Need for transverse strain data for fitting constitutive models of arterial tissue to uniaxial tests.

The study deals with the process of estimation of material parameters from uniaxial test data of arterial tissue and focuses on the role of transverse strains. Two fitting strategies are analyzed and their impact on the predictive and descriptive capabilities of the resulting model is evaluated. The standard fitting procedure (strategy A) based on longitudinal stress-strain curves is compared with the enhanced approach (strategy B) taking also the transverse strain test data into account. The study is performed on a large set of material data adopted from literature and for a variety of constitutive models developed for fibrous soft tissues. The standard procedure (A) ignoring the transverse strain test data is found rather hazardous, leading often to unrealistic predictions of the model exhibiting auxetic behaviour. In contrast, the alternative fitting method (B) ensures a realistic strain response of the model and is proved to be superior since it does not require any significant demands of computational effort or additional testing. The results presented in this paper show that even the artificial transverse strain data (i.e., not measured during testing but generated ex post based on assumed Poisson's ratio) are much less hazardous than total disregard of the transverse strain response.

Biaxial stretch can overcome discrepancy between global and local orientations of wavy collagen fibres.

Most frequently used structure-based constitutive models of arterial wall apply assumptions on two symmetric helical (and dispersed) fibre families which, however, are not well supported with histological findings where two collagen fibre families are seldom found. Moreover, bimodal distributions of fibre directions may originate also from their waviness combined with ignoring differences between local and global fibre orientations. In contrast, if the model parameters are identified without histological information on collagen fibre directions, the resulting mean angles of both fibre families are close to ±45°, which contradicts nearly all histologic findings. The presented study exploited automated polarized light microscopy for detection of collagen fibre directions in porcine aorta under different biaxial extensions and approximated the resulting histograms with unimodal and bimodal von Mises distributions. Their comparison showed dominantly circumferential orientation of collagen fibres. Their concentration parameter for unimodal distributions increased with circumferential load, no matter if acting uniaxially or equibiaxially. For bimodal distributions, the angle between both dominant fibre directions (chosen as measure of fibre alignment) decreased similarly for both uniaxial and equibiaxial loads. These results indicate the existence of a single family of wavy circumferential collagen fibres in all layers of the aortic wall. Bimodal distributions of fibre directions presented sometimes in literature may come rather from waviness of circumferentially arranged fibres than from two symmetric families of helical fibres. To obtain a final evidence, the fibre orientation should be analysed together with their waviness.

Poisson's ratio and compressibility of arterial wall - Improved experimental data reject auxetic behaviour.

Poisson's ratio of fibrous soft tissue is analyzed in this paper on the basis of direct experimental measurements of porcine arterial wall layer under uniaxial tension and immersed in tempered saline bath. The current study follows the previously published testing methodology but with a new totally redesigned testing apparatus allowing more credible and precise evaluation of arterial wall behaviour. The new results confirm most of previous findings focused on positivity/negativity of Poisson's ratio playing a crucial role in (in)validation aspects of some constitutive models widely used in recent computational vascular mechanics. The effect of frozen & thawed conditions is also evaluated in comparison with fresh specimens. The in-plane Poisson's ratio of arterial wall was identified in the range of 0.3-0.4, whereas its out-of-plane component is much higher ranging from 0.5 to 0.7. These results contrast with predictions of some frequently used constitutive models. The volumetric (in)compressibility of arterial specimens is also analyzed, quantified and discussed in the paper, as a key property of soft tissues closely related to the topic of their constitutive modelling.

Compressibility of arterial wall - Direct measurement and predictions of compressible constitutive models.

Volumetric compressibility and Poisson's ratios of fibrous soft tissues are analyzed in this paper on the basis of constitutive models and experimental data. The paper extends the previous work of Skacel and Bursa (J Mech Behav Biomed Mater, 54, pp. 316-327, 2016), dealing with incompressible behaviour of constitutive models, to the area of compressibility. Both recent approaches to structure-based constitutive modelling of anisotropic fibrous biomaterials (based on either generalized structure tensor or angular integration) are analyzed, including their compressibility-related aspects. New experimental data related to compressibility of porcine arterial layer are presented and compared with the theoretical predictions of analyzed constitutive models. The paper points out the drawbacks of recent models with distributed fibres orientation since none of the analyzed constitutive models seems to be capable to predict the experimentally observed Poisson's ratios and volume change satisfactory.

