Magn Reson Med. 2018;1–11. © 2018 International Society for Magnetic Resonance in Medicine | 1wileyonlinelibrary.com/journal/mrm

Received: 9 July 2018 | Revised: 27 September 2018 | Accepted: 2 October 2018

DOI: 10.1002/mrm.27579

N O T E

Integrated motion correction and dictionary learning for free‐breathing myocardial T1 mapping

Yanjie Zhu1,2 | Jinkyu Kang1 | Chong Duan1 | Maryam Nezafat1 | Ulf Neisius1 | Jihye Jang1,3 | Reza Nezafat1

1Department of Medicine (Cardiovascular Division), Beth Israel Deaconess Medical Center and Harvard Medical School, Boston, Massachusetts2Paul C. Lauterbur Research Center for Biomedical Imaging, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China3Department of Computer Science, Technical University of Munich, Munich, Germany

CorrespondenceReza Nezafat, Beth Israel Deaconess Medical Center, 330 Brookline Avenue, Boston, MA 02215.Email: rnezafat@bidmc.harvard.edu

Funding informationNational Institutes of Health (NIH), Grant/Award Number: 1R21HL127650, 1R01HL129185 and 1R01HL129157; American Heart Association (AHA), Grant/Award Number: 15EIA22710040; National Natural Science Foundation of China, Grant/Award Number: 61771463 and 81830056

Purpose: To develop and evaluate an integrated motion correction and dictionary learning (MoDic) technique to accelerate data acquisition for myocardial T1 mapping with improved accuracy.Methods: MoDic integrates motion correction with dictionary learning–based re-construction. A random undersampling scheme was implemented for slice‐inter-leaved T1 mapping sequence to allow prospective undersampled data acquisition. Phantom experiments were performed to evaluate the effect of reconstruction on T1 measurement. In vivo T1 mappings were acquired in 8 healthy subjects using 6 dif-ferent acceleration approaches: uniform or randomly undersampled k‐space data with reduction factors (R) of 2, 3, and 4. Uniform undersampled data were recon-structed with SENSE, and randomly undersampled k‐space data were reconstructed using dictionary learning, compressed sensing SENSE, and MoDic methods. Three expert readers subjectively evaluated the quality of T1 maps using a 4‐point scoring system. The agreement between T1 values was assessed by Bland‐Altman analysis.Results: In the phantom study, the accuracy of T1 measurements improved with in-creasing reduction factors (−31 ± 35 ms, −13 ± 18 ms, and −5 ± 11 ms for reduction factor (R) = 2 to 4, respectively). The image quality of in vivo T1 maps assessed by subjective scoring using MoDic was similar to that of SENSE at R = 2 (P = .61) but improved at R = 3 and 4 (P < .01). The scores of dictionary learning (2.98 ± 0.71, 2.91 ± 0.60, and 2.67 ± 0.71 for R = 2 to 4) and CS‐SENSE (3.32 ± 0.42, 3.05 ± 0.43, and 2.53 ± 0.43) were lower than those of MoDic (3.48 ± 0.46, 3.38 ± 0.52, and 2.9 ± 0.60) for all reduction factors (P < .05 for all).Conclusion: The MoDic method accelerates data acquisition for myocardial T1 map-ping with improved T1 measurement accuracy.

K E Y W O R D Scompressed sensing, dictionary learning, motion correction, myocardial T1 mapping

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1 | INTRODUCTION

Myocardial T1 mapping allows noninvasive assessment of diffuse myocardial fibrosis.1 In myocardial T1 mapping, most of the techniques use a single‐shot acquisition scheme.2-7 The T1‐weighted images are acquired at a quiescent period of the cardiac motion in middiastole2-6 or systole.8 The limited quiescent period leads to the need to shorten the acquisition window for each T1‐weighted image, especially in applica-tions with fast heart rates.8-11 The duration of the acquisition window can affect the accuracy of myocardial T1 mapping. The continuous application of RF pulses in the imaging se-quence alters the shape of the recovery curve and underes-timates the T1 value.6,12 By reducing the number of applied RF pulses, there is less saturation of the recovering signal, ultimately improving T1 estimation.12 T1 mapping using seg-mented acquisition has also been proposed13-18 to improve the spatial resolution of the image and reduce the acquisition window. However, these techniques require lengthy scanning time, making it less appealing for routine clinical scanning. Therefore, there is still a need to accelerate data acquisition for each T1‐weighted image.

