Derivation of a simplified relation for assessing aortic root pressure drop incorporating wall compliance

Abstract

Aging and some pathologies such as arterial hypertension, diabetes, hyperglycemia, and hyperinsulinemia cause some geometrical and mechanical changes in the aortic valve microstructure which contribute to the development of aortic stenosis (AS). Because of the high rate of mortality and morbidity, assessing the impact and progression of this disease is essential. Systolic transvalvular pressure gradient (TPG) and the effective orifice area are commonly used to grade the severity of valvular dysfunction. In this study, a theoretical model of the transient viscous blood flow across the AS is derived by taking into account the aorta compliance. The derived relation of the new TPG is expressed in terms of clinically available surrogate variables (anatomical and hemodynamic data). The proposed relation includes empirical constants which need to be empirically determined. We used a numerical model including an anatomically 3D geometrical model of the aortic root including the sinuses of Valsalva for their identification. The relation was evaluated using clinical values of pressure drops for cases for which the modified Gorlin equation is problematic (low flow, low gradient AS).

Appendix

For a distensible wall with fixed EOA, the flow disturbance downstream of the stenosis causes a change in the cross-sectional area by an amount dA.Then, for the compliant vessel, the convective pressure loss becomes:

$$\Delta p_{\text{C}} = \frac{{\rho Q^{2} }}{{2A_{\text{d}}^{2} }}\left( {\frac{{A_{\text{d}} + {\text{d}}A}}{\text{EOA}} - 1} \right)^{2}$$

(32)

If the vessel is modeled with a thin-walled cylinder obeying Hooke’s law (assuming physiological deformation), since the longitudinal stress is much smaller that the circumferential one, then

$$e_{{\uptheta \uptheta }} = \frac{{{\text{d}}a}}{a} = \frac{{a{\text{d}}p}}{Eh}$$

(33)

where e _θθ is the circumferential strain, a the radius of the vessel, a ₀ the initial radius, E the Young’s modulus of the wall material, and h the wall thickness. Then, the cross-sectional variation in Eq. (32) can be expressed as:

$${\text{d}}A = \frac{{2A_{\text{d}} \sqrt {A_{\text{d}} } }}{\sqrt \pi }\frac{{{\text{d}}p}}{Eh}$$

(34)

Hence, by expanding Eq. (32), neglecting the small higher-order terms and using Eqs. (33) and (34), the effect of the wall compliance in pressure loss can be expressed as:

$$\Delta p_{\text{C}} =\Delta p_{\text{Rigid}} +\Delta p_{\text{Co}} = \frac{{\rho Q^{2} }}{2}\left[ {\left( {\frac{1}{\text{EOA}} - \frac{1}{{A_{\text{AO}} }}} \right)^{2} + \frac{{4\sqrt {A_{\text{d}} } {\text{d}}p}}{{Eh{\kern 1pt} {\kern 1pt} \sqrt \pi }}\left( {\frac{1}{\text{EOA}}} \right)\left( {\frac{1}{\text{EOA}} - \frac{1}{{A_{\text{d}} }}} \right)} \right]$$

(35)

The first term in the right-hand side (∆p _Rigid) is pressure loss caused by convective inertia in the rigid vessel, while the second term (∆p _Co) corresponds to the contribution of the vessel compliance to convective pressure loss as the following:

$$\Delta p_{\text{Co}} = 2\rho Q^{2} \frac{{\sqrt {A_{\text{d}} } {\text{d}}p}}{{Eh{\kern 1pt} {\kern 1pt} \sqrt \pi }}\left( {\frac{1}{\text{EOA}}} \right)\left( {\frac{1}{\text{EOA}} - \frac{1}{{A_{\text{d}} }}} \right)$$

(36)

where dp is the pressure variation in the aorta. This term can be scaled as a fraction of the pulse pressure by introducing an empirical constant k _p which has the dimension of the pressure. Hence, by including the effect of all constant coefficients of Eq. (36) in k _p, it can be simplified as:

$$\Delta p_{\text{Co}} = \frac{{\rho k_{\text{p}} Q^{2} }}{Eh}\left( {\frac{{\sqrt {A_{\text{d}} } }}{\text{EOA}}} \right)\left( {\frac{1}{\text{EOA}} - \frac{1}{{A_{\text{d}} }}} \right)$$

(37)