Mathematical Blueprint of a Mitral Valve

Abstract

Mathematical modeling tries to simplify understanding and proposes a fundamental mechanism that governs the motion and function of a complex biological system such as a mitral valve (MV) motion which represents a dynamic interplay between papillary muscle (PM) position in the context of left ventricular (LV) shape dynamics. Current therapeutic strategies to intervene on the MV may not have exploited these relationships due to lack of understanding of the interactions. We present a MV 3D mathematical model characterized by LV shape dynamics to understand fundamental working principles of ventriculo-papillary-mitral complex. A complex 3D functional unit of MV apparatus was mathematically modeled based on a principle of dynamics. The model comprises of primary components including the annulus, anterior leaflet, posterior leaflet, chordae tendineae, anterior and posterior PM, and LV wall based on normal anatomical reference values from published series. Simulations based on Carpentier's classification of MV disease were created as well as based on LV shape dynamics and presented graphically. Autodesk Inventor (Autodesk Inc., San Rafael, CA) and Matlab (Mathworks, Natick, MA) were used for modeling and analysis. A stepwise analysis and mathematical models of the annulus, leaflets, chords, PMs, and LV were obtained by combining finite element analysis and computerized model creations. The model was then applied to Carpentier's functional classification. PM positions extrapolated based on different LV deformation in normal and mitral regurgitation (MR) model resulted in a different degree of MV leaflet coaptation with regurgitation (presented numerically and graphically). Abnormal MV coaptation was amended by manipulating PM positions independent with LV size or shape deformation, demonstrating that PM positioning maneuver may improve leaflet coaptation. LV dilation combined with increased interpapillary muscle distance turned out to intensify the level of leaflet prolapse, creating even greater regurgitation volume. Our mathematical model may provide a clue to complex interactions in play within a mitral, papillary, and LV complex. The model offers a possibility of manipulating various variables to obtain the desired outcome.

Keywords: 3D model; Left ventricle; Mathematical model; Mitral regurgitation; Mitral valve; Papillary muscle.