostate Imaging - Reporting and Data System (PI-RADS) calls for 2D T2SE imaging with 3D methods as a possible adjunct [16, 17].

Another strategy for obtaining finer resolution in the slice select (Z) direction is to acquire data with some redundancy and treat the estimation of thinner slices as an inverse problem. An approach initially studied in [18] is to acquire 2D slices which have some slice-to-slice overlap along Z and account for the slice profile in the inversion process. An alternative approach is to acquire 2D multislice data sets of the object in two or more orientations and then leverage the redundant data to estimate the object with resolution higher than in one set [19–24]. This use of multiple low resolution images to estimate a high resolution image set is sometimes referred to as “super resolution.” Recently, investigators have used the single orientation strategy of [18] to generate high through-plane resolution images along that orientation. Applications have included 2D time-of-flight MR angiography [25] and T2SE prostate MRI [26]. In both cases the inverse problem of deconvolution of the slice profile was addressed using model-based linear reconstruction with Tikhonov regularization (TK).

With other acquisition parameters held constant, as slice thickness is reduced in MRI the signal-to-noise ratio (SNR) is reduced by a similar factor. For the studies of Refs. [25, 26], the Tikhonov regularization parameter was used to trade off the degree of resolution improvement with SNR loss. However, this resulted in performance at the limit of acceptable SNR. The motivation for this current work was to investigate if an alternative reconstruction approach, specifically one which is wavelet-sparsity-regularized (WS) could continue to restore the recovery of high through-plane spatial frequencies similar to TK but with improved SNR. Such methods have been widely used in MRI for reduction of scan time, e.g. [27]. We next describe the model-based WS reconstruction and its implementation, application, and evaluation in comparison with TK.

METHODS

Model-Based Wavelet-Sparsity-Regularized (WS) Reconstruction

The acquired data is comprised of (k_X, k_Y, coils) samples for each Z_acq slice. The received signal y (rasterized from 4D: K_x, K_y,acq, Z_acq, N_coils) is modeled as:

$$y=\text{Γ}F\text{Φ}BCx+\text{ϵ}=Ax+\text{ϵ}$$ (1)

where x is the (3D; rasterized from X,Y,Z) object to be estimated, C, formally blockwise diagonal of size X Y Z N_coils × X Y Z, is practically implemented via elementwise multiplication by the (3D) coil responses, B (diagonal matrix; X Y Z N_coils) likewise performs elementwise modulation by the slice selection profile (along Z), Φ (X Y Z_acq N_coils × X Y Z N_coils) selects the (3D) subset (removing slice-direction padding) of the reconstruction volume encoded within y, F is the two-dimensional Fourier transform operation along X and Y, and Γ (K_x K_y,acq Z_acq N_coils × K_x K_y Z N_coils) selects those points actually sampled in k-space, specifically allowing undersampling along Y. ϵ is assumed additive white Gaussian noise. The entire 4D (K_x K_y,axq Z_acq N_coils) signal y and 3D (X Y Z) object x are considered simultaneously, not in a slice-by-slice manner. For this work the data acquired with the 25 to 30 coil elements was prewhitened and then compressed into 12 virtual coils using the method of [28]. Any residual cross correlation was not accounted for. We seek to minimize

$$J\left(x\right)=\frac{\lambda }{\left|\text{Ω}\right|}\sum _{n\in \text{Ω}}\Vert {\text{ΨΔ}}_{n}x{\Vert }_1+\Vert Ax-y{\Vert }_2^2$$ (2)

where the sparsifying transform Ψ is a lifted [29] 3D Daubechies-4 wavelet transform [30] and Δ_n is a 3D spatial shifting or cycle-spinning operator, applied for a shift from set Ω, which approximates shift-invariance for the wavelet transform [31, 32]. This minimization is currently performed with composite splitting projected gradient descent [33], which comprises the following iteration:

$$x_{t+1}==\frac{1}{\left|\text{Ω}\right|}\sum _{n\in \text{Ω}}{\text{Δ}}_n^{\text{*}}{\text{Ψ}}^{\text{*}}{S}_{\lambda \tau /2}\left\{{\text{ΨΔ}}_n\left(x_t-\tau A^{\text{*}}\left(A{x}_t-y\right)\right)\right\}$$ (3)

where * denotes the Hermitian transpose operator, τ = 1/max(eigenvalues(A*A)) as determined using the power iteration method, and S is the soft thresholding operator:

$$S_{\gamma }\left(x\right)=\{\begin{array}{cc}x\frac{\left|x\right|-\gamma }{\left|x\right|},& \left|x\right|\ge \gamma \\ 0,& otherwise\end{array}$$ (4)

To retain contrast, the original lowpass wavelet subband intensities (without shrinkage) are retained during the soft-thresholding process. While incoherent sampling could be used – and may provide a performance improvement, it is beyond the scope of this work. As discussed later, the regularization term λ (or thresholding parameter γ where γ = λ / 2) is selected to balance sharpness and noise suppression.