Predictive capabilities of various constitutive models for arterial tissue.

INTRODUCTION: Aim of this study is to validate some constitutive models by assessing their capabilities in describing and predicting uniaxial and biaxial behavior of porcine aortic tissue. METHODS: 14 samples from porcine aortas were used to perform 2 uniaxial and 5 biaxial tensile tests. Transversal strains were furthermore stored for uniaxial data. The experimental data were fitted by four constitutive models: Holzapfel-Gasser-Ogden model (HGO), model based on generalized structure tensor (GST), Four-Fiber-Family model (FFF) and Microfiber model. Fitting was performed to uniaxial and biaxial data sets separately and descriptive capabilities of the models were compared. Their predictive capabilities were assessed in two ways. Firstly each model was fitted to biaxial data and its accuracy (in term of R2 and NRMSE) in prediction of both uniaxial responses was evaluated. Then this procedure was performed conversely: each model was fitted to both uniaxial tests and its accuracy in prediction of 5 biaxial responses was observed. RESULTS: Descriptive capabilities of all models were excellent. In predicting uniaxial response from biaxial data, microfiber model was the most accurate while the other models showed also reasonable accuracy. Microfiber and FFF models were capable to reasonably predict biaxial responses from uniaxial data while HGO and GST models failed completely in this task. CONCLUSIONS: HGO and GST models are not capable to predict biaxial arterial wall behavior while FFF model is the most robust of the investigated constitutive models. Knowledge of transversal strains in uniaxial tests improves robustness of constitutive models.

Poisson׳s ratio of arterial wall - Inconsistency of constitutive models with experimental data.

Poisson׳s ratio of fibrous soft tissues is analyzed in this paper on the basis of constitutive models and experimental data. Three different up-to-date constitutive models accounting for the dispersion of fibre orientations are analyzed. Their predictions of the anisotropic Poisson׳s ratios are investigated under finite strain conditions together with the effects of specific orientation distribution functions and of other parameters. The applied constitutive models predict the tendency to lower (or even negative) out-of-plane Poisson׳s ratio. New experimental data of porcine arterial layer under uniaxial tension in orthogonal directions are also presented and compared with the theoretical predictions and other literature data. The results point out the typical features of recent constitutive models with fibres concentrated in circumferential-axial plane of arterial layers and their potential inconsistence with some experimental data. The volumetric (in)compressibility of arterial tissues is also discussed as an eventual and significant factor influencing this inconsistency.

Numerical implementation of constitutive model for arterial layers with distributed collagen fibre orientations.

Several constitutive models have been proposed for the description of mechanical behaviour of soft tissues containing collagen fibres. Some of the commonly used approaches accounting for the dispersion of fibre orientations are based on the summation of (mechanical) contributions of differently oriented fibre families. This leads to the need of numerical integration on the sphere surface, and the related numerical consumption is the main disadvantage of this category of constitutive models. The paper is focused on the comparison of various numerical integration methods applied to a specific constitutive model applicable for arterial walls. Robustness and efficiency of several integration rules were tested with respect to application in finite element (FE) codes. Among all the analysed numerical integration rules, the best results were reached by Lebedev quadrature; the related parameters for the specific constitutive model are presented in the paper. The results were implemented into the commercial FE code ANSYS via user subroutines, and their applicability was demonstrated by an example of FE simulation with non-homogenous stress field.

Comparison of constitutive models of arterial layers with distributed collagen fibre orientations.

Several constitutive models have been proposed for description of mechanical behaviour of soft tissues containing collagen fibres. The model with aligned fibres is modified in this paper to take the dispersion of fibre orientations into account through angular integration and it is compared with the model that is defined through generalized structure tensor. The paper is focused on the effect of fibre dispersion on the resulting stress-strain behaviour predicted by both models analyzed. Analytical calculations are used for the comparison of the mechanical behaviour under a specific biaxial tension mode. The two models have been implemented into commercial finite element code ANSYS via user subroutines and used for numerical simulation resulting in a non-homogeneous stress field. The effects of the fibre dispersion predicted by both models being compared differ significantly, e.g., the resulting stress difference between both models is lower than 10% only in the case of extremely small dispersion of collagen fibres orientation (κ< (0.01 to 0.03)). These results are consistent with those of other related literature. The applicability of the model defined through the generalized structure tensor is discussed.

