1. Introduction
Compressed sensing (CS) is one way to speed up magnetic resonance imaging (MRI) by collecting less data while simultaneously retaining high image quality. Essentially, CS theory states that only a small subset of k-space samples is required for accurate image reconstruction [
1,
2]. In an MRI, CS can be realized by undersampling the k-space incoherently, which produces noise-like artifacts in the image domain. Then, by utilizing sparsifying transforms, compressibility of the MRI images is enforced in the image reconstruction, which minimizes image noise and artifacts, allowing for the original image to be mostly recovered [
3,
4].
One of the MRI methods most in need of fast acquisition strategies is the 3D quantitative MRI (qMRI) approach. qMRI methods offer extra information that can be valuable in certain situations, but they are also multiple times slower than the standard contrast-weighted MRI. The benefits of 3D imaging include more comprehensive coverage of the anatomy of interest, which in most cases is a welcome addition, but at a further cost to acquisition times. Thus, combining these two time-consuming methods is problematic and necessitates an acceleration of the acquisition methods. Coincidentally, 3D qMRI is a good candidate for CS—undersampling in three dimensions spreads the arising artifacts better when compared to 2D, and the added parametric dimension can be exploited in various ways for better image quality and more aggressive undersampling [
5,
6,
7,
8,
9,
10].
For all CS methods, significant attention should be paid to the undersampling scheme of the acquisition, as it heavily influences how the artifacts manifest in the images and, thus, how well the original image can be recovered [
3,
11]. Practical considerations and scanner hardware limit the possible trajectories. From the standard k-space sampling schemes, non-cartesian radial trajectories are well suited for accelerated 3D imaging. For radial acquisitions, even for uniform undersampling, the undersampling artifacts are inherently spread out in all three dimensions, and the k-space center is densely sampled, making the acquisition resilient to motion and undersampling.
The downside of the non-cartesian trajectories is that the image reconstruction is computationally much more demanding since the standard fast Fourier transform needs to be replaced by a non-uniform fast Fourier transform (NUFFT) [
12]. In addition, usually iterative reconstruction methods are utilized for non-cartesian acquisitions, further increasing the computational cost. Since the large-scale optimization problem of the 3D qMRI CS is usually solved with first-order optimization methods, which require many iterations, the computational aspect of 3D radial qMRI CS needs to be considered. Recently, preconditioned primal-dual proximal splitting was demonstrated to significantly speed up the convergence of CS image reconstructions without the increased noise amplification of the traditional density correction preconditioner approach [
13]. The novel preconditioner combined with algorithm implementation on a graphics processing unit (GPU) seems to offer feasible reconstruction times for 3D qMRI CS, negating a large part of the downsides of the non-cartesian CS.
In this study, we approach the topic of CS qMRI by demonstrating the performance of multiple CS models with variable flip angle 3D radial T
1 mapping. The T
1 relaxation time, for example, shows promise as a potential biomarker for lung disease [
14] and myocardial fibrosis [
15]. Additionally, a specific variable flip angle acquisition shows promise in proton resonance frequency shift thermometry [
16]. T
1 mapping has also been combined with compressed sensing to, for example, make single-breath-hold multiparametric MRI feasible [
17].
The CS models tested include spatial and contrast dimension total variation, spatial and contrast dimension finite differences with the Huber penalty function, and locally low rank and locally low rank combined with spatial total variation. These sparsifying priors are well studied, but their performance with true 3D radial acquisitions is not well defined. The large-scale image reconstruction problems are solved in a feasible time frame with the primal-dual proximal splitting (PDPS) algorithm [
18], as it permits preconditioning for faster convergence [
13,
19] and allows the implementation of the sparsifying priors [
20]. Reconstructions are computed with various amounts of data to investigate the undersampling tolerance of the methods. Ex vivo and simulation data are used, as the former provides evidence of real-world performance and the latter gives us the possibility to measure performance with respect to the known ground truth.
4. Results
The significantly faster convergence of the preconditioned PDPS algorithm was verified and is visualized in
Figure 2, where the objective function value of Equation (3) for fully sampled simulation data reconstruction is plotted. With the preconditioner, the PDPS algorithm reached the stop criterion in 72 iterations. The non-preconditioned variant that utilized step sizes of
[
18,
20] ran 5000 iterations (500 shown in the image) and did not achieve the same objective function value as the preconditioned version. Similar gains in convergence were achieved for all the reconstruction models. Slight oscillations that are visible in the objective function value were largely unaffected by the choice of the step sizes.