Numerous techniques have been proposed to reconstruct images using reduced k‐space data in cardiac MR.19-23 The SENSE24 or GRAPPA25 techniques are used widely in myo-cardial T1 mapping. However, the acceleration rate is only limited to 2.12,26 Recently, compressed sensing (CS)27 has shown potential in fast myocardial T1 mapping.13-15,28,29 The redundancy in the parametric direction is used to achieve a higher acceleration rate. The fingerprinting‐based method has also been proposed to accelerate myocardial parametric mapping.30 Relative to traditional global sparsifying trans-forms, the dictionary learning–based reconstruction method (DL) achieves much sparser representation31-33 by combin-ing elements in the predefined dictionary (i.e., dictionary atom) to represent a signal curve along the parametric direc-tion.28,34 To achieve similarity between the acquired signal curve and the dictionary atom, the dictionary is trained based on a mono‐exponential decay model.34 However, the existing DL method is not applicable for myocardial T1 mapping be-cause of the effect of motion. Motion induces fluctuations in the signal curve along the parametric direction and corrupts the similarity between the acquired signal curves and the dic-tionary atoms. The loss of similarity adversely affects the performance of the DL method. Therefore, any dictionary‐based reconstruction approaches for myocardial T1 mapping should consider respiratory and/or cardiac induced motion.

In this study, we propose an integrated motion correc-tion and dictionary learning–based reconstruction method (MoDic) to allow highly accelerated data acquisition for myocardial T1 mapping. This method reduces the effect of physiological motion and increases measurement accuracy. It also integrates motion correction and image reconstruction,

thereby eliminating the need for stand‐alone image registra-tion, a necessary step in myocardial T1 mapping. The perfor-mance of MoDic was evaluated using phantom and in vivo experiments.

2 | METHODS

2.1 | Reconstruction algorithmThe block diagram of the proposed MoDic method is shown in Figure 1. T1‐weighted images are first reconstructed using the zero‐filling method for rigid registration. Global translational motion of the left ventricle is estimated using a region‐matching algorithm35 and corrected in k‐space using Fourier shift theorem. The corrected k‐space data are then applied to the DL‐based reconstruction using the for-mulation in Appendix A. Second, the T0‐term dictionary approximations in each reconstruction iteration are used to compose raw images

{Ip

}Np

p=1, where Np is the number of

T1‐weighted images. A group of synthetic images {

MIp

}Np

p=1

can then be estimated by minimizing the energy function as follows36:

where SIp is the pth signal image that is calculated from the mono‐exponential model using the T1 map. The initial T1 map is estimated from the rigid motion‐corrected images in the first step. Ψ is the spatial partial derivative constraint applied to MIp, and β1 and β2 are the regularization param-eters. The values for β1 and β2 are chosen empirically (β1 = 1 and β2 = 0.02). Because the synthetic images have similar contrast to the raw DL‐reconstructed images, robust nonrigid motion alignment can be achieved by registering the 2 groups frame by frame.36 The synthetic image, MIp, is the reference image and the raw image, Ip, is registered to MIp. Here, the optical‐flow algorithm37 is used as the nonrigid motion align-ment method. Finally, the motion‐aligned images are used to update both the k‐space data and the T1 map used for gen-erating signal images. Note that updating k‐space with the motion‐aligned images can lead to inconsistencies with the acquired data if the motion‐aligned images are in different motion states relative to the original images. The reconstruc-tion pseudo code is summarized in Appendix B.

We built a relaxation curve‐based dictionary to enable re-construction of images using DL methods. The dictionary is trained from a data set using the K‐SVD algorithm.38 The training data set is generated using Bloch simulation with im-aging parameters and expected T1 and T2 ranges. Compared with the mono‐exponential model,34 Bloch simulation takes into account the effect of balanced SSFP (bSSFP) readout

(1)

argminM

�p

�‖Ip−MIp‖2

2+�1‖SIp−MIp‖2

2+�2Ψ

�MIp

��

| 3ZHU et al.

and offers a better approximation of the signal evolution curve. Because the TI in myocardial T1 mapping depends on patient heart rate, the training data set and dictionary should be regenerated for every slice, leading to a long computation duration. To minimize the computation burden, we used a 2‐step procedure in the Bloch simulation. First, the longitu-dinal magnetization at TI is calculated using the following relaxation model:

where M0 =[0 0 1]T is the equilibrium magnetization. Second, the signal of the k‐space center using bSSFP readout is calculated as

where A and B represent the transformation matrix of the bSSFP acquisition, which are fixed for all T1‐weighted images and can be precalculated according to the imaging

parameters. The derivation and formulas for A and B are shown in Appendix C. Only Mp,TI is updated for each image, which is computationally less expensive. The time used for regenerating the training data set is under 10 seconds.

