Trans-stenotic pressure gradient estimation using a modified Bernoulli equation

TL;DR

This study introduces a modified Bernoulli model that accounts for flow regime-dependent pressure losses, improving non-invasive pressure gradient estimation in stenosis, validated through in-vitro experiments and MRI analysis.

Contribution

The paper presents a novel Bernoulli formulation incorporating Reynolds-number-dependent losses, enhancing accuracy over traditional models in estimating pressure drops.

Findings

MB model outperforms SB and EB in accuracy

Coarse MRI sampling underestimates pressure drops

Peak velocity measurements are less sensitive to pixel size

Abstract

Accurate non-invasive estimation of trans-stenotic pressure gradients remains a challenge. In clinical practice, pressure gradients are often estimated from velocity measurements using Bernoulli-based formulas, but these simplified relations do not explicitly account for how pressure losses change with the flow regime. Here, we introduce a modified Bernoulli (MB) formulation that incorporates regime-dependent pressure losses through a Reynolds-number-dependent loss coefficient. Steady in-vitro experiments were performed in an idealized stenosis model over physiologically relevant flow rates (0.65-3.9 L/min), combining direct pressure measurements with ultrasound imaging velocimetry and phase-contrast magnetic resonance imaging (PC-MRI) to measure velocities. The MB model was calibrated from the measured pressure drops and then evaluated against the simplified Bernoulli (SB) and extended Bernoulli (EB) formulations. Over the tested flow regime, MB agreed best with the measurements. SB and EB showed larger biases, with errors of roughly 10-55% (SB) and -15 to 25% (EB), and overestimated the pressure drop in the clinically relevant range. We additionally quantified the effect of PC-MRI in-plane pixel size on MRI-based pressure estimates. Coarse sampling of the stenosis throat led to systematic underestimation of flow rate and bulk velocity and, consequently, of the MB-predicted pressure drop. In contrast, the peak throat velocity was substantially less sensitive to pixel size, resulting in smaller estimation errors when used as input for the MB. Overall, the results demonstrate that accounting for flow-regime-dependent loss mechanisms enhances pressure drop estimation, and that sufficient sampling of the stenotic throat is crucial for MRI-based flow rate and pressure drop estimation. In addition, peak-velocity-based MB pressure drop estimations are less sensitive to pixel size.