Abstract
Lagrangian-Euler Equations, Navier-Stocks Equations Combined with Bernoulli Equation and the Lagrangian Coherent Structure Methods to Assess the Best Mitral Valve Modeling of the Heart and Blood Fluid Movements in the Mitral Valve
Highlights
The mitral valve opens during diastole to allow the blood flow from the left atrium (LA) to the left ventricle (LV)
The normal function of the mitral valve depends on its 6 components, which are [1] the left atrial wall, [2] the annulus, [3] the leaflets, [4] the chordae tendineae, [5] the papillary muscles, and [6] the left ventricular wall (Figure 1) [1-5]
A muscle volume sample on the mitral valve leaflets is realized as a geometrical point in the 3D space (Figure 2)
Summary
This article reviews and combines different mathematical methods and techniques in order to model the mitral valve of heart and blood fluid movements in the vicinity of the valve. The body force interacted between the fluid and mitral valve leaflet tissues, combined with the general Lagrangian-Euler equations, gives a fibered model of the mitral valve. This fibered model presents a boundary condition for the solution of the Navier-Stocks equation combined with Bernulli’s equation of the blood fluid around the valve. Resulting from the velocity field data due to the convolution of the Euleran velocity from Navier-Stocks equation combined with the Bernoulli’s equation with 3D Delta function and the other velocity vector field of the blood flow movements due to the convolution of the solution of Lagrange-Euler equations with 3D Delta function, we can apply Lagrangian coherent structure (a vector flow mapping method) in order to study topology of blood transport in the mitral valve per cardiac cycle
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