**1. Introduction**

Cardiovascular diseases are the leading cause of death worldwide, with 17.9 million deaths per year, which represent 31% of the demises [1]. The majority of the coronary events are related to heart attack or cerebral strokes, which are commonly trigged by atherosclerotic plaque rupture [2]. The atherosclerotic plaque is the result of lipid deposition in the artery wall, which creates a lipid core surrounded by fibrotic tissue. The fibrotic tissue that separates the lipid core from the lumen is called the fibrous cap [3]. The rupture of the fibrous cap induces a thrombus in the artery that obstructs the blood flow, leading to an acute coronary event [4]. The vulnerability of the plaque is related to the risk of fibrous cap rupture and the thrombus formation. There are some geometrical parameters that are important for the vulnerability characterization. Some studies have suggested that atherosclerotic plaques with fibrous cap thicknesses (FCT) thinner than 65 μm and lipid cores with a large area are vulnerable and prone to rupture [3,5]. On the other hand, a

**Citation:** 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

Academic Editor: Eva H. Dulf

Received: 26 September 2022 Accepted: 25 October 2022 Published: 29 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

large FCT usually indicates that the plaque is stable. However, the prediction of the plaque rupture is not only based on geometrical features, but also on the mechanical properties of the tissues [6,7]. Nowadays, intravascular ultrasound (IVUS) images are the gold standard for clinical diagnosis of atherosclerotic plaques in coronary arteries. IVUS images show a cross section of the artery wall in greyscale, and the segmentation usually depends on the cardiologist's experience. Each plaque tissue has different echo reflectivity characteristics, so its appearance within an IVUS image can be distinguished [8]. The segmentation can be performed manually; nevertheless, it requires clinical expertise, a high amount of time and, therefore, cost, and it depends on the image quality [9]. In order to solve this problem, new clinical techniques such as virtual histology intravascular ultrasound (VH-IVUS) have emerged. VH-IVUS is a clinical method for visualizing color-coded tissue maps, which provides an automated plaque characterization [10]. However, there are some limitations to this technique: first, it has a poor recognition of the FCT due to false detection of lipid core tissue and limitations in the plaque type classification (thin FCT, calcified plaque, or stable plaque) [11]. Second, only one computation can be performed per cardiac cycle, which reduces the number of IVUS frames used to characterize the plaque [12]. Third, the clinics have to be equipped with VH software.

That is why new techniques, mostly based on machine learning, have been developed to segmen<sup>t</sup> or characterize the atherosclerotic plaque tissues from IVUS images [9]. Methods based on Random Forest were used to classify IVUS image pixels into different tissues (dense calcium, necrotic, fibrotic tissue, and fibrofatty tissue) [12,13]. Although these strategies have achieved high classification accuracy (70–85%), the validation was performed with VH-IVUS and the results were unstable [13]. Other techniques, such as the the Neuro Fuzzy classifier, showed potential results in detecting fibrotic, lipidic and calcified tissues by classifying different pixels of the IVUS image [14]. Supporting vector machines have been used with IVUS and VH-IVUS images to classify the vulnerability of the plaques depending on the FCT (thin FCT vs. normal/stable FCT) [11,15] or to detect calcifications [16,17]. Recently, convolutional neural networks (CNN) have emerged strongly as a good classifier. CNN has also been used to classify plaque into thin or stable FCT [18], to detect calcifications in the IVUS frames [19,20], or to segmen<sup>t</sup> the lumen and outer contours [21,22]. Newer studies presented CNNs that detect different tissues of the atherosclerotic plaque with high accuracy [8,23]. However, the accuracy of the majority of the machine learning methods does not include the actual measure of the FCT, which plays a key role in the plaque vulnerability. Although some studies analyzed the FCT, they only used it as a classifier to characterize the plaque as thin or normal FCT [11,15,18]. The main limitation of these machine learning techniques arises from on the lack of a large public database to train and test the models [9]. This entails that, usually, each study proved the efficiency of their technique with less than 12 patients [8,11–13,15,20] and human plaque geometries vary greatly in each patient.

These machine learning methodologies give morphological information of the composition of the atherosclerotic plaque; however, the vulnerability also depends on the mechanical properties of the tissues. For this reason, another line of research focuses on segmentation by using mechanical properties such as strain or elasticity maps. For vulnerability characterizations, elastography is commonly used to obtain the elasticity map of the arterial wall [24–27]. Therefore, the main objective of many studies was to segmen<sup>t</sup> and characterize the mechanical properties of the different plaque tissues at the same time [25,28,29]. Different speckle estimators or optical flow methods can be used to track the pixels' motion or estimate the strains in IVUS images [27,30]. Then, the segmentation and mechanical property estimation procedures are linked and usually consist of an iterative optimization problem. Segmentation results depend on the number of inclusions evaluated at each iteration [28,29,31]. With this optimization process it is possible to estimate the mechanical properties of the arterial wall and, furthermore, the segmentation of the plaque geometry. This methodology allows to take measurements of the FCT, lipid area, and the stiffness of the tissues. These types of processes have been tested in silico with finite

element (FE) models [25,28,32], in vitro with polyvinyl acetate (PVA) phantoms [24], and in vivo with IVUS images from patients. The main disadvantages of these techniques are the high computational cost and the fact that the result depends on the number of inclusions evaluated.

Our work continues the study of the state-of-the-art of atherosclerotic plaque vulnerability by separating the segmentation procedure from the estimation of the mechanical properties. The main contribution of this paper is the definition of a new intuitive segmentation tool to segmen<sup>t</sup> the atherosclerotic plaque tissues without iterative or optimization steps, thus reducing computational costs. In addition, the method allows segmentation based on the representation of a large number of variables. By knowing the exact number of tissues, this technique opens the opportunity to obtain mechanical properties in future studies. This is a theoretical framework to lay the groundwork for future research; therefore, the methodology was developed and validated with in silico data. We have simulated the estimated strains that could be obtained from IVUS images with speckle estimators, with FE models, and adding some noise to the strain fields. We have defined this process as simulated IVUS data, as we are recreating the type of data that could be extracted from IVUS images. Our segmentation process is based on the representation of the modulus of the strain gradients and Watershed and Gradient Vector Flow (W-GVF) algorithms. The results are mainly focused on the lipid core segmentation, because of the importance of measuring the FCT and the lipid area for plaque vulnerability. This methodology was studied by using different strain variables in the segmentation process with different geometries. We have modeled three idealized geometries to analyze the FCT influence on the segmentation and three real IVUS patient geometries. In all of the analyzed cases, the proposed method was able to segmen<sup>t</sup> the lipid core and to measure the lipid area and FCT with enough accuracy.

### **2. Materials and Methods**

The structure of the methodology was divided into five steps and it is schematized in Figure 1. The first step was to simulate IVUS data by computing different FE models, and then the FE results were analyzed mimicking two consecutive pictures taken by an IVUS. In the second step, some noise was added to the FE strains to mimic the intrinsic noise of the IVUS images. After that, the different strain gradient variables (SGVs) were computed in order to use them for the lipid segmentation process. Finally, after the segmentation we analyzed the performance of the results.

**Figure 1.** Scheme of the five steps that define the methodology.

### *2.1. Simulating IVUS Data*