With the simulation data, we observed good tolerance for undersampling for all the regularized models. T
1 maps, with AFs up to 20, remain of high image quality. The sTV + cTV and sTV + LLR models outperform every other method, with the latter achieving the lowest nRMSE scores while being almost identical in performance SSIM-wise. Staircasing is increasingly visible as AF grows, with the sTV + LLR model generally suffering less from it. The sH + cH model clearly leaves more noise in the reconstructions and subsequently performs slightly worse nRMSE-wise and a lot worse SSIM-wise when compared to the TV model. The LLR model demonstrates some residual noise in areas where the blocks overlap multiple image details, e.g., around the outer ring of the phantom, and the reconstructions also look over-regularized with faint block artifacts (
Figure 3a and
Figure 4).
Generally, for the S
0 component, the small details on the phantom are visible for all models for all Afs but with varying amounts of blur, and the nRMSE values are overall larger than with the T
1 maps. The sTV + LLR model is again the best model, with the sTV + cTV model a close second, as agreed by both error metrics. The sH + cH model suffers from image blur the most and is the second-worst model here. The LLR model has the most noise and thus the highest nRMSE and the worst SSIM values (
Figure 3b and
Figure 4).
The ex vivo data do not tolerate undersampling as well as the phantom data. For all regularized models, T
1 maps with Afs up to four remain visually similar and under 10% in nRMSE when compared to the respective reference. Afs larger than four suffer from significant image blur, and image details are consequently lost. This is clearly visible in the space between the vertebra, where the areas with high T
1 values are lost. All the regularized models are relatively close in performance, with the sTV + LLR outperforming others for all Afs tested nRMSE-wise and being the optimal model for the Afs up to four SSIM-wise. The LLR model outperforms the sH + cH and sTV + cTV models in nRMSE, but the reconstructions are clearly noisier, and the SSIM suffers greatly. The SSIM favors the Huber model as it outperforms the LLR and sTV + cTV models for all Afs. Additionally, the Huber model is the optimal one, SSIM-wise, for Afs of 10 and 20. Note that the error metrics are calculated with respect to the corresponding full-data reference reconstructions (
Figure 5a and
Figure 6).
For Afs of four and below, all the reconstruction models perform similarly and produce low nRMSE and high SSIM values for the S
0-component, but further data reduction quickly deteriorates the image quality. The sTV + LLR model is the best below fourfold acceleration by a small margin in both error metrics. However, the LLR model retains the most image details and noise as AF increases, while all others suffer from significant image blur (
Figure 5b and
Figure 6).
5. Discussion
We demonstrated the performance of multiple CS image reconstruction models in a qMRI setting and how the preconditioned PDPS algorithm, relying additionally on fast GPU implementation, allows for a computationally fast solution of the chosen models. Without the preconditioner, computation times for the reconstructions would have been prohibitively long, even with the GPU implementation. By nRMSE and SSIM of the T1 maps, generally the best-performing model was the sTV + LLR for both phantom and specimen data. However, the differences between all the tested models are not necessarily large.
The similar performance of the models is not very surprising, as they do promote sparsity in a reasonably similar manner. For the models utilizing contrast dimensions TV and Huber, the minimum solution, regularization-wise, would be an image series of constant contrast, as the contrast differences would then be zero. Similarly, for the low-rank model, the minimum solution would be a rank-one matrix, meaning that the matrix can be represented as a matrix product between a contrast dimension vector and a spatial dimension vector (vectorized image). That is, the minimum solution is a matrix where only specific (though not necessarily constant) signal evolution is allowed in the contrast dimension. For locally low rank, this holds block-wise. These slight fundamental differences seem not to significantly affect the performance of the models.
The best-performing model was the sTV + LLR, with the TV model being a close second, indicating that LLR regularization is slightly better than cTV for the VFA acquisition. The combination of large (from a flip angle of one to two) and small (from a flip angle of seven to eight) signal jumps is known to be non-optimal for TV. Thus, discarding the flip angle of one from the acquisition might close this performance gap.
The Huber model was tuned so that most of the small signal changes were below the Huber parameter value, i.e., flat image areas would be reconstructed piecewise-smooth. However, here the models are compared after a non-linear fitting procedure, and the effect of the Huber penalty function is mostly seen as increased noise in the phantom T1 maps. A positive feature of the Huber model is that it also permits classical gradient-based reconstruction methods (e.g., conjugate gradient) to be used, but in our case, the loss of the preconditioner with the classical methods makes these still slower than the PDPS. However, as the Huber and TV models were close in performance, Huber could still be an interesting choice for cartesian reconstructions where no preconditioner is needed.
