# Elastic AlignedSENSE for Dynamic MR Reconstruction: A Proof of Concept in Cardiac Cine

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

^{2}, slice thickness 8 mm, 20 cardiac phases, and FOV 320 × 320 mm

^{2}.

^{2}, slice thickness 8 mm, and FOV 320 × 320 mm

^{2}. Twelve short-axis slices were acquired in a single 9.23 s breath-hold scan.

#### 2.2. Reconstruction Problem: Elastic AlignedSENSE

#### 2.3. Methods Used for Performance Comparison

- Transformations are defined in opposite directions, as illustrated in Figure 2. In GWCS, the coordinate space ${\mathcal{X}}_{cr}\subset {\mathbb{R}}^{2}$ is defined in the common reference image, and each frame ${\mathbf{m}}_{n}$ ($1\le n\le N$, N being the number of frames) is transformed so that it fits into such ${\mathcal{X}}_{cr}$, i.e., we calculated ${\mathbf{m}}_{n}\left(\right)open="("\; close=")">{\U0001d4e3}_{{\mathbf{\Theta}}_{n}}\left(\mathbf{x}\right)$ with $\mathbf{x}\in {\mathcal{X}}_{cr}$. Thus, in the optimization problem described in Equation (8), we aimed to find that ${\mathbf{m}}_{p}\left(\right)open="("\; close=")">{\U0001d4e3}_{{\mathbf{\Theta}}_{p}}\left(\mathbf{x}\right)$, with $p\ne q$. In the case of EAS, the coordinate space ${\mathcal{X}}_{n}\subset {\mathbb{R}}^{2}$ is defined in each frame ${\mathbf{m}}_{n}$—and coincides for all frames (${\mathcal{X}}_{n}\equiv \mathcal{X}$, $1\le n\le N$)—so that each frame ${\mathbf{m}}_{n}$ is a deformed version of the pattern image $\mathbf{m}$, i.e., ${\mathbf{m}}_{n}=\mathbf{m}\left(\right)open="("\; close=")">{\mathbf{T}}_{{\mathbf{\Theta}}_{n}}\left(\mathbf{x}\right)$. In summary, the transformations have their origin in the space in which the coordinate system is defined, and the direction is the opposite of what “common sense” dictates. The reason for this is because the transformation defined in that way makes the underlying interpolation process more convenient.
- The common reference image in GWCS is the average of the registered images, following [26], while in EAS, the reference arises as a result of the optimization subproblem in Equation (4a), which is transformed to create the images of the final sequence and does not necessarily correspond to any pre-selected cardiac phase.

#### 2.4. Combination of Elastic AlignedSENSE and Group-Wise Motion-Compensated Compressed Sensing

#### 2.5. Performance Analysis and Hyperparameter Selection

- For each of the K datasets, the value of the parameter that maximizes the IQM is determined by sweeping in a range of candidate values; let ${\mu}_{k}^{ds},1\le k\le K$, denote this value for the k-th dataset.
- The K datasets are split into P datasets for training and $(K-P)$ for testing. Let ${\mathbf{c}}_{i}$ be the i-th training set and ${\mathbf{d}}_{i}$ its corresponding test set, $1\le i\le \left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{K}{P}$. Let ${\left[{\mathbf{c}}_{i}\right]}_{j}$ denote the index within the set $\{1,\dots ,K\}$ of the j-th element of ${\mathbf{c}}_{i}$, with $1\le j\le P$. The purpose of this stage is to determine the optimum parameter for each ${\mathbf{c}}_{i}$. To this end, we accumulated the IQM for all datasets within ${\mathbf{c}}_{i}$, but dataset ${\left[{\mathbf{c}}_{i}\right]}_{j}$, using the parameter ${\mu}_{{\left[{\mathbf{c}}_{i}\right]}_{j}}^{ds}$ from the previous stage. The optimal value is the one that provides the maximum accumulated IQM out of the P accumulated quantities. Let ${\mu}^{{\mathbf{c}}_{i}}$ denote that value.
- The final stage pursues finding which of the ${\mu}^{{\mathbf{c}}_{i}},1\le i\le \left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{K}{P}$, is the optimum. This is accomplished by calculating the accumulated IQM in the datasets within ${\mathbf{d}}_{i}$, using ${\mu}^{{\mathbf{c}}_{i}}$; the optimal parameter ${\mu}_{opt}$ is the value that maximizes this quantity out of the $\left(\right)$ accumulated IQM values.