The MoDic method was implemented in MATLAB R2017a (The MathWorks, Natick, MA). The ranges of T1 and T2 values used for simulating the training data set were 250 ms to 1600 ms and 35 ms to 250 ms, respectively. The parameters for DL reconstruction were initially chosen based on the study by Doneva et al34 and were adjusted em-pirically to be optimal for our data set. The maximum itera-tion numbers were 16, 20, and 25 for reduction factor (R) = 2, 3, and 4, respectively. The threshold for stopping criterion ε = 0.0001. Dictionary atom number K = 1000. The sparsity level T0 = 3. Regularization parameter ν = 125.

2.2 | EvaluationAll MR data were acquired on a 1.5T Philips Achieva (Philips Healthcare, Best, Netherlands) system with a 32‐channel

(2)Mp,TIz =M0

(1−2e−TI∕T1

)

(3)Mp =AMp,TI+B

F I G U R E 1 Block diagram of the proposed motion correction and dictionary learning (MoDic) method. Translational motion is first corrected in k‐space using the Fourier shift theorem; the corrected k‐space data are then used for the dictionary learning (DL)‐based reconstruction iteration. In each iteration, dictionary approximation is used to compose raw images. Synthetic images are then estimated by minimizing the energy function of the raw and signal images generated by the model. The optical‐flow algorithm is used to register the raw and synthetic images frame by frame. Finally, motion‐aligned images update both the k‐space data and the T1 map used for generating signal images

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cardiac coil. A random undersampled k‐space data acquisi-tion scheme39 at R = 2, 3, and 4 (Supporting Information Figure S1) was implemented for the slice‐interleaved T1 map-ping sequence (STONE) to allow for prospectively randomly undersampled data.6 A simulation of myocardial T1 map-ping was used to assess the MoDic performance (Supporting Information Figures S1 and S2).

2.3 | Phantom studyPhantom experiments were performed to assess the accuracy and precision of T1 measurements related to the reconstruc-tion method. The phantom consisted of 9 vials containing NiCl2‐doped agarose gel with different concentrations mim-icking different cardiac compartments, including the native myocardium (T1 = 1120 ms, T2 = 53 ms) and blood (T1 = 1563 ms, T2 = 273 ms).40 Phantom data were acquired with this image acquisition scheme. Imaging parameters were as follows: bSSFP readout with TR/TE/flip angle = 3.12 ms/1.56 ms/70º, FOV = 240 × 228 mm2, voxel size = 1.36 × 1.36 × 8 mm3, matrix size = 176 × 167, 10 linear ramp‐up pulses, and simulated electrocardiogram of 60 bpm. The DL method was used to reconstruct phantom T1‐weighted im-ages without motion correction. Both Bloch simulation and mono‐exponential model were used to generate the DL train-ing data set. The T1 maps were obtained by pixel‐wise fitting of the reconstructed images.

Reference T1 values of the vials were measured using an inversion‐recovery spin‐echo sequence (IR‐SE). Sixteen 2D IR‐SE images were acquired using the following parame-ters: TE/TR = 10 ms/10 seconds, FOV = 140 × 140 mm2, voxel size = 1.1 × 1.1 × 8 mm3, TI = 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1250, 1500, 1750, 2000, and 3000 ms. The T1 values were obtained by fitting a 3‐parame-ter model.6 The T1 map obtained with the STONE sequence using SENSE at R = 2 are also shown for reference.

2.4 | In vivo studyThe study was approved by the institutional review board and written informed consent was obtained for 8 healthy subjects (2 females, 47 ± 22 years old). The T1 maps were acquired for all subjects using STONE. In each subject, we acquired 6 data sets using prospective undersampling at R = 2 through 4 with both random and uniform undersampling masks. The T1 maps were acquired using the STONE sequence with the following parameters: TR/TE/flip angle = 2.8 ms/1.4 ms/70º, FOV = 360 × 342 mm2, voxel size = 2.1 × 2.1 × 8 mm3, matrix size = 176 × 167, and 10 linear ramp‐up pulses. The acquisition windows were 232, 154, and 118 ms for R = 2 through 4, respectively. All data sets were acquired during free breathing using a slice‐tracking approach.41 Each data set contained 5 slices covering the left ventricle from base

to apex in the short axis. The DL, CS‐SENSE,42 and MoDic methods were used to reconstruct T1‐weighted images with random undersampling masks. The DL method was applied to the original k‐space before rigid motion correction, while CS‐SENSE was applied to the rigid motion‐corrected k‐space data followed by the motion‐correction method pro-posed by Xue et al.36 Images with uniform undersampling masks were reconstructed using the SENSE method avail-able by the vendor, followed by image registration.35 All T1 maps were calculated using a 2‐parameter model6 from the T1‐weighted images.

