Generalized low‐rank nonrigid motion‐corrected reconstruction for MR fingerprinting

Develop a novel low‐rank motion‐corrected (LRMC) reconstruction for nonrigid motion‐corrected MR fingerprinting (MRF).

This is achieved by integrating low-rank dictionary-based compression into the generalized MC model to reconstruct MC singular images, reducing motion artifacts in the resulting parametric maps. The proposed LRMC reconstruction was applied for cardiac motion correction in 2D myocardial MRF (T 1 and T 2 ) with extended cardiac acquisition window (~450 ms) and for respiratory MC in free-breathing 3D myocardial and 3D liver MRF. Experiments were performed in phantom and 22 healthy subjects. The proposed approach was compared with reference spin echo (phantom) and with 2D electrocardiogram-triggered/breathhold MOLLI and T 2 gradient-and-spin echo conventional maps (in vivo 2D and 3D myocardial MRF). Results: Phantom results were in general agreement with reference spin-echo measurements, presenting relative errors of approximately 5.4% and 5.5% for T 1 and short T 2 (<100 ms), respectively. The proposed LRMC MRF reduced residual blurring artifacts with respect to no MC for cardiac or respiratory motion in all cases (2D and 3D myocardial, 3D abdominal). In 2D myocardial MRF, leftventricle T 1 values were 1150 ± 41 ms for LRMC MRF and 1010 ± 56 ms for MOLLI; T 2 values were 43.8 ± 2.3 ms for LRMC MRF and 49.5 ± 4.5 ms for T 2 gradient and spin echo. Corresponding measurements for 3D myocardial MRF were 1085 ± 30 ms and 1062 ± 29 ms for T 1 , and 43.5 ± 1.9 ms and 51.7 ± 1.7 ms for T 2 . For 3D liver, LRMC MRF measured liver T 1 at 565 ± 44 ms and liver T 2 at 35.4 ± 2.4 ms.

| INTRODUCTION
Quantitative MRI is a powerful tool for tissue characterization. Measuring the physical properties of tissue (such as T 1 and T 2 ) can objectively identify underlying disease, offering the potential for standardized clinical diagnosis. In cardiac MR, T 1 is used to characterize a wide range of disease including myocardial infarction, amyloidosis, or fibrosis 1 ; similarly, T 2 is sensitive to several diseases including myocardial edema, myocarditis, or transplant rejection. 2 In liver MR, T 1 and T 2 perform similar roles in the detection of fibrosis or inflammation. 3,4 Physiological motion is a common source of error in both of these applications; therefore respiratory and/or cardiac motion must be addressed to achieve good-quality parametric maps. Conventional T 1 and T 2 cardiac parametric mapping [5][6][7][8] is achieved with electrocardiogram (ECG)-triggered acquisitions performed under breathhold to minimize cardiac and respiratory motion, respectively. Consequently, scan time is limited to the achievable breath-hold (usually less than 20 seconds), resulting in limited 2D coverage. Furthermore, the acquisition window at each heartbeat is limited to the cardiac resting period (~200 ms), further limiting scan efficiency. Similarly, conventional T 1 and T 2 liver parametric mapping 9,10 is performed in 2D under multiple breath-holds to achieve sufficient coverage. Free-breathing acquisitions have enabled the acquisition of 3D parametric maps, for both cardiac and liver applications. 9,10 However, these approaches usually rely on diaphragmatic respiratory-gating approaches that lead to long and unpredictable scan times. Respiratory translational motion-compensation methods have been recently introduced to reduce the scan times of 3D T 1 and T 2 parametric mapping [11][12][13] ; however, these approaches do not account for nonrigid motion and are limited to steadystate encoding approaches.
Magnetic resonance fingerprinting (MRF) 14 has been introduced for simultaneous, multi-parametric mapping. It continuously samples the transient state magnetization of an acquisition with varying sequence parameters to simultaneously encode multiple parameters, estimating these parameters through dictionary matching. This approach leads to co-registered, jointly modeled, multiparametric maps from a single scan. Cardiac 15,16 and liver 17,18 applications, relying on cardiac triggering and breath-holds to minimize motion, have been proposed. Rigid motion correction (MC) has been proposed for brain MRF. [19][20][21] More recently, translation MC has enabled freebreathing 3D myocardial MRF 22 with predictable scan time, although only facilitating translational MC, whereas nonrigid cardiac or respiratory MC reconstruction with MRF has not been yet investigated. Image-based nonrigid cardiac motion alignment has been considered in 2D T 1 mapping by applying cardiac motion fields to dynamic contrast images, 23 and a similar strategy has also been considered for 2D cardiac MRF 24 ; however these approaches simply average the registered images and do not incorporate the nonrigid motion in the iterative reconstruction process. [25][26][27][28] Global low-rank models derived from the MRF dictionaries [29][30][31] have been proposed for undersampled MRF reconstruction, enabling reconstruction of transient magnetization from highly undersampled data. However, these low-rank matrix formulations inherently share information among all acquired k-space data, assuming a static image. In the presence of motion, these reconstructions produce motion artifacts in the resulting parametric maps, determined by the underlying motion, k-space trajectory, and parametric encoding (ie, pulse sequence). 19 In this work, we propose a novel low-rank nonrigid motion-corrected (LRMC) reconstruction for transient imaging applications with varying contrast, such as MRF. The proposed framework integrates low-rank matrix dictionary-based subspace models 29 into a generalized MC reconstruction to produce a time series of motioncompensated varying contrast images from highly undersampled data. Moreover, the reconstruction is regularized with a patch-based low-rank tensor approximation (HD-PROST) 32 for improved performance. To evaluate this LRMC reconstruction (regularized with HD-PROST), we considered three applications: (1) cardiac motion in 2D extended cardiac window 2D myocardial MRF (where increased scan efficiency can improve SNR); (2) respiratory motion in 3D free-breathing myocardial MRF Conclusion: The proposed LRMC reconstruction enabled generalized (nonrigid) MC for 2D and 3D MRF, both for cardiac and respiratory motion. The proposed approach reduced motion artifacts in the MRF maps with respect to no motion compensation and achieved good agreement with reference measurements.

