Dynamic contrast enhanced magnetic resonance imaging (DCEMRI) involves the continuous acquisition of
For the acceleration of dynamic MR images, a common strategy used to balance the tradeoff between spatial and temporal resolution is to subsample the data (also known as “
Compressed sensing (CS) [
CS has great potential in cancer MRI [
Before CS can be used clinically in such a critical area of care as cancer imaging, its effect on the reliability and accuracy must be understood. The aim of this paper partially addresses this by quantitatively evaluating five common temporal sparse regularizers for breast DCEMRI:
Total variation (TV),
Secondorder total generalized variation (
Nuclear norm (NN)
We hypothesize that one of these regularizers will produce more accurate reconstructed images than the others and that one regularizer (not necessarily the same one) will produce the most accurate quantitative parameters.
We applied all models retrospectively to in vivo breast DCEMRI data [
The data that were acquired were fully sampled with a Cartesian geometry and then retrospectively undersampled according to a range of random sampling patterns. While this was less realistic than prospective undersampling, it was necessary because acquiring hundreds of different sampling patterns prospectively would be impractical.
To narrow the focus of the paper, we looked only at the slice that passed through the center of the tumor. For this particular dataset, the center slice was the sixth slice. Also, for all reconstructions, we cropped the image posterior of the chest wall to improve sparsity. The beating heart caused aliasing artifacts in the firstphase encode direction (superiorinferior). The final cropped dimensions of our test data were
We generated 200 distinct Cartesian sampling masks by choosing 200 different random seeds before pattern generation. The dimensions of the masks were
An example sampling pattern for all dynamics (a) and a 2D sampling for one dynamic (b). The 2D mask on (b) is generated by repeating one column of the mask on (a). For each dynamic, the central lines are fully sampled and the periphery is randomly sampled. The total undersampling factor is around 4.5.
We denote the reconstructed spatiotemporal image slice as the matrix
The reconstruction problem solved was
In this paper, all the sparse models are solved by the Fast Iterative ShrinkageThresholding Algorithm (FISTA) [
Here
The key step of FISTA is to find an efficient algorithm to solve the proximal map subproblem. As in most compressed sensing models, we use
Dynamic MRI often shows temporal redundancy (repeated features of the signal in time), so the 1D Fourier transform applied along the temporal dimension can be used to sparsify the signal. The more periodic the signal, the sparser the representation in the Fourier domain. A nonperiodic, noisy signal can still be compressed in the Fourier domain due to the denoising effect of downsampling, although this is a minor effect. For example, in cardiac cine imaging, the image can be efficiently sparsified using a temporal Fourier transform due to the periodic motion of the heart. In
Wavelets are a family of sparsifying transforms commonly used in CS reconstructions. In particular for MRI, Haar wavelets have two features of interest. First, piecewise constant signals are known to be sparse under the Haar wavelet transform. And second the Haar wavelet in particular has a high mutual incoherence [
In the first application of compressed sensing in MRI, Lustig et al. [
However, real MR images are not piecewise constant, and constraining the gradient of these images produces “staircase” artifacts. Staircase artifacts manifest when higherorder variations in the signal intensity are reduced to piecewise constant regions. This occurs because the smallest gradient coefficients are being reduced to zero in the reconstruction in order to sparsify the image under the gradient transform.
TV or finite difference based strategies, which were originally designed for image denoising [
As mentioned above, finite difference operators are well suited to sparsify MR images, but the firstorder finite difference can introduce staircase artifacts. Total generalized variation (TGV) [
The discrete secondorder TGV is defined as follows:
As can be seen from the definition,
Like the case in total variation, we denote
Lowrank matrix completion has been applied to dynamic MRI by considering each temporal frame as a column of a spatiotemporal matrix, where the spatiotemporal correlations produce a lowrank matrix. The combination of compressed sensing and lowrank matrix completion [
In dynamic MRI, previous work on this combination proposed a solution that is both low rank and sparse. In
The choice of parameters for each sparse model was crucial, as comparisons were only possible if each model were optimized. In the experiments, the data were rescaled to
FISTA parameters.
Regularizer 

Iter 




FT  0.059  35  —  —  — 
WT  0.008  60  —  —  — 
TV  0.5  100  0.2  0.2  0.5 

0.5  100  0.2  0.2  0.5 
NN  0.3  40  —  —  — 
One of the most commonly used pharmacokinetic models is the standard ToftsKety model [
We used two methods to quantitatively evaluate the temporal transforms. For consistency with previous paper [
The second assessment was the concordance correlation coefficients (CCCs) of the parametric maps
To visually evaluate the accuracy in determining time profiles, we computed the difference in signal intensity curves with respect to the fully sampled data in specific regions within the tumor. For consistently plotting these curves, we first manually generated a mask for the tumor and applied the mask to all reconstructions. Additional visual assessment of parameter agreement was conducted using BlandAltman plots of the average of the undersampled parameters and the fully sampled parameters relative to the fully sampled parameter values.
The CS reconstruction was written in MATLAB, and the DCE analysis software used was
We first evaluated the performance of the five constraints on image error. The first column of Table
Mean SER and CCC across all sampling patterns.
Constraints  SER (dB)  CCC 
CCC 

