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Abstract

Arterial tissue is commonly assumed to be incompressible. While this assumption is convenient for both experimentalists and theorists, the compressibility of arterial tissue has not been rigorously investigated. In the current study we present an experimental-computational methodology to determine the compressibility of aortic tissue and we demonstrate that specimens excised from an ovine descending aorta are significantly compressible. Specimens are stretched in the radial direction in order to fully characterise the mechanical behaviour of the tissue ground matrix. Additionally biaxial testing is performed to fully characterise the anisotropic contribution of reinforcing fibres. Due to the complexity of the experimental tests, which entail non-uniform finite deformation of a non-linear anisotropic material, it is necessary to implement an inverse finite element analysis scheme to characterise the mechanical behaviour of the arterial tissue. Results reveal that ovine aortic tissue is highly compressible; an effective Poisson’s ratio of 0.44 is determined for the ground matrix component of the tissue. It is also demonstrated that correct characterisation of material compressibility has important implications for the calibration of anisotropic fibre properties using biaxial tests. Finally it is demonstrated that correct treatment of material compressibility has significant implications for the accurate prediction of the stress state in an artery under in vivo type loading.

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Acknowledgments

The authors wish to acknowledge funding from Science Foundation Ireland under project SFI-12/IP/1723. Furthermore we acknowledge funding from the Irish Research Council and the College of Engineering and Informatics at NUI, Galway. The authors wish to thank Noel Reynods and Prof. Michel Destrade for insightful discussions on this topic.

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Correspondence to J. P. McGarry.

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Associate Editor Estefanía Peña oversaw the review of this article.

Appendices

Appendix A: Investigation of Different Forms of \(\Psi _\text {iso}\)

In the "Compressible Anisotropic Constitutive Model" section the strain energy potential for the MA model is additively split into isotropic and anisotropic parts \(\Psi = \Psi _\text {iso} + \Psi _\text {aniso}\). The MA model uses a common neo–Hookean strain energy potential, \(\Psi _\text {iso} = \frac{\mu }{2}(\bar{I}_1 -3) + \frac{\kappa }{2}(J - 1)^2\). Here the use of two additional hyperelastic strain energy potentials for use as the isotropic term \(\Psi _\text {iso}\) are investigated. Firstly a modified version of the standard neo–Hookean model which was developed for scenarios involving large compressive stresses is examined. Its strain energy potential is given as:

$$\begin{aligned} \Psi _\text {iso} = \frac{\mu }{2}(\bar{I}_1 -3) + \frac{\kappa }{2}\left( \frac{J^2 - 1}{2} - \text {ln}(J)\right) . \end{aligned}$$
(A.1)

In this model the volumetric term has been modified to include the function \(\text {ln}(J)\). This has the effect of penalising very large volume changes and prevents \(J \rightarrow 0\).

Secondly a hyperfoam model38 developed for highly compressible hyperelastic polymer foams is examined. Its strain energy function is given as:

$$\begin{aligned} \Psi _\text {iso} = \sum _{i=1}^{N} \frac{2\mu _i}{\alpha _{i}^{2}} \left[ \lambda _1^{\alpha _{i}} + \lambda _2^{\alpha _{i}} + \lambda _3^{\alpha _{i}} - 3 + \frac{1}{\beta _i} (J^{- \alpha _{i} \beta _{i} }-1) \right] , \end{aligned}$$
(A.2)

where N is the order of the function, \(\lambda _m\) (\(m=1,2,3\)) are the principal stretches, \(\alpha _i\) determines the non-linearity for each term in the function, \(\beta _i\) determines the compressiblity for each term in the strain energy function and is related to the Poisson’s ratio \(\nu _i\) through the expression \(\beta _i = \nu _i / (1-2 \nu _i)\), and the initial bulk modulus \(\kappa _0 = \sum _{i=1}^{N} 2\mu _i [(1-3\beta _i)/3]\).

Equation (A.1) is added to the anisotropic potential of the MA model to form the MA_LNH model and Eq. (A.2) is added to the anisotropic potential of the MA model to form the MA_HYF model. The simulations from "Assessment of Error Generated by the Incompressibility Assumption" are repeated using the MA_LNH and MA_HYF models for the artery wall. The MA_LNH model uses the mean bulk and shear modulus from Table 1 and the MA_HYF model uses \(N=1\), \(\alpha _i=2\), \(\nu _i=0.44\), and \(\mu _i=7.04\) kPa.