The coil response C was extracted from the auto-calibration lines of the acquisition, as described in ESPIRiT [34], utilizing the central 24×24 region of the calibration signal, a 6×6 kernel size and a singular value cut-off of 10⁻³ and eigenvalue threshold of 0.999, followed by (Gaussian) filtering performed after extraction to improve slice-to-slice consistency within the calibration maps in keeping with the slow physical change in sensitivity along the slice direction for an axial acquisition with surface coils. The Δ_n operator set contained 24 shifts (6 × 2 × 2 in slice, phase, and frequency). Six slice-shifts were used to match the number of distinct groups of slices as defined by their sampling [35], and was found to provide superior results compared to symmetric (e.g. 3 × 3 × 3) neighborhoods. The slice profile B was measured in previous experiments [35] using the same pulse sequence as used in this work. The solver was implemented in MATLAB with compiled C++ code for the lifted wavelet transform. A fixed count of iterations (200) was performed, with a representative penalty curve shown in Supporting Information Figure S1. Total processing time for a volume was approximately 3 hours, with iterations consuming 45 minutes, and the bulk of the remainder consisting of ESPIRiT processing.

Data Acquisition

All imaging was performed on a 3 T whole body MRI system (Premier v28.0, General Electric, Waukesha WI USA). The model-based WS reconstruction method was evaluated in studies in phantoms and humans, the latter performed under an IRB-approved protocol for which all participants provided written informed consent. Data acquisition used the 2D T2SE k_Z-multislice (KZM) parameters of the prostate MRI study of Ref. [35], shown in Table 1. TR, TE, and inplane resolution were selected to be consistent with the recommendations of PI-RADSv2.1 [16]. The R=1.5 k-space-based acceleration [36] with auto-calibration reduces the number of acquired k_Y encodes per slice by 1.39× from 560 (2 × 280) to 403, requiring 20 repetitions using ETL 21, or about one minute to fully sample 13 slices at TR 3000 msec. This scales to about 6 minutes for 78 slices. If >78 slices are desired, TR can be slightly increased to sample >13 slices per TR interval. In this work the acquired slice thickness was 3 mm with a 1-mm slice-to-slice increment, or equivalently, a 2-mm overlap slice to slice. The nominal acquisition orientation was axial, but for human studies the slice select direction was adjusted on a patient-specific basis to be aligned with the interface between the posterior aspect of the prostate capsule and the anterior rectal wall, as determined from sagittal localizer images. This tended to keep the prostate within the central region of the axial slices. Signal reception was done using anterior and in-table posterior coil arrays, with linear five-element groups of both selected automatically by the scanner software based on the technologist-prescribed FOV.

Table 1.

Acquisition parameters for axial T2-weighted spin-echo (T2SE) of prostate.

Parameter	2D kZ-Multislice (KZM)	3D CUBE
Field-of-View (A/P (freq) × L/R (phase))	20×20 cm2	20×20 cm2
Inplane Sampling (Frequency × Phase)	320×280	320×180
Inplane Resolution (Frequency × Phase)	0.62×0.71 mm2	0.62×1.1 mm2
Slice Thickness	3 mm	1.2 mm
Slice-to-Slice Increment	1 mm (2 mm overlap)	abutting
Number of Slices (range, median)	78 – 102, median 84	70
Coverage along Z (mm)	typically 84	84
Repetition Time (TR) (range, mean (msec))	3000 – 3650, mean 3179	3000
Effective Echo Time (TEEFF, msec)	100	68
Echo Train Length	21	60
Signal Bandwidth	64 kHz (±32 kHz)	256 kHz (±128 kHz)
Acceleration (RY × RZ)	1.5	2.0 × 1.75
Number of Averages	2	2
*No Phase Wrap (NPW)	enabled	enabled
Number of Receiver Coil Elements	typically 25 or 30	30
Scan Time (range, mean (m:s))	6:00 – 7:18, mean 6:20	6:20

A/P: anterior/posterior; L/R: left/right; S/I: superior/inferior;

^*enabling NPW doubles the phase FOV but halves the Number of Averages

Low Contrast Phantom

After it was implemented computationally the model-based WS algorithm was evaluated using a low contrast phantom. The phantom was designed to test the ability of the T2SE sequence to detect small negative contrasts with no identifying containment or chemical shift vs. background, similar to the task of detection of malignancies in prostate T2SE MRI. The phantom consisted of a 17 cm diameter, 15 cm long cylindrical glass container into which was poured a 70%/30% mix of poly-vinyl-chloride (PVC) resin / PVC resin softener (LureCraft Industries, LaGrange IN). After heating to 300°F the mixture solidifies upon subsequent cooling, providing background signal. During the pouring several solid 3mm, 6mm, and 10mm diameter glass rods were inserted along the long axis of the container but not touching the bottom. Shortly before the PVC mixture hardened the glass rods were removed, preserving the long, small-diameter cylindrical channels. These were then filled with slightly modified mixtures of 72.5/27.5, 75/25, and 80/20 PVC/softener in subsequent pours, which provided progressively increased negative contrast inclusions vs. the background signal in T2SE images. T2 values for the mixtures were in the range of 25 to 30 msec. The phantom was imaged with the T2SE acquisition, and images were reconstructed using the standard TK and the new WS reconstruction methods.

