Transversely isotropic membrane shells with application to mitral valve mechanics. Constitutive modelling and finite element implementation

Abstract

SUMMARYThe present study addresses constitutive modelling and implementation of transversely isotropic hyperelastic material models for the analysis of the mitral valve. This valve separates the left atrium and left ventricle in the heart. Two convex strain energy potentials are employed in derivation of stress tensors and elasticity tensors. The plane stress and incompressibility conditions are accounted for directly. The relationships are implemented in an implicit code (ABAQUS) via the user-defined interface. Numerical simulations of the valve motion during a part of the heart cycle are carried out and compared to ultrasound measurements of a healthy human valve. The significance of placement of chordae tendinae is illustrated. The implementation provides a tool for simulations of both healthy and pathological mitral valve conditions.

Mitral valve finite-element modelling from ultrasound data: a pilot study for a new approach to understand mitral function and clinical scenarios

“…These assumptions are present in the first-generation structural model by Kunzelman's research group (Kunzelman et al 1993a), in the studies we have performed (Votta et al 2002(Votta et al , 2007Maisano et al 2005), in the one by Prot et al (2007) and in the model adopted by Dal Pan et al (2005), who also verified that the action of marginal chordae can be implicitly accounted for by imposing proper kinematical boundary conditions on the leaflets' free margin. As for other modelling aspects, the most sophisticated approach was adopted by Einstein et al (2005a,b), who accounted for the presence of all types of chordae, marginal, basal and strut, as well as for their branched insertions into the leaflets.…”

“…Currently, only two models include a realistic description of the leaflets' stress-strain behaviour: the structural model by Prot et al (2007) and the fluid-structure interaction model by Einstein et al (2005a). In both cases, the stress-strain relationship is modelled by means of a strain energy function j, in accordance with the theory of hyperelasticity.…”

“…The load increased up to a systolic peak value of 120 mm Hg, with a time dependence already adopted by other authors (Prot et al 2007). The time-scale of the simulated event was scaled: in vivo it lasts for approximately 0.15 s, while in the simulation the systolic peak was reached in 1 s. This choice allowed extremely abrupt contact phenomena to be avoided and convergence to be obtained more easily.…”

“…Kunzelman and co-workers were the first to use this approach and have adopted it to the widest extent, either to mimic normal mitral function (Kunzelman et al 1993a;Einstein et al 2005a) or to understand the biomechanics underlying valvular diseases (Kunzelman et al 1997(Kunzelman et al , 1998aEinstein et al 2005b) and surgical corrections (Reimink et al 1995(Reimink et al , 1996Cochran & Kunzelman 1998;Kunzelman et al 1998b). More recently, two other research groups have focused their attention on normal mitral function (Lim et al 2005;Prot et al 2007). Finally, the group at Politecnico di Milano have used FE models to analyse the effects of annuloplasty procedures (Maisano et al 2005;Votta et al 2007) and to simulate the biomechanical response of the valve to the Alfieri stitch technique (Votta et al 2002).…”

Constitutive modelling of arteries

“…Rodríguez et al (2009) compare the models (2.29) and (4.2) and an isotropic model on the basis of two patient-specific AAA geometries. Modelling and analysis of the mechanical behaviour of mitral valves with their attached chordae tendineae on the basis of the models (4.1) and (4.2) was conducted by Prot et al (2007Prot et al ( , 2010 and Prot & Skallerud (2009). An application to a membrane shell using a finite element implementation was illustrated, and a comparative study of healthy and diseased valves was documented.…”