A modified Holzapfel-Ogden law for a residually stressed finite strain model of the human left ventricle in diastole

Abstract

In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging–derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, incompressible material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.

Computational modelling of left-ventricular diastolic mechanics: Effect of fibre orientation and right-ventricle topology

“…The second limitation of this study is the assumption of an initial stress-free state, which is present in all previous simulations of the heart based on in-vivo images. However, a recent study shows that the effects of residual stresses are relatively small in diastole (Wang et al, 2014). Finally the current model used simplified boundary conditions without accounting the basal movement of the ventricles.…”

Modelling of residually stressed materials with application to AAA

“…Using a numerical approach in an analysis of the elasticity of the human left ventricle in diastole, Wang et al (2014) adopted the Holzapfel and Ogden (2009) constitutive formulation modified to include residual stresses. Based on the analysis of Holzapfel and Ogden (2010), the effect of residual stresses was also incorporated into the finite element formulation of Pierce et al (2015), which was used to determine the wall stress distribution of patientspecific abdominal aortic aneurysms (AAAs).…”

Towards a unified approach in the modeling of fibrosis: A review with research perspectives

“…Moreover the neglecting of one invariant in a nonlinear treatment may induce wrong predictions or fail to recover an important feature. In the anisotropic material case, the state-of-the-art is very limited; the case of the heart [218,239] and arteries [220] have been studied in details. Few models have been proposed, and the most famous is the Hopzafel-Gasser-Ogden model [221].…”

“…The mechanical model can predict the loose of stability of the implant surface due to the growth in conjunction with active cells contained in the tissue as a function of the capsule thickness and the two shear modulus ratios: implant/capsule, implant/healthy tissue for a 3 layer combination. Such analysis [239] gives an estimation of the stresses, an evaluation of the growth arrest due to homeostasis. The calculation is based on the multiplicative decomposition gradient [240,241].…”