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

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

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

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^{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 ${\mathbf{m}}_{n}\left({\U0001d4e3}_{{\mathbf{\Theta}}_{n}}\left(\mathbf{x}\right)\right)$ with $\mathbf{x}\in {\mathcal{X}}_{cr}$. Thus, in the optimization problem described in Equation (8), we aimed to find that ${\mathbf{m}}_{p}\left({\U0001d4e3}_{{\mathbf{\Theta}}_{p}}\left(\mathbf{x}\right)\right)\cong {\mathbf{m}}_{q}\left({\U0001d4e3}_{{\mathbf{\Theta}}_{q}}\left(\mathbf{x}\right)\right)$, 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., ${\mathbf{m}}_{n}=\mathbf{m}\left({\mathbf{T}}_{{\mathbf{\Theta}}_{n}}\left(\mathbf{x}\right)\right)$. 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, $1\le i\le \left(\genfrac{}{}{0pt}{}{K}{P}\right)$. 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 ${\mu}^{{\mathbf{c}}_{i}},1\le i\le \left(\genfrac{}{}{0pt}{}{K}{P}\right)$, 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 $\left(\genfrac{}{}{0pt}{}{K}{P}\right)$ 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

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**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 $\left[1,6\right]$, 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 |

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**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