**3. Results**

This section presents the results of the lipid segmentation. It is divided into the results in idealized geometries, results in real IVUS geometries, and an analysis of the best SGVs after considering all of the geometries.The results were analyzed with the FE strains and the strains with 20 dB of SNR.

### *3.1. Idealized Geometries*

The process is summarized in Figure 5, where the idealized geometries with 65, 150, and 300 μm of fibrous cap thickness are represented. A combination of the SGVs |*<sup>ε</sup>min*| and |*FA*| was chosen due to its high segmentation performance. Both SGV variables are shown in each idealized geometry in Figure 5a,b, respectively. The combination of both is represented in Figure 5c and it was used as an input for the W-GVF process. The final result is shown in Figure 5d, where every segmented part has a different color. The segmented lipid core is represented before and after the smooth treatment in Figure 5e, and it is shown as an overlap between the actual and segmented lipid core. In this representation, the white area displays the well-segmented area, the purple one is the actual lipid that is not segmented, and inversely, the green area is the extra area wrongly segmented by the procedure.

In the idealized geometries, the lipid area was similar in all scenarios, so the number of successful SGV combinations depended only on the fibrous cap thickness. The box-plot in Figure 6 represents the SI distribution of the 105 SGV combinations in each geometry. The geometries were segmented using the FE strains (clean strains) and the strains with 20 dB of noise. By analyzing only the influence of the FCT, the results show that the mean (represented by an asterisk) and the median of the SI value increased with the FCT. In addition, the interquartile range decreased with greater thicknesses. This fact can be observed in Figure 6 for the segmentation with or without noise. On the other hand, the noise addition led to an increase of the outliers and the interquartile range. The mean and median decreased after considering the noise. Despite having different SI results in each geometry, there were some combinations of SGVs with proper results for all thicknesses. This was the case of *<sup>ε</sup>yy* + |*εθθ* |, |*<sup>ε</sup>vMises*| + |*<sup>ε</sup>r<sup>θ</sup>* |, |*FA*| + |*εθθ* | or |*<sup>ε</sup>max*| + |*<sup>ε</sup>rr*|. It was also possible to yield good results with only one variable such as *<sup>ε</sup>yy*, |*FA*|, *dW*, |*<sup>ε</sup>Tresca*| and *<sup>ε</sup>xy*. However, there were variables with low SI values for all cases, such as *dWsimpli fied*, |*<sup>ε</sup>r<sup>θ</sup>* |, so they were discarded.

**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 μm of FCT. The segmentation process consists of the combination of two SGVs, in these cases |*<sup>ε</sup>min*| and |*FA*| in (**<sup>a</sup>**,**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 μm 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 μm FCT with and without noise; 94.88% and 93.42% for geometry of 150 μm FCT; and 95.23% and 94.17% for 300 μm. Mean values are represented with asterisks. Outliers are represented with circles. Some outliers were below 65% but are not shown.

### *3.2. Real IVUS Geometries*

The proposed methodology was tested with the three real IVUS plaque geometries. The lipid cores had different shapes, areas (1.65, 5.53, and 1.93 mm2), and FCTs (330, 175, and 209 μm). Figure 7 is an example of lipid segmentation with the SGV combination of the invariants |*<sup>ε</sup>min*| and |*FA*| for the case of clean strains. The performance of the segmentation is represented in the box-plot shown in Figure 8, where the segmentation was not only affected by the FCT, but also by the lipid core area. In all cases, the noise addition increased the interquartile range and decreased the mean and the median (except in the first plaque, where the median increased after the 20 dB). A single SGV such as |*<sup>ε</sup>min*| , |*dW*|, |*<sup>ε</sup>rr*|, *dW*, |*<sup>ε</sup>Tresca*| or combinations such as *<sup>ε</sup>yy* + |*<sup>ε</sup>rr*|, *<sup>ε</sup>yy* + |*<sup>ε</sup>min*|, or |*<sup>ε</sup>min*| + |*<sup>ε</sup>Mises*| still had promising results for these geometries. As what happened in the idealized cases, the SGV |*<sup>ε</sup>r<sup>θ</sup>* | did not show any adequate SI for any plaque.