Resolution Phantom

A phantom was designed to assess spatial resolution ranging from 0.152 to 0.625 line pairs (lp)/mm. It is composed of polylactic acid (PLA) and was constructed using a 3D printer. The phantom base had area 13 x 3.5 cm² above which vanes were printed to a thickness of 1.5 cm. Cutouts in the base identify specific spatial frequencies. Three copies of the phantom were fixed in orthogonal orientations within a 9 × 20 × 30 cm³ water-filled plastic container, allowing resolution assessment in three directions. The phantom was imaged using the KZM T2SE sequence of Table 1, and a 3D FSE “CUBE” commercial sequence available on the scanner (parameters also shown in Table 1). For the latter a minimum TR of 3000 msec was specified to provide adequate T2-weighted contrast and the maximum allowed TE_EFF (68 msec). Parameters were further selected to provide equivalent coverage along Z (84 mm) and acquisition time as for the 2D T2SE KZM. Also used was a standard clinical T2SE multislice sequence (26 3mm thick abutting slices, 4 averages, other parameters identical to T2SE KZM, typical scan time 4:20).

Human Studies

Based on the favorable results observed in phantom studies, it seemed appropriate to evaluate the performance of the WS reconstruction vs. standard TK in human studies. The subjects were all men for whom prostate MRI was clinically indicated, and data acquisition was done using the technique of Table 1. Although anti-spasmodic drugs were not used, all subjects were instructed to fast for three hours before the MRI exam and to empty the rectum and void immediately prior to the exam.

Six calibration studies were done at the outset to allow two uroradiologist reviewers (8 and >15 years experience in prostate MRI) to set the WS regularization parameter λ which would then be fixed for the subsequent comparison studies. Selection of this parameter determines the radiologically acceptable tradeoff between noise reduction and blurred or otherwise patchy signal. Demographics of these calibration studies were age (58 – 84 years, mean 64.3), Body Mass Index (BMI) (23.5 – 29.8 kg/m², mean 27.5), prostate volume (23 – 84 cc, mean 41.8). For these cases the WS reconstruction was performed using several values of λ: 0.00005, 0.00010, 0.00015, 0.00020, 0.00030, 0.00040. The two reviewers independently reviewed the cases, and both settled on λ = 0.00015 as providing the best tradeoff between resolution enhancement with SNR.

Comparison studies were next done in 16 consecutive subjects using the T2SE sequence of Table 1. Demographics for these were age (63 – 82 years, mean 70.3), BMI (22.5 – 29.6 kg/m², mean 25.4), prostate volume (20 – 99 cc, mean 59.1). PI-RADS scores as determined from the companion clinical MRI exams were 3 subjects with PI-RADS 2, 4 with PI-RADS 3, 5 with PI-RADS 4, 1 with PI-RADS 5, and 3 post focal therapy with no applicable PI-RADS score. For each study reference Tikhonov and wavelet-sparsity-enforced-reconstructed stacks of 1mm thick images were generated, the former using the Tikhonov Regularization method of Refs. [26, 35, 37] and the latter using the aforementioned λ = 0.00015.

For each comparison study the two resultant image series (TK and WS) were presented independently to the two reviewers in random order (Series A, B) with identifying information removed. The two series were compared (A vs. B) using multiple criteria. These included relative SNR, contrast, fidelity of contrast (lack of patchiness or blur), and other artifact. They were also compared for sharpness of: prostate capsule, interface between the transition zone (TZ) and peripheral zone (PZ), any structures within the TZ or PZ, and seminal vesicles. Comparisons were done using a five-point scale (−2: Series A markedly superior to Series B; −1: Series A superior to Series B; 0: Series A equivalent to Series B; +1: Series A inferior to Series B; +2: Series A markedly inferior to Series B). After derandomization, results of the comparative evaluations were tested for significance using the Wilcoxon signed-rank test [38] with p<0.05 taken as significant. Finally, each series was evaluated individually for overall diagnostic quality on a three-point scale (0: non-diagnostic, 1: adequate for diagnosis; 2: more than adequate for diagnosis).

