Synopsis

Keywords: Heart, MR Fingerprinting

In this work, T1/T2/PDFF mapping with rosette cardiac MRF is improved by exploiting the idea of virtual coils along with subspace constrained reconstruction. The Hermitian symmetry of k-space is explicitly included in the forward model via virtual coils and combined with global low-rank models. This model is further combined with a regularizer leveraging prior information from dictionaries, patch-similarity, and locally low-rank. This virtual-coil + low-rank + patch-based approach is used to jointly reconstruct multi-echo rosette MRF data for improved quality of the final T1, T2, and PDFF maps.

INTRODUCTION:

Global low-rank methods have been widely used in MR, for a variety of applications. In the context of MR Fingerprinting (MRF)¹, global subspace constrained reconstructions^2,3,4 help suppress residual aliasing commonly present in property maps derived from highly undersampled acquisitions. Partial Fourier⁵ reconstruction is another longstanding method that exploits the Hermitian symmetry of a (real) object for undersampled acquisitions, which has been studied under various formulations like PFPP,⁶ LORAKS,⁷ and Virtual Coil Concept.⁸ The first contribution in this work is combining virtual coils with global low-rank methods to leverage both types of a priori information.

Regularization methods are also effective at suppressing aliasing and noise. Compressed Sensing,⁹ locally low-rank regularizers¹⁰ and dictionary based regularization,^4,11 among others, have been proposed for MRF applications. Redundant information between similar image patches has been widely used in other fields (e.g. Non-Local-Means¹², Block-Matching-3D¹³), and have also been investigated for MR in methods like LOST¹⁴ and (HD)-PROST¹⁵. The second contribution in this work is developing an MRF-tailored regularizer, incorporating ideas from,^4,10,15 and combining it with the aforementioned virtual-coil + low-rank model. The proposed approach was evaluated for T1/T2/PDFF (Proton Density Fat Fraction) cardiac MRF in five healthy subjects.

METHODS:

A common way of incorporating low-rank constraints into the forward model is via the so-called Low-rank Inversion (LRI) expressed by the following problem:
$$\boldsymbol{\hat{y}=argmin_y}\begin{Vmatrix}\boldsymbol{SU_{TR}Fy-s}\end{Vmatrix}_2^2,[eq.1]$$
where $$$\boldsymbol{S}$$$ is the sampling trajectory, $$$\boldsymbol{F}$$$ is the Fourier transform, $$$\boldsymbol{C}$$$ are coil sensitivities, $$$\boldsymbol{U_{TR}}$$$ is the low-rank subspace, $$$\boldsymbol{y}$$$ are the so-called singular images and $$$\boldsymbol{s}$$$ are the acquired data. In (single-echo) MRF, $$$\boldsymbol{U_{TR}}$$$ is commonly derived from the dictionary itself, used to compress the data along the TR dimension, and can be applied in k-space for computational advantages.
Virtual coils are a convenient way of leveraging Hermitian symmetry into the forward model, and may be exploited within a low-rank model via:
$$\boldsymbol{\hat{y}=argmin_y}\begin{Vmatrix}\boldsymbol{SU_{TR}F\begin{bmatrix}\boldsymbol{P}\\\boldsymbol{P*}\end{bmatrix}\begin{bmatrix}\boldsymbol{C}\\\boldsymbol{C*}\end{bmatrix}y-\begin{bmatrix}\boldsymbol{s}\\\boldsymbol{s'}\end{bmatrix}}\end{Vmatrix}_2^2,[eq.2]$$
where $$$\boldsymbol{P}$$$ is the image phase, and $$$\boldsymbol{s'(k)=s^*(-k)}$$$ (where $$$\boldsymbol{k}$$$ denotes k-space coordinate). For multi-echo data, we can further consider a low-rank compression $$$\boldsymbol{U_{TE}}$$$ along the echo-time dimension. Since eq. [2] reconstructs real-valued images, this basis could potentially be lower rank than for complex valued images; the corresponding model (also considering regularization) would be:
$$\boldsymbol{\hat{y}=argmin_y}\begin{Vmatrix}\boldsymbol{SU_{TR}F\begin{bmatrix}\boldsymbol{P}\\\boldsymbol{P*}\end{bmatrix}\begin{bmatrix}\boldsymbol{C}\\\boldsymbol{C*}\end{bmatrix}U_{TE}y-\begin{bmatrix}\boldsymbol{s}\\\boldsymbol{s'}\end{bmatrix}}\end{Vmatrix}_2^2+R(\boldsymbol{y}),[eq.3]$$
where $$$R$$$ is some regularizer. In practice, $$$\boldsymbol{U_{TE}}$$$ can be derived via a preliminary reconstruction from eq. [2]. The forward model in eq. [3], virtual-coil + low-rank (Fig.1-top), exploits Partial Fourier and redundant information along the TR and TE dimensions.
Additionally, we propose a novel regularizer, combining ideas from locally low-rank, geometric augmentation, patch-based and dictionary-based approaches, defined as:
$$R(\boldsymbol{y})= \sum_b \lambda_b\begin{Vmatrix}\boldsymbol{Q_bADD^Hy}\end{Vmatrix}_*,[eq.4]$$
where $$$\boldsymbol{D}$$$ is the MRF dictionary, $$$\boldsymbol{A}$$$ creates replicas of mirrored and rotated about the center, and $$$\boldsymbol{Q_b}$$$ finds a set of self-similar patches relative to pixel b and arranges them into a Casorati matrix. $$$\boldsymbol{DD^H}$$$ projects each fingerprint into its’ (compressed) dictionary entry of length N_r, producing a similar denoising effect to standard template matching used in MRF for parameter estimation. $$$\boldsymbol{A}$$$ creates eight copies of via basic (interpolation-less) mirror/rotation operations. This is relevant for the following step, as it creates additional redundant patches for self-similarity. Finally (and for each pixel b), $$$\boldsymbol{Q_b}$$$ finds a set of similar image blocks (of size N_pxN_p) via inner product, and arranges them into a N_p²N_rxN_b (low-rank) matrix, which can be efficiently denoised via SVD. This patch-based regularizer (Fig.1-bottom) is combined with the virtual-coil + low-rank forward model, and applied to T1/T2/PDFF mapping in cardiac MRF.

