2.1.1. Geometries

As this was a theoretical study, the IVUS data were simulated by using FE models with idealized and real patient geometries. The idealized geometries consisted of a 3D geometry with a 13 mm long atheroma plaque and with a lipid core length of 6.5 mm and different FCTs [33]. We analyzed three different FCTs, trying to cover the different geometric possibilities. A FCT of 65 μm was considered to represent a vulnerable case [5],

300 μm represented a stable plaque, and finally, 150 μm was an intermediate value between the two extremes. The geometry was reconstructed following the Glagov results [34] and the Finet law [6]. The model was constructed with symmetric conditions, so only a quarter of the geometry is shown in Figure 2a. The model had different tissues: adventitia, healthy media and intima, fibrotic tissue, and lipid core. As one of the aims of this study was to check the influence of the FCT on the segmentation process, three thicknesses, 65, 150, and 300 μm, were considered, and they are represented in Figure 2b–d. Despite the use of a 3D model, we only analyzed the section of maximum stenosis with the plane strain assumption in order to reproduce the IVUS technique. On the other hand, the real patient geometries were obtained from three IVUS images of human coronary plaques that were manually segmented by an expert cardiologist in a previous study [35]. Both IVUS images and the cardiologist segmentation of the three plaques are shown in Figure 3. When there was lack of information in the axial direction, in these cases the FE models were 2D. In two IVUS geometries, only the fibrotic tissue and the lipid core were considered, and on the third plaque a calcification was also included.

**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.

### 2.1.2. Modeling of Tissue Behavior

All tissues were modeled as hyperelastic, non-lineal, and incompressible with the constitutive model proposed by Gasser et al. [36]. The healthy tissues (adventitia, media, and intima) were modeled as anisotropic with two families of fibers. Conversely, the unhealthy tissues (fibrotic tissue and lipid core) were considered as an isotropic behavior model by the use of parameter *κ* = 1/3 in the *Gasser* model. All tissue was assumed to be fully incompressible (D = 0) in the idealized geometries and quasi-incompressible (D = 0.49) in the real IVUS geometries. The material parameters of the equation in (1) were fitted from experimental curves obtained from the bibliography [37,38] using the software Hyperfit [39]. All the parameters of (1) are reflected in Table 1, where *α* is the angle of the fibers with respect to the circumferential direction. The calcification of the third IVUS plaque was modeled with an isotropic neo-Hookean material model [35].

$$\Psi = \frac{1}{D} \cdot \left[ I - 1 \right]^2 + \mu \left[ I\_1 - 3 \right] + \frac{k\_1}{2k\_2} \sum\_{i=4,6} \left( \exp \left( k\_2 \left[ \kappa \left[ I\_1 - 3 \right] + \left[ 1 - 3 \kappa \right] \left[ I\_i - 1 \right] \right]^2 \right) - 1 \right), \tag{1}$$

**Table 1.** Fitted parameters for the hyperelastic model.


### 2.1.3. FE Models

The FE models were created in the commercial software Abaqus [40], where the boundary conditions and loads were imposed. In the 3D idealized FE models, not only were the symmetrical conditions imposed, but also the contact with the heart was mimicked by avoiding displacement in a contour line on the outside of the adventitia [33]. The blood pressure imposed inside the artery was 115 mmHg, which is the average pressure in patients with high normal pressure and grade 1 hypertension [41]. On the other hand, the FE models of real IVUS geometries were 2D, so they were solved following the plain strain assumption and the rigid body motion was constrained by two contour points with zero displacements. Furthermore, the IVUS geometry was previously reconstructed from pressurized images (third column in Figure 3). Therefore, it was necessary to obtain the zero-pressure geometry to be used as the initial geometry. For this purpose, we assumed that IVUS images were taken with an internal blood pressure of 110 mmHg, and the zeropressure geometry was recovered using the pull back algorithm defined by Raghavan et al. [42]. After applying the pull back method, the resulted geometry was extracted as the initial geometry (fourth column in Figure 3). Finally, it was possible to impose the pressure of 115 mmHg in the lumen and achieve the final pressurized geometry (fifth column in Figure 3). In all of the FE models (idealized and real IVUS geometries), the origin of the coordinate system was located in the center of the lumen in order to simulate the position of the IVUS catheter.

A sensitivity mesh analysis was performed to assure good precision and low computational cost. In the 3D idealized geometries, the maximum stenosis section, which was the most important part of the study, was meshed with small-sized elements. The fibrous cap between the lipid and the lumen had a smaller size depending on the thickness (e.g., 0.02 mm in the thickness of 65 microns that is on the order of IVUS technique precision). The rest of the 3D model had larger-sized elements, because of the lack of importance in the segmentation process. The element type selected for 3D was the hybrid quadratic tetrahedral elements with hybrid formulation to avoid numerical problems due to incompressibility (C3D10H), with at least three elements in the FCT in each case. In the 2D IVUS models, the element type was the plain strain hybrid three-node linear element (CPE3H) with additionaly, at least, three elements between the lumen and the lipid core.