K E Y W O R D S
2D cardiac, 3D liver, 3D cardiac, low rank, MR fingerprinting, nonrigid motion correction (enabling free breathing without gating); and (3) respiratory motion in 3D free-breathing liver MRF (demonstrating free-breathing 3D liver MRF without gating). The proposed approach was evaluated in a total of 22 healthy subjects, in comparison with conventional methods. To the best of our knowledge, this corresponds to the first demonstration of the feasibility of free-breathing 3D liver MRF.

| METHODS
The proposed framework is summarized in Figure 1. The framework can be divided into three steps: (1) data acquisition and dictionary generation to estimate lowrank dictionary-based compression, (2) auxiliary motionresolved reconstruction to enable nonrigid motion estimation (via image registration), and (3) LRMC reconstruction. The low-rank compression operator is derived from the MRF dictionary (step 1), whereas the motion operator is obtained from the actual acquired data (step 2). Nonrigid motion estimation is performed in a bin-tobin basis using image registration and relying on a 1D motion surrogate to bin the data in different motion states. Electrocardiogram and respiratory bellow signals are used as surrogate for cardiac and respiratory motion, respectively. In the case of respiratory motion, the relative bellows signal is used to drive a localized autofocus algorithm, [33][34][35] producing an absolute measurement of 1D respiratory translational motion. 22 Auxiliary motion-resolved reconstructions (ie, at different cardiac or respiratory motion states) are used to estimate dense, nonrigid motion fields that are then incorporated (together with low-rank compression) into the proposed LRMC reconstruction.

| Acquisition
The LRMC approach was validated in three MRF applications in which nonrigid cardiac or respiratory motion pose a challenge: (1) 2D myocardial MRF with higher cardiac acquisition efficiency using prolonged acquisition window (cardiac MC); (2) 3D myocardial MRF; and (3) 3D liver MRF with free-breathing, nongated acquisitions (respiratory MC). The acquisition protocol for each of these applications is described subsequently. F I G U R E 1 Diagram of the proposed MR fingerprinting (MRF) framework combining low-rank and motion-correction models. The MRF sequence is simulated through Bloch simulations (or extended phase graphs) to create a MRF dictionary. A singular value decomposition (SVD) of this dictionary-based matrix reveals the low-rank basis U r for reconstruction. The acquired MRF k-space data can be binned into multiple motion states (eg, different respiratory or cardiac phases) to reconstruct auxiliary motion-resolved images. Image registration is used to estimate the nonrigid motion fields M n from the auxiliary motion-resolved images. Together with U r , the nonrigid motion fields M n are used in the proposed low-rank motion-corrected (LRMC) reconstruction to generate motioncorrected T 1 and T 2 MRF maps 2.1.1 | Two-dimensional myocardial MRF Two-dimensional ECG-triggered myocardial MRF scans were performed under breath-hold with an extended acquisition window of 450 ms (at each heartbeat), capturing midsystole through diastole using a constant density spiral trajectory. This strategy minimizes through-plane motion in 2D; however considerable in-plane motion is expected to remain. The acquisition protocol used three inversion-recovery (IR) preparation pulses and six T 2 preparation (T2p) pulses over the course of 16 heartbeats ( Figure 2A) to encode T 1 and T 2 , respectively. The preparation pulse scheme was based on previous work on 3D myocardial MRF 22 ; for each heartbeat, a different preparation pulse (or no pulse) was applied: IR15, 3× NP, IR130, 3× NP, IR260, NP, 3× T2p20, and 3× T2p50. In this notation, NP denotes "no preparation pulse"; the numerical values after IR or T2p denote the corresponding TI or T2p TE, respectively; and "N×" represents the number of times the same preparation is applied consecutively. Spectral presaturation with inversion recovery 36 was used in heartbeats with a TI superior to 300 ms to suppress fat signal, and gradient and RF-spoiled readouts (FLASH) were used to reduce sensitivity to field inhomogeneities. Fixed TE, TR, and flip angle (FA) were used.

