Consideration of stiffness of wall layers is decisive for patient-specific analysis of carotid artery with atheroma

RESEARCH ARTICLE Consideration of stiffness of wall layers is decisive for patient-specific analysis of carotid artery with atheroma Ondřej Lisicky ´ID 1☯ *, Aneta Mala ´ 2 , Zdeněk Bednařı ´k 3 , Toma ´s ˇNovotny ´ 4 , Jiřı ´Burs ˇa 1☯ 1Institute of Solid Mechanics, Mechatronics and Biomechanics, Brno University of Technology, Brno, Czech Republic, 2Institute of Scientific Instruments, The Czech Academy of Science, Brno, Czech Republic, 31st Department of Pathology, St. Anne’s University Hospital Brno and Faculty of Medicine, Masaryk University, Brno, Czech Republic, 42nd Department of Surgery, St. Anne’s University Hospital Brno and Faculty of Medicine, Masaryk University, Brno, Czech Republic ☯These authors contributed equally to this work. *[email protected] Abstract The paper deals with the impact of chosen geometric and material factors on maximal stresses in carotid atherosclerotic plaque calculated using patient-specific finite element models. These stresses are believed to be decisive for the plaque vulnerability but all applied models suffer from inaccuracy of input data, especially when obtained in vivo only. One hundred computational models based on ex vivo MRI are used to investigate the impact of wall thickness, MRI slice thickness, lipid core and fibrous tissue stiffness, and media anisotropy on the calculated peak plaque and peak cap stresses. The investigated factors are taken as continuous in the range based on published experimental results, only the impact of anisotropy is evaluated by comparison with a corresponding isotropic model. Design of Experiment concept is applied to assess the statistical significance of these investigated factors representing uncertainties in the input data of the model. The results show that consideration of realistic properties of arterial wall in the model is decisive for the stress evaluation; assignment of properties of fibrous tissue even to media and adventitia layers as done in some studies may induce up to eightfold overestimation of peak stress. The impact of MRI slice thickness may play a key role when local thin fibrous cap is present. Anisotropy of media layer is insignificant, and the stiffness of fibrous tissue and lipid core may become significant in some combinations. Introduction Atherosclerosis is a cardiovascular disease causing local intimal thickening of artery wall and plaque formation. Vulnerable atherosclerotic plaques, characterized by lipid accumulation under a thin fibrous cap (FC), have attracted attention of researchers for more than two decades [1,2]. A rupture of the plaque may cause blood clot formation and leads to the stroke in case of carotid arteries [3]. The rupture occurs when stresses induced by mechanical loading PLOS ONE PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 1 / 18 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Lisicky ´O, Mala ´A, Bednařı ´k Z, Novotny ´T, Burs ˇa J (2020) Consideration of stiffness of wall layers is decisive for patient-specific analysis of carotid artery with atheroma. PLoS ONE 15(9): e0239447. https://doi.org/10.1371/journal. pone.0239447 Editor: Fang-Bao Tian, University of New South Wales, AUSTRALIA Received: February 17, 2020 Accepted: September 7, 2020 Published: September 29, 2020 Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: https://doi.org/10.1371/journal.pone.0239447 Copyright: ©2020 Lisicky ´et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All supporting data (models, statisticas) are available on DOI: https:// doi.org/10.6084/m9.figshare.12937196. exceed the tissue strength. Thus, the peak cap stress (PCS) within the FC is believed to be the most valuable indicator of the plaque vulnerability [4–7]. Recent progress in imaging techniques and in computational modelling enables us to use patient-specific (PS) geometries for assessment of stresses. Finite element (FE) analysis [6,8] or fluid-structure interaction (FSI) [9–12] are mostly used to analyse the impact of key factors like geometrical parameters or mechanical characteristics of the plaque and arterial wall on the stresses. Like any diseased arterial wall, the thickened intima layer called fibrous tissue (FT) is very heterogeneous. Hence its mechanical properties vary locally within a plaque as well as among different plaques [7,13,14]. Therefore, analyses of plaque composition and