# Atherosclerotic Plaque Segmentation Based on Strain Gradients: A Theoretical Framework

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Simulating IVUS Data

#### 2.1.1. Geometries

#### 2.1.2. Modeling of Tissue Behavior

#### 2.1.3. FE Models

#### 2.1.4. Strain Variables

#### 2.2. Adding Noise

#### 2.3. Computing SGVs

#### 2.4. Segmentation Process

- Plaque-related variables: We analyzed the influence of considering the fibrotic tissue as fully incompressible or with different degrees of quasi-incompressibility. We have also considered four different fibrotic tissues (default, stiff, medium, and soft tissues). Furthermore, some inclusions were added to the FE model, mimicking the presence of micro calcifications. These inclusions were simplified as spheres with calcification properties presented in Table 1, and four diameters were studied (10, 50, 150, and 300 $\mathsf{\mu}$m).
- IVUS-related variables: The influence of the catheter position was studied by changing the origin and orientation of the coordinate system in the FE models. It was also important to check if the segmentation methodology was affected by the blood pressure. In addition, the pressure increment between both steps was also studied.

#### 2.5. Geometrical Measures

## 3. Results

#### 3.1. Idealized Geometries

#### 3.2. Real IVUS Geometries

#### 3.3. SGV Candidates

#### 3.4. Sensitivity Analysis

## 4. Discussion

#### 4.1. Segmentation Analysis

#### 4.1.1. Idealized Geometries

#### 4.1.2. IVUS Geometries

#### 4.2. SGV Candidates

#### 4.3. Sensitivity analysis

#### 4.4. Relevance for Clinical Applications

- A segmentation process based on strain representation was presented to extract the different tissues of an atherosclerotic plaque. This methodology achieved high accuracy in measuring FCT and the lipid core area. These measurements play a key role in the vulnerability of the plaque.
- Unlike other segmentation processes, this method does not require a database to be trained or an optimization process, as it relies on image processing rather than machine learning or analysis of the mechanical properties of the tissues. In addition, it could be performed with many different strain variables instead of a single one [27,28,31,47]. Thus, there are different possibilities to obtain the segmentation using only one variable or combining different SGVs.
- The results show that the performance of the segmentation was linked to the plaque geometry and the selected SGVs. However, there were some SGVs with good results regardless of the geometry. The method also showed good robustness in sensitivity analysis, providing accurate results with different catheter positions, pressures, and noise addition.

#### 4.5. Limitations

- Since this work was a theoretical framework, the methodology was only tested with computational models of in silico data. Therefore, the next step would be to prove the segmentation methodology with in vitro and in vivo IVUS data from patients with coronary atherosclerotic plaques. After analyzing the methodology with noise, which simulates the intrinsic noise of IVUS data, the results for segmentation are expected to be valid on real IVUS data.
- In the finite element analysis we only have considered the load of the blood pressure. We have disregarded the residual stress and the influence of heart motion. As the methodology is based on gradients and not on absolute strain/stress values, we could expect a minimum influence of the residual stress on this segmentation methodology.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IVUS | Intravascular Ultrasound |

FCT | Fibrous Cap Thickness |

VH | Virtual Histology |

FE | Finite Element |

CNN | Convolutional Neural Networks |

SGV | Strain Gradient Variable |

SNR | Signal-to-Noise Ratio |

SI | Segmentation Index |

W-GVF | Watershed-(Gradient Vector Flow) |

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**Figure 2.**(

**a**) 3D Idealized geometry; (

**b**) fibrous cap thickness of 65 microns; (

**c**) fibrous cap thickness of 150 microns; (

**d**) fibrous cap thickness of 300 microns.

**Figure 3.**The first column presents IVUS images [35] of the three different plaques; the second column is the manual segmentation performed by a cardiologist [35]; the third column is the IVUS reconstruction in Abaqus of the pressurized geometry; the fourth column shows the plaque models with zero-pressure geometry in Abaqus. These geometries were used to initiate the FE simulations. The fifth column is the final FE model after applying an internal pressure of 115 mmHg to the previous geometry.

**Figure 4.**Comparison between the real and the segmented lipid core. The lumen is represented in gray color, the true positive area in white, the false negative area in green, the actual area that was not segmented in purple, and the measure of the FCT is the red line.

**Figure 5.**Influence of the FCT on the segmentation procedure analyzed with clean strains. The rows represent the segmentation process with the geometries of 65, 150, and 300 $\mathsf{\mu}$m of FCT. The segmentation process consists of the combination of two SGVs, in these cases $\left|\u25bd{\epsilon}_{min}\right|$ and $\left|\u25bdFA\right|$ in (

**a**,

**b**), respectively. The combination of both is represented in (

**c**). This representation is the input for the W-GVF and its results are represented in (

**d**); finally, the overlap between the actual and the segmented lipid core before and after the smooth treatment is represented in (

**e**), where the true positive area is in white, the false negative area in green and the actual area that is not segmented in purple.