| Three-dimensional myocardial MRF
Three-dimensional ECG-triggered free-breathing myocardial MRF data were acquired with a stack of variable density spirals using the acquisition scheme described in previous work 22 ( Figure 2B): IR15, 4× NP, IR150, 4× NP, IR300, NP, T2p20, T2p30, T2p40, T2p50, T2p60, and T2p80. Gradient spoiled readouts, fixed TE/TR, and sinusoidally varying FA in the range of 5° to 10° were used. The preparation pulse scheme described previously was used for each slice encoding of the stack of spirals with a minimum of 4-second waiting period between slice encodings to allow for longitudinal magnetization (Mz) to recover.
F I G U R E 2 Sequence diagrams of the MRF protocols used in this study. A, The sequence for 2D myocardial MRF used inversion-recovery (IR) pulses, T 2 preparation pulses (of 20 ms and 50 ms), fat suppression (spectral presaturation with inversion recovery [SPIR]) pulses, and cardiac-triggered readouts, in 18 heartbeats. B, The sequence for 3D myocardial MRF used a similar encoding scheme to the 2D myocardial MRF, except T 2 preparation pulses were varied over 20 ms to 80 ms. This sequence of 18 heartbeats was repeated for each slice encoding. C, The sequence for 3D liver MRF was acquired continuously, with preparation pulses interrupting an otherwise "free-running" acquisition. The sequence depicted was repeated for each slice encoding 2.1.3 | Three-dimensional liver MRF Three-dimensional free-breathing liver MRF was acquired with a similar encoding strategy to previous work in 2D liver MRF. 18 Data were acquired with a free-running acquisition, otherwise interrupted by IR, T2p, and spectral presaturation with inversion-recovery pulses. Additionally, pause sections of 400 ms (no RF pulses applied) were used after T2p shots to improve SNR. Data were acquired with stack of spirals. The preparation scheme used for each slice encoding was as follows ( Figure 2C): IR15, 19× NP, IR15, 9× NP, 15× T2p20, and 15× T2p50 for a total of 60 preparation blocks. Radiofrequency-spoiled readouts were used with fixed TE/TR and sinusoidally varying flip angle in the range of 8° to 12°. After each set of 38 preparation blocks, a pause of 3 seconds was introduced to allow for Mz to recover similar to 3D myocardial MRF; the same preparation scheme was repeated for the following slice encoding.

| Motion estimation
Auxiliary motion-resolved reconstructions (to estimate nonrigid motion fields) were obtained using soft-weighted low-rank inversion (LRI) 37 with HD-PROST regularization (LRI-HD-PROST) 38 : where y n are the reconstructed singular images for the Nth motion state (or bin); W n are soft weights; A n corresponds to k-space sampling; U r is the dictionary-based low-rank compression (to rank r) operator derived from the MRF dictionary; F is the nonuniform Fourier transform; C are the coil sensitivities; k ′ n is the k-space or the translationally corrected k-space for the Nth bin (if intrabin MC is considered as described subsequently); and Q b generates a 3D tensor n b of voxels associated with the bth voxel (and Nth bin) by concatenating local voxels (within a local patch) along the first dimension, nonlocal voxels (from patches that exhibit structural similarity with the patch around b), and contrast voxels (along the compressed singular value domain). Image registration based on free-form deformations 39,40 is then applied to the auxiliary motion-resolved reconstructions to retrieve the corresponding nonrigid motion fields. When a binning approach is considered to define motion states; this reconstruction is relatively well-posed, leading to images with sufficient quality for motion estimation. However, the underlying parametric encoding may vary between motion states and may not be optimal for immediate template matching without incorporating information from all motion states (through LRMC).

| Two-dimensional myocardial MRF
Auxiliary cardiac motion-resolved images were obtained by binning the data within each 450-ms acquisition window into six cardiac bins of equal size. To improve contrast in this reconstruction (Equation 1) and facilitate image registration, only data with T2p (ie, the last six heartbeats) with good blood/myocardium contrast was used for this purpose. In this case, because a single contrast is used for the reconstruction, U r = I in Equation 1 and a 2D matrix (instead of 3D tensor) is used for the HD-PROST regularization.

| Three-dimensional myocardial MRF
Auxiliary respiratory-resolved images were obtained by binning the acquired data according to the signal provided by the respiratory bellows. Before this binning, a localized autofocus algorithm was used to determine an absolute estimate of the translational motion of the heart due to breathing, as described in previous work. 22 Briefly, the respiratory motion in each dimension is assumed to be proportional to the acquired bellows signal, r (t). A set of translationally corrected images x are obtained by reconstructing the data with different motion signals r (t). The optimal scaling ̂ (that determines an absolute estimate of respiratory motion) is obtained by minimization of the localized gradient entropy where h is the normalized spatial gradient. After estimating respiratory translational motion, data are grouped into three equally sized respiratory bins; for each bin, k-space is translationally corrected toward the bin center (intrabin MC) to minimize remaining intrabin motion. Similar to the case of 2D myocardial MRF, only the T2p data are selected for the auxiliary motion-resolved reconstruction.

| Three-dimensional liver MRF
Auxiliary respiratory-resolved images were obtained in a similar fashion to the case of 3D cardiac MRF previously. Respiratory bellows were used to drive an autofocus algorithm and estimate translational motion, which was used to both bin the data into three respiratory-motion states and translationally correct each bin k-space toward the center of its bin. Instead of selecting the subset of T2p data, all data were considered for the auxiliary respiratory-resolved reconstruction (as it provided better contrast between abdominal organs), and nonrigid motion estimation was performed from the reconstructed first singular image.

| Low-rank MC reconstruction
The generalized matrix formulism for MC introduced by Batchelor et al 28 is characterized by the following problem: where x is the MC image; M n is a sparse matrix that encodes the motion transformation for the Nth motion state; and k is the acquired (MC) k-space. This formulation incorporates MC directly into the reconstruction process and is not limited to affine MC unlike k-space-based corrections.
Initially proposed for single-contrast imaging, this formulation would be highly ill-posed if applied to a problem where x includes a dimension of varying contrast. It has been shown that varying contrast MRF time series are correlated in time, lying in a low-rank subspace. Several methods have demonstrated that casting the MRF time-series reconstruction problem under these conditions is both faster and better posed. [29][30][31] These approaches usually estimate the low-rank subspace from a singular value decomposition of the MRF dictionary by truncating the left singular vectors to the appropriate rank r. However, these approaches will produce motion artifacts if the acquired data are corrupted by motion. 19 Here we propose a novel LRMC reconstruction that integrates the low-rank subspace model (here we adopt the LRI formulism) within the generalized MC matrix formulism. Furthermore, the proposed LRMC considers HD-PROST regularization, resulting in the following reconstruction: where y are MC singular images, and k ′ is the sampled kspace (or the translationally corrected k-space in case intrabin correction is considered). All remaining operators are defined in Equations 1 and 2.