EXPERIMENTS:

The proposed approach was evaluated in five healthy subjects (age = 31.0±13.1 years, 4 females) at 1.5T (Magnetom Sola, Siemens Healthineers, Erlangen, Germany) using an 18-channel cardiac coil. Imaging parameters included one short axis slice; field of view (FOV) = 300 mm²; 8 mm slice thickness; resolution = 1.56×1.56 mm²; TE/ΔTE/TR = 1.39/0.94/9.7 ms; flip angle = 4-25º; FISP readout; rosette trajectory; 15-heartbeat breath-hold. Data were reconstructed with LRI (eq. [1]) and with the proposed virtual-coil + low-rank + patch-based recon (eqs. [3,4]). Parametric values were measured in ROIs in the interventricular septum.

RESULTS:

Residual aliasing and noise amplification are present in parametric maps reconstructed with LRI; however these artefacts are considerably reduced with the proposed approach (Fig.3, Fig.4). Additionally, a better delineation of structures, particularly around water/fat interfaces, was observed with the proposed approach. Mean T1 for the cohort was (in format [LRI; proposed]): [1127±23; 1150±25] ms, T2 was [42.0±1.5; 43.0±2.2] ms, PDFF was [2.8±3.8; 1.5±1.6] %. Corresponding standard deviations were [89±32; 48±16] ms, [6.8±2.6; 4.9±1.7] ms, [7.5±2.0; 4.2±1.8] %. T1/T2/PDFF values were in general agreement between LRI and virtual-coil + low-rank + patch-based, however a higher apparent precision (lower standard deviation) was observed for the latter (Fig.5). Slightly higher T1/T2 values were observed with the proposed method and are presumably due to the improved water/fat separation (and improved PDFF mapping with the proposed approach), which reduces bias from fat signal leaking into the water signal.

CONCLUSION:

A novel forward model for multi-echo MRF is developed, combining virtual coils with global low-rank; furthermore a novel MRF denoiser is developed, combining ideas from dictionary-regularization, geometric image augmentation, patch-similarity and locally low-rank. The proposed approach outperformed conventional global low-rank and will be investigated further in comparison to gold standard methods.

Acknowledgements

This work was supported by the NIH (R01 HL153034, R01HL163991) and Siemens healthcare.