Mathematical analysis of mitral valve leaflets

Effects of hemodynamic alterations on anterior mitral leaflet curvature during systole

OBJECTIVES

The application of repair techniques to treat mitral valve incompetence has increased progressively during the past 20 years. Unfortunately, recent reports have demonstrated the longevity of these repairs to be less than previously believed. Most repair failures are stress related. Computational models to optimize valve repair are in development, but to be brought to fruition, a better understanding of dynamic leaflet geometry is necessary. In this study, sonomicrometry was used in an ovine model to compute systolic leaflet curvature at varying afterloads and states of contractility.

METHODS

The anterior leaflet of 12 sheep was instrumented with 5 piezoelectric transducers in a cruciate array. Systolic blood pressure ranged from 90 to 200 mm Hg with increasing phenylephrine hydrochloride infusion. Epinephrine was used to vary contractile state. Leaflet curvature was calculated continuously (200 Hz) during systole.

RESULTS

Anterior leaflet curvature in the septolateral direction was double that in the intercommisural direction. There were also significant changes in leaflet curvature during systole. Curvature in neither direction was affected by afterload. Epinephrine augmented intercommisural curvature in a dose-independent fashion, whereas it had no effect on curvature in the septolateral direction.

CONCLUSIONS

Dynamic mitral anterior leaflet geometry was found to be amazingly constant over a wide range of hemodynamic conditions. These data provide information about leaflet geometry that will aid in the construction of realistic computational models. Such models may facilitate the design of annuloplasty rings and surgical techniques that minimize leaflet stress and increase mitral valve repair longevity.

The relationship of normal and abnormal microstructural proliferation to the mitral valve closure sound

BACKGROUND

Many diseases that affect the mitral valve are accompanied by the proliferation or degradation of tissue microstructure. The early acoustic detection of these changes may lead to the better management of mitral valve disease. In this study, we examine the nonstationary acoustic effects of perturbing material parameters that characterize mitral valve tissue in terms of its microstructural components. Specifically, we examine the influence of the volume fraction, stiffness and splay of collagen fibers as well as the stiffness of the nonlinear matrix in which they are embedded.

METHODS AND RESULTS

To model the transient vibrations of the mitral valve apparatus bathed in a blood medium, we have constructed a dynamic nonlinear fluid-coupled finite element model of the valve leaflets and chordae tendinae. The material behavior for the leaflets is based on an experimentally derived structural constitutive equation. The gross movement and small-scale acoustic vibrations of the valvular structures result from the application of physiologic pressure loads. Material changes that preserved the anisotropy of the valve leaflets were found to preserve valvular function. By contrast, material changes that altered the anisotropy of the valve were found to profoundly alter valvular function. These changes were manifest in the acoustic signatures of the valve closure sounds. Abnormally, stiffened valves closed more slowly and were accompanied by lower peak frequencies.

CONCLUSION

The relationship between stiffness and frequency, though never documented in a native mitral valve, has been an axiom of heart sounds research. We find that the relationship is more subtle and that increases in stiffness may lead to either increases or decreases in peak frequency depending on their relationship to valvular function.

A constitutive law for mitral valve tissue

Biaxial mechanical testing and theoretical continuum mechanics analysis are employed to formulate a constitutive law for cardiac mitral valve anterior and posterior leaflets. A strain energy description is formulated based on the fibrous architecture of the tissue, accurately describing the large deformation, highly nonlinear transversely isotropic material behavior. The results show that a simple three-coefficient exponential constitutive law provides an accurate prediction of stress-stretch behavior over a wide range of deformations. Regional heterogenity may be accommodated by spatially varying a single coefficient and incorporating collagen fiber angle. The application of this quantitative information to mechanical models and bioprosthetic development could provide substantial improvement in the evaluation and treatment of valvular disease, surgery, and replacement.

Time-frequency analysis of the first heart sound: Part 3: Application to dogs with varying cardiac contractility and to patients with mitral mechanical prosthetic heart valves

The cone-kernel distribution (CKD) is first applied to the analysis of the intracardiac and the thoracic first heart sound (S1) of dogs in various cardiac contractile states, and secondly to the S1 of patients with mitral mechanical prosthetic heart valves. The CKD of native S1 in dogs shows that the dominant components of S1 are generally concentrated in a band at around 50 Hz with a horizontal flat or a semi-lunar shape, independently of the myocardial contractile state. There is no significant systematic rising frequency component. The instantaneous frequency of S1 shows a good cross-correlation with the time derivative of the left ventricular pressure (dP/dt), but the maximum frequency is not proportional to the maximum of dP/dt. The CKD of S1 in patients with mitral mechanical prosthetic heart valves showed a pulse-like component with a high-frequency bandwidth, which is distinct from the low constant-frequency components of S1 produced by native heart valves.

Time-frequency transforms: a new approach to first heart sound frequency dynamics

This study employed a new analytical tool, the Binomial joint time-frequency transform, to test the hypothesis that first heart sound frequency rises during the isovolumic contraction period. Cardiac vibrations were recorded from eight open chest dogs using an ultralight accelerometer cemented directly to the epicardium of the anterior left ventricle. The frequency response of the recording system was flat +/- 3 dB from 0.1 to 400 Hz. Three characteristic time-frequency spectral patterns were evident in the animals investigated: 1) A frequency component that rose from approximately 40-140 Hz in a 30-50 ms interval immediately following the ECG R-wave. 2) A slowly varying or static frequency of 60-100 Hz beginning midway through the isovolumic contraction period. 3) Broad-band peaks occurring at the time of the Ia and Ib high frequency components. The presence of rapid frequency dynamics limits the usefulness of stationary analysis techniques for the first heart sound. The Binomial transform provided much better resolution than the spectrograph or spectrogram, the two most common non-stationary signal analysis techniques. By revealing the onset and dynamics of first heart sound frequencies, time-frequency transforms may allow mechanical assessment of individual cardiac structures.

A mathematical model of aortic valve vibration

It has been shown experimentally that the second heart sound is produced by diastolic vibrations of the closed aortic valve. In the present paper a mathematical model of this vibration is developed from first principles. The model assumes a one dimensional but non-linear fluid behavior. The problem is coupled through a non-linear, planar valve. A solution is obtained using the method of characteristics developed in finite difference form. The resulting valve frequency and amplitude are in good agreement with patient data. The model predicts a strong dependency of response on the valve forcing function and valve stiffness; and a weaker dependency of response on valve mass.

Stresses in the closed mitral valve: a model study

In the present model study on the closed mitral valve, tensile force in the chordae tendineae is related to transvalvular pressure using a mathematical model of mechanics of the closed mitral valve. Circumferential stress as well as bending stress in the valve leaflets were neglected. Without precisely knowing the mechanical properties of the leaflet material, geometry of the leaflets was estimated by applying Laplace's law, which relates leaflet stress to leaflet curvature. Independent of shape of the mitral valve orifice, under all circumstances tensile force in the chordae tendineae was calculated to be equal or greater than half the force exerted on the mitral valve orifice by the transvalvular pressure.