| Simulations
Digital phantoms based on realistic nonrigid motion and parametric maps were used to investigate the performance of the proposed LRMC approach. Three digital phantom simulations were carried out: (1) myocardial MRF with cardiac motion, (2) myocardial MRF with respiratory motion, and (3) liver MRF with respiratory motion. Ground truth for the digital phantoms (T 1 , T 2 , M 0 , coil sensitivities, and motion fields) was initially obtained from in vivo MRF experiments. For each simulated sequence, associated Bloch simulations were performed producing the corresponding k-space from a simulated MRF acquisition (considering k-space undersampling, motion, and coil sensitivities). In all cases, a single 2D slice was simulated. Data acquisitions were simulated in each case using the parameters described in section 2.5.2. Reconstructions were performed using the same parameters as described in section 2.6. In each case, simulated data were reconstructed with (1) no MC using LRI-HD-PROST (Equation 3 with M n = I), (2) the proposed LRMC using nonrigid motion estimated using image registration from auxiliary motion-resolved images, (3) LRMC using the known (simulated) motion, and (4) without motion corruption (reconstructed using LRI-HD-PROST). For the simulation, coil sensitivity maps and motion fields were assumed to be known. To investigate the behavior of the reconstruction in the presence of large motion errors, an additional simulation was performed using a cardiac digital phantom containing respiratory and cardiac motion, where errors were introduced to the motion model by scaling it to 105%, 110%, 115%, 120%, 125%, and 130%.

| Experiments
The proposed LRMC was evaluated in a standardized phantom (static) and in a total of 22 healthy subjects (12 males, age 31 ± 3 years) on a 1.5T Ingenia MR system (Philips, Best, The Netherlands) using a 28-channel cardiac coil. Eight subjects were scanned for 2D myocardial MRF; 8 subjects were scanned for 3D myocardial MRF; and 6 subjects were scanned for 3D liver MRF. The study was approved by the institutional review board, and written informed consent was obtained from all subjects according to institutional guidelines.
All MRF phantom scans were compared with 2D reference IR spin echo and spin echo for T 1 and T 2 , respectively. Key parameters for the IR spin echo included TE/TR = 15/15 000 ms and 15 TIs in the range of 50 to 5000 ms; key parameters for spin echo included 8 TEs in the range of 10 to 640 ms.

Two-dimensional myocardial MRF
Eight healthy subjects were scanned in mid-short axis with the ECG-triggered 2D myocardial MRF protocol with an extended cardiac acquisition window of 450 ms as previously described. Additionally, a conventional 2D cardiac MRF 15 protocol was acquired for comparison. The same parameters were used, except a shorter cardiac window of 150 ms was used, leading to the acquisition of only 336 time points (Supporting Information Table S1). Additionally, MOLLI 5 and T 2 -GRASE 8 were acquired for comparison using the same FOV and resolution. The MOLLI (5 [3]3 variant) key parameters included FOV = 256 × 256 mm 2 ; resolution = 1.6 × 1.6 mm 2 ; TE/TR = 1.4/2.8 ms; FA = 35°; SENSE factor = 2; and acquisition window = 224 ms. The T 2 -GRASE (with black-blood preparation) key parameters included nine TEs equally spaced from 9.3 ms to 83.7 ms, FA = 90°; EPI factor = 7, SENSE factor = 3, and acquisition window = 84 ms.

Three-dimensional liver MRF
Six healthy subjects were scanned with the same 3D liver MRF protocol as in the phantom experiments. Respiratory bellows were used to monitor respiratory motion.

