Constituent-based quasi-linear viscoelasticity: A revised quasi-linear modelling framework to capture non-linear viscoelasticity in arteries

Abstract

Arteries exhibit fully non-linear viscoelastic behaviours (i.e., both elastically and viscously non-linear). While elastically non-linear arterial models are well established, effective mathematical descriptions of non-linear viscoelasticity are lacking. Quasi-linear viscoelasticity (QLV) offers a convenient way to mathematically describe viscoelasticity, but its viscous linearity assumption is unsuitable for whole-wall vascular applications. Conversely, application of fully non-linear viscoelastic models, involving deformation-dependent viscous parameters, to experimental data is impractical and often reduces to identifying specific solutions for each tested loading condition. The present study aims to address this limitation: By applying QLV theory at the wall constituent rather than at the whole-wall level, the deformation-dependent relative contribution of the constituents allows to capture non-linear viscoelasticity with a unique set of deformation-independent model parameters. Five murine common carotid arteries were subjected to a protocol of quasi-static and harmonic, pseudo-physiological biaxial loading conditions to characterise their viscoelastic behaviour. The arterial wall was modelled as a constrained mixture of an isotropic elastin matrix and four families of collagen fibres. Constituent-based QLV was implemented by assigning different relaxation functions to collagen- and elastin-borne parts of the wall stress. Non-linearity in viscoelasticity was assessed via the pressure-dependency of the dynamic-to-quasi-static stiffness ratio. The experimentally measured ratio increased with pressure, from 1.03 ± 0.03 (mean ± standard deviation) at 80–40 mmHg to 1.58 ± 0.22 at 160–120 mmHg. Constituent-based QLV captured well this trend by attributing the wall viscosity predominantly to collagen fibres, whose recruitment starts at physiological pressures. In conclusion, constituent-based QLV offers a practical and effective solution to model arterial viscoelasticity.