The mean radial strain – pressure curves for both of these models, as well as that of the original MA model are given in Fig. A.1. Use of the MA_LNH model results in no difference in arterial compliance compared to the MA model. The MA_HYF model computes a small difference in compliance. For the hyperfoam model it is recommended that the effective Poisson’s ratio does not exceed 0.45.

Figure A.1
figure 9

Mean radial strain with increasing lumen pressure for (i) the MA model (Eq. (1)), (ii) the MA_LNH model which uses a modified version of the neo–Hookean model to represent the ground matrix; this version uses a logarithmic penalty term \(\text {ln}(J)\) to prevent excessive volumetric deformations, and iii) the MA_HYF model which uses a hyperfoam model to represent the ground matrix.

Appendix B: Calibration of Anisotropic Constitutive Models

Calibrations are performed for each of the six individual specimens tested using their unique force–displacement curve and the specimen thickness measured using the technique outlined in "Inverse FE Analysis for Calibration of Anisotropic Fibres" section. The results of these individual calibrations are presented in Table B.1.

Table B.1 Anisotropic material parameters for the MA model defined in Eq. (1) uniquely calibrated for each individual specimen, as well as the individual specimen thickness.

Appendix C: Quasi-3D Method for Volume Change Measurement

Volume change is calculated using an alternative method by measuring two orthogonal axes/diameters of the specimen at each segmentation plane. The volume is computed by integrating the resultant cross-sectional areas over the height of the specimen using a trapezoidal type method. A schematic outlining this method is given in Fig. C.1 This alternative methodology results in a volume change of 10.11 ± 4.61% (mean ± SD). A t test indicates that there is no statistically significant difference between the mean volume change calculated using the axi-symmetric or the quasi-3D method.

Figure C.1
figure 10

Quasi-3D calculation to determine the volume change in cylindrical specimens. Orthogonal experimental images measure the orthogonal diameters, 2a and 2b, of the specimen at each segmentation plane. The volume, V , is calculated by integrating cross-sectional areas over the height of the specimen.

Appendix D: Inclusion of Residual Stress

The influence of residual stresses on the stress-state in the vessel wall is assessed in this appendix. It is well established that residual stresses are present in the unpressurised vessel wall. Simulations of lumen inflation were performed, following the computational method outlined by Raghavan et al. 31 and using the experimental data of Vaishnav and Vossoughi42 to determine stress-free geometry. The vessel has a stress-free internal radius \(R_i=21.92\) mm, an external radius \(R_e=24.94\) mm, and an opening angle of \(80.22^{\circ }\). The material parameters for both the HGO and MA models are identical to those used in "Assessment of Error Generated by the Incompressibility Assumption" section.

Figure D.1 shows a contour plot of the residual von Mises stress in the vessel wall in the unpressurised configuration. The vessel moves from a state of compression on the inner face to tension on the external face. As the anisotropic component of both the HGO and MA models are inactive in compression, this explains the higher stress on the external face of the vessel. Figure D.2 shows a contour plot of the ratio of the axial stress to the circumferential stress (\(\sigma _{zz}/\sigma _{\theta \theta }\)) for the HGO and MA models. This figure further illustrates the dependence of the stress-state on the treatment of compressibility. Differences between the HGO and MA models are similar to those shown in Fig. 8 without residual stress. This illustrates that the important influence of compressibility on triaxial artery stress is significant regardless of the inclusion of residual stress.

Figure D.1
figure 11

Residual (von Mises) stress in the vessel wall under zero lumen pressure for (a) the incompressible HGO model, and (b) the compressible MA model.

Figure D.2
figure 12

Ratio of the axial stress to the circumferential stress (\(\sigma _{zz}/\sigma _{\theta \theta }\)) in a pressurized vessel for (a) the incompressible HGO model, and (b) the compressible MA model. Results are similar to those presented in Fig. 8

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Nolan, D.R., McGarry, J.P. On the Compressibility of Arterial Tissue. Ann Biomed Eng 44, 993–1007 (2016). https://doi.org/10.1007/s10439-015-1417-1

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