experimental investigation of mechanical properties of its components are needed. In particular, little information is known about mechanical behaviour of lipid core (LC). Frequently its very soft behaviour is assumed in computational modelling [4,6,15], except for a study applying the other extreme of very high stiffness [16]. In contrast, mechanical behaviour of the vessel wall layers appears to be well supported by experiments. Uniaxial and biaxial mechanical tests of separated wall layers [8,17,18] enable us to describe its anisotropic behaviour even accounting for the wall structure. Simplified 2D models used earlier in FE analyses were found to overestimate the stresses [6, 10,19], therefore the research should focus on 3D models [20]. However, the level of these models varies. Simpler models with idealized geometry use to consider either homogeneous or layered arterial wall [4,21], although the omission of the wall is also presented frequently [7, 10,12]. Nowadays PS models are preferred but they suffer from lack of geometrical data and large dispersion of mechanical properties. While 2D histology sections or ex vivo MRI scans of autopsy samples enable us to reconstruct completely both the lumen and the vessel wall [4,6, 8], the MRI images taken either in vivo [9,12] or with samples from endarterectomy (where the plaque is resected without media and adventitia) do not enable us to capture wall boundaries. Consequently, the 3D models based on in vivo imaging give often contradictory results depending, for instance, on consideration of real mechanical properties of outer wall layers or FT stiffness [4,7] or quality of the geometrical data [6,22]. In this study, 3D PS FE models of carotid atherosclerotic artery for two patients accounting for both media and adventitia layers are used to investigate the significance of several factors in the calculation of extreme stresses in the atheroma; this should indicate their significance in further vulnerability diagnosis and enable us to propose a reasonable level of computational models. In total, five different factors related to mechanical properties or geometry of the atheroma are investigated; application of the design of experiment (DoE) concept, especially central composite design (CCD), resulted in one hundred solved computational models. Methods Acquiring data on geometry and structure Ethics statement. This study was approved by the medical ethical committee of St. Anne’s University Hospital in Brno (reference number 12V/2017). Written informed consent was obtained from the subject. Magnetic resonance imaging (MRI). Atherosclerotic plaques were harvested (sample 1: man, 61 years, sample 2: man 71 years) during endarterectomy in St. Anne’s University Hospital in Brno which met the following requirements: (1) the presence of at least one large LC with calcifications within, (2) undamaged and entire external surface of the FT and (3) an overall length which could be covered with a sufficient number of images using standard in vivo axial resolution. The samples were classified as an atherosclerotic lesion of Type VII [23]. PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 2 / 18 Funding: This work was supported by Czech Science Foundation (https://gacr.cz/en/) project No. 18-13663S. The MRI service was funded by project LM2015062 "National Infrastructure for Biological and Medical Imaging (CzechBioImaging)" of the Ministry of Education, Youth and Sports of the Czech Republic (http://www. msmt.cz/?lang=2). Competing interests: No authors have competing interests. Abbreviations: 2D, two-dimensional; 3D, threedimensional; CCD, central composite design; CT, computed tomography; DoE, design of experiment; FC, fibrous cap; FE, finite element; FSI, fluid structure interaction; FT, fibrous tissue; LC, lipid core; MB, media behaviour; MRI, magnetic resonance imaging; PCS, peak cap stress; PPS, peak plaque stress; PS, patient specific; SEDF, strain-energy density function; ST, slice thickness; WT, wall thickness. Post-operative imaging of the samples was performed with a 9.4 T MRI system (Bruker BioSpec 94/30 USR, Ettlingen, Germany). The images were acquired in two series: the first one with the RARE technique (repetition time TR = 2500 ms, effective echo time TEeff = 10.77 ms), the second one with IR-RARE (TR = 3000 ms, TEeff = 6.66 ms, inversion time TI = 950 ms). The slice thickness (ST) was set to 1.5 mm, 1 mm and 0.25 mm for the purpose of comparison. An in-plane resolution of 78 μm was constant for all measurements with the matrix size of 256x256 (see Fig 1). Histology. After MRI the samples were fixed in 10% neutral formalin for 24 hours and then subjected to decalcification, dehydrated and embedded in paraffin. Subsequently, 3–5 μm thick tissue slices were cut off and stained with hematoxylin-eosin. Then the slices were scanned and compared with the individual MRI images to distinguish the tissue components properly (see Fig 1). However, they could not be directly used for model segmentation due to their unavoidable deformation after the histological processing [22]. 