Quantitative Assessment of Noise Performance

In prostate T2SE MRI the signal magnitude of the muscle surrounding the prostate is near zero owing to its relatively short, 30-35 msec T2 relaxation time [39, 40] vs. TE_EFF ≈ 100 msec. Although already dark in the TK images, the muscle signal was consistently observed to further subtly diminish in the WS images. We sought to quantify this. For materials whose signal levels are small relative to the noise, the noise level can contribute appreciably to the reconstructed magnitude values; e.g. for a single receiver the noise is Rician [41]. The WS thresholding (Eq. 4) eliminates small signal oscillations in the wavelet domain, presumed to be noise, but the unity slope for |x| ≫ γ as well as the lack of thresholding for the lowpass wavelet subband tend to preserve overall signal level and contrast. This is distinct from TK reconstruction in which no noise thresholding is done. The effect of the WS reconstruction in reducing the noise level in low signal regions was evaluated as follows. For each of the 16 comparison studies a mid-gland axial section was selected, and large (>750 pixel) regions-of-interest (ROIs) were identified within the prostate and either the left or right obturator internus muscle. The same ROIs were used for both the TK- and WS-reconstructed images for a given study. The mean values were measured in each ROI, and the ratio of mean values of prostate to muscle was used as a metric of noise performance.

Contrast Studies

In addition to the above-described comparison of resolution, 2D T2SE multislice and 3D CUBE were compared with respect to contrast at physiologically relevant T2 times; i.e. larger than those of the PVC phantom. The standardized (NIST) contrast phantom [42] containing vials with known T1 and T2 relaxation times was imaged using the KZM and CUBE acquisitions of Table 1. Signal levels and noise measurements were made in multiple vials.

RESULTS

Figure 1A–C shows 1mm thick axial images of the contrast phantom formed using the reference Tikhonov (TK) reconstruction (A) and the wavelet sparsity (WS) approach (B, C) all from the same acquired data. The 6mm and 10mm diameter inclusions are shown in the left and right columns of each image with smaller contrast in the second vs. top row. The improvement in apparent SNR over TK (A) provided by WS allows portrayal of the 10mm lesion in the second row using λ = 0.02 (purple arrow, B) and further portrayal of the 6mm lesion using λ = 0.05 (blue arrow, C). Sagittal reformats 2.5 mm thick through the 10 mm diameter inclusion as identified (A, dashed line) are shown in (D-F). These were chosen to include the bottom of the inclusion within the phantom. Similar progressive improvement in apparent SNR is observed in (D-F) as in (A-C), but the sharpness of the inferior edge of the inclusion is artifactually blunted for the λ = 0.05 case (red arrow, F) vs. (D) and (E).

Figure 1.

1mm thick axial images (A-C) of the low contrast phantom from the same raw data set reconstructed using the reference Tikhonov (TK) regularization (A) and the wavelet-sparsity (WS) reconstruction using λ = 0.02 (B) and λ = 0.05 (C). Note in (B) the improved visualization of the larger diameter lower contrast inclusion (purple arrow) vs. (A) and in (C) the further improved visualization of the smaller inclusion (blue arrow) vs. (A) and (B). 2.5 mm thick sagittal reformats (D-F) made along the higher-contrast, larger-diameter inclusion (dashed line, A) show similar improvement of visualization. However, the sharp inferior edge of the inclusion is blunted for the λ = 0.05 case (red arrow, F) vs. (D) and (E).

Results of the studies of resolution are shown in Figure 2. (A) is a photograph of the phantom with values of resolution at the notches identified. (B-D) show images of the phantom in the acquired axial orientation. The frequency and phase encoding directions identified in (B) were the same for (B-D). (B) was acquired using standard T2SE multislice with abutting 3mm thick slices, (C) using the KZM T2SE, and (D) using 3D CUBE. For (B-C) the highest spatial frequency pattern (0.625 lp.mm) is well portrayed for both directions. The acquired KZM and TK reconstructed views matching (C) are essentially indistinguishable from (C) and not shown. For 3D CUBE (D), the phase encoding resolution is somewhat coarser than 1.0 mm. (E-I) show sagittal reformats of the axial images. (E) was formed from abutting 3mm thick 2D multislice data and shows the expected limited slice resolution. (F-H) were all formed from the same acquired data set of overlapped 3mm thick slices (Table 1 KZM). (F) is a reformat of the acquired 84 axial slices. Modulation at spatial frequencies higher than the cutoff of (E) is clearly observed (blue arrow) as well as a region of negligible modulation owing to the zero crossing along k_Z of the rect-like excitation (red arrow). (G) shows the reformat of axial images reconstructed using the Tikhonov (TK) regularization of [26], providing increased modulation at all frequencies vs. (F). (H) shows the reformat formed from axial images reconstructed using the new wavelet-sparsity (WS) method with further improved modulation vs. (G) and frequency content approaching the notch at 0.5 lp/mm (orange arrow). Finally, results using CUBE (I) have a cutoff at a frequency consistent with the 1.2 mm partition thickness. We note parenthetically that TK (G) and WS (H) both undo the intensity inversion caused by the excitation at k_Z values above the zero crossing, seen in that peaks and valleys remain as such before and after the zero crossing (yellow line, H). Frequency response profiles based on (F-I) are presented in Supporting Information Figure S2.

Figure 2.