2.5 | Data analysisIn the phantom study, the T1 value for each vial was meas-ured as the average over a manually drawn region of inter-est. The accuracy of the T1 measurement was defined as the difference between the T1 value of the DL method and the IR‐SE T1 value. The precision was defined as the SD of the estimated T1 measurements in the region of interest.

For the in vivo study, the performance of MoDic was eval-uated using both subjective and objective assessments. The subjective assessment was performed by 3 readers (with 7, 3, and 1 year(s) of experience in cardiac MR, respectively). The presence of aliasing artifacts on each T1 map was first assessed as a binary answer (presence or absence). The image quality of each T1 map was subsequently assessed using a 4‐point score: score 1 = nondiagnostic (the T1 map at the myocardium was completely distorted); score 2 = fair (the myocardial T1 map was partially distorted, but the map can still be used for diagnosis in some segments); score 3 = good (the myocardial T1 map was complete but slightly blurred); score 4 = excellent (the myocardial T1 map was complete with sharp edges). Supporting Information Figure S3 shows sample T1 maps with the corresponding scores. The T1 maps from different reconstruction methods and reduction factors (total 120 T1 maps per method) were displayed on the screen in random order and readers were blinded to the acquisition/reconstruction methods and reduction factors. Readers per-formed evaluations independently. Mann‐Whitney U test was conducted to compare image quality scores between methods. Statistical significance was set to P < .05. Intraclass correla-tion coefficient (ICC) analysis with consistency agreement was used to assess interobserver agreement.

For objective assessment, MoDic T1 measurements at R = 2 through 4 were extracted to compare with SENSE at R = 2 (as the reference).43 The endocardial and epicardial contours of the left ventricle were drawn manually on each T1 map. For each subject, 5 short‐axial slices were used for the 16‐segment model (1 apex slice, 2 midventricular, and 2 basal). The T1 val-ues of the global and segmented myocardium were calculated. Bland‐Altman analysis was performed to assess agreement be-tween T1 values acquired by MoDic versus the reference.

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3 | RESULTS

3.1 | Phantom studyFigure 2 shows the phantom DL‐reconstructed T1 maps for R = 2 through 4 and the reference maps. The T1 maps of DL using both Bloch simulation and the mono‐exponential model were visually comparable to the reference map of IR‐SE at R = 2 and 3. However, artifacts appeared in the T1 map reconstructed using the mono‐exponential model at R = 4 (black arrow). The results were consistent with the DL method theory that mismatch between the training data set and the image to be reconstructed degrades the performance of the DL method.

The T1 values of references and DL using Bloch simula-tion are given in Supporting Information Table S1. The accu-racy of the T1 measurement (−31 ± 35 ms, −13 ± 18 ms, and −5 ± 11 ms for R = 2 to 4) improved with increasing reduc-tion factors compared with the gold standard (i.e., IR‐SE T1 values). This does not mean that undersampling improves T1 measurement accuracy, but rather reduces the intrinsic bias of the myocardial T1 mapping method. The precision of the T1 measurement was similar for R = 2 (19 ± 13 ms) and R = 3 (19 ± 8 ms), but decreased when R reached 4 (22 ± 14 ms). Differences between DL T1 values and IR‐SE T1 values nar-rowed with increasing reduction factors for the native myo-cardium (vial 2: −60, −27, and −16 ms for R = 2 to 4) and the blood (vial 4: 21, 14, and 8 ms for R = 2 to 4). Compared with the T1 values obtained by SENSE at R = 2, T1 differences of DL increased with increasing reduction factors (7 ± 11, 25 ± 21, and 33 ± 27 ms for R = 2 to 4) due to the larger bias in the T1 of SENSE at R = 2.

3.2 | In vivo studyAll images were reconstructed with MoDic successfully. Figure 3 shows representative reconstructed T1‐weighted

images and T1 maps at R = 2 through 4 using MoDic and SENSE methods. Aliasing artifacts can be seen in SENSE‐reconstructed images (yellow arrows) at R > 2. Figure 4A illustrates the T1‐weighted images and their corresponding T1 maps estimated by DL and MoDic methods at R = 2 and 3. Artifacts appeared (yellow arrows) in the DL‐reconstructed images, as motion‐corrupted signal curves were approxi-mated by the dictionary trained from the ideal curves. Motion artifacts were present in the T1 map of the DL method (black arrows). The corrupted segments of the myocardial T1 map with motion artifacts were restored after integrating motion correction into reconstruction using MoDic. Figure 4B shows the T1‐weighted images and their corresponding T1 maps es-timated by the CS‐SENSE and MoDic methods at R = 2 and 3. Small motion artifacts (black arrow) were present on the T1 maps of CS‐SENSE at R = 2 in some cases. At R = 3, the aliasing artifacts (yellow arrow) shown in the T1‐weighted images of CS‐SENSE led to artifacts in the T1 map (black arrow).

