New Image Reconstruction Methods for Accelerated Quantitative Parameter Mapping and Magnetic Resonance Angiography

Advanced MRI techniques often require sampling in additional (non-spatial) dimensions such as time or parametric dimensions, which significantly elongate scan time. Our purpose was to develop novel iterative image reconstruction methods to reduce amount of acquired data in such applications using prior knowledge about signal in the extra dimensions. The efforts have been made to accelerate two applications, namely, time resolved contrast enhanced MR angiography and T1 mapping. Our result demonstrate that significant acceleration (up to 27x times) may be achieved using our proposed iterative reconstruction techniques.


Introduction
Magnetic Resonance Angiography (MRA) is an attractive non-invasive imaging choice that has shown utility in visualizing vasculature and blood haemodynamics. MRA is general term comprising a number of imaging techniques, including both single image and time-resolved imaging with or without the use of contrast agents. While many of MRA techniques may benefit from increased spatial and/or temporal resolution, time-resolved contrast-enhanced (CE) imaging often is in a particular need of acceleration. Indeed, in many diseases useful information can be gleaned from separate images of arterial and venous enhancement, hence rapid passage of contrast puts a bound on the acquisition time of a single frame, depending on application. For instance, such pathologies as intracranial arterial-venous malformations may require temporal resolution on the order of 0.5 -1 s [1], while imaging of upper and lower extremities would benefit from 3-7 second frames [2]. At the same time, inherently low speed of MR acquisitions limits the number of k-space samples that can be acquired within a given time period. This duality creates a typical for MRA need for a tradeoff between spatial resolution/coverage on one hand and temporal resolution on the other hand. An attempt to maintain both high spatial and temporal resolution inevitably leads to the need to reconstruct images from incomplete data. Quantitative parameter mapping is another area of MRI that can often benefit from acceleration, albeit for different reasons. The need to acquire k-space data for different values of control parameters often makes total acquisition time too long, increasing probability of patient motion. Since conventional reconstruction from undersampled data produces aliased images and SNR degradation, advanced imaging techniques are needed to address this problem.
The required acceleration can be achieved on either acquisition side (e.g., by using efficient non-Cartesian trajectories [3][4][5] or specialized Cartesian acquisitions [6][7][8]) or reconstruction side (e.g. by using constrained reconstruction). Often, the two are developed jointly to maximize the benefits, as is done in the recently developed compressed sensing (CS) technique [9], which provide a way to reconstruct images that admit a sparse representation from incomplete incoherently sampled data.

The problem of reconstructing image series is a system of linear equations
where is the encoding matrix, and is the vector of measured k-space data for all time frames. If k-space data is incomplete, the above equation has infinitely many solutions. To isolate a single solution, prior information about the image series is invoked to constrain the solution. Standard CS approaches exploit sparsity of the image series after application of general transforms , such as wavelets or spatial/temporal gradients [10][11][12], implemented through minimization of a cost function: Equation (1), whose size is measured by lp norms ( ), with the balance between the data consistency term (first summand) and the penalty term (second summand) adjusted by the size of regularization parameter λ. Still, if the underlying image series is not sparsified efficiently by the transform , the reconstruction may be biased towards the assumed behavior or suffer from unresolved aliasing artifacts. Therefore, proper choice of the transform is an important factor. In this work, we propose to consider two different types of transforms: a general transform equal to the second discrete derivative in temporal dimension and a data-driven one obtained by performing principal component analysis (PCA) of the training data. The first type of transform relies on the smoothness of temporal/parametric curves, which can be efficiently sparsified by double differentiation [13]. The second approach, termed Model Consistency COndition (MOCCO) [14], relies on linearized representation of non-linear operators mapping between image space and space of MR parameters. Such linearization is obtained by applying PCA to compress a set of analytical curves derived from a theoretical signal model equation or a low-resolution estimate of a time series obtained from a more fully sampled k-space center. Similar to other principal component basis (PCB) techniques [15,16], MOCCO assumes that temporal behaviour of most pixels can be approximated well by a linear combination of several waveforms. However, it allows for deviations from the chosen model as follows. Let be the set of chosen representative waveforms that span a linear subspace of waveforms satisfying the model assumption. Let be a synthesis operator mapping a set of coefficients to a linear combination ; and be its adjoint analysis operator yielding coefficients of projection of an arbitrary waveform onto , . If the model assumption is satisfied for a given pixel, then its temporal waveform is "close" to , so . Thus, the choice of provides requisite sparsity of the image series, while the use of norm provides robustness of the reconstruction to outliers, for instance, due to pathologies.

Methods
Fully sampled datasets were acquired using hybrid radial (in-plane)/Cartesian (through-plane) SPGR sequence (TR = 5.5 ms, BW = 125kHz, 20 slices, 8 coil receivers) for a set of 12 flip angles . The acquired data for a single central slice was retrospectively undersampled to simulate an interleaved undersampled radial acquisition with an acceleration factor of 20. The twelve interleaves were scheduled according to bit-reverse ordering. The images were obtained from the undersampled data by solving equation (1) with . For comparison purposes, we also reconstructed images from the undersampled data using gridding with density compensation and iterative SENSE. Also, we used a pair of ideal flip angles for this case (calculated to be 3° and 16°) and the number of projections that could be acquired within the same scan time (100 projections per image) to assess the improvement in T1 estimation accuracy for the given scan time. As a reference, images from fully sampled data for all 12 flip angles were reconstructed. T1 values were then estimated from each set of the reconstructed images using non-linear minimization [17].
MOCCO reconstruction was validated in a CE exam from an intracranial aneurysms patient conducted according to the IRB at our institution. The patient was scanned on a 3.0 T clinical scanner (Discovery TM MR750, GE Healthcare, Waukesha, WI) with an 8-channel head coil using a hybrid radial (inplane)/Cartesian (through-plane) acquisition during a contrast injection. The scan parameters were TE/TR=1.5/4 ms, FA=25o, BW=125 kHz, 20 slices, voxel size 0.86x0.86x2 mm3. The data were reconstructed from 15 radial projections per slice per 1.2 s time frame (acceleration factor R=27) using iterative SENSE [18], strictly constraining PCB different number of principal components and the proposed method with both -norm and -norm in the penalty term. Reconstructed images were compared for image quality and temporal waveform fidelity.

Results
Results of R1 (1/ T1 ) estimation are given in figure 1. The proposed approach provided the best accuracy due to lower noise level, better artefact reduction and higher spatial resolution in the reconstructed images. The increase in the standard deviation measured in a region in white matter was measured to be 2.13 times for the proposed reconstruction versus 3.82 in the SENSE reconstruction from 2 ideal flip angles (both relative to the reference fully sampled data).
Images in figure 2 show limited maximum intensity projections of two early time frames for the four reconstruction techniques. Note that while all three constrained reconstruction techniques provide good spatial resolution and adequate SNR, the algorithms relying on -norm regularization exhibit premature enhancement of some vessels. This observation is further confirmed by examining temporal waveforms of the aneurysm and its feeding artery in figure 3.

Conclusions
The proposed approaches exploiting sparsity of the image series in temporal/parametric dimension showed ability to improve image quality of accelerated acquisitions compared to parallel imaging/quadratic norm-based methods. In CE MRA applications, temporal dynamics in both normal and pathological vessels was preserved. In quantitative parameter mapping, the proposed method improved accuracy of parametric maps.