Evaluation of spatial resolution. (A) Photograph of the resolution phantom. (B-D) Images of two replicates of the phantom placed to evaluate resolution in the frequency and phase encoding directions. For all four (two shown) 2D cases (B-C) the resolution in both directions is finer than 0.8mm, consistent with Table 1. Acquired KZM and Reconstructed TK, not shown, are essentially indistinguishable from (B). For the 3D CUBE case (D) the phase resolution is slightly coarser than 1.0mm, also consistent with Table 1. (E-I) Images showing the resolution along the slice select direction for the 2D cases (E-H) and the slice encoding direction for 3D (I). (E) was formed from abutting 3mm thick slices. (F-H) were all formed from the same axial data set in which overlapped 3mm thick slices were acquired (Table 1 KZM). For the 3D CUBE case (I) the observed resolution (≈1.2mm) is consistent with that specified in Table 1. The orange arrow in (H) indicates the furthest extent of modulation, nearing the 1.0mm notch, while the yellow line highlights the correction of modulation inversion in (G) and (H) compared to the acquired (F) beyond the initial null in the slice profile’s frequency response.

The ability of WS reconstruction to provide improved apparent SNR vs. TK (Figure 1) while retaining and perhaps slightly improving the improvement in Z resolution of TK vs. conventional 3mm abutting slices (Figure 2H vs. 2E) provided the incentive for evaluation in human studies. Figure 3 shows images from one of the six calibration studies, illustrating dependence of image appearance on parameter λ in the wavelet-sparsity (WS) regularized reconstruction (B-E) as compared to reference Tikhonov regularization (A). All images (A-E) were formed from the same raw data set. In this example the nodule (B, orange arrow) is portrayed with improved contrast in WS with λ=0.0001 vs. its appearance in the TK reference (A, yellow arrow). However, as λ further increases the nodule signal level artifactually becomes more patchy and uniform (e.g. E, blue arrow), and the prostate margin becomes less distinct (E, green arrows).

Figure 3.

Example from an in vivo calibration study of the dependence of image appearance in wavelet-sparsity (WS) reconstruction on regularization parameter λ. (A-E) were all formed from the same raw data. (A) 1mm thick axial image of prostate formed using reference Tikhonov (TK) regularization. A nodule (yellow arrow) is identified in the transition zone for reference. (B-E) Corresponding images formed using WS reconstruction with the λ value shown. The reference nodule (B, orange arrow) appears to have improved SNR vs. (A), but as λ increases further the nodule signal artifactually appears progressively more patchy and smeared (e.g. E, blue arrow). At large λ the prostate margin becomes less distinct (E, green arrows).

Figure 4 shows results of the comparative studies. For criteria not related to sharpness, WS was evaluated as significantly superior to TK for relative SNR (A), contrast (B), fidelity of contrast (C), and equivalent in artifact (D). No artifact specifically associated with WS was observed. For sharpness-related criteria, WS was significantly superior to TK for PZ and TZ structure sharpness (E-F). For sharpness of the prostate capsule, PZ/TZ interface, and seminal vesicles, WS was scored superior to TK but not significantly so (Supporting Information Figure S3).

Figure 4.

Results of radiological evaluation of reference Tikhonov regularization (TK) vs. wavelet-sparsity (WS)-regularized reconstruction for (A) relative SNR, (B) contrast, (C) fidelity of contrast, (D) artifact, and sharpness of structures in the peripheral zone (PZ) (E) and transition zone (TZ) (F) by two radiologists on a five-point scale. Results from each of the radiologists are shown in stacked columns in a different color. WS was preferred to TK in terms of relative SNR, contrast, fidelity of contrast, and sharpness of structures in the peripheral zone and transition zone (p<0.05). No significant difference was observed regarding artifact. (N = total number of samples, μ = mean value)

Figure 5 shows the evaluations of overall diagnostic quality. As seen, 6 of 32 evaluations were upgraded in WS vs. TK with none downgraded.

Figure 5.

Results of radiologist assessment of overall diagnostic quality on a three-point scale. Scores for the reference Tikhonov regularization (TK) and wavelet-sparsity (WS)-regularized reconstruction for a given study are connected by a line. As seen, scores for six studies were upgraded from Adequate for TK to More than Adequate for WS (p<0.05). Mean scores were 0.97 for TK and 1.16 for WS.

Images in Figure 6 illustrate performance in the PZ. Supporting Information Figure S4 illustrates results in the TZ. Figure 7 shows the ability of the 1mm slice resolution of both TK (A) and WS (B) to portray a lesion in thin axial slices. Figure 7 additionally illustrates the potential to form reformats in sagittal (E) and coronal (I) orientations from the 1mm thick axial WS images, along with comparisons to TK, T2SE, and native acquisitions.

Figure 6.