3D PS model of carotid artery with atheroma Semi-automatic segmentation of the recorded MR images was applied using RETOMO (BETA CAE Systems) to distinguish the plaque components. We focused on segmentation of the FT and LC (including the regions of calcification). As biological structures are characterized by smooth surfaces [8], the geometry was smoothed [24] with the same parameters for all the models. Since the individual vessel wall layers cannot be distinguished in the in vivo MRI scans and the sample harvested at endarterectomy cannot include the media and adventitia layers of the wall, the outer surface of the obtained geometry was used to create the volume of the missing layers. For this purpose, the outer surface of the specimen was offset equidistantly by 0.5 mm [11,21] to create the outer surface of the artery wall. Although it is known the media may be degraded and non-uniformly thick under the plaque, the assumption of constant wall thickness (WT) was adopted here as an appropriate simplification and the thickness Fig 1. Example of carotid plaque bifurcation. From the histology cross-section (A) and the high-resolution MRI 9.4 T with ST = 0.25 mm (B). https://doi.org/10.1371/journal.pone.0239447.g001 PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 3 / 18 was included into DoE as one of the investigated parameters. The reconstructed model is depicted in Fig 2. Material models Media was modelled as a fibre-reinforced anisotropic layer, composed of a non-collagenous matrix and two families of collagen fibres. Based on mean material response [25], its anisotropic behaviour was characterized by the following strain-energy density function (SEDF) according to [26]: C¼Ciso þCaniso ¼m 2þk1 2k2 ek2ð1rÞðI13Þ2þrðI41Þ21 � � ð1Þ where μ>0 is a stress-like parameter describing the isotropic response of the tissue, k 1 >0 is a stress-like parameter related to the collagen fibre stiffness, the dimensionless parameter k 2 >0 refers to the level of fibre strain stiffening,I1¼l2 rþl2 yþl2 zand I4¼l2 ycos2φþl2 zsin2φare invariants of the right Cauchy-Green deformation tensor C(I 4 related to stretches of two fibre families), and ρ<0;1>is a concentration parameter representing a "degree of anisotropy". In the applied model, + or -φdenotes the angle between the circumferential direction and direction of each fibre family located symmetrically in the tangential plane of the tube. As the applied FE software Ansys does not offer this material option, the model was implemented via a user material subroutine. Fig 2. Illustration of the 3D PS model of the atherosclerotic carotid plaque (sample 1) obtained from endarterectomy. Carotid wall (red) shares the surface with FT. https://doi.org/10.1371/journal.pone.0239447.g002 PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 4 / 18 The other wall layers and plaque components were modelled as incompressible, hyperelastic and isotropic since their anisotropy is low [17] compared to the inter-patient variability; this simplification is generally accepted [27]. To fit the biaxial response of the adventitia (and also of the media when modelled as isotropic for comparison), 3 rd order Yeoh type SEDF [28] was used in the following form: C¼X 3 i¼0 ci0ðI13Þið2Þ where c i0 are material parameters. Material responses of both layers were taken from experimental biaxial testing of each carotid component [17]. Material parameters for the wall layers were identified with considering residual stresses and stretches observed in the load-free geometry [25]. Uniaxial test data of FT [7,13,14] were fitted with the 2 nd order Yeoh SEDF while the LC was modelled with Neo-Hookean (i.e. 1 st order Yeoh) constitutive model. Material responses can be seen in Fig 3 with the used material parameters presented in Table 1. FE model setup and boundary conditions FE analysis was