Instability in Computational Models of Vascular Smooth Muscle Cell Contraction

Abstract

Purpose

Through their contractile and synthetic capacity, vascular smooth muscle cells (VSMCs) can regulate the stiffness and resistance of the circulation. To model the contraction of blood vessels, an active stress component can be added to the (passive) Cauchy stress tensor. Different constitutive formulations have been proposed to describe this active stress component. Notably, however, measuring biomechanical behaviour of contracted blood vessels ex vivo presents several experimental challenges, which complicate the acquisition of comprehensive datasets to inform complex active stress models. In this work, we examine formulations for use with limited experimental contraction data as well as those developed to capture more comprehensive datasets.

Methods

First, we prove analytically that a subset of constitutive active stress formulations exhibits unstable behaviours (i.e., a non-unique diameter solution for a given pressure) in certain parameter ranges, particularly for large contractile deformations. Second, using experimental literature data, we present two case studies where these formulations are used to capture the contractile response of VSMCs in the presence of (1) limited and (2) extensive contraction data.

Results

We show how limited contraction data complicates selecting an appropriate active stress model for vascular applications, potentially resulting in unrealistic modelled behaviours.

Conclusion

Our data provide a useful reference for selecting an active stress model which balances the trade-off between accuracy and available biomechanical information. Whilst complex physiologically motivated models’ superior accuracy is recommended whenever active biomechanics can be extensively characterised experimentally, a constant 2nd Piola-Kirchhoff active stress model balances well accuracy and applicability with sparse contractile data.