Importance of material model in wall stress prediction in abdominal aortic aneurysms.

BACKGROUND: Results of biomechanical simulation of the abdominal aortic aneurysm (AAA) depend on the constitutive description of the wall. Based on in vitro and in vivo experimental data several constitutive models for the AAA wall have been proposed in the literature. Those models differ strongly from each other and their impact on the computed stress in biomechanical simulation is not clearly understood. METHODS: Finite element (FE) models of AAAs from 7 patients who underwent elective surgical repair were used to compute wall stresses. AAA geometry was reconstructed from CT angiography (CT-A) data and patient-specific (PS) constitutive descriptions of the wall were derived from planar biaxial testing of anterior wall tissue samples. In total 28 FE models were used, where the wall was described by either patient-specific or previously reported study-average properties. This data was derived from either uniaxial or biaxial in vitro testing. Computed wall stress fields were compared on node-by-node basis. RESULTS: Different constitutive models for the AAA wall cause significantly different predictions of wall stress. While study-average data from biaxial testing gives globally the same stress field as the patient-specific wall properties, the material model based on uniaxial test data overestimates the wall stress on average by 30 kPa or about 67% of the mean stress. A quasi-linear description based on the in vivo measured distensibility of the AAA wall leads to a completely altered stress field and overestimates the wall stress by about 75 kPa or about 167% of the mean stress. CONCLUSION: The present study demonstrated that the constitutive description of the wall is crucial for AAA wall stress prediction. Consequently, results obtained using different models should not be mutually compared unless different stress gradients across the wall are not taken into account. Highly nonlinear material models should be preferred when the response of AAA to increased blood pressure is investigated, while the quasi-linear model with high initial stiffness produces negligible stress gradients across the wall and thus, it is more appropriate when response to mean blood pressure is calculated.

Impact of poroelasticity of intraluminal thrombus on wall stress of abdominal aortic aneurysms.

BACKGROUND: The predictions of stress fields in Abdominal Aortic Aneurysm (AAA) depend on constitutive descriptions of the aneurysm wall and the Intra-luminal Thrombus (ILT). ILT is a porous diluted structure (biphasic solid-fluid material) and its impact on AAA biomechanics is controversially discussed in the literature. Specifically, pressure measurements showed that the ILT cannot protect the wall from the arterial pressure, while other (numerical and experimental) studies showed that at the same time it reduces the stress in the wall. METHOD: To explore this phenomenon further a poroelastic description of the ILT was integrated in Finite Element (FE) Models of the AAA. The AAA model was loaded by a pressure step and a cyclic pressure wave and their transition into wall tension was investigated. To this end ILT's permeability was varied within a microstructurally motivated range. RESULTS: The two-phase model verified that the ILT transmits the entire mean arterial pressure to the wall while, at the same time, it significantly reduces the stress in the wall. The predicted mean stress in the AAA wall was insensitive to the permeability of the ILT and coincided with the results of AAA models using a single-phase ILT description. CONCLUSION: At steady state, the biphasic ILT behaves like a single-phase material in an AAA model. Consequently, computational efficient FE single-phase models, as they have been exclusively used in the past, accurately predict the wall stress in AAA models.

Material parameter identification of arterial wall layers from homogenised stress-strain data.

Multilayer structure of the artery can have significant effects on the resulting mechanical behaviour of the artery wall. Separation of the artery into individual layers is sometimes performed to identify the layer-specific parameters of constitutive model proposed by Holzapfel, Gasser and Ogden (HGO model). Inspired by this single-layer model, a double-layer model was formulated and used for identification of material parameters from homogenised stress-strain data (of non-separated artery wall). The paper demonstrates that the layer-specific parameters of the double-layer constitutive model can be identified without the need of artery separation. The resulting double-layer model can credibly describe the homogenised stress-strain behaviour of the real artery wall including large-strain stiffening effects attributed to multilayer nature of the artery.