Images from Study #8. (A) 1mm thick axial section at the mid-gland of the prostate reconstructed using the reference Tikhonov regularization (TK). (B) 1mm thick reconstruction using the same acquired data as for (A) but with the wavelet-sparsity (WS)-regularized reconstruction. WS image (B) in general has less noise than LR image (A). The peripheral and transition zone differentiation (A, yellow arrow) and a small triangular-shaped lesion in the left posterior lateral peripheral zone (A, red arrow) are better defined with WS (B) than TK (A) reconstruction. (C) Corresponding 3mm thick axial image acquired using conventional T2SE sequence (similar parameters to those in Table 1 for KZM but with 4 averages, 26 abutting 3mm thick slices acquired in 4:40.)

Figure 7.

Images from Study #16. (A) Image from 1mm thick slice in the apex region formed using Tikhonov regularization (TK). Suspicious focal lesion in the left peripheral zone is identified (A, yellow arrow). Linear area of hypointensity is present in the right posterior lateral peripheral zone (A, green arrow). (B) Corresponding image of the same slice as (A) using the same acquired data but formed using wavelet-sparsity (WS)-regularized reconstruction. SNR appears improved vs. (A). The aforementioned left-sided focal lesion and right-sided linear lesion are better defined with WS (B) than TK (A). (C) Corresponding 3mm thick section of the same region formed using conventional T2SE acquisition. This was the only slice portraying the suspicious lesions, while they were present in three slices for TK and WS (Supporting Information Figure S5). (D-G) Comparison of sagittal images created by reformatting 1mm thick TK reconstruction (D), 1mm thick WS reconstruction (E), and 3mm thick abutting T2SE (F) with direct sagittal T2SE acquisition (G). S/I resolution of (E) is superior to (F) but is not as fine as the 0.8mm S/I resolution of (G). Subtle motion also causes scalloping artifact in (D-F). (H-K) Comparison of coronal images created by reformatting 1mm thick TK reconstruction (H), 1mm thick WS reconstruction (I), and 3mm thick abutting T2SE (J), with direct coronal T2SE acquisition (K). S/I resolution of (I) is superior to (J) but inferior to the 0.43mm S/I resolution of (K).

Figure 8 illustrates measurement of the noise performance metric in TK (A, C) and WS (B, D) reconstruction. As shown, the metric values are 2.22 for TK and 3.16 for WS, indicating an increase (improvement) of 1.42× in this metric for this study. The metric exceeded unity for all 16 comparison studies, ranging from 1.20 to 1.52 with a mean of 1.37. Fig. 8E compares pixel values for TK (ordinate) vs. WS (abscissa) for the same slice, illustrating the skewing in TK for low signal values.

Figure 8.

Illustration of computation of noise performance metric from Study #4. (A) 1mm thick axial image formed using Tikhonov regularization (TK). (B) 1mm thick axial image formed from the same data as (A) using wavelet-sparsity (WS)-regularized reconstruction. Yellow arrow in (B) identifies suspicious lesion, and orange star identifies right obturator internus muscle. Signal within muscle visually appears slightly darker in (B) than in (A). The prostate capsule (A, red arrows) and transition and peripheral zone differentiation (A, blue arrows) appear more conspicuous with WS (B) than TK (A). (C) Same as (A) but with additional yellow (prostate) and orange (muscle) ROIs shown used for measurement of mean signals. Ratio of mean values of prostate to muscle ROIs = 2.22. (D) Same as (B) but with identical ROIs as used in (C). Ratio of mean values of prostate to muscle ROIs = 3.16. Improvement in metric provided by WS over TK = 1.42. (E) Comparison of signal values in TK (ordinate) vs. WS (abscissa) in the slice shown in (A-D).

Figure 9 shows axial images of the NIST phantom as acquired with the overlapped multi-slice sequence of Table 1 (A) and subsequently reconstructed using standard TK reconstruction (B) and WS-based reconstruction (C) as well as an image acquired using the 3D FSE CUBE sequence of Table 1 (D). In (A) Vials 1-10 are numbered, and shown next to Vials 4-8 are the T1 and T2 relaxation times [42]. The T2 values for Vials 4-8 are similar to those relevant for prostate T2SE MRI: normal peripheral zone (≈100-160 msec) transition zone (≈70-110 msec), and malignancy (generally similar to or slightly smaller than for the transition zone) [43]. Figure 9E shows the relative intensity in these vials compared to the bright background. Figure 9F shows the contrast-to-noise (CNR) defined as the difference in mean signal in a target vial vs. background divided by the standard deviation of signal within the vial. The CNRs between any pair of Vials 4-8, relevant in distinguishing malignancy vs. normal prostate, tend to be at least 30% smaller for CUBE than the KZM methods. Supporting Information Figure S6 shows in vivo results from an additional subject and compares axial images formed with the KZM acquisition with WS reconstruction (A) and CUBE (D). Also shown are sagittal and coronal reformats from WS (B,C) and CUBE (E,F), respectively. The lesion (yellow arrow, A,B) is arguably better seen in the WS result. The full clinical MRI exam was evaluated as PI-RADS 5, and subsequent pathology showed this to be a Gleason 3 + 4.