### 2.1.4. Strain Variables

After simulating all of the FE models, the post-processing of the data was performed in Matlab 2021b [43]. The nodal coordinates ( *X* and *Y*) and displacements (*ux* and *uy*) of the steps at 110 mmHg and 115 mmHg were collected. Then, the relative node displacements between both pressure steps were computed, as we can see in the example of Equation (2). This data processing attempted to simulate the data obtained by speckle estimators on two consecutive pictures taken by an IVUS in a 5 mmHg pressure increment [24,25,28,31]. Afterwards, the strains were calculated under the infinitesimal strain theory. Despite having the displacement field through the entire idealized 3D FE models, we analyzed the maximum stenosis section with the hypothesis of plane strain to obtain results that are closer to what happens in the IVUS. Different strain variables were computed: strains referring to Cartesian coordinates (*<sup>ε</sup>xx*,*εyy* and *<sup>ε</sup>xy*); strains in cylindrical coordinates (*<sup>ε</sup>rr*,*εθθ* and *<sup>ε</sup>r<sup>θ</sup>*); principal strains (*<sup>ε</sup>max* and *<sup>ε</sup>min*), and equivalent strains of the von Mises, Tresca, and Anisotropic index (FA) [44], defined in Equation (3). Some variables, such as principal strains [45] or equivalent strains, do not depend on the coordinate system. Thus, their value will be the same regardless of the positions of the IVUS catheter.

$$
u\_{\mathbf{x}} = \boldsymbol{u}\_{\mathbf{x}}^{115mmH\mathbf{g}} - \boldsymbol{u}\_{\mathbf{x}}^{110mmH\mathbf{g}},\tag{2}$$

$$FA = \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{(\lambda\_1 - \lambda)^2 + (\lambda\_2 - \lambda)^2 + (\lambda\_3 - \lambda)^2}}{\sqrt{\lambda\_1^2 + \lambda\_2^2 + \lambda\_3^2}},\tag{3}$$

$$= \frac{\lambda\_1 + \lambda\_2 + \lambda\_3}{3}, \qquad \lambda\_i = \begin{pmatrix} \varepsilon\_{\text{max}} \text{ or } \ \varepsilon\_{\text{med}} \text{ or } \ \varepsilon\_{\text{min}} \end{pmatrix} + 1,$$

Sumi et al. [46] developed a method to obtain a relationship between the vector gradient of the Young modulus and the strain tensor components for the plane stress approach. This criterion was adapted by Le Floc'h et al. [28] under the plane strain assumption, and they developed the elastic gradient of the material (*dWsimpli fied*) by neglecting the shear strains and computing only with the radial strains; Equation (4). This variable was selected due to the good results when marking the lipid core contour shown in different studies [28,29].

$$dW\_{simplified} = -\frac{1}{\varepsilon\_{rr}} \left( \frac{\partial \varepsilon\_{rr}}{\partial r} + \frac{2\varepsilon\_{rr}}{r} \right) dr - \frac{1}{\varepsilon\_{rr}} \frac{\partial \varepsilon\_{rr}}{\partial \theta} d\theta,\tag{4}$$

In this work, the parameter *dW* was also calculated without any type of simplification, and it is developed in Equations (5)–(7). Furthermore, the absolute value of this parameter was computed for segmentation purposes, and it was represented as |*dW*|.

$$H\_{r} = \frac{-1}{\varepsilon\_{rr}^{2} + \varepsilon\_{r\theta}^{2}} \cdot \left[ \varepsilon\_{rr} \cdot \left( \frac{\partial \varepsilon\_{rr}}{\partial r} + \frac{1}{r} \cdot \frac{\partial \varepsilon\_{r\theta}}{\partial \theta} + \frac{2 \cdot \varepsilon\_{rr}}{r} \right) + \varepsilon\_{r\theta} \cdot \left( \frac{\partial \varepsilon\_{r\theta}}{\partial r} - \frac{1}{r} \cdot \frac{\partial \varepsilon\_{r}}{\partial \theta} + \frac{2 \cdot \varepsilon\_{r\theta}}{r} \right) \right], \tag{5}$$

$$H\_{\theta} = \frac{-1}{\varepsilon\_{rr}^{2} + \varepsilon\_{r\theta}^{2}} \cdot \left[ \varepsilon\_{r\theta} \cdot \left( \frac{\partial \varepsilon\_{rr}}{\partial r} + \frac{1}{r} \cdot \frac{\partial \varepsilon\_{r\theta}}{\partial \theta} + \frac{2 \cdot \varepsilon\_{rr}}{r} \right) - \varepsilon\_{rr} \cdot \left( \frac{\partial \varepsilon\_{r\theta}}{\partial r} - \frac{1}{r} \cdot \frac{\partial \varepsilon\_{rr}}{\partial \theta} + \frac{2 \cdot \varepsilon\_{r\theta}}{r} \right) \right], \tag{6}$$

$$d\mathcal{W} = \vec{H} \cdot d\vec{\mathbf{x}} = \left[ H\_{r} H\_{\theta} \right] \cdot \left[ \frac{dr}{r \cdot d\theta} \right], \tag{7}$$

