Trans-stenotic pressure gradient estimation using a modified Bernoulli equation

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.

Figures (6)

• Figure 1: Overview of experimental approach and methodology: (a) closed-loop flow rig; (b) the FDA nozzle geometry and dimensions; (c) ultrasound imaging configuration at the nozzle throat; (d) UIV approach; (e) image of the MRI system and part of the experimental facility; (f) MRI approach; (g) MB formulation, where the pressure-loss coefficient $K_\text{MB}$ is calibrated using the experimental coefficient $K_\text{EXP}$.
• Figure 2: (a) Pressure-loss coefficient, $K$ as function of $Re$. (b) Predicted pressure drop, $\Delta P_{{Model}}$, versus measured trans-stenotic pressure drop, $\Delta P_{{EXP}}$. MB (green; dashed) uses $C_\text{mean}$; the grey band shows the range for $C=$ 1.37-1.47. SB = simplified Bernoulli; EB = extended Bernoulli; FIT = fitted correlation of experimental data (EXP). Pressure is reported in mmHg (bottom/left) and Pa. The dotted line in (b) is the identity line $y=x$. The yellow shaded band indicates clinically relevant $Re$ number range for stenotic flows.
• Figure 3: Effect of in-plane pixel size on the masked velocity profile at $z =$ -0.036 m in the throat. (a) Normalized mean velocity profiles $V/V_\text{bulk}$ from PC-MRI for five different pixel sizes $0.13-1.33\ $ mm/px for $Re=$ 667, compared with the reference profile at the same $Re$ number Manchester:2020. (b) Velocity maps at different pixel sizes, showing increasing PVE and a bias toward lower $V_{\text{bulk}}$ as pixel size increases.
• Figure 4: Heatmaps showing the effect of in-plane pixel size on MRI-based throat velocities in the idealized stenosis model. (a) Percentage error in mean throat velocity versus pixel size and flow rate $Q$, with respect to the reference; negative values indicate underestimation. (b) Percentage error in peak throat velocity versus pixel size and flow rate $Q$, computed relative to the peak velocity measured at the smallest pixel size (0.13 mm/px).
• Figure 5: The effect of the in-plane pixel size on MRI-based mean velocity in the throat of the idealized stenosis model. Percentage error in mean throat velocity versus the number of pixels across the throat radius ($N_{\text{pixels}}$) for five steady flow rates ($Q=1.5 - 3.5\ $ L/min) with respect to the reference flowrate measured by the flowmeter (negative indicates underestimation).