Figure 9.

Study of contrast. (A) Axial 3mm thick T2SE image of the NIST phantom acquired using 2D T2SE KZM multislice sequence of Table 1. Vials (each 15mm diameter) are numbered. T1 and T2 relaxation times are noted for Vials 4-8 the values of which are relevant to prostate T2SE MRI. (B) and (C) are 1mm thick slices of the phantom following Tikhonov regularization (B) and wavelet-sparsity reconstruction. (D) 1.2 mm thick slice of phantom acquired using 3D CUBE sequence of Table 1. (E) Plot of mean signal level of each vial as a percentage of the mean signal of a 5mm wide annulus immediately surrounding each vial. (F) Plot of CNR of each vial as defined in text. Note general reduced CNR between any pairs of vials for 3D CUBE vs. 2D KZM.

DISCUSSION

We have presented a model-based image reconstruction based on wavelet-sparsity (WS) regularization for providing improved retention of SNR in thin, 1mm-thick, T2SE axial images of the prostate as generated from overlapping 3mm thick slices using deconvolution of the slice profile. Such retention is desirable to account for the loss of SNR in going from the typical 3mm to the finer 1mm slice thickness. The reconstruction is based on the use of 3D Debauchies-4 wavelets as applied in the 3D image space formed by the acquired data.

Once the method was developed algorithmically it was evaluated in phantoms. Studies using low contrasts suggested the WS method could provide clear improvement in apparent CNR (Figure 1) while maintaining the approximate 1mm through-plane resolution of the original Tikhonov approach (Figure 2). The WS algorithm was then calibrated radiologically and evaluated in clinical prostate MRI exams. Calibration in six studies were used to determine that the regularization parameter λ which best consistently balanced SNR restoration with potential artifactual, patchy appearance of signal was λ = 0.00015. This value was used in 16 subsequent studies in which WS reconstruction was compared radiologically with linear reconstruction using Tikhonov (TK) Regularization, both reconstructions for a given subject using the same acquired data set. Relative SNR, contrast, fidelity of contrast, and sharpness of structures in the peripheral zone (PZ) and transition zone (TZ) were all superior statistically in WS vs. TK reconstructions. For all other evaluation criteria WS and TK were statistically equivalent. Overall diagnostic image quality was upgraded in WS vs. TK for 6 of the 32 comparisons and downgraded in none.

In addition to the qualitative radiological comparisons, the TK and WS methods were compared quantitatively using the metric of signal ratio of the prostate-to-periprostatic muscle. Due to its low T2 relaxation time, the muscle signal is small in prostate T2SE images, and consequently the noise level contributes significantly to the mean magnitude signal. Thus, methods for noise reduction can tend to reduce this mean, causing the prostate-to-muscle signal ratio to increase. Such an increase in WS vs. TK was observed in all 16 studies (range 1.20 – 1.52, mean 1.37), confirming this expected trend. We emphasize that this is not a direct measure of SNR improvement but rather simply an objective manifestation that WS provides reduced noise level.

In the radiological evaluation the superiority of WS over TK was most significant statistically for relative SNR (Figure 4A), with WS preferred in 23 of the 32 evaluations. However, WS was also statistically superior in other categories, including sharpness of structures in the peripheral and transition zones and contrast. The signal of suspicious lesions in prostate T2SE is generally lower than surrounding normal prostate tissue, and although not as pronounced as for the near-zero peri-prostatic muscle signal described above, the noise reduction provided by WS can conceivably cause reduction in the mean magnitude lesion signal, possibly providing subtle contrast improvement.

Previous work using the Tikhonov regularization reconstruction method for prostate T2SE MRI showed that the 1mm thick super resolution images consistently provided improved sharpness vs. the 3mm thick images of commonly used prostate T2SE [26]. However, this was offset by reduced SNR due to the thinner slice. The WS method presented here provides improved retention of SNR in the thin slices while retaining the high through-plane (1mm) resolution as verified in the phantom studies. Although rigorous comparison with commonly used T2SE was outside the scope of this work, 3mm thick images were provided for comparison for several of the cases to illustrate relative performance of the WS method.