**Figure 6.**Box-plots of the SI values of the idealized geometries. From left to right: 65, 150, and 300 $\mathsf{\mu}$m of fibrous cap thickness. Each geometry was analyzed with clean strains and with 20 dB of SNR. The median values were represented with a horizontal line. The median values were 93% and 90.14% for the idealized geometry with 65 $\mathsf{\mu}$m FCT with and without noise; 94.88% and 93.42% for geometry of 150 $\mathsf{\mu}$m FCT; and 95.23% and 94.17% for 300 $\mathsf{\mu}$m. Mean values are represented with asterisks. Outliers are represented with circles. Some outliers were below 65% but are not shown.

**Figure 7.**Segmentation procedure in the IVUS geometries with clean strains. The rows represent the segmentation process with the plaques 1, 2, and 3; (

**a**) representation of $\left|\u25bd{\epsilon}_{min}\right|$; (

**b**) representation of $\left|\u25bdFA\right|$; (

**c**) the combination of $\left|\u25bd{\epsilon}_{min}\right|$+$\left|\u25bdFA\right|$; (

**d**) W-GVF results; and (

**e**) segmented lipid before and after the smooth treatment, where the true positive area is in white, the false negative area in green and the actual area that is not segmented in purple.

**Figure 8.**Box-plots of the SI values of the real IVUS geometries. The three plaques were analyzed with clean strains and with 20 dB of SNR. The median values are represented with a horizontal line. The median values were 92.02% and 93.74% for the first IVUS geometry, with and without noise; 92.60% and 88.82% for the second IVUS geometry; and 94.30% 91.84% for third IVUS geometry, respectively. Mean values are represented with asterisks. Outliers are represented with circles. Some outliers were below 65% but are not shown.

**Figure 9.**Idealized geometry with 150 $\mathsf{\mu}$m thickness with the different $dW$ represented without noise. (

**a**) $d{W}_{simplified}$, (

**b**) $dW$ computed with the 2D strain tensor, and (

**c**) $\left|dW\right|$.

**Figure 10.**Graphic summary of the influence of incompressibility, changing fibrotic tissues, addition of inclusions, different catheter positions, different pressures and pressure increments, and noise addition on the segmentation process. The considered SGVs are $\left|\u25bd{\epsilon}_{Mises}\right|$+$\left|\u25bd{\epsilon}_{r\theta}\right|$,$\left|\u25bd{\epsilon}_{yy}\right|$+$\left|\u25bd{\epsilon}_{rr}\right|$, $\left|\u25bd{\epsilon}_{yy}\right|$+$\left|\u25bd{\epsilon}_{min}\right|$, $\left|\u25bd{\epsilon}_{min}\right|$+$\left|\u25bd{\epsilon}_{Tresca}\right|$, and $dW$. Each SGV has different variables to analyze, divided by colors. Each plaque- or IVUS-related variable had different cases that were differentiated by shape markers.

Tissue | $\mathit{\mu}$ [kPa] | ${\mathit{k}}_{1}$ [kPa] | ${\mathit{k}}_{2}$ [-] | $\mathit{\kappa}$[-] | $\mathit{\alpha}$[°] |
---|---|---|---|---|---|

Adventitia | 4.22 | 547.67 | 568.01 | 0.26 | ±61.80 |

Media | 0.7 | 206.16 | 58.55 | 0.29 | ±28.35 |

Intima | 3.41 | 109.10 | 101.04 | 0.21 | ±52.72 |

Fibrotic | 4.79 | 17,654.91 | 0.51 | 1/3 | - |

Lipid Core | 0.025 | 956.76 | 70 | 1/3 | - |

Calcification | 1875 | - | - | - | - |

**Table 2.**Summary of the results for the best five single SGVs and fifteen SGV combinations for the lipid segmentation of all of the 105 possible combinations based on the results of the idealized and IVUS geometries with FE strains. The SGVs with the SI value in dark green mean a perfect segmentation; light green stands for good segmentations; yellow for SGV indicates some problems in segmenting the fibrous cap or the lipid area. $\overline{SI}$ and $\overline{SInoise}$ represent the mean SI value without and with noise, respectively.

Segmentation Index (SI) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Idealized Geometry | Real IVUS Geometry | $\overline{\mathit{SI}}$ | $\overline{{\mathit{SI}}_{\mathit{noise}}}$ | ||||||

65 $\mathsf{\mu}$m | 150 $\mathsf{\mu}$m | 300 $\mathsf{\mu}$m | Plaque 1 | Plaque 2 | Plaque 3 | ||||

One SGV | $dW$ | 95.20 | 97.13 | 94.31 | 96.98 | 92.06 | 92.50 | 94.70 | 90.62 |

$\left|dW\right|$ | 92.33 | 94.55 | 96.02 | 93.99 | 94.66 | 96.23 | 94.63 | 86.04 | |

$\left|\u25bf{\epsilon}_{vMises}\right|$ | 93.49 | 93.60 | 97.65 | 90.74 | 94.99 | 97.08 | 94.59 | 94.47 | |

