Euler’s Elastica-Based Biomechanics of the Papillary Muscle Approximation in Ischemic Mitral Valve Regurgitation: A Simple 2D Analytical Model

Figures

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  Figure 1. (A). Distribution and incidence of valve annular stress after ischemic mitral regurgitation IMR. Valvular region is organized in anterior and posterior leaflet. Each leaflet is divided in three segments or scallops that are named A1, A2, A3 and P1, P2, P3 relating to anterior or posterior cusp, respectively. The leaflets are attached to fibrous circular annulus and join in the anterior and posterior commissure, respectively, giving to the mitral valve the appearance of a curtain that is closed during systole. Tethering greatly affects the A3 and P3 segments and the posterior commissure (PC); (B) Segmentation and morphological types of papillary muscles; I, single uniform unit; II, groove with two apexes; III, fenestrations with muscular bridges; IV, complete separation in two adjacent heads; V, complete separation with two distant heads. Division can occur according to two directions: (A small) Division in a sagittal plane leading to a separate posterior leaflet head. (B small) Division in a coronal plane leading to a separate commissural head. A, anterior leaflet; C, commissure; P, posterior leaflet; (C) Mechanisms of ischemic mitral valve regurgitation for chordae stress and…

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  Figure 2. Schematic representation of symmetric and asymmetric pattern of mitral valve tethering. (Left) Symmetric tethering pattern. With apical and lateral vector components, the posteromedial papillary muscle and both mitral leaflets are medially displaced, resulting in symmetric leaflet tethering. Of note, primarily apical displacement of both PMs favours less leaflet tethering, while a more medial shift results in increased tethering of both leaflets. (a–b) Posterior PM tethering, distance between the posteromedial PM tip and the saddle horn; (a–d) anterolateral PM tethering, distance between the anterolateral PM tip and the saddle horn; (a–c) anterior component of posterior PM tethering; (b–c) posterior (inferior) component of posteromedial PM tethering; (a–d) anterolateral PM tethering with its components (a–e) and (e–d). (Right) Asymmetric tethering pattern. A posterior vector imparts a posterior (major) and apical (minor) displacement of the postero medial papillary muscle (PMPM) resulting in asymmetric tethering. (a–b) Posterior PM tethering, distance between the posteromedial PM tip and the saddle horn; (a–d) anterolateral PM tethering, distance between the anterolateral PM tip…

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  Figure 3. Hand-made sketch of the in-plane mechanical modelling of the mitral valve nonlinear bending combined with tethering due to the interaction with hyperelastic chordae anchored to the moving PM site.

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  Figure 4. Top Panel. Symmetric pattern of mitral valve tethering determines biomechanical change in mitral valve configuration depicted on two- and three-dimensional echocardiography. (Left) Symmetric mitral valve leaflet tethering primarily in the apical direction results in a central ischemic mitral regurgitant jet. (Right) En face (surgeon’s view) of the mitral valve exemplifies a central, crescentic-shaped regurgitant orifice and MR jet. Middle Panel. Asymmetric pattern of mitral valve tethering determines biomechanical change in mitral valve configuration depicted on two- and three-dimensional echocardiography. (Left) Asymmetric mitral valve leaflet tethering in the inferior/posterior direction (yellow arrow) results in posteriorly-directed eccentric ischemic mitral regurgitation (IMR). (Right) En face (surgeons view) of the mitral valve exemplifies the resultant regurgitant orifice, which is more medially located, and the eccentric MR. Bottom panel. Two- and three-dimensional reconstruction of biomechanical change in mitral valve configuration. (Left) Representation of tenting area and tenting height. (Right) 3D TEE (transthoracic echography) reconstruction. AL = anterolateral…

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  Figure 5. Preoperative configuration determined from the clinical dataset, with the illustration of the considered geometric factors and markers characterizing IMR grade.

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  Figure 6. Conceptual scheme of the solution strategy. Starting from random search, the postoperative configuration is determined by adopting a minimum-stress criterion.

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  Figure 7. Analytically determined postoperative configuration showing mitral valve (MV) restriction, leaflet coaptation, and optimal interpapillary muscle distance (IPMD) ensuring the minimum stress within the tethering chordae.

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  Figure 8. Comparison of the optimized postoperative outcome with both absent PMA and complete PMA. Different outcomes were obtained by starting from the same pre-operative configuration. In each case, effectiveness of the outcomes was measured by means of non-dimensional indicators in terms of i) valvular annulus stress ratio (black bars), measured as the ratio between post-operative and pre-operative bending moments predicted in the leaflets roots Mri=Mpoi/Mprei. ii) Chordae stress ratios Tri=Tpoi/Tprei, evaluated as the ratio between the post-operative and the pre-operative nominal axial stresses of the chordae tendinae (Gray bars). iii) Stretch ratios of the chordae tendinae λri=λpoi/λprei (white bars), denoting the ratio between post-operative and pre-operative lengths of the chordae. Bars with dashed contours refer to posterior leaflets.