performed in FE software ANSYS 19.2. (Ansys Inc., PA, USA). The vessel wall was meshed with four/eight elements across the WT using linear 8 node hexahedral elements (SOLID185) in ICEM 19.2. (Ansys Inc. PA, USA) see Fig 5. A linear tetrahedral element (SOLID285) mesh performed in ANSA (Beta CAE) was used for the FT and LC. A bonded contact was used to join the FT with the vessel wall, while the other components were connected by shared nodes. A constant pressure value of 13 kPa (mean arterial pressure) was applied on the luminal surface of the geometry. Free ends of the vessel wall were fully Fig 3. Stress-stretch equi-biaxial responses of the chosen constitutive models. The dotted lines show the isotropic fit of the circumferential (solid) and axial (dash dotted) anisotropic responses. The soft (dashed) and the hard (solid) isotropic responses for the FT and LC are also presented for comparison. https://doi.org/10.1371/journal.pone.0239447.g003 PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 5 / 18 constrained. A mesh convergence analysis was performed to obtain a suitable element size for all models. A typical mesh density resulted in 24k elements for the vessel wall and 95 k for FT with LC. Material orientation. Consideration of media layer anisotropy drew the problem of the orientation of principal material axes in each point of the model. Each element coordinate system needs to be rotated to the preferred orientation of the collagen fibres. In this study, we take advantage of the applied locally structured hexahedral mesh which enables us to define a centroid of each section and subsequently also the central line (see Fig 4). Two points of this Table 1. Material parameters. Tissue Type Material constants a Media A μ= 122.3 k 1 = 24.7 k 2 = 16.5 [-] φ= 6.9 [˚] ρ= 0.8 [-] I c 10 = 122.3 c 20 = 0 c 30 = 337.7 Adventitia I c 10 = 88.7 c 20 = 0 c 30 = 45301.4 Lipid core I soft c 10 = 0.1 I hard c 10 = 50 Fibrous tissue I soft c 10 = 2.7 c 20 = 20 I hard c 10 = 342.1 c 20 = 20 a The above values are in kPa if not otherwise specified. Symbols I and A denote the isotropic and anisotropic behaviour, respectively. https://doi.org/10.1371/journal.pone.0239447.t001 Fig 4. Computational model with principal material directions. (A) The discretized model of the vessel wall of sample 1 with hexahedral elements; the central line (white dashed) serves for definition of the axial base e z . (B) Detailed illustration of the coordinate system of an individual element. (C) Detail of element-specific material directions within the vessel wall model; the blue, green and red arrows indicate the circumferential, axial and radial directions, respectively. https://doi.org/10.1371/journal.pone.0239447.g004 PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 6 / 18 line close to the section were used to define the local axial basis e z . A vector normal to the inner surface of the element located in its centroid was used as the local radial basis e r . After that, a cross product of those two bases resulted in the circumferential basis e θ . The resulting distribution of bases determinates the principal material directions when the anisotropic constitutive model is applied. The investigated factors and their statistical analysis In biomechanics, all the input data for computational modelling suffer from large uncertainties or inaccuracies. This variance of input data may influence the character of the deformation and consequently, the stresses evaluated as a potential indicator of plaque vulnerability. Similarly to another biomechanical study [29], DoE with 95% confidence level was applied for this statistical analysis; specifically the CCD was chosen, to analyse (using Minitab 15 statistical software) the possible non-linear dependence of PCS on some of the five selected factors. This resulted in overall 100 computational models for the two patients, differing in their material and geometrical factors; these models afforded the input data for statistical analysis. In the DoE methodology, a 2-factorial design was performed and thereafter augmented with a set of axial points resulting in the CCD, specifically the face-centred design. Here, the middle level of each factor is used (centre of a factorial face) enabling to estimate a possible curvature of the response. The model suitability was checked for all the fitted responses via normal distribution of residuals and a sufficient coefficient of determination (R 2 >0.8). The