# Semi-Automatic 3D Reconstruction of Atheroma Plaques from Intravascular Ultrasound Images Using an ad-hoc Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Materials

#### 2.2. Participants

#### 2.3. Algorithms/Methods

- Reading.
- Preprocessing.
- Extraction of the atheroma at each cross-sectional image.
- Extraction of the atheroma border.
- Postprocessing.

#### 2.3.1. Preprocessing

- Before starting the algorithm, the specialist is requested to provide the intensity of the atheroma ${c}_{s}$. If the atheroma is not the brightest region of the image (i.e., ${c}_{s}<255$), then each pixel of the image is transformed following a tent-like map given by the function:$${T}_{{c}_{s}}\left(c\right)=255-\left(\right)open="|"\; close="|">{c}_{s}-c$$In such way, it is ensured that the brightest region is the region that needs to be selected. Figure 2 gives an example of such map with a value ${c}_{s}=160$. Once the transformation is used, the algorithm extracts the whitest regions in the next step.
- Binarisation of the image. The image is binarised by following an adaptive threshold based on the local mean intensity. In such way, the resulting image is smoother and less dependent on difference of light due to the noise that can be introduced by the capture device. Other methods have been implemented and tested, such as the Otsu filter or a fixed threshold value in order to decide whether the pixel is white or black. However, the extraction obtained is not as precise as with the adaptative threshold when comparing against the extraction made by a specialist. Thanks to the tent-like transformation, the intensity corresponding to the atheroma is always captured and is never lost due to the effects of the local mean intensity.

#### 2.3.2. Detection of the Atheroma at Each Cross-Sectional Image

#### 2.3.3. Extraction of the Atheroma Edge

- Compute $l={\sum}_{j=1}^{m}\parallel {x}_{i,j}-{x}_{i,j+1}\parallel $, where we set ${x}_{i,m+1}={x}_{i,1}$ to close the boundary. l gives the total length of the boundary following a piecewise connection.
- Compute the step $\delta =l/N$.
- Fix ${\delta}_{a}=0$, ${\tilde{x}}_{1}={x}_{1}$ and $k=1$.
- For $j=1,\dots ,m$,
- (a)
- Compute the accumulated distance ${\delta}_{a}={\delta}_{a}+\parallel {x}_{i,j}-{x}_{i,j+1}\parallel $.
- (b)
- While ${\delta}_{a}>k\delta $ a new point should be added. Use the following steps:
- Use linear interpolation to add the next boundary point to the uniformised boundary:$${\tilde{x}}_{k+1}={x}_{i,j+1}-\epsilon ({x}_{i,j}-{x}_{i,j+1}).$$Where $\epsilon =(k\delta -{\delta}_{a})/\delta $.
- Increase $k=k+1$.

#### 2.3.4. Postprocessing

- Triangulation.After the last step for each image ${I}_{i}$, the points ${\tilde{x}}_{i,j}$ at the boundary of the atheroma have been selected. We follow the triangulation (see flowchart in Figure 4b), as follows:
- Connect each layer $i=1,\dots ,n-1$ with the layer $i+1$ by adding triangles using the vertices:
- –
- ${\tilde{x}}_{i,j},{\tilde{x}}_{i,j+1}$ and ${\tilde{x}}_{i+1,j}$.
- –
- ${\tilde{x}}_{i,j+1},{\tilde{x}}_{i+1,j}$ and ${\tilde{x}}_{i+1,j+1}$.

- Once layer i is connected, proceed with the next layer, $i=i+1$.

In such way, each layer is connected to the previous one by triangulation. The boundary of the atheroma is a 2-dimensional manifold; therefore, it can be covered by a mesh of triangles to join triplets of points. We note to the reader that here triangulation does not refer to the location of a point in the 3D space given two or more images but to cover a shape by a mesh of triangles. For each set of four points (${\tilde{x}}_{i,j}$, ${\tilde{x}}_{i,j+1}$, ${\tilde{x}}_{i+1,j}$ and ${\tilde{x}}_{i+1,j+1}$). The triangles $\widehat{{\tilde{x}}_{i,j}{\tilde{x}}_{i,j+1}{\tilde{x}}_{i+1,j}}$ and $\widehat{{\tilde{x}}_{i,j+1}{\tilde{x}}_{i+1,j}{\tilde{x}}_{i+1,j+1}}$ are stored in order to reconstruct the 3D figure. With this, the atheroma is reconstructed. - Cubic spline surface reconstruction.Following the previous method, an additional attempt was made to reconstruct the surface of the atheroma from a set of splines that approximated its contour. Thus, using the initial vertical discretization (i.e., the 15 contours with 100 points per level), clusters of 15 points in the z-direction were used to fit a set of 100 cubic splines that had individually described functions that fit the lateral surface and allowed an improved discretization. Final atheroma’s reconstruction is achieved by joining adjacent points, which are the result of a denser sampling based on the previously obtained functions, with straight lines.

#### 2.4. Reconstruction of the Artery

## 3. Results

- First, the initial figure is obtained by the specialist.
- Then, the binarised image is shown.
- The user select the red square in the third.
- Finally, the boundaries of the possible regions are highlighted by the method. The user must select the one of interest to start the process.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 5.**Steps to extract the information for the first layer according to the proposed algorithm: (

**a**) Initial image; (

**b**) Binarised image; (

**c**) Selected zone for the analysis; (

**d**) Boundaries detected by the Moore-Neighbour algorithm; (

**e**) Atheroma spot selected by the pathologist.

**Figure 6.**(

**a**) Binarised image in the first cross sectional image with the ateroma selected in red; (

**b**) Binarised second image in the search region; (

**c**) Binarised image with the pixels selected in the previous steps overlapped (red: the pixel is binarised and selected in the previous step, blue: the pixel is not binarised and was selected in the previous step); (

**d**) Selected region according to the maximum number of coincidences.

**Figure 7.**Border obtained the Moore-Neighbour tracing algorithm (blue) and the reparametrization of the border to 100 border points (red): (

**a**) Application for the first image; (

**b**) application for the second image.

**Figure 8.**Reconstruction of the atheroma through a triangulation (

**a**,

**c**) and cubic spline surface approximation (

**b**,

**d**).

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**MDPI and ACS Style**

Martínez, J.; Pérez-Palau, D.; Cilla, M.; Garrido, N.; Larrañaga, A.; Pérez-Rey, I.
Semi-Automatic 3D Reconstruction of Atheroma Plaques from Intravascular Ultrasound Images Using an ad-hoc Algorithm. *Mathematics* **2023**, *11*, 537.
https://doi.org/10.3390/math11030537

**AMA Style**

Martínez J, Pérez-Palau D, Cilla M, Garrido N, Larrañaga A, Pérez-Rey I.
Semi-Automatic 3D Reconstruction of Atheroma Plaques from Intravascular Ultrasound Images Using an ad-hoc Algorithm. *Mathematics*. 2023; 11(3):537.
https://doi.org/10.3390/math11030537

**Chicago/Turabian Style**

Martínez, Javier, Daniel Pérez-Palau, Myriam Cilla, Neus Garrido, Ana Larrañaga, and Ignacio Pérez-Rey.
2023. "Semi-Automatic 3D Reconstruction of Atheroma Plaques from Intravascular Ultrasound Images Using an ad-hoc Algorithm" *Mathematics* 11, no. 3: 537.
https://doi.org/10.3390/math11030537