With the LLR model, optimal regularization would threshold out the singular vectors corresponding to pure noise, with over-regularization then affecting the signal evolution itself and under-regularization leading to residual noise. Thus, by under-regularizing, noise can be selectively retained without imposing strict a priori information (i.e., piecewise smoothness) on the image estimate, which might explain why the LLR model performs well with the noisy ex vivo specimen. For the phantom data, the LLR minimum nRMSE T1 maps suffer from block artifacts, which could be alleviated with 50% block overlap, with substantial increases in computation time or with an sTV as shown. The block artifacts are also amplified in the signal fitting, as no clear artifacts are visible on the magnitude images (not shown here).
The model combining spatial TV and LLR alleviates the block artifacts and further suppresses the leftover noise of the LLR regularization, making it the best model with the simulation data. With the ex vivo specimen, the sTV + LLR is not a significant improvement over the LLR model, and the LLR reconstructions can retain even more image details, as is particularly visible in the S0 maps. This behavior can probably be admitted to the noise and more complex structure of the reference reconstruction since, with the simulation data, the sTV + LLR clearly outperform the LLR model.
In this study, we are missing one widely acknowledged regularization model, the total generalized variation (TGV) [
38]. TGV was not included because the memory footprint of the model is much larger than that of the TV approach. The larger memory footprint prevented us from doing the reconstruction using the GPU, as all other reconstructions were completed, and the CPU implementation would have been infeasibly slow. As the 3D TV demonstrated some staircasing artifacts, it is a shame that no comparison to TGV could be performed, as minimizing the staircasing artifacts is what TGV was originally proposed for.
Ex vivo data were utilized to tie the performance of the tested models to a more realistic case compared to the ideal phantom data. Although the ex vivo data cannot be used to address the possible motion and complications of multiple coils of clinical in vivo data, the ex vivo data at least introduce slight k-coordinate inconsistencies that are inherent in radial acquisitions. Additionally, the noise might also differ slightly from the ideal assumed complex white Gaussian noise, as in ex vivo acquisition, there are multiple possible sources of noise and artifacts, such as eddy currents.
The comparison of the methods between the simulation and ex vivo data is not straightforward because of the lack of real ground truth for the ex vivo specimen. However, the ex vivo specimen can be thought to be the more “correct” comparison, as it represents the image that we are usually trying to match with CS methods. With our ex vivo specimen, the reference happens to be quite noisy, which opens up the topic that the fully sampled reference could also be improved with stronger regularization (suppressing more noise). Doing this would likely increase the performance of the tested models, as the CS methods inherently denoise images. Thus, trying to optimize denoised CS reconstructions for a noisy reference with error metrics that have their own problems does impose some biases. However, as it is unknown which parts of the (reference) image are noise and which are actual details, and further considering that the regularizations impose a priori information on the reconstruction, tuning the optimal reference reconstruction is not clear. Here, we thus chose to use the noisy reference.
Clear differences between the behavior of nRMSE and SSIM error metrics were observed. Especially SSIM seems to greatly punish the presence of any image noise, even if, to the eye of the reader, the slightly noisy reconstruction looks more “natural”. nRMSE, on the other hand, does not discriminate where the error comes from, i.e., whether it is image noise or loss of detail, and just measures the global error. Using the lowest nRMSE of the T1 maps seemed to produce images that matched with the expectation of natural-looking reconstructions; hence, that criterion was used. If the same was completed with SSIM, the individual contrast images and the T1 maps looked over-regularized.
We chose not to report any computation time metrics in the results, as the absolute performance is greatly related to the hardware used and implementation details. We note here that for our Python implementation, the reconstruction times in minutes on a computational server with an RTX A6000 GPU ranged between 2.2 and 13.9; 2.3 and 14.3; 26.4 and 111.5; 28.0 and 113.5; and 0.9 and 7.7 for the sTV + cTV, sH + cH, LLR, sTV + LLR, and LS models, respectively. The models utilizing LLR regularization are considerably slower than the rest, as multiple singular value decompositions are needed per iteration. Still, from the reconstruction times, it can be concluded that, with the preconditioner, the reconstruction times are feasible for the large-scale radial 3D qMRI problem. This still ignores the problem of how to set the regularization parameters without a reference reconstruction, which remains an issue for most models. There are some methods that utilize automatic or data-driven regularization parameters, for example, with the wavelet regularization [
39] and with TV regularization [
40], but to our knowledge, no such methods exist for the models tested here in the qMRI setting.
Lastly, with the chosen VFA acquisition, the ex vivo results are affected by the uniformity of the B1 + field, which we did not correct in any way. However, it does not affect the results error-metric-wise since the reference contains the same error.