Zerofilled  15.1  0.694  0.636 
FT  26.4  0.763  0.575 
WT  21.8  0.878  0.733 
TV  27.7  0.974  0.916 
TGV  27.8  0.974  0.917 
NN  29.1  0.842  0.799 
Figure
Reconstruction using the first mask for all five temporal constraints. (a) The 105th dynamic; (b) the first dynamic. In each subfigure, the first row shows the fully sampled, zerofilled, WT, FT, TV,
Figure
Figure
Boxplots of SER over 200 different sampling patterns. In the boxplot, the results of the five temporal regularizers are statistically different from each other except the ones between TV and
The second and third columns of Table
A zoomed image of
Zoomed
BlandAltman plots of
BlandAltman plots of
Figure
Difference plots of tumor mean intensity curves (a) and voxel intensity curves (b) in tumor area. Here the coordinate of the voxel is (96, 96). Panel (a) shows that the zerofilled reconstruction underestimated the mean intensity after CA injection. WT underestimated the mean intensity over all the dynamics. FT failed to capture the mean intensity curve in the first and last 5 dynamics. TV and
Since predictable accuracy is important in breast CS DCEMRI, standard boxplots of CCCs and the tumor mean
Boxplots of CCCs and tumor means over 200 different sampling patterns. In the first two boxes, the results of the five temporal regularizers are statistically different from each other except the ones between TV and
The quantitative comparisons of the five temporal constraints showed that NN was most capable of suppressing background artifacts and thus produced the highest SER. We believe this is due to its better artifact suppression ability and better edge preservation. The reason is that the minimization of the nuclear norm will suppress features that are not the same in all dynamics, such as interference and noise, while keeping static features, such as the breast and the tumor. Thus, at least in compressed sensing DCEMRI of the breast, if one needs relatively higher image quality (SER in our case), nuclear norm would be a reasonable choice.
On the other hand, TV and
The boxplots in Figure
The results of the quantitative comparisons presented here should inform clinical and research imaging reconstruction methods. For techniques that value image fidelity above accurate quantitative parameters, the best temporal regularizer may be the nuclear norm. For techniques that value quantitative parameter accuracy above image quality, TV or TGV should be preferred. The same data may be reconstructed multiple ways, of course.
Prior to this work, no measurements of the quantitative accuracy obtained from common temporal regularizers across a range of Cartesian sampling patterns had been made. This work addresses that gap, but with three major caveats. First, only one breast DCEMRI data set was used: it is possible that the results will vary across subjects, although they are not expected to significantly. Second, though we obtained a fairly normal distribution of errors, the entire space of sampling patterns is astronomically large compared to the 200 tested ones here. It is improbable, but still possible, that a nonrepresentative set of sampling patterns was selected. Third, tuning the FISTA parameters to create the best reconstruction for each constraint is an inexact process. It is possible that slightly different results could be obtained with different FISTA parameters, but we have no reason to think that the general patterns would change.
In this paper, we compare the quantitative performance of five temporal regularizers for CS DCEMRI of the breast. We find that the Fourier transform is the least suitable regularizer because of the nonperiodic behavior of breast DCEMRI data. The Haar wavelet transform was average in performance but was the least consistent in accuracy across the range of sampling patterns. The nuclear norm best suppressed background artifacts caused by undersampling, thus maximizing SER, but was less accurate in the recovery of pharmacokinetic parameters. Total variation and total generalized variation retrieved the most accurate pharmacokinetic parameters, with
Since the goal of CS DCEMRI is to accurately measure tumor properties, we recommend using TV or
Future work includes testing on the full 3D breast DCEMRI datasets instead of only a single slice. Since performing the computations required for this work on the whole 3D data in MATLAB would be prohibitively slow, we intend to use stateoftheart GPU acceleration techniques to reduce the computation time. We will also examine prospective nonCartesian sampling schemes such as radial and spiral to explore the effect of higher acceleration on the quality of reconstructions achieved with the temporal regularizers examined here.
Thomas E. Yankeelov is the W.A. “Tex” Moncrief Professor of Computational Oncology and is a CPRIT Scholar in Cancer Research.
The authors declare that there are no conflicts of interest regarding the publication of this article.
Financial support from NCI/NIH K25 CA176219, NCI/NIH U01 CA142565, CPRIT RR160005, and the National Nature Science Foundation of China (Grants 91330101 and 11531005) is appreciated.