Mathematical Blueprint of a Mitral Valve

Arrhythmic Mitral Valve Prolapse: Introducing an Era of Multimodality Imaging-Based Diagnosis and Risk Stratification

“…In patents with MAD and MVP both the annular height and annular height-to-commissural width ratio (Figure 1) decreased, resulting in paradoxical annular flattening during systole [68], while the systolic annular diameter increased [67]. Although annular alterations have not been investigated as a risk factor for SCD, both in vivo [5] and advanced computational [72] evidence demonstrate the significance of the physiological saddle-shaped annulus in maintenance of normal MV leaflet stress, suggesting that MAD might be one of the precursor mechanisms leading to MR in MVP [63] but also for the occurrence of VA.…”

Geometric description for the anatomy of the mitral valve: A review

“…The annular saddle shape has been mathematically defined through paraboloid equations, which are quadric surfaces with one axis of symmetry and no centre of symmetry (Salgo et al , 2002; Stevanella et al , 2009; Park et al , 2019). In Salgo et al (2002), for example, hyperbolic paraboloids were employed (Salgo et al , 2002), whereas Park et al .…”

“…used hyperbolic paraboloids for the saddle shape definition. In addition, they designed a ring‐shaped structure, mathematically described as a curved toroid and with empirical relationships derived below (Park et al , 2019):where α L and β L are transformation parameters in lateral directions in the xy plane that define the saddle point, α V is a transformation parameter in the vertical direction in the xz plane, and β V is the transformation parameter in the vertical direction in the yz plane. These parameters vary between 0 and 1, and a and b are determined by annular dimensions:…”

“…Hyperboloids representing the mitral valve (MV) saddle shape and defined by Equation (Park et al , 2019) are displayed, created from literature values for CW (33.3 mm), AP diameter (28 mm), AL–PM diameter (31 mm) and AHCWR (0.237). Different commissural heights (b) and posterior annular depth (c) in comparison with a pure hyperboloid (a) can be observed, shown by changes in transformation parameters…”

“…With the evolution of medical imaging modalities, in vivo procedures, such as three‐dimensional (3D) transoesophageal echocardiography, have been used quantitatively to evaluate human MV morphology in healthy and diseased cases (Sonne et al , 2009; Delgado et al , 2009; Lee et al , 2013; Mihaila et al , 2013; Jolley et al , 2017), proving crucial for 3D printing purposes or transcatheter MV implantation, for example. On the other hand, computational studies have mathematically represented the MV (Salgo et al , 2002; Stevanella et al , 2009; Park et al , 2019), including the annulus (Salgo et al , 2002; Stevanella et al , 2009), leaflet shape (Stevanella et al , 2009; Shen et al , 2017; Park et al , 2019) and chordae architecture (Kaiser et al , 2019) from both human and porcine dimensions and characteristics. Although many sources of MV anatomy quantification exist, a single study bringing together the current best knowledge around its morphometry, including related mathematical definitions, is lacking.…”