We also attempted to illustrate some of the tradeoffs of the 2D overlapped-slice KZM-based approach developed here with a commercial, direct 3D FSE acquisition (CUBE). The two acquisitions were normalized: axial orientation, equivalent Z coverage, minimum TR 3000 msec for adequate T2 weighting, and equivalent scan time. For CUBE, for the level of acceleration used (R_Y × R_Z = 2.00 × 1.75) the numbers of Y and Z encodings were adjusted to fit the target scan time, resulting in somewhat reduced phase and slice resolution vs. WS (Figure 2). Higher acceleration or additional means for undersampling could address this. An additional challenge for direct 3D FSE is to provide adequate contrast at the relevant T2 times (Figure 9), a challenge given the long echo train length (ETL) of 60 of CUBE vs. ETL 21 for WS. This can conceivably be addressed for 3D FSE with methods such as T2 shuffling [10]. Although the scan times of the 2D WS and 3D CUBE were the same, the manifestation of motion is arguably more severe in the scalloping or slice-to-slice mismatch in 2D vs. diffuse unsharpness in a 3D approach. With successful implementation, scans done in ≈6 min using either approach could provide a time savings for acquiring ≤1mm isotropic images vs. the typical ≈11 min required for 2D T2SE prostate MRI using three orientations [13, 14]. Compared to standard T2SE with 3mm thick slices used for Fig. 2B, using KZM acquisition and WS reconstruction provides 1mm thick slices with some SNR reduction in ≈50% increased scan time (6:40 vs. 4:20)

This work has limitations. The reconstruction time, currently several hours long, makes the current implementation impractical for routine clinical use. Some of the processing can be converted from the current MATLAB to C++ or potentially benefit from hardware accelerators (GPUs). The number of subjects studied was limited. Also, in 7 of the 32 evaluations the WS (as well as the TK) results were evaluated as being inadequate for diagnosis. This was primarily due to motion. Anti-spasmodic drugs (which were not used here) may alleviate this. Motion detection and correction methods similar to [37] may reduce the impact of motion, but have not yet been included as part of this iterative reconstruction.

In summary, we have developed a model-based image reconstruction method based on wavelet-sparsity (WS)-regularization which provides improved retention of SNR in high through-plane resolution (1mm thick) axial prostate T2SE MRI. In a cohort of 16 subjects undergoing clinical prostate MRI, WS was shown to be significantly superior to the previously developed reconstruction method using Tikhonov (TK) regularization in SNR and several other evaluation criteria and with no specific artifact and no loss of spatial resolution.

Supplementary Material

supinfo

Supporting Information Figure S1

Convergence of search algorithm for wavelet-sparsity (WS)-regularized solution. Data are taken from Study #7.

Supporting Information Figure S2

Relative through-plane spatial frequency response of KZM Acquired, TK reconstructed, WS reconstructed, and CUBE acquisition of the resolution phantom described in Figure 2.

Responses were calculated directly from the reformatted reconstructions of the resolution phantom (Figure 2 F–I) by taking the magnitude of the sum of each column of pixels within the imaged resolution phantom multiplied by the complex exponential matched to the primary frequency of the resolution phantom at that location. For each column this sum is completed over the available whole periods. This real portion of this complex exponential is displayed as the Detection Basis. This is equivalent to performing the length-matched (to a multiple of the period) FFT at each column of the resolution phantom data, and extracting the magnitude of the response at the expected primary frequency.

Supporting Information Figure S3

Results of radiological evaluation of Tikhonov regularization (TK) vs. wavelet-sparsity (WS)-regularized reconstruction for sharpness of: (A) prostate capsule, (B) transition zone / peripheral zone interface, and (C) seminal vesicles. Results from each of the radiologists are shown in stacked columns in a different color. (N = total number of samples, μ = mean value)

Supporting Information Figure S4

Images from Study #6. (A) 1mm thick axial section mid-gland reconstructed using reference Tikhonov regularization (TK). (B) 1mm thick reconstruction using the same acquired data as (A) but with wavelet-sparsity (WS)-regularized reconstruction. Multiple nodules (A, red stars) in transition zone were evaluated as having superior sharpness with WS in comparison with TK (A). Also, the peripheral and transition zone differentiation (A, green arrows) and prostate capsule (A, yellow arrow) are better defined with WS (B) than TK (A) reconstruction. (C) 3mm thick axial image acquired using conventional T2SE sequence.

Supporting Information Figure S5

Images from Study #16. (A-C) Images from three consecutive 1mm thick slices in the apex region formed using Tikhonov regularization (TK). Suspicious focal lesion in the left peripheral zone is identified (A, yellow arrow). Linear area of hypointensity is present in the right posterior lateral peripheral zone (A, green arrow). (D-F) Corresponding images of the same slices as (A-C) using the same acquired data but formed using wavelet-sparsity (WS)-regularized reconstruction. SNR appears improved vs. (A-C). The aforementioned left-sided focal lesion and right-sided linear lesion are better defined with WS (D) than TK (A).

Supporting Information Figure S6

1mm thick axial images from an additional subject meeting the enrolment criteria but recruited outside the group evaluated radiologically in this work. Acquisitions were done using KZM T2SE multislice (A, with 3mm-thick sagittal and coronal reformats B and C) and CUBE (D, likewise E and F), both specified in Table 1, with the KZM data subsequently processed using the wavelet sparsity (WS) method to form (A-C). Suspicious lesion is identified in (A, B yellow arrow), arguably better seen than in (D, E). The full clinical MRI exam was evaluated as PI-RADS 5, the subject went on to MRI-informed biopsy, and the lesion identified in (A) was evaluated pathologically as Gleason 3+4.