Abstract
PurposeResidual stress tensor has an essential influence on the mechanical behaviour of soft tissues and can be particularly useful in evaluating growth and remodelling of the heart and arteries. It is currently unclear if one single radial cut using the opening angle method can accurately estimate the residual stress. In many previous models, it has been assumed that a single radial cut can release the residual stress in a ring of the artery or left ventricle. However, experiments by Omens et al. (Biomech Model Mechanobiol 1:267–277, 2003) on mouse hearts, have shown that this is not the case. The aim of this paper is to answer this question using a multiple-cut mathematical model.MethodsIn this work, we have developed models of multiple cuts to estimate the residual stress in the left ventricle and compared with the one-cut model. Both two and four-cut models are considered. Given that the collagen fibres are normally coiled in the absence of loading, we use the isotropic part of the Holzapfel-Ogden strain energy function to model the unloaded myocardium.ResultsThe estimated residual hoop stress from our multiple-cut model is around 8 to 9 times greater than that of a single-cut model. Although in principle infinite cuts are required to release the residual stress, we find four cuts seem to be sufficient as the model agrees well with experimental measurements of the myocardial thickness. Indeed, even the two-cut model already gives a reasonable estimate of the maximum residual hoop stress. We show that the results are not significantly different using homogeneous or heterogeneous material models. Finally, we explain that the multiple cuts approach also applies to arteries.ConclusionWe conclude that both radial and circumferential cuts are required to release the residual stress in the left ventricle; using multiple radial cuts alone is not sufficient. A multiple-cut model gives a marked increase of residual stress in a left ventricle ring compared to that of the commonly used single-cut model.
Highlights
Living tissues in the heart continuously interact with their bio-environment, reshape and rearrange their constituents under chemical, mechanical or genetic stimuli during their life cycles
From B2 in the one-cut model, we assume that the axial stretch has the constant value kðz3Þ 1⁄4 1:14, and RðiÞ 1⁄4 2:06, RðoÞ 1⁄4 3:20 a 1⁄4 65 are estimated from the experiments.[21]
We assume that kðZ1nÞ 1⁄4 kðZ2nÞ 1⁄4 1 for the deformation from B0 to B1, i.e. no axial deformation occurs as a result of the circumferential cuts
Summary
Living tissues in the heart continuously interact with their bio-environment, reshape and rearrange their constituents under chemical, mechanical or genetic stimuli during their life cycles. The tissues of a healthy heart remain in a homeostatic state. Heart diseases disrupt this balance, and cause the tissues to grow and remodel. An important ingredient in evaluating the mechanics involved in the cardiovascular system is knowledge of the solid mechanical properties of the soft tissues involved, including the components of the heart, such as the left ventricle, abbreviated as LV. A particular aspect is that the tissues of the heart are residually stressed, so that when the external loading is removed, residual stresses remain in the material. Residual stresses, which are generally assumed to result from growth and remodeling, are imprecisely characterized (experimentally) at present, and how best to include the important effects of residual stresses in cardiovascular applications presents a modelling challenge
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