In the subjective assessment, percentages of T1 maps with aliasing artifacts using SENSE (50.3%, 47.3%, and 31.0% for 3 readers) were higher than those for MoDic (34.9%, 14.7%, and 5.4% for 3 readers). The ICC scores indicated modest agreement among readers for both SENSE (ICC: 0.64, 95% confidence interval [CI]: 0.52‐0.74) and MoDic (ICC: 0.55, CI: 0.39‐0.67). The subjective MoDic score (Supporting Information Figure S4) was similar to that of SENSE at R = 2 (MoDic: 3.48 ± 0.46 versus 3.49 ± 0.60, P = .61) but higher than SENSE at R = 3 (3.38 ± 0.52 versus 2.98 ± 0.53, P < .01) and R = 4 (2.98 ± 0.60 versus 2.31 ± 0.71, P < .01). The scores of DL (2.98 ± 0.71, 2.91 ± 0.60, and 2.67 ± 0.71 for R = 2 to 4, respectively) and CS‐SENSE (3.32 ± 0.42, 3.05 ± 0.43, and 2.53 ± 0.43 for R = 2 to 4, respectively) were lower than those of MoDic for all reduction factors (P < .05 for all). In terms of the interobserver agreement, SENSE showed strong agreement (ICC: 0.91, CI: 0.87‐0.93). The DL (ICC: 0.88, CI: 0.84‐0.91), CS‐SENSE (ICC: 0.84, CI: 0.79‐0.89),

F I G U R E 2 Reference phantom T1 maps using inversion recovery spin echo (IR‐SE) and STONE with SENSE at reduction factor (R) = 2 and DL using Bloch simulation and mono‐exponential modeling to generate the training data set for R = 2 through 4. The Bloch simulation and mono‐exponential DL T1 maps were visually comparable to the IR‐SE reference map at R = 2 and 3. However, artifacts appeared in the T1 map reconstructed using the mono‐exponential model at R = 4 (black arrow)

6 | ZHU et al.

and MoDic (ICC: 0.80, CI: 0.74‐0.86) methods showed good agreement.

Figure 5 shows a Bland‐Altman plot of T1 values for SENSE at R = 2 and MoDic at R = 2 through 4 in the global T1 (Figure 5A‐C) and 16‐segment model (Figure 5D‐F).

There was no significant difference between global T1 values for SENSE and MoDic at R = 2 (−3 ± 24 ms, P = .48, CI: −43~43 ms). As expected, global T1 values for the MoDic method increased as the reduction factor increased (18 ± 34 ms, P < .05, CI: −49~84 ms for R = 3, and 30 ± 33 ms, P <

F I G U R E 3 Representative reconstructed T1‐weighted images and corresponding T1 maps at R = 2, 3, and 4 using SENSE and MoDic methods for a healthy subject. Aliasing artifacts are observed in T1‐weighted images (yellow arrows) and T1 maps (black arrows) with SENSE at reduction factors greater than 2

F I G U R E 4 A, Reconstructed T1‐weighted images and their corresponding T1 maps using DL and MoDic methods in free‐breathing myocardium T1 mapping at R = 2 and 3. Yellow arrows indicate artifacts in the DL‐reconstructed images due to inconsistencies of the heart position in the presence of bulk motion. The T1 map of the conventional DL method was corrupted by motion (black arrows). The corrupted segments of the myocardial T1 map with motion artifacts were restored using MoDic. B, T1‐weighted images and their corresponding T1 maps estimated by compressed sensing (CS) SENSE and MoDic methods at R = 2 and 3. Small motion artifacts (black arrow) were present on the T1 maps of CS‐SENSE at R = 2 due to imperfect image registration. At R = 3, aliasing artifacts (yellow arrow) shown in the T1‐weighted image of CS‐SENSE led to artifacts in the T1 map (black arrow)

| 7ZHU et al.

