Mathematical representation and nonlinear modelling of the Wheatley mitral valve

Abstract

This study is concerned with the Wheatley design of the mitral valve. A mathematical description, in terms of elementary functions, is provided for the S-shaped leaflets. This is based on a level set containing symmetric circles (or more generally ellipses) which allow parametrisation. A geometric nonlinear mechanical model subjected to a uniform pressure gradient and in the absence of inertial forces is introduced. The model results in a system of nonlinear equations that is solved using iterative incremental techniques. Under normal pressure loads, the S-shaped geometries induce internal forces which manifest themselves in two combined effects: bending and torsion. As a consequence, the supports are subject to periodic bending actions that tend to deform the support frame towards the interior of the valve. Providing resistance becomes vital for maintaining stable equilibrium. It is also observed that for circular base shape geometries, the mechanism for transmitting the equilibrium forces remains unchanged when the height/diameter ratio is kept below 2.