| Magnetic resonance fingerprinting reconstruction
Auxiliary motion-resolved reconstructions to enable nonrigid motion estimation through image registration (Equation 1) and the proposed LRMC (Equation 3) reconstruction were solved with the alternating direction method of multipliers (ADMM) 42 ; conjugate gradient (CG) was used to solve the internal least-square problem. For comparison, data were also reconstructed with no MC using LRI-HD-PROST (Equation 3 with M n = I), using otherwise corresponding parameters to the ones used for the proposed LRMC. Coil sensitivities were derived from ESPIRiT 43 and density-compensation functions estimated using voronoi diagrams. NiftyReg 39 was used for image registration to estimate nonrigid motion fields, using local normalized cross correlation for similarity metric. All dense motion fields were cast as sparse matrices (M n ), considering linear interpolation.
The following key parameters were used for the auxiliary motion-resolved reconstruction (Equation 1): CG iterations = 4, ADMM iterations = 2, number of similar patches = 20, patch search window size = 41 pixels, λ = 5 × 10 −2 , LRI rank = 2 (3D liver MRF only), linear soft weighting with bin's width effectively increased by 50% for cardiac motion and 25% for respiratory motion. For respiratory-motion estimation, the unit-normalized respiratory bellows signal was scaled by = [0: 0.1: 1] , where was the maximum amplitude observed over all subjects for the autofocusing step. Data binning along cardiac (for 2D myocardial MRF) and respiratory (for 3D myocardial and liver MRF) allowed for successful auxiliary motionresolved reconstructions in a modest number of motion states. Six motion states were considered for cardiac motion (effective resolution of ~75 ms), and three motion states were considered for respiratory motion (in both 3D myocardial MRF and 3D liver MRF).
Reconstructions were performed in a Linux workstation with 12 Intel Xeon X5675 (3.07 GHz) and 200 GB RAM. Acquired raw data were about 0.1 GB for 2D MRF and about 1-2 GB for 3D MRF; corresponding memory burden of the full pipeline was bout 10 GB for 2D MRF and about 100 GB for 3D MRF. We estimate the computational cost of the LRMC reconstruction as [aM + bMlogM]2rcnN cg + (psr) 2 M N ADMM , where a and b are gridding parameters; M is the total number of data points; r is the rank; c is the number of coils; n is the number of motion states; N cg is the number of conjugate gradient iterations; p is the number of self-similar patches; s is the patch size; and N ADMM is the number of ADMM iterations. The reconstruction times for 2D myocardial MRF were 1 hour and 40 minutes, with about 10 minutes for motion-resolved reconstruction/image registration (step 1) and about 1.5 hours for LRMC (step 2). Corresponding results for 3D myocardial MRF were 6 hours and 50 minutes, with a corresponding split of about 50 minutes (step 1) and about 6 hours (step 2); corresponding results for 3D liver MRF were 11 hours and 50 minutes with a corresponding split of approximately 90 minutes (step 1) and approximately 10 hours (step 1). High-dimensionality PROST (denoising step), NiftyReg (image registration step), ESPIRiT, and the nonuniform fast Fourier transform ran as C++ compiled code; the remaining code was implemented in MATLAB (R2018b; The MathWorks, Natick, MA).

| Magnetic resonance fingerprinting dictionaries
The MRF dictionaries were computed using extended phase graphs (EPGs). 44 The extended phase graph model did not consider B 0 /B 1 imperfections, slice profile, diffusion, or magnetization transfer effects. The MRF sequences used here employed gradient spoiling (and RF spoiling) with low FAs, which are relatively insensitive to B 0 , B 1 , and slice profile errors as show in previous works. 46 Mean values in different myocardial segments (or liver) were used as surrogates for accuracy; SDs of these measurements were used as surrogates for precision.

| Simulations
Simulated nonrigid cardiac and respiratory motion resulted primarily in blurring artifacts in the parametric maps. The proposed LRMC led to a substantial reduction in motion artifacts as seen in Supporting Information Figures S1-S3 for cardiac motion in myocardial MRF, respiratory motion in myocardial MRF, and respiratory motion in liver MRF, respectively. Large deviations in parametric error maps and elevated mean square errors (MSEs) measured in regions of interest around the heart and liver (~115 ms for T 1 , ~27 ms for T 2 ) were observed without MC and subsequently reduced with the proposed (estimated motion) LRMC (~40 ms for T 1 , ~9 ms for T 2 ), similar to LRMC using known motion (~20 ms for T 1 , ~5 ms for T 2 ) (Supporting Information Figures S1-S3). When evaluating the behavior of LRMC in the presence of large motion errors, we can observe that errors in the motion model propagate into the parameter maps (Supporting Information Figures S7-S10). In the presence of artificial errors in the motion model, we can observe some errors in the maps (error 46-77 ms for T 1 and 13.0-15.7 seconds for T 2 ), although much smaller than with no MC. In contrast, LRMC from accurate image registration (error ~45 ms for T 1 and 13.0 ms for T 2 ) achieves similar quality to LRMC using the true motion (error ~35 ms for T 1 and 11.7 ms for T 2 ), and both are of similar quality to the ground truth (where no motion exists).

| Phantom acquisitions
Phantom measurements performed for 2D myocardial MRF (150-ms and 450-ms windows), 3D myocardial MRF, and 3D liver MRF were in general agreement with conventional spin-echo IR and spin-echo values; however, a slight underestimation of T 2 was generally observed (Supporting Information Figures S4-S6, respectively). Larger T 2 errors were observed for long T 2 values, particularly for the 3D myocardial MRF sequence that used gradient spoiled readouts. For the 2D myocardial MRF with 150-ms acquisition window, normalized RMS errors for T 1 , short T 2 , and long T 2 were 4.9%, 5.4% and 17.2%, respectively; for 2D myocardial MRF with 450-ms acquisition window, the corresponding values were 6.4%, 6.3%, and 17.3%. For 3D myocardial MRF, the corresponding values were 3.7%, 5.3% and 36.3%, whereas for 3D liver MRF, they were 7.1%, 4.8%, and 12.8%.

| In vivo 2D myocardial MRF
Singular images from 2D MRF (450-ms window) presented blurring artifacts around the myocardium and papillary muscles when cardiac motion was not accounted for (no MC); these artifacts were considerably reduced with the proposed LRMC (representative subject A; Figure 3). These motion artifacts propagated into the parametric maps and were again considerably reduced with LRMC (representative subject A; Figure 4). Conventional MRF (150-ms window) showed slightly lower precision than LRMC MRF (450-ms window), likely due to only acquiring one third of the data (representative subject A; Figure 4).
The MOLLI technique showed lower T 1 values and higher noise amplification in the lateral wall than LRMC MRF. A myocardial segment analysis of mean and SD of T 1 and T 2 revealed higher T 1 values and higher apparent precision in the septal wall (compared with the lateral wall) for all methods; for T 2 , values were slightly higher in the septal wall (compared with the lateral wall) for the MRF approaches ( Figure 5, showing aggregated results over the subject cohort). No significant differences in T 1 and T 2 values were observed with/without MC for MRF with a 450-ms acquisition window. Two-dimensional myocardial MRF T 1 values were significantly elevated relative to MOLLI (more so for LRM MRF with 450-ms window); T 2 values were significantly lower than T 2 -GRASE. T 1 precision was significantly improved with LRMC MRF (450-ms window), and T 2 precision for LRMC MRF (450-ms window) and MRF (150-ms window) was significantly higher than T 2 -GRASE. These values are presented in Supporting Information Table S2, aggregated over the respective subject cohorts.