## 3. Experiments

#### 3.1. Experiment 1: Cartesian Acquisition

#### 3.2. Experiment 2: Radial Acquisition

## 4. Results

#### 4.1. Results of Experiment 1

#### 4.2. Results of Experiment 2

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Menchón-Lara, R.M.; Simmross-Wattenberg, F.; Casaseca-de-la Higuera, P.; Martín-Fernández, M.; Alberola-López, C. Reconstruction techniques for cardiac cine MRI. Insights Imaging
**2019**, 10, 1–16. [Google Scholar] [CrossRef] [PubMed] - Bluemke, D.A.; Boxerman, J.L.; Atalar, E.; McVeigh, E.R. Segmented K-space cine breath-hold cardiovascular MR imaging: Part 1. Principles and technique. Am. J. Roentgenol.
**1997**, 169. [Google Scholar] [CrossRef] - Larson, A.C.; White, R.D.; Laub, G.; McVeigh, E.R.; Li, D.; Simonetti, O.P. Self-gated cardiac cine MRI. Magn. Reson. Med.
**2004**, 51, 93–102. [Google Scholar] [CrossRef] [Green Version] - Liu, J.; Spincemaille, P.; Codella, N.; Nguyen, T.; Prince, M.; Wang, Y. Respiratory and cardiac self-gated free-breathing cardiac CINE imaging with multiecho 3D hybrid radial SSFP acquisition. Magn. Reson. Med.
**2010**, 63, 1230–1237. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Krämer, M.; Herrmann, K.H.; Biermann, J.; Reichenbach, J. Retrospective reconstruction of cardiac cine images from golden-ratio radial MRI using one-dimensional navigators. J. Magn. Reson. Imaging
**2014**, 40, 413–422. [Google Scholar] [CrossRef] [PubMed] - Usman, M.; Ruijsink, B.; Nazir, M.; Cruz, G.; Prieto, C. Free breathing whole-heart 3D CINE MRI with self-gated Cartesian trajectory. Magn. Reson. Imaging
**2017**, 38, 129–137. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Seo, H.; Kim, D.; Oh, C.; Park, H. Self-gated cardiac cine imaging using phase information. Magn. Reson. Med.
**2017**, 77, 1216–1222. [Google Scholar] [CrossRef] [PubMed] - Tsao, J.; Boesiger, P.; Pruessmann, K.P. k-t BLAST and k-t SENSE: Dynamic MRI with high frame rate exploiting spatiotemporal correlations. Magn. Reson. Med.
**2003**, 50, 1031–1042. [Google Scholar] [CrossRef] - Huang, F.; Akao, J.; Vijayakumar, S.; Duensing, G.R.; Limkeman, M. k-t GRAPPA: A k-space implementation for dynamic MRI with high reduction factor. Magn. Reson. Med.
**2005**, 54, 1172–1184. [Google Scholar] [CrossRef] - Lustig, M.; Donoho, D.; Pauly, J.M. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med. Off. J. Int. Soc. Magn. Reson. Med.
**2007**, 58, 1182–1195. [Google Scholar] [CrossRef] - Jung, H.; Ye, J.C. Motion estimated and compensated compressed sensing dynamic magnetic resonance imaging: What we can learn from video compression techniques. Int. J. Imaging Syst. Technol.
**2010**, 20, 81–98. [Google Scholar] [CrossRef] - Asif, M.S.; Hamilton, L.; Brummer, M.; Romberg, J. Motion-adaptive spatio-temporal regularization for accelerated dynamic MRI. Magn. Reson. Med.