 ·

> *r* ·

·

### *2.2. Adding Noise*

*λ*

The strain information was obtained from the FE models (clean strains). Nevertheless, the in vivo IVUS images have some noise and the speckle estimated strains will also contain that noise. Therefore, to reproduce more realistic strains we added white Gaussian noise to the different FE strain fields. We used an SNR of 20 dB in the FE strains [24] . However, the segmentation procedure was studied in all geometries with and without noise so as to analyze the robustness of the process.

### *2.3. Computing SGVs*

Once all of the strain variables were obtained (with and without noise), the next step was to obtain the modulus of their gradient. For instance, |*<sup>ε</sup>rr*| represents the modulus of the gradient of the radial strains. These SGVs allowed to highlight the contours of the different tissues of the plaque. Each SGV marked different parts of the tissue contours; this is why the segmentation procedure could use one or two combined SGVs to extract the entire lipid contour. We computed the modulus of the gradient of all strain variables, except for *dW*. By definition, *dW* showed the contours of the areas with different stiffness. The use of this single variable marked the tissue contours. At the end, there were 14 single SGVs and 91 possible combinations of two SGVs.

### *2.4. Segmentation Process*

The methodology was based on the representation of two combined SGVs or a single SGV and image segmentation algorithms. The W-GVF processes were imposed on the SGVs representation to extract the lipid core. The watershed process used the contour and the grayscale representation to treat a set of pixels as a topography separating the lipid core. The W-GVF algorithm allowed to segmen<sup>t</sup> different tissues as the lipid core

or the calcifications. In this work, only the results of the lipid core are presented due to their relevance to FCT measurements and plaque vulnerability. After the segmentation, the lipid was smoothed in order to reduce the sharp areas of the segmentation. The method was tested in all geometries (three idealized and three real geometries) with all of the 105 SGVs. A sensitivity analysis of different relevant variables in the segmentation process was performed. For this analysis only the idealized geometry with 150 μm of FCT was considered. These variables were related to the plaque morphology or related to the IVUS technology:


Although the methodology was mainly focused on the lipid core segmentation, different areas were segmented as well. Lumen was segmented using the W-GVF technique in each geometry in order to measure the FCT. Large calcifications, such as the one in the third real IVUS geometry, were segmented by using the same segmentation process. On the other hand, fibrotic tissue could be easily segmented as the difference of the whole plaque minus the segmented lipid and lumen. Finally, adventitia and media could be segmented with the W-GVF technique; however, this segmentation has no clinical application due to the fact that IVUS images provide little information on the outermost tissues.

### *2.5. Geometrical Measures*

After the lipid and the lumen were segmented, it was possible to assess the FCT as the minimum distance between them. The area of the lipid core was also computed. Both measurements are closely related to the risk of plaque rupture [5]. The indices *I*1, *I*2, and *I*3 were defined in order to quantify the accuracy of the segmentation for each SGV or combination of two SGVs. The first index (*I*1) in Equation (8) is the relative error between the real and the measured FCT (*treal* and *tmeasure*). The second index (*I*2) in Equation (9) defines the percentage of the lipid area that was correctly segmented (true positive area). This index could be represented as the white area in Figure 4. The third index (*I*3) in Equation (10) corresponds to the extra lipid area that was segmented (false positive area). It could be represented as the green area in Figure 4. The second and third indices were defined to quantify not only the lipid area value, but also the correct segmentation of its shape. In order to quantify the segmentation using only one index, we defined the Segmentation Index (SI) as a linear combination of the previous indices. The final SI parameter was defined in Equation (11) and its value was directly related to the performance of the segmentation. SGV combinations with an *SI* ≥ 90% provided measurements of the lipid area and the fibrous cap thickness with high precision. An SI between 90–85% meant that the segmentation had trouble with one measure, normally the fibrous cap thickness. SI values in the range of 85–75% indicated a poor lipid segmentation. Finally, values of *SI* ≤ 75% were for those SGVs with a high measurement error or those that could not segmen<sup>t</sup> the lipid. 

$$I\_1 = \left| \frac{t\_{\text{measurre}} - t\_{\text{real}}}{t\_{\text{real}}} \right| \cdot 100 \,, \tag{8}$$

$$I\_2 = \frac{Area\_{RealSegmented}}{Area\_{Real}} \cdot 100,\tag{9}$$

$$I\_3 = \frac{Area\_{FalesSegmented}}{Area\_{Real}} \cdot 100,\tag{10}$$

$$SI = \frac{(100 - I\_1) + I\_2 + (100 - I\_3)}{3},\tag{11}$$

**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.