.05, CI: −35~94 ms for R = 4, compared with SENSE for R = 2). The variation of global T1 using MoDic increased relative to SENSE at R = 2 (61 ± 15 ms, 60 ± 11 ms, and 67 ± 14 ms for R = 2 to 4 versus 52 ± 11 ms, P < .05). The segmented myocardial T1 value for MoDic was not significantly differ-ent than that of SENSE at R = 2 (−6 ± 39 ms, P = .11, CI: −84~72 ms); however, it increased at higher reduction factors (19 ± 38 ms, P < .05, CI: −56~94 ms for R = 3, and 31 ± 48 ms, P < .05, CI: −64~130 ms for R = 4, compared with SENSE for R = 2). The variation for segmented myocardial T1 values using MoDic also increased relative to SENSE at R = 2 (53 ± 18 ms, 56 ± 19 ms, and 59 ± 26 ms for R = 2 to 4 versus 48 ± 11 ms, P < .05).

4 | DISCUSSION

In this study, we proposed a novel fast myocardial T1 map-ping reconstruction method to obtain T1 maps from highly undersampled data. Compared with the conventional SENSE method, T1 maps obtained by MoDic showed fewer aliasing artifacts and overall higher image quality at R > 2. Artifacts in DL‐reconstructed and CS‐SENSE‐reconstructed images were removed under MoDic, thus improving the image qual-ity. The T1 map can be directly estimated from the recon-structed MoDic T1‐weighted image without an additional image registration step.

The advantage of combining motion correction and re-construction has already been shown in cardiac perfusion,21 in which only rigid motion correction is used. The MoDic method has a similar rigid motion correction approach as in Ref 21; additionally, nonrigid motion alignment is explic-itly integrated into the reconstruction iteration to further compensate for motion. The robustness of motion align-ment is improved by the data fidelity step in reconstruction. Furthermore, residual signal disturbances on the recovery curve, caused by small motion or noise, are suppressed im-plicitly by the DL method itself. Because the dictionary is designed from the ideal simulated T1 relaxation data set, the approximated dictionary signal is closer to the ideal relax-ation curve. MoDic achieves superior performance relative to SENSE and DL by combining these effects.

MoDic adopts both the same energy function to obtain synthetic images and a similar nonrigid motion alignment ap-proach as Xue et al.36 Theoretically, both methods should have similar performance. Xue et al’s method requires additional steps to reconstruct T1‐weighted images from undersampled k‐space data. However, image quality affects the performance of the image registration and may lead to misregistration. In MoDic, k‐space updating during the reconstruction iteration alleviates the misregistration issue and improves the robust-ness of the motion alignment method. However, this strategy introduces blurring artifacts when the k‐space data of the mo-tion‐aligned images are in different motion states relative to

F I G U R E 5 Bland‐Altman plot of T1 values for SENSE at R = 2 and MoDic for R = 2 through 4 in the global myocardium (Figure 5A–C) and 16‐segment model (Figure 5D–F)

8 | ZHU et al.

the acquired k‐space data. The tradeoff between the 2 effects is controlled by the regularization parameter ν.

In myocardial T1 mapping with single‐shot acquisition, MoDic does not reduce the total scan time but shortens the acquisition window in each cardiac cycle. A shorter acqui-sition window is beneficial in myocardial T1 mapping of patients with fast heart rates8,9 and in animal studies.11,14 Shorting the acquisition window also improves the accuracy of T1 measurements, as shown in phantom studies. In vivo studies show elevated myocardial T1 values (i.e., improved accuracy) with a shorter acquisition window. MoDic has the potential to be used in myocardial T1 mapping with seg-mented acquisitions to reduce the total scan time, and war-rants further investigation.

One limitation of the proposed technique is its long com-putation time. In our study, we implemented the reconstruc-tion in MATLAB, resulting in a reconstruction time of about 30 minutes per slice. As described, the dictionary needs to be rebuilt for each individual patient, which contributes to its long reconstruction time. The computation time can be reduced by parallel computing44,45 and a fast iteration method,46,47 which should be further investigated.

5 | CONCLUSIONS

The MoDic method combines motion correction and image reconstruction in myocardial T1 mapping. The higher accel-eration factor relative to conventional parallel imaging tech-niques enables a shorter acquisition window and fewer RF saturation pulses, thereby increasing the accuracy of the na-tive T1 measurement.

ACKNOWLEDGMENT

The authors wish to thank Jennifer Rodriguez for her edito-rial corrections.

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SUPPORTING INFORMATION

Additional supporting information may be found online in the Supporting Information section at the end of the article.

FIGURE S1 Random sampling masks for R = 2 through 4 used in this study. The x‐axis is the parametric direction (p) (i.e., different TIs) and the y‐axis is the phase‐encoding directionFIGURE S2 A, Simulated T1‐weighted image with TI =10 000 ms. B‐E, Area containing the heart on the simulated T1‐weighed images. F, Reference T1 map fitted from T1‐weighted images after image registration. G‐I, Reconstructed T1 maps using MoDic at R = 2 through 4FIGURE S3 Illustration of the 4‐point scoring system used for assessment of T1 map quality: 1 = nondiagnostic (A); 2 = fair (B); 3 = good (C); and 4 = excellent (D)

10 | ZHU et al.