**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 |*<sup>ε</sup>min*|; (**b**) representation of |*FA*|; (**c**) the combination of |*<sup>ε</sup>min*|+|*FA*|; (**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.

### *3.3. SGV Candidates*

Finally, the mean SI value (*SI*¯ ) was computed by considering all of the geometries together. All of the SGVs from FE strains, alone or in combination with others, were analyzed, and the best five single SGVs and fifteen SGV combinations were collected in Table 2. This table shows the SGVs or SGV combinations with better performance in the segmentation process, allowing to have good accuracy not only in the area, but also in the FCT. The mean SI value of those combinations with 20 dB of noise was included (*SI* ¯*noise*) to visualize the noise influence. Single SGVs, such as *dW* or |*<sup>ε</sup>rr*|, presented good segmentation results. Furthermore, |*<sup>ε</sup>vMises*| or |*<sup>ε</sup>min*| had good performance as well, and in these cases they had the advantage of being invariants, so they will be not affected by the catheter position or coordinate system. However, the invariants have the disadvantage of needing the entire strain tensor. On the other hand, there were a grea<sup>t</sup> number of SGV combinations with high segmentation accuracy regardless of the analyzed case, such as |*<sup>ε</sup>vMises*|+|*<sup>ε</sup>r<sup>θ</sup>* |, or *<sup>ε</sup>yy* <sup>+</sup>|*<sup>ε</sup>rr*|. Additionally, combination of invariants appeared to have good results, such as |*<sup>ε</sup>min*|+|*<sup>ε</sup>Tresca*|. Table 2 shows that SGV combinations achieved higher SI values than single SGVs.


**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. *SI* ¯ and *SInoise* ¯ represent the mean SI value without and with noise, respectively.

The elastic gradient of the material (*dW*) was calculated in a simplified way, following Le Floc'h et al. [29] and with the whole 2D strain tensor. In order to have the lipid contour marked and to achieve a better combination with other SGVs, the absolute value |*dW*| was computed. These variables are shown in Figure 9. The variable *dWSimpli fied* had a mean SI value of 32.13% in the six geometries without noise and 28.48% with noise, whereas *dW* without simplifications achieved a mean SI value of 94.70% without noise and 90.62% with 20 dB of noise and |*dW*| 94.63% and 86.04% without and with noise, respectively.

**Figure 9.** Idealized geometry with 150 μm thickness with the different *dW* represented without noise. (**a**) *dWsimpli fied*, (**b**) *dW* computed with the 2D strain tensor, and (**c**) |*dW*|.

### *3.4. Sensitivity Analysis*

Once the SGV candidates were analyzed, we selected the best four SGV combinations and the best single SGV of Table 2 to evaluate the robustness of the proposed segmentation methodology. For this purpose, we studied the influence of plaque and IVUS variables in the segmentation results only within the idealized geometry with 150 μm of FCT. Figure 10 shows all of the results of the sensitivity study. Each SGV had seven variables to analyze (incompressibility, different material behavior of fibrotic tissues, addition of inclusions, different catheter positions, total pressures and pressure increments, and noise addition), and all are color-coded in Figure 10. The SI values obtained with the FE strains were taken as a reference. In this reference case, the model was analyzed as fully incompressible, with a stiffer tissue, the catheter placed in the center of the lumen, with a pressure analysis of 110–115 mmHg and without noise. The SI values achieved with that model are represented in Figure 10 with a horizontal dotted line.

**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 |*<sup>ε</sup>Mises*|+|*<sup>ε</sup>r<sup>θ</sup>* | ,*<sup>ε</sup>yy*+|*<sup>ε</sup>rr*| , *<sup>ε</sup>yy*+|*<sup>ε</sup>min*| , |*<sup>ε</sup>min*|+|*<sup>ε</sup>Tresca*|, 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.