**2013**, 70, 800–812. [Google Scholar] [CrossRef] - Lingala, S.G.; DiBella, E.; Jacob, M. Deformation Corrected Compressed Sensing (DC-CS): A Novel Framework for Accelerated Dynamic MRI. IEEE Trans. Med. Imaging
**2015**, 34, 72–85. [Google Scholar] [CrossRef] [PubMed] - Usman, M.; Atkinson, D.; Heathfield, E.; Greil, G.; Schaeffter, T.; Prieto, C. Whole left ventricular functional assessment from two minutes free breathing multi-slice CINE acquisition. Phys. Med. Biol.
**2015**, 60, N93–N107. [Google Scholar] [CrossRef] - Royuela-del Val, J.; Cordero-Grande, L.; Simmross-Wattenberg, F.; Martín-Fernández, M.; Alberola-López, C. Nonrigid group-wise registration for motion estimation and compensation in compressed sensing reconstruction of breath-hold cardiac cine MRI. Magn. Reson. Med.
**2016**, 75, 1525–1536. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Royuela-del Val, J.; Cordero-Grande, L.; Simmross-Wattenberg, F.; Martín-Fernández, M.; Alberola-López, C. Jacobian weighted temporal total variation for motion compensated compressed sensing reconstruction of dynamic MRI. Magn. Reson. Med.
**2017**, 77, 1208–1215. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Batchelor, P.G.; Atkinson, D.; Irarrazaval, P.; Hill, D.L.G.; Hajnal, J.; Larkman, D. Matrix description of general motion correction applied to multishot images. Magn. Reson. Med.
**2005**, 54, 1273–1280. [Google Scholar] [CrossRef] [PubMed] - Cordero-Grande, L.; Teixeira, R.P.A.; Hughes, E.J.; Hutter, J.; Price, A.N.; Hajnal, J.V. Sensitivity encoding for aligned multishot magnetic resonance reconstruction. IEEE Trans. Comput. Imaging
**2016**, 2, 266–280. [Google Scholar] [CrossRef] [Green Version] - Rueckert, D.; Sonoda, L.I.; Hayes, C.; Hill, D.L.G.; Leach, M.O.; Hawkes, D.J. Nonrigid registration using free-form deformations: Application to breast MR images. IEEE Trans. Med. Imaging
**1999**, 18, 712–721. [Google Scholar] [CrossRef] - De Boor, C. A Practical Guide to Splines; Springer: New York, NY, USA, 1978. [Google Scholar]
- Sun, W.; Niessen, W.J.; Klein, S. Free-form deformation using lower-order B-spline for nonrigid image registration. In Medical Image Computing and Computer-Assisted Intervention–MICCAI 2014; Golland, P., Hata, N., Barillot, C., Hornegger, J., Howe, R., Eds.; Springer International Publishing: Cham, Switzerland, 2014; pp. 194–201. [Google Scholar]
- Menchón-Lara, R.M.; Royuela-del-Val, J.; Simmross-Wattenberg, F.; Casaseca-de-la-Higuera, P.; Martín-Fernández, M.; Alberola-López, C. Fast 4D elastic group-wise image registration. Convolutional interpolation revisited. Comput. Methods Programs Biomed.
**2021**, 200, 105812. [Google Scholar] [CrossRef] [PubMed] - Nocedal, J.; Wright, S.J. Numerical Optimization; Springer: New York, NY, USA, 1999. [Google Scholar]
- Beatty, P.J.; Nishimura, D.G.; Pauly, J.M. Rapid gridding reconstruction with a minimal oversampling ratio. IEEE Trans. Med. Imaging