FIGURE S4 Subjective assessment of image quality of T1 maps obtained by SENSE, DL, and MoDic using 4‐point scoring (1 = nondiagnostic; 2 = fair; 3 = good; 4 = excel-lent). Mann‐Whitney U test was used to assess the differences between different methods. The MoDic scores were higher than those of DL and CS‐SENSE for all reduction factors (P <.05) and higher than SENSE for R = 3 and 4 (P <.01)TABLE S1 T1 values (mean ± SD) for each vial using the DL method (Bloch simulation was used for generating the training data set) and reference T1 values using IR‐SE and STONE with SENSE at R = 2. Note: Using IR‐SE T1 val-ues as the gold standard, the accuracy of the T1 measurement (‐31 ± 35 ms, ‐13 ± 18 ms, and ‐5 ± 11 ms for R = 2 to 4) improved with increasing reduction factors. Compared with the T1 values obtained by SENSE at R = 2, the T1 differences of DL increased with increasing reduction factors (7 ± 11, 25 ± 21, and 33 ± 27 ms for R = 2 to 4) due to the larger bias in T1 with SENSE at R = 2. Undersampling itself does not improve T1 measurement accuracy but rather reduces the intrinsic bias of the myocardial T1 mapping method, which is observed in SENSE at R = 2

How to cite this article: Zhu Y, Kang J, Duan C, et al. Integrated motion correction and dictionary learning for free‐breathing myocardial T1 mapping. Magn Reson Med. 2018;00:1–11. https://doi.org/10.1002/mrm.27579

APPENDIX AThe dictionary learning–based reconstruction method formulation used in the reconstruction of the T1‐weighted images can be expressed as34

where Ip is the pth vectorized T1‐weighted image to be reconstructed, with p = 1, …, Np; Np is the total frame number; Ωp is the undersampling operator for Ip; F denotes the discrete Fourier transform; Sc is the sensitivity map of the cth coil and kc,p is the corresponding undersampled k‐space data; Rn is the operator that extracts the pixels at the same spatial location along the parametric direction; D is the relaxation curve–based dictionary with K atoms; �n is the sparse representation of the nth curve with respect to D and T0 is the predefined sparsity level; and ν is the regulari-zation parameter.

APPENDIX BAlgorithm1: Pseudocode of the reconstruction iteration

Inputs: kc,p: Undersampled k‐space data for cth coil and pth T1‐weighted image Sc: Sensitivity map of the cth coil F: Discrete Fourier transform Np: Total frame number D: Trained dictionary T0: Predefined sparsity level Outputs: Ip: Reconstructed T1‐weighted image

1. Initialize k1c,p

=kc,p and I1p=

∑c SH

cF

Hk

1

c,p for P = 1, …, Np.

2. Estimate the T1 map T11 from I1

p.

3. For iteration i = 1, 2, …, perform the following until the relative change of energy in Ip gets smaller than a given threshold ∑

p ‖Iip−Ii−1

p‖

2∑p ‖Ii

p‖

2

<𝜖 or the iteration reaches the predefined maximum

iteration number.

[1] Iip=

∑c SH

cF

Hk

i

c,p for P = 1, …, Np.

[2] Calculate the T0‐term estimation Rn

[Ii1,… ,Ii

Np

]=D�i

n,

‖�in‖

0≤T0 using orthogonal matching pursuit.48

[3] Compute the signal images with

SIi(TI)=SI0

(1−2e−TI∕Ti

1

).

[4] Estimate the synthetic image using Eq. 1 with Iip and SIi.

[5] Run the pairwise registration between Iip and the synthetic

image. [6] Update Ti+1

1with Ii

p.

[7] Using k̃c,p =FScIip, update the k‐space data as follows:

ki+1c,p

=

{k̃c,p

(kx,ky

)kc,p

(kx,ky

)∉ acquireddata

k̃c,p(kx,ky)+𝜐kc,p(kx,ky)

1+𝜐kc,p

(kx,ky

)∈ acquireddata

.

APPENDIX CIn the STONE sequence, the magnetization before bSSFP readout is a pure T1 relaxation. Assuming that the equilib-rium magnetization is M0 =[0,0,1]T, the magnetization just before bSSFP readout can be expressed as

Assuming that the excitation pulse in the bSSFP imaging sequence is applied in the x direction with a flip angle of θ, the magnetization just after the first excitation pulse of the bSSFP sequence is

minimizeIp,�n

�n

‖D�n−Rn[I1,… ,INP]‖22+�

�c,p

‖kc,p−Ωp(FScIp)‖22

subject to ∀n, ‖�n‖0 ≤T0

M−

rf,1=Mp,TI =

[0,0,1−2e−TI∕T1

]T.

