The Mathematical Model of a Procedure for Percutaneous Annuloplasty

Existing mathematical models of the mitral valve allow the simulation of ring open-heart annuloplasty procedures intended to reduce the lumen of the valve. Using these models, only a posteriori effects can be predicted. With the advent of novel percutaneous annuloplasty approaches, there is a need to describe a priori effects; in particular, this paper focuses on a technique which consists of sequentially installing interconnected anchors around the mitral annulus, whose lumen is reduced by the tightening of the tethered wire. We develop here a static mathematical model of the mitral annulus that takes into account the mechanical response of its tissue and the surrounding muscular tissue. A number of roughly coplanar points corresponding to anchor positions, at about equal distantes, are identified on the annulus. Each of these points is then attached to a linearly elastic spring of a given stiffness, The spring-end is connected to a fixed pinned support, the other end supporting the wire, that forms a loop. With this model we estimate the anchor-points position vectors after lumen reduction and the wire tension that is needed to reduce the perimeter of the polygon defined by the anchor points to a given value, which, for each patient, is related to the desired lumen. This formulation leads to the minimization of the potential energy of the mechanical system over the position vectors of the anchor points after tightening, which are the design variables. These are found by solving the first-order normality conditions of the equality-constrained optimization problem. Preliminary experimental data obtained on cadaveric swine hearts validate the model: it can be used to predict, for a given perimeter size reduction, the wire tension as well as the anchor position after repair.