first (maximal) principal stresses were adopted as stress response indicators, especially PPS and PCS [4,6]; for all the 3D models analysed, they were evaluated in cross-sections with a 0.2 mm span. Lastly, the analysis of variance was performed within the DoE to find the significance of the individual factors and of their combinations. Except for media behaviour (MB), all the factors were set as continuous. A specific choice of factors can be found below with their range of values summarized in Table 2. Lipid core stiffness. The LC plays a key role in the plaque vulnerability through its size or location in the plaque where the resulting distance from the lumen to the LC is characterized as FC thickness [7,21]. However, the LC composition almost disables its mechanical testing. Thus, the stiffness of the LC was chosen as a variable factor ranging from 0.1 to 50 kPa. The lower value (related to lipids) is mostly used in computational models while the upper value was chosen to reflect the hypothesis that micro-calcifications occurring in the extracellular matrix of the LC may cause its stiffening by orders. Fibrous tissue stiffness. Experimental data of carotid plaques [7,13,14] were analysed to find samples with extreme stiffness which were then fitted with the second-order Yeoh SEDF (see Fig 3). Nearly linear stress-strain response, i.e. a very low strain stiffening, was found for both the most and the least stiff plaques, thus the number of variables could be reduced. The second material parameter c 20 (related to strain stiffening) was fixed and only the first constant c 10 (initial stiffness) was fitted giving a sufficient quality of the fit (R 2 >0.8). Table 2. DoE factors. Selected factors for DoE with their chosen upper and lower limits. Factor Minimum Maximum Wall thickness [mm] 0 0.5 Slice thickness [mm] 0.25 1.5 Lipid core μ[kPa] 0.1 50 Fibrous tissue c 10 [kPa] 2.7 342.1 Media behaviour isotropic anisotropic https://doi.org/10.1371/journal.pone.0239447.t002 PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 7 / 18 Media behaviour. The biaxial response of carotid media [25] was used to fit both anisotropic and isotropic constitutive models (see Fig 3). The anisotropic description was applied with the locally varying orientation of the element coordinate system. Slice thickness. In vivo and ex vivo imaging methods differ substantially in their resolution (both axial and in-plane). Histological slices are often used for reconstruction of 2D and 3D PS plaque geometries [4,6] since they provide a suitable in-plane resolution. However, it is very laborious to achieve a sufficient axial resolution and the deformation caused by slice preparation [20,22] may result in distortion of the shape. Therefore, ex vivo MRI was used for the 3D reconstruction in this study. As only in vivo imaging has potential in clinical assessment of plaque vulnerability, the impact of different resolutions of the MR images was chosen as another investigated factor, since the lower axial resolution of the in vivo MRI may cause loss of geometrical information significant for the computational analysis [6]. This factor was set as continuous with values ranging from 0.25 (resolution of the applied ex vivo MRI) to 1.5 mm (a typical resolution of in vivo MRI). Vessel wall thickness. The vessel WT was set from 0 to 0.5 mm [11,21] where the lower limit represents models with identical mechanical properties for the wall and the FT [7,12, 19], neglecting thus the impact of different properties of the outer arterial layers. This means the volume of the wall was always included in the model but with different material properties. In all the models the thickness of the wall was halved between media and adventitia. Fig 5 shows the used configurations where the vessel wall is represented with 4 elements per thickness even when the half thickness t = 0.25 mm (axial point) is considered (see Fig 5). Since the CCD requires only continuous variables, the discrete MA factor was fixed and the DoE was performed twice with four continuous factors. The DoE results were compared using a paired t-test to investigate the significance of the MB factor. Results The factors significant for the statistical model are summarized in the Table 3. An example of the numerical results of the PS model of the atherosclerotic carotid artery is presented in the form of displacements and first principal stresses for sample 1 in Fig 6. CrossFig 5. Cross-sections