**2005**, 24, 799–808. [Google Scholar] [CrossRef] - Knoll, F.; Schwarzl, A.; Diwoky, C.; Sodickson, D.K. gpuNUFFT-An Open Source GPU Library for 3D Regridding with Direct Matlab Interface. In Proceedings of the 22nd Annual Meeting of ISMRM, Milan, Italy, 10–16 May 2014; p. 4297. [Google Scholar]
- Polfliet, M.; Klein, S.; Huizinga, W.; Paulides, M.M.; Niessen, W.J.; Vandemeulebroucke, J. Intrasubject multimodal group-wise registration with the conditional template entropy. Med. Image Anal.
**2018**, 46, 15–25. [Google Scholar] [CrossRef] - Becker, S.; Bobin, J.; Candès, E.J. NESTA: A fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci.
**2011**, 4, 1–39. [Google Scholar] [CrossRef] [Green Version] - Cruz, G.; Atkinson, D.; Buerger, C.; Schaeffter, T.; Prieto, C. Accelerated motion corrected three-dimensional abdominal MRI using total variation regularized SENSE reconstruction. Magn. Reson. Med.
**2016**, 75, 1484–1498. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process.
**2004**, 13, 600–612. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Godino-Moya, A.; Royuela-del Val, J.; Usman, M.; Menchón-Lara, R.M.; Martín-Fernández, M.; Prieto, C.; Alberola-López, C. Space-time variant weighted regularization in compressed sensing cardiac cine MRI. Magn. Reson. Imaging
**2019**, 58, 44–55. [Google Scholar] [CrossRef] - Feng, L.; Srichai, M.B.; Lim, R.P.; Harrison, A.; King, W.; Adluru, G.; Dibella, E.V.R.; Sodickson, D.K.; Otazo, R.; Kim, D. Highly accelerated real-time cardiac cine MRI using k-t SPARSE-SENSE. Magn. Reson. Med.
**2013**, 70, 64–74. [Google Scholar] [CrossRef] [Green Version] - Feng, L.; Grimm, R.; Block, K.T.; Chandarana, H.; Kim, S.; Xu, J.; Axel, L.; Sodickson, D.K.; Otazo, R. Golden-angle radial sparse parallel MRI: Combination of compressed sensing, parallel imaging, and golden-angle radial sampling for fast and flexible dynamic volumetric MRI. Magn. Reson. Med.
**2014**, 72, 707–717. [Google Scholar] [CrossRef] [Green Version] - Feng, L.; Axel, L.; Chandarana, H.; Block, K.T.; Sodickson, D.K.; Otazo, R. XD-GRASP: Golden-angle radial MRI with reconstruction of extra motion-state dimensions using compressed sensing. Magn. Reson. Med.
**2016**, 75, 775–788. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Metz, C.; Klein, S.; Schaap, M.; van Walsum, T.; Niessen, W. Nonrigid registration of dynamic medical imaging data using nD+t B-splines and a group-wise optimization approach. Med. Image Anal.
**2011**, 15, 238–249. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The scheme of the EAS reconstruction as an alternating minimization approach. If the deformations ${\mathbf{T}}_{\mathbf{\Theta}}$ are assumed to be known, the best possible $\mathbf{m}$ in terms of fidelity to the measured data $\mathbf{y}$ can be obtained. Likewise, assuming $\mathbf{m}$ to be known, the best possible ${\mathbf{T}}_{\mathbf{\Theta}}$ can be obtained. The final image sequence is obtained by applying each of the transformations ${\mathbf{T}}_{\tilde{\mathbf{\Theta}}}$ to the pattern image $\tilde{\mathbf{m}}$. The input to the reconstruction method is the shaded circle. Outputs are enclosed by a dashed line rectangle.