| In vivo 3D myocardial MRF
Singular images for free-breathing 3D myocardial MRF with (LRMC) and without (no MC) respiratory MC were compared, revealing minor blurring artifacts with no MC, which were reduced after LRMC (representative subject B; Figure 6). Corresponding effects were observed in the parametric maps: residual blurring in T1 and T2 was observed with no MC and considerably reduced with LRMC, achieving parametric quality more in line with conventional MOLLI and T 2 -GRASE (representative subject B; Figure 7). A myocardial segmental analysis (Figure 8, showing aggregated results over the subject cohort) revealed slightly higher septal values (compared with lateral) for T 1 in all methods; slightly higher septal values were also observed with MRF for T 2 . The T 1 values with MRF were higher than MOLLI (significantly so for no MC); both no-MC MRF and LRMC MRF presented significantly improved precision relative to MOLLI. The T 2 values with MRF were significantly lower than T 2 -GRASE, and a significant difference in T 2 values was observed between no MC and LRMC. These results are compiled in Supporting Information Table S2.

F I G U R E 7 T 1 and T 2 maps obtained with MOLLI (and T 2 -GRASE for T 2 ) and
with 3D myocardial MRF with no MC and with the proposed LRMC, for subject B. Residual respiratory motion artifacts are visible with no MC, although these are reduced with LRMC (arrows), achieving more comparable quality to conventional methods

| In vivo 3D liver MRF
Free-breathing 3D liver MRF was also reconstructed without (no MC) and with (LRMC) respiratory MC, leading to considerable differences in residual motion artifacts in the singular images (representative subject C; Figure 9). Similar to the two previous cases, considerable motion artifacts were observed in the parametric maps with no MC; however, a considerable reduction of these artifacts was achieved with LRMC (representative subject C; Figure 10). Mean and SD of T 1 and T 2 measured in a region of interest in the liver presented similar values with and without MC. T 1 values were in agreement with those reported in literature; however, T 2 values were considerably F I G U R E 8 Myocardial segmental analysis for 3D myocardial MRF. Higher T 1 values and lower SDs (precision surrogate) were observed for MRF when compared with MOLLI. For T 2 , a difference between septal and lateral values was observed with MRF (not with T 2 -GRASE), and lower precision was obtained (particularly with 150-ms window myocardial MRF). Slightly higher values in the septal wall (compared with the lateral wall) were observed for all T 1 methods, as well as for T 2 with myocardial MRF. Lower T 2 values were obtained with myocardial MRF compared with T 2 -GRASE F I G U R E 9 Three-dimensional liver MRF singular images for subject C, reconstructed with LRI-HD-PROST without MC and with the proposed LRMC. Motion artifacts can be observed with no MC, and subsequently reduced with LRMC (arrows) lower than literature values. 47 These results are included in Supporting Information Table S2.

