Modeling of the Instantaneous Transvalvular Pressure Gradient in Aortic Stenosis

Abstract

The simplified and modified Bernoulli equations break down in estimating the true pressure gradient across the stenotic aortic valve due to their over simplifying assumptions of steady and inviscid conditions as well as the fundamental nature in which aortic valves are different than idealized orifices. Nevertheless, despite having newer models based on time-dependent momentum balance across an orifice, the simplified and modified Bernoulli continue to be the clinical standard because to date, they remain the only models clinically implementable. The objective of this study is to (1) illustrate the fundamental considerations necessary to accurately model the time-dependent instantaneous pressure gradient across a fixed orifice and (2) propose empirical corrections when applying orifice based models to severely stenotic aortic valves. We introduce a general model to predict the time-dependent instantaneous pressure gradient across an orifice that explicitly model the Reynolds number dependence of both the steady and unsteady terms. The accuracy of this general model is assessed with respect to previous models through pulse duplicator experiments on a round orifice model as well as an explanted stenotic surgical aortic valve both with geometric areas of 0.6 cm² (less than 1 cm² which is the threshold for stenosis determination) over cardiac outputs of 3 and 5 L/min and heart rates of 60, 90 and 120 bpm. The model and the raw experimental data corresponding to the orifice showed good agreement over a wide range of cardiac outputs and heart rates (R² exceeding 0.91). The derived average and peak transvalvular pressure gradients also demonstrated good agreement with no significant differences between the respective peaks (p > 0.09). The new model proposed holds promise with its physical and closed form representation of pressure drop, however accurate modeling of the time-variability of the valve area is necessary for the model to be applied on stenotic valves.

Appendix A: Non-dimensional Analysis

From non-dimensional analysis, we choose to select 6 different variables $\left( {\Delta P, \rho , V, D, \nu , \frac{\text{d}}{{{\text{d}}t}}\left( {\frac{{\rho V^{2} }}{2}} \right)} \right)$ where $\nu$ is the kinematic viscosity.

The π groups obtained can be summarized:

$$\pi_{1} = \frac{\Delta P}{{\rho V^{2} }}$$

(19)

$$\pi_{2} = \frac{1}{Re}$$

(20)

$$\pi_{3} = \frac{\text{d}}{{{\text{d}}t}}\left( {\frac{{V^{2} }}{2}} \right)\frac{D}{{V^{3} }} = \frac{D}{{V^{2} }}\frac{{{\text{d}}V}}{{{\text{d}}t}}$$

(21)

Stemming from Eq. (4), the relationship between the π groups can be written as follows:

$$\pi_{1} = f\left( {\pi_{2} } \right) + g\left( {\pi_{3} } \right)$$

(22)

$$\frac{\Delta P}{{\rho V^{2} }} = f\left( {\frac{1}{Re}} \right) + g\left( {\frac{D}{{V^{2} }}\frac{{{\text{d}}V}}{{{\text{d}}t}}} \right)$$

(23)

$$\Delta P = f\left( {\frac{{\rho V^{2} }}{Re}} \right) + g\left( {\frac{\mu Re}{V}\frac{{{\text{d}}V}}{{{\text{d}}t}}} \right)$$

(24)

$$\Delta P = f\left( {\frac{{\rho Q^{2} }}{{ReA^{2} }}} \right) + g\left( {\frac{\mu Re}{VA}\frac{{{\text{d}}Q}}{{{\text{d}}t}}} \right)$$

(25)

Equations (25) and (16) compare.