M+

rf,1=

⎡⎢⎢⎢⎣

1 0 0

0 cos� sin�

0 −sin� cos�

⎤⎥⎥⎥⎦M−

rf,1.

(C1)

(C2)

(A1)

| 11ZHU et al.

After relaxation in the TR of bSSFP, the magnetization just before the second excitation pulse of bSSFP sequence is

The combination of Eqs. 3 and 4 gives

Considering phase cycling in the bSSFP sequence (i.e., ex-citation pulse phases are 0, π, 0, π, …), the flip angle of the third excitation pulse can be treated as −θ. The magnetization just before the third excitation pulse can be expressed as

and T denotes matrix transposition) and �=

⎡⎢⎢⎢⎣

0

0

1−E1

⎤⎥⎥⎥⎦. �

and � are determined by the flip angle and relaxation times T1

and T2. The formation of the magnetization dynamics becomes

The magnetization at the TE of n + 1 pulse in the bSSFP sequence would be

where

Therefore, the simulated magnetization using bSSFP read-out can be expressed as

where A and B are

A and B depend only on the bSSFP readout parameters, including the acquired phase‐encoding number, TR, TE, and the bSSFP readout flip angle, as well as relaxation times T1 and T2.

M−

rf,2=

⎡⎢⎢⎢⎣

E2 0 0

0 E2 0

0 0 E1

⎤⎥⎥⎥⎦

M+

rf,1+

⎡⎢⎢⎢⎣

0

0

1−E1

⎤⎥⎥⎥⎦

,

where E2 = e−TR∕T2 and E1 = e−TR∕T1.

M−

rf,2=

⎡⎢⎢⎢⎣

E2 0 0

0 cos�E2 sin�E2

0 −sin�E1 cos�E1

⎤⎥⎥⎥⎦M−

rf,1+

⎡⎢⎢⎢⎣

0

0

1−E1

⎤⎥⎥⎥⎦.

M−

rf,3=

⎡⎢⎢⎢⎣

E2 0 0

0 cos�E2 −sin�E2

0 sin�E1 cos�E1

⎤⎥⎥⎥⎦M−

rf,2+

⎡⎢⎢⎢⎣

0

0

1−E1

⎤⎥⎥⎥⎦.

Let �=

⎡⎢⎢⎢⎣

E2 0 0

0 cos�E2 sin�E2

0 −sin�E1 cos�E1

⎤⎥⎥⎥⎦(�T

=

⎡⎢⎢⎢⎣

E2 0 0

0 cos�E2 −sin�E2

0 sin�E1 cos�E1

⎤⎥⎥⎥⎦

M−

rf,2=�M−

rf,1+�

M−

rf,3=�TM

−

rf,2+�=�T�M−

rf,1+�T�+�

M−

rf,4=�M−

rf,3+�=�T�2M

−

rf,1+�T��+��+�

M−

rf,5=�TM−

rf,4+�=

(�T

)2�2M

−

rf,1+

(�T

)2��+�T��+�T�+�

M−

rf,n=

� ��T

�l�lM

−

rf,1+

∑l

i=1

��T

�i�i−1�+

∑l−1

i=0

��T

�i�i� n=2l+1�

�T�l−1

�lM−

rf,1+

∑l−1

i=1

��T

�i−1�i�+

∑l−1

i=0

��T

�i�i� n=2l

, l = 1, 2, 3,…

MTE,n+1 =�TEM−

rf,n+�TE or MTE,n+1 =�T

TEM−

rf,n+�TE

�TE =

⎡⎢⎢⎢⎣

e−TE∕T2 0 0

0 cos�e−TE∕T2 sin�e−TE∕T2

0 −sin�e−TE∕T1 sin�e−TE∕T1

⎤⎥⎥⎥⎦and �TE =

⎡⎢⎢⎢⎣

0

0

1−e−TE∕T1

⎤⎥⎥⎥⎦.

Mp =AMp,TI+B

�A=�TE

��T

�l�l,B=�TE�(

∑l

i=1

��T

�i�i−1

+

∑l−1

i=0

��T

�i�i)+�TE n=2l+1

A=�TTE

��T

�l−1�l,B=�T

TE�(

∑l−1

i=1

��T

�i−1�i+

∑l−1

i=0

��T

�i�i)+�TE n=2l

, l=1,2,3,…

(C3)

(C6)

(C4)

(C7)(C5)