representing the model configurations. Different WT of 0.5 mm (A), 0.25 mm (B) and 0 mm (C). The adventitia (dark blue) is always represented with 2 elements, as well as the media (light green). FT (light blue) increases its volume with decreasing WT. LC representation (pink) is immutable in the same axial position and with the same ST. Details of the wall are presented in the white lumen area. https://doi.org/10.1371/journal.pone.0239447.g005 PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 8 / 18 sections in the selected locations show typical stress distributions focused on the PPS and PCS. Results near the boundary conditions and the bifurcation were not considered since they may cause non-realistic artefacts (stress concentrations due to boundary conditions). The discrete media-related factor MB was found to be insignificant (probability value p = 0.99) with stresses showing a mean difference of 1 kPa between the models with isotropic and anisotropic media for both patients. Therefore, this factor was not included in the evaluations below. Peak plaque stress The statistical model fitted to the PPS response showed the same significance of the WT for both samples. Unlike in sample 1, ST was found significant in sample 2. Also, the interaction between FT stiffness and WT (FT�WT) was found significant in both cases. The squared effect, evaluated by the additional face-centred axial points, was significant for sample 1 in case of WT factor. The maximum was mostly localized in the narrow part of the FT component between the luminal surface and the inner surface of the vessel wall, except for the cases with lower stress concentration where the maximum was located within the FC (see detail D in Fig 6). The cube plot in Fig 7 shows the PPS responses at low and high levels of each factor. In the model with soft FT, the PPS increased on average up to five times (from 29 to 160 kPa for sample 1 and 44 to 230 kPa for sample 2) when the wall was not included. In contrast, an opposite tendency occurred when the hard FT was considered; the PPS decreased by some 30% when the wall was not included. The highest stress concentrations (168 kPa and 313 kPa) were found in the models without the vessel wall and with the soft atheroma components, while the lowest stresses (22 kPa and 26 kPa) were found in the models with the artery wall included and with the soft FT and hard LC. The results show that similarly to the MB factor, neither LC stiffness nor FT stiffness itself has a significant impact on the PPS minimum/maximum as long as interactions are not considered. For all the responses with the artery wall included, however, Fig 7 shows 3 up to 5 Table 3. Analysis of variance for the PPS and PCS. Sample 1 Peak Plaque Stress Linear Square 2-Way interaction Model R 2 Factor WT FT WT�WT WT�FT 0.894 p-value <0.001 0.48 0.08 <0.001 Peak Cap Stress Linear Square 2-Way interaction Model R 2 Factor WT LC FT LC�LC WT�LC WT�FT LC�FT 0.845 p-value <0.001 0.01 0.117 0.124 0.072 <0.001 0.038 Sample 2 Peak Plaque Stress Linear 2-Way interaction Model R 2 Factor ST WT LC FT WT�FT LC�FT 0.874 p-value 0.025 0.001 0.113 0.115 <0.001 0.146 Peak Cap Stress Linear Square 2-Way interaction Model R 2 Factor ST WT LC FT LC�LC WT�LC WT�FT 0.823 p-value 0.026 <0.001 <0.001 0.015 0.018 0.122 <0.001 The table shows the statistical significance of factors included in the statistical model represented by their p-values and R 2 . The statistically significant or nearly significant factors and their combinations are highlighted in bold. https://doi.org/10.1371/journal.pone.0239447.t003 PLOS ONE Stiffness of wall layers is decisive for the atheroma stresses PLOS ONE | https://doi.org/10.1371/journal.pone.0239447 September 29, 2020 9 / 18 (if being non-zero) are less significant for maximum stresses in the fibrous cap; this fact appears encouraging for further studies based on in vivo MRI. Supporting information S1 Appendix. Verification of Ansys user subroutine for media material model. (DOCX) S1 Fig. (TIF) S2 Fig. Geometry model of patient 2. Section of sample 2 model with two large lipid cores. (TIF) S1 File. Ethical agreement. (PDF) Acknowledgments We thank to Pavel Ska ´cel who implemented the constitutive model needed for our study into the ANSYS software. Author Contributions Conceptualization: Ondřej Lisicky ´, Jiřı ´Burs ˇa. Formal analysis: Jiřı ´Burs ˇa. Funding acquisition: Jiřı ´Burs ˇa. 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