| DISCUSSION
A reconstruction method was introduced to enable generalized nonrigid MC of MRF imaging with varying contrast. This approach incorporates low-rank (dictionary-based) and nonrigid MC (LRMC) models into a regularized iterative reconstruction. The feasibility of this approach was investigated for cardiac and respiratory MC in three different applications: 2D myocardial MRF, 3D myocardial MRF, and 3D liver MRF. The average relative errors in phantom over the three applications were about 5% for T 1 and about 6% for T 2 (for T 2 values less than 100 ms).
The study on 2D myocardial MRF demonstrated that the proposed approach is capable of correcting for nonrigid in-plane cardiac motion. Cardiac parametric mapping often relies on ECG triggering to minimize cardiac motion, at the expense of scan efficiency. This limitation can impose longer scans and/or limit spatial resolution. Extending the cardiac acquisition window beyond the middiastolic period (~150 ms) increases the amount of data collected at each heartbeat at the expense of motion artifacts. Here, the proposed LRMC was used to increase the scan efficiency by extending the acquisition to about 450 ms, while correcting for cardiac motion. This approach achieved higher apparent precision and parametric map quality than MOLLI and myocardial MRF with middiastolic acquisition window. The T 2 -GRASE technique achieved the highest apparent precision, and middiastolic myocardial MRF presented the lowest apparent precision for T 2 . T 1 measured with the proposed MC 2D myocardial MRF were higher than those obtained with middiastolic myocardial MRF, particularly in blood, potentially due to flow effects. The LRMC MRF myocardium T 1 was generally higher than MOLLI (bias of +140 ms), and T 2 was generally lower than T 2 -GRASE (bias of −5.7 ms), in line with previous studies in cardiac MRF. 22 These biases are likely associated with magnetization-transfer effects 48 and diffusion, 49 in addition to flow, all of which are not considered in the current model. When comparing parametric maps without (no MC) and with (LRMC) cardiac MC, a considerable reduction of motion artifacts could be observed.
The study on free-breathing 3D myocardial MRF demonstrated the feasibility of the proposed approach to correct for 3D nonrigid respiratory motion. Threedimensional imaging is often impossible to perform within a breath-hold, requiring diaphragmatic respiratory gating or MC solutions. For MRF, however, respiratory gating can affect the underlying parametric encoding in addition to increased and unpredictable scan times. Free-breathing 3D myocardial MRF with translational respiratory MCs F I G U R E 1 0 T 1 and T 2 maps obtained 3D liver MRF with no MC and with the proposed LRMC, for subject C. Multiple motion artifacts are present in the maps without MC, which are considerably reduced after MC (arrows) has been recently proposed 22 ; however that approach does not enable nonrigid motion compensation. Here, LRMC was used to enable MC of remaining nonrigid motion components in free-breathing 3D myocardial MRF, resulting in comparable parametric maps to 2D breath-held conventional methods. A slight positive bias relative to MOLLI (+23 ms) and negative bias relative to T 2 -GRASE (−8.2 ms) was observed. Apparent precision for LRMC 3D cardiac MRF (61 ms for T 1 and 4.7 ms for T 2 ) was better than MOLLI (77 ms) and comparable to T 2 -GRASE (4.9 ms). When comparing the results before and after MC, an increase in sharpness and delineation of myocardium wall and papillary muscles can be appreciated.
In the third study, LRMC was used to enable for the first time (to the best of our knowledge) free-breathing 3D liver MRF T 1 and T 2 mapping with 100% respiratory scan efficiency (ie, no respiratory gating). Due to the complex nature of motion, simpler affine models may not fully capture the elastic respiratory induced motion of the liver. 50 Our experiments showed that in the absence of MC, blurring artifacts are present in both T 1 and T 2 3D-MRF maps, obscuring vessels in the liver and other small structures. The LRMC technique led to a considerable reduction in motion artifacts, allowing the visualization of these structures and removing biases in regions of large motion (eg, diaphragm dome) on the T 1 and T 2 maps. No significant differences for T 1 and T 2 (in terms of mean values and SDs) were observed before/after MC in homogenous regions of interest in the liver. T 1 values were in agreement with literature values (bias of −9 ms); however, T 2 values were underestimated relative to literature (bias of −10.6 ms).
All acquisition sequences used in this work were heuristic, guided by previous preliminary works on sequence design for MRF. For cardiac MRF, a choice of small FAs 46 (and FLASH readouts for 2D cardiac MRF and 3D liver MRF) were considered to minimize sensitivity to B 1 and slice profile. Previous work on liver MRF noted considerable sensitivities to field errors 17 and the need for B 1 correction. In an effort to minimize this problem, here we opted to generate all parametric encoding through preparation pulses with adiabatic elements 18 and low-FA FLASH readouts. Additionally, fat-suppression modules were used to reduce potential aliasing signal originating from fat. Further investigation on optimal sequence design in the presence of field inhomogeneities for specific cardiac and liver applications is warranted, similarly to previous work done on brain MRF. [51][52][53] Motion estimation is a key challenge for the proposed framework. Similarly to previous work on nonrigid MC steady-state imaging, 25,54 dense motion fields were obtained through image registration of auxiliary motion-resolved images. Here we use a soft-weighted LRI-HD-PROST reconstruction to produce these images; however, other motion-resolved approaches like XD-GRASP 55 or multitasking 56 could be considered for similar purposes. Elastic motion fields have also been produced using localized autofocus ideas 57 ; however, that approach has not yet been validated in dynamic contrast applications, like MRF. These images are used here for motion estimation and thus can suffer from insufficient diagnostic image quality. Poor contrast, low SNR, and residual artifacts in the auxiliary images can all affect the accuracy of image registration. Because non-Cartesian trajectories are used, it is possible to improve apparent SNR/residual aliasing by reconstructing lower resolution for the purposes of motion estimation, as motion is generally smooth. Preliminary experiments (not shown) indicated better motion estimation using T2p data in cardiac; however, this was not the case for the liver. Simulations demonstrated that accurate motion estimation was obtained from the auxiliary motion-resolved images with the proposed framework. In the presence of errors in the motion model, residual blurring is present in the resulting maps; however, map quality is still considerably higher than no MC. Additional simulations showed that larger errors in the motion model (up to 30%) can introduce errors in the parameter maps (increase in MSE from 35 ms to 77 ms for T 1 , increase in MSE from 11.7 ms to 15.7 ms in T 2 ), although these errors remain considerably lower than the case in which motion is not corrected (MSE of 171 ms for T 1 , MSE of 37.0 ms for T 2 ). Finally, the binning step before reconstructing motion-resolved images assumes periodic motion. Irregular motion like arrythmia may have to be discarded, reducing scan efficiency and potentially affecting map quality.
Alternatively, motion could be resolved instead of corrected. Magnetic resonance multitasking 56 has recently been proposed to resolve motion in quantitative MRI. This is a promising method, although with considerable computational complexity. Multitasking uses self-navigation signals to estimate cardiac and respiratory-motion states, casting the image reconstruction of the dynamic motion and contrast data as a low-rank tensor. The estimation of the associated motion subspaces for multitasking is expected to require less information than for dense motion fields in LRMC; however, the reconstruction costs are also considerably different. We note that although the LRMC produces a single MC state, multiple LRMC reconstructions (toward different motion states) could be used to achieve motion-resolved imaging (through multiple MC reconstructions), albeit at increased computational cost. Comparisons between motion-resolved and MC approaches would be interesting in future work.
This work features several limitations. Each study included a small number of subjects to evaluate feasibility.
More in-depth studies with larger cohorts should be considered in the future to further characterize the performance of the proposed LRMC in specific applications. The in vivo study for liver application focused on demonstrating the feasibility of free-breathing 3D liver MRF and in comparing LRMC against no motion compensation. The accuracy of this approach was demonstrated in phantom experiments; however, further validation of T 1 and T 2 accuracy in vivo will be considered in dedicated studies in the future. Cardiac MC was only evaluated within an approximate 450-ms acquisition window (in 2D) to minimize through-plane motion, as it can affect T 1 and (especially) T 2 values. 19 Nonetheless, the increased acquisition window in 2D cardiac MRF provides increased scan efficiency (3 times) that could be exploited to achieve higher spatial resolution, shorter scan time, or potentially map additional parameters. Investigating the feasibility of LRMC for 3D cardiac imaging would be of interest in future studies. Future works should also validate this approach in specific patient populations.