$\left|\u25bf{\epsilon}_{rr}\right|$ | 97.65 | 94.40 | 93.07 | 98.48 | 86.78 | 96.32 | 94.45 | 92.27 | |

$\left|\u25bf{\epsilon}_{min}\right|$ | 86.07 | 93.77 | 97.86 | 97.73 | 93.43 | 97.46 | 94.39 | 93.29 | |

Combination of two SGVs | $\left|\u25bf{\epsilon}_{vMises}\right|+\left|\u25bf{\epsilon}_{r\theta}\right|$ | 95.87 | 97.63 | 97.09 | 98.61 | 96.14 | 97.28 | 97.10 | 95.22 |

$\left|\u25bf{\epsilon}_{yy}\right|+\left|\u25bf{\epsilon}_{rr}\right|$ | 95.74 | 97.93 | 94.79 | 98.53 | 97.51 | 96.76 | 96.88 | 95.68 | |

$\left|\u25bf{\epsilon}_{yy}\right|+\left|\u25bf{\epsilon}_{min}\right|$ | 95.74 | 98.21 | 94.23 | 98.53 | 95.55 | 98.28 | 96.75 | 94.28 | |

$\left|\u25bf{\epsilon}_{min}\right|+\left|\u25bf{\epsilon}_{Tresca}\right|$ | 96.97 | 96.32 | 96.09 | 97.36 | 95.98 | 97.43 | 96.69 | 93.67 | |

$\left|\u25bf{\epsilon}_{max}\right|+\left|\u25bf{\epsilon}_{rr}\right|$ | 97.73 | 94.08 | 98.58 | 97.76 | 92.33 | 98.03 | 96.42 | 92.88 | |

$\left|\u25bf{\epsilon}_{min}\right|+\left|\u25bfFA\right|$ | 97.85 | 93.01 | 96.17 | 97.46 | 95.24 | 97.47 | 96.20 | 94.76 | |

$\left|\u25bf{\epsilon}_{min}\right|+\left|\u25bf{\epsilon}_{vMises}\right|$ | 95.17 | 93.28 | 96.87 | 97.43 | 95.98 | 98.37 | 96.18 | 95.10 | |

$\left|\u25bf{\epsilon}_{rr}\right|+\left|\u25bf{\epsilon}_{r\theta}\right|$ | 92.81 | 97.49 | 97.95 | 98.94 | 93.13 | 96.43 | 96.13 | 88.68 | |

$\left|\u25bf{\epsilon}_{Tresca}\right|+\left|\u25bf{\epsilon}_{rr}\right|$ | 93.30 | 96.25 | 95.81 | 97.34 | 96.12 | 97.77 | 96.10 | 93.67 | |

$\left|\u25bf{\epsilon}_{yy}\right|+\left|\u25bf{\epsilon}_{Tresca}\right|$ | 93.06 | 97.64 | 95.17 | 98.41 | 93.55 | 97.94 | 95.96 | 93.09 | |

$\left|\u25bfFA\right|+\left|\u25bf{\epsilon}_{rr}\right|$ | 92.87 | 94.88 | 96.53 | 97.51 | 95.96 | 97.49 | 95.87 | 92.88 | |

$\left|\u25bf{\epsilon}_{max}\right|+\left|\u25bf{\epsilon}_{min}\right|$ | 97.53 | 94.46 | 92.74 | 96.34 | 95.40 | 97.96 | 95.74 | 93.65 | |

$\left|\u25bfFA\right|+\left|\u25bf{\epsilon}_{r\theta}\right|$ | 93.68 | 97.40 | 95.04 | 98.56 | 90.99 | 97.53 | 95.53 | 92.69 | |

$\left|\u25bf{\epsilon}_{xx}\right|+\left|\u25bfFA\right|$ | 95.93 | 95.79 | 95.51 | 92.44 | 95.29 | 98.23 | 95.53 | 92.03 | |

$\left|\u25bf{\epsilon}_{xy}\right|+\left|\u25bfFA\right|$ | 94.32 | 94.21 | 94.84 | 96.49 | 95.12 | 97.16 | 95.36 | 93.04 |

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**MDPI and ACS Style**

Latorre, Á.T.; Martínez, M.A.; Cilla, M.; Ohayon, J.; Peña, E.
Atherosclerotic Plaque Segmentation Based on Strain Gradients: A Theoretical Framework. *Mathematics* **2022**, *10*, 4020.
https://doi.org/10.3390/math10214020

**AMA Style**

Latorre ÁT, Martínez MA, Cilla M, Ohayon J, Peña E.
Atherosclerotic Plaque Segmentation Based on Strain Gradients: A Theoretical Framework. *Mathematics*. 2022; 10(21):4020.
https://doi.org/10.3390/math10214020

**Chicago/Turabian Style**

Latorre, Álvaro T., Miguel A. Martínez, Myriam Cilla, Jacques Ohayon, and Estefanía Peña.
2022. "Atherosclerotic Plaque Segmentation Based on Strain Gradients: A Theoretical Framework" *Mathematics* 10, no. 21: 4020.
https://doi.org/10.3390/math10214020