**Figure 2.**The scheme of spatial transformations in GWCS (

**left**) and EAS (

**right**) for 2D cardiac cine MRI.

**Left**: points to be transformed $\mathbf{x}\in {\mathcal{X}}_{cr}\subset {\mathbb{R}}^{2}$ are defined on the common reference coordinate space.

**Right**: points to be transformed $\mathbf{x}\in {\mathcal{X}}_{n}\equiv \mathcal{X}\subset {\mathbb{R}}^{2}$, $1\le n\le N$ are defined on each image coordinate space, which coincides for all images.

**Figure 3.**The scheme of the MIX reconstruction method as a combination of, at least, two EAS phases followed by a GWCS phase. The output of EAS, $\tilde{\mathbf{m}}$ and ${\mathbf{T}}_{\tilde{\mathbf{\Theta}}}$, is fed to GWCS. Since EAS provides directly a set of transformations ${\mathbf{T}}_{\tilde{\mathbf{\Theta}}}$ that maps the pattern image ${\mathbf{m}}_{0}$ to each cardiac state, there is no need for the registering stage within GWCS. Thus, only the MC stage within GWCS is applied to obtain the final reconstruction. The input to the whole reconstruction method is the shaded circle. Outputs are enclosed by a dashed line rectangle.

**Figure 4.**The scheme of the registrations performed for motion quality assessment. Note that a periodic extension is considered (represented with dotted lines), so that the first frame is registered to the last one.

**Figure 5.**Temporal profiles along radial directions every 45 degrees (the center of which coincides with the center of the left ventricle) are concatenated to form an image. The NCC between such images is used to assess motion quality.

**Figure 6.**Comparison of EAS and MIX reconstructions with other methods from the literature for a representative case with $R=8$. The fully sampled reconstruction is included in the top line as a reference. Diastole and systole frames are shown in the two leftmost columns, respectively. Two temporal profiles of the horizontal and vertical lines—marked in the reference image with white lines—are shown in the rightmost columns for all the methods. Arrows point to significant locations.

**Figure 7.**Comparison of EAS and MIX reconstructions with other methods from the literature for a representative case with $R=8$. The fully sampled reconstruction is included in the top line as a reference. Diastole and systole frames are shown in the two leftmost columns, respectively. Two temporal profiles of the horizontal and vertical lines—marked in the reference image with white lines—are shown in the rightmost columns for all the methods. Arrows point to significant locations.

**Figure 8.**Results for EAS and MIX reconstructions. The average values across slices and volunteers for the HFSER (

**a**), SSIM (

**b**), NCC (

**c**), and RMSE (

**d**) and the average time needed to reconstruct one slice (

**e**) are provided for different values of R.

**Figure 9.**Results for EAS and MIX reconstructions distributed according to the 17-segment AHA model. The average values across volunteers are provided for $R=8$.

**Figure 10.**EAS radial reconstructions in comparison with iGRASP and GWCS ($R=19.33$). Reconstructions from the MIX method are also included. Arrows point to significant locations.

**Table 1.**The mean value ± the standard deviation of the scores given by the expert to each reconstruction method. The scores vary in the range $\left(\right)$, 6 being the method that provides reconstructions with the highest image quality.

R = 8 | R = 10 | R = 14 | |
---|---|---|---|

$\mathit{sPICS}$ | $5.14\pm 0.69$ | $4.86\pm 0.69$ | $3.43\pm 1.72$ |

$\mathit{GWCS}$ | $5.00\pm 1.83$ | $5.71\pm 0.49$ | $4.86\pm 1.68$ |

$\mathit{EAS}$ | $2.57\pm 1.13$ | $2.57\pm 0.79$ | $3.29\pm 1.38$ |

**Table 2.**Mean values of the execution times for reconstructing one slice using the EAS and MIX radial approaches in comparison with iGRASP and GWCS.

Mean Running Time (min) | |
---|---|

$\mathit{iGRASP}$ | 1.9513 |

$\mathit{GWCS}$ | 6.4263 |

$\mathit{EAS}$ | 2.2940 |

$\mathit{MIX}$ | 3.7792 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Godino-Moya, A.; Menchón-Lara, R.-M.; Martín-Fernández, M.; Prieto, C.; Alberola-López, C.
Elastic AlignedSENSE for Dynamic MR Reconstruction: A Proof of Concept in Cardiac Cine. *Entropy* **2021**, *23*, 555.
https://doi.org/10.3390/e23050555

**AMA Style**

Godino-Moya A, Menchón-Lara R-M, Martín-Fernández M, Prieto C, Alberola-López C.
Elastic AlignedSENSE for Dynamic MR Reconstruction: A Proof of Concept in Cardiac Cine. *Entropy*. 2021; 23(5):555.
https://doi.org/10.3390/e23050555

**Chicago/Turabian Style**

Godino-Moya, Alejandro, Rosa-María Menchón-Lara, Marcos Martín-Fernández, Claudia Prieto, and Carlos Alberola-López.
2021. "Elastic AlignedSENSE for Dynamic MR Reconstruction: A Proof of Concept in Cardiac Cine" *Entropy* 23, no. 5: 555.
https://doi.org/10.3390/e23050555