| CONCLUSIONS
A novel low-rank (dictionary-based) MC reconstruction was proposed to enable generalized nonrigid MC of MRF imaging with varying contrast. This approach enabled 2D myocardial MRF with higher cardiac scan efficiency with cardiac MC and free-breathing 3D myocardial and demonstrated for the first time free-breathing 3D liver MRF with respiratory MC.

SUPPORTING INFORMATION
Additional Supporting Information may be found in the online version of the article at the publisher's website.

FIGURE S1
Simulated 2D myocardial MRF with cardiac motion occurring in a 450 ms window, reconstructed with LRI-HDPROST (no MC) and with the proposed Low Rank Motion Corrected (LRMC), in comparison to the simulated case without motion (no motion). Cardiac motion introduces blurring artefacts in T1 and T2, particularly in the borders of the myocardium (as visible in the error maps) which are removed with LRMC. A considerable reduction in mean squared error (MSE, computed in a region of interest around the heart) is also observed after LRMC FIGURE S2 Simulated 2D myocardial MRF with respiratory motion reconstructed with LRI-HDPROST (no MC) and with the proposed Low Rank Motion Corrected (LRMC), in comparison to the simulated case without motion (no motion). Respiratory motion introduces minor blurring artefacts in T1 and T2, particularly in the borders of the myocardium (as visible in the error maps) which are removed with LRMC. A considerable reduction in mean squared error (MSE, computed in a region of interest around the heart) is also observed after LRMC FIGURE S3 Simulated 2D liver MRF with respiratory motion reconstructed with LRI-HDPROST (no MC) and with the proposed Low Rank Motion Corrected (LRMC), in comparison to the simulated case without motion (no motion). Respiratory motion introduces considerable blurring artefacts in T1 and T2, obscuring vessels and other image structures (as visible in the error maps) which are removed with LRMC. A considerable reduction in mean squared error (MSE, computed in a region of interest around the liver) is also observed after LRMC FIGURE S4 2D myocardial MRF measurements in a parametric phantom in comparison to reference inversion recovery spin echo and spin echo for T1 and T2, respectively. T1 measurements are in general agreement with the reference (with a slight overestimation at high values), however a slight underestimation of T2 is observed (larger for high T2 values) FIGURE S5 3D myocardial MRF measurements in a parametric phantom in comparison to reference inversion recovery spin echo and spin echo for T1 and T2, respectively. T1 measurements are in general agreement with the reference (with a slight underestimation at high values), however a slight underestimation of T2 is observed (larger for high T2 values) FIGURE S6 3D liver MRF measurements in a parametric phantom in comparison to reference inversion recovery spin echo and spin echo for T1 and T2, respectively. T1 measurements are in general agreement with the reference (with a slight underestimation at high values), however a slight underestimation of T2 is observed (larger for high T2 values) FIGURE S7 T1 maps obtained from a realistic MRF simulation including respiratory and cardiac motion. Different degrees of error in the motion fields were added to the estimated motion (ranging from 5-30%); the proposed LRMC was used to reconstruct each map. Motion artefacts are present in the case with no motion correction and minor blurring appears in some cases where the artificial motion errors are high. LRMC using the real estimation motion produces similar quality to LRMC using true motion and to the ground truth. Corresponding errors are shown in Figure B FIGURE S8 T1 errors obtained from a realistic MRF simulation including respiratory and cardiac motion corresponding to the maps shown in Figure A. Considerable errors are present in the case of no motion correction (>170 ms). Artificial motion errors (range of 5-30%) produce errors in the range of 46-77 ms, considerably lower than no motion correction, but still higher than LRMC with estimated motion (45 ms) and LRMC with true motion (35 ms) FIGURE S9 T2 maps obtained from a realistic MRF simulation including respiratory and cardiac motion. Different degrees of error in the motion fields were added to the estimated motion (ranging from 5-30%); the proposed LRMC was used to reconstruct each map. Motion artefacts are present in the case with no motion correction and minor blurring appears in some cases where the artificial motion errors are high. LRMC using the real estimation motion produces similar quality to LRMC using true motion and to the ground truth. Corresponding errors are shown in Figure C FIGURE S10 T2 errors obtained from a realistic MRF simulation including respiratory and cardiac motion corresponding to the maps shown in Figure A. Considerable errors are present in the case of no motion correction (>37 ms). Artificial motion errors (range of 5-30%) produce errors in the range of 13.1-15.7 ms, considerably lower than no motion correction, but still higher than LRMC with estimated motion (130.0 ms) and LRMC with true motion (11.7 ms)