Method for estimating pulsatile wall shear stress from one-dimensional velocity waveforms

Abstract

Wall shear stress (WSS)-a key regulator of endothelial function-is commonly estimated in vivo using simplified mathematical models based on Poiseuille's flow, assuming a quasi-steady parabolic velocity distribution, despite evidence that more rapidly time-varying, pulsatile blood flow during each cardiac cycle modulates flow-mediated dilation (FMD) in large arteries of healthy subjects. More exact and accurate models based on the well-established Womersley solution for rapidly changing blood flow have not been adopted clinically, potentially because the Womersley solution relies on the local pressure gradient, which is difficult to measure non-invasively. We have developed an open-source method for automatic reconstruction of unsteady, Womersley-derived velocity profiles, and WSS in conduit arteries. The proposed method (available online at https://doi.org/10.5281/zenodo.7576408) requires only the time-averaged diameter of the vessel and time-varying velocity data available from non-invasive imaging such as Doppler ultrasound. Validation of the method with subject-specific computational fluid dynamics and application to synthetic velocity waveforms in the common carotid, brachial, and femoral arteries reveals that the Poiseuille solution underestimates peak WSS 38.5%-55.1% during the acceleration and deceleration phases of systole and underestimates or neglects retrograde WSS. Following evidence that oscillatory shear significantly augments vasodilator production, it is plausible that mischaracterization of the shear stimulus by assuming parabolic flow leads to systematic underestimates of important biological effects of time-varying blood velocity in conduit arteries.

Keywords: Womersley solution; flow-mediated dilation; reduced-order model; shear rate; velocity profile.

FIGURE 1

Representative 1D velocity waveforms (U _1D(t)) output from the reduced‐order model presented in Muskat et al. (Muskat et al., 2021) for the right common carotid (a), brachial (b), and femoral (c) arteries. Computed waveforms for rest (70 beats/min), fear (100 beats/min), and exercise (150 beats/min) states are displayed in black, blue, and red curves, respectively. These idealized data provided inputs for our initial tests of the Poiseuille‐ and Womersley‐derived estimates of wall shear stress.

FIGURE 2

Demonstration of methods using an analytical solution for pulsatile flow in a straight pipe. (a) input velocity waveform as a cosine function with time‐averaged mean velocity of 20 cm/s and period of 1.0 s. (b) Womersley velocity profiles (solid lines) at 0.25 s (left subpanel) and 0.75 s (right subpanel) with steady (dotted lines) and unsteady (dash‐dotted lines) components displayed separately. Steady and unsteady components are combined to produce the Womersley solution as illustrated in each subpanel. The steady component defined by Poiseuille's flow neglects the effects of acceleration and deceleration which are shown to modulate the radial velocity profile across the cardiac cycle.

FIGURE 3

Morphological and hemodynamic data acquired for comparison to computational fluid dynamics (CFD). (a) Volume rendering of brachial artery time‐of‐flight magnetic resonance angiography (MRA) in healthy adult male subject. MRA localized to 18 × 20 × 15 cm of upper arm proximal to the antecubital fossa. Brachial artery segmentation overlaid in red. Region of interest (dashed lines) where (b) color and (c) pulsed‐wave Doppler were acquired. (d) One‐dimensional time‐varying velocity waveform extracted from ultrasound data at the midsection of the region of interest. Physiologic brachial artery waveform indicated by triphasic blood flow.

FIGURE 4

Validation of Womersley theory with computational fluid dynamics (CFD). (a) time‐varying wall shear stress (WSS) distributions with respect to subject‐specific CFD (median, dotted line; interquartile Q1–Q3 range, dark gray; and total range, light gray), Poiseuille's flow (dash‐dotted lines), and Womersley theory (solid lines). (b) Absolute WSS difference from ground truth CFD simulations for Womersley and Poiseuille solutions. The deviation of the Poiseuille solution from CFD is increased during periods of high‐velocity acceleration.

FIGURE 5

Reconstruction of transient radial velocity profiles using Poiseuille (P., dash‐dotted lines) and Womersley (W., solid lines) solutions for right common carotid (a–c), brachial (d–f), and femoral (g–i) arteries during rest (black), fear (blue), and exercise (red) states. Representative velocity profiles are shown for both methods at maximum (WSS_max, left subpanels) and minimum (WSS_min, right subpanels) WSS as determined via Womersley theory; therefore, the velocity fields shown here illustrate instantaneous variation between methods at matching time points. Negative slopes at the vessel wall in right subpanels mirror physiologic levels of oscillatory shear stress.

FIGURE 6

The effect of underlying flow assumptions on transient wall shear stress (WSS) in the major conduit arteries. WSS waveforms based on Poiseuille's flow (P., dash‐dotted lines) and Womersley theory (W., solid lines) for right common carotid (a), brachial (b), and femoral (c) arteries. Solutions for rest, fear, and exercise are displayed in black, blue, and red lines, respectively. Relative to the Womersley solution, Poiseuille's flow underestimated systolic WSS, overestimated diastolic WSS, and failed to capture negative values of WSS during the systolic deceleration phase. Differences are attributed to the unsteady component or rate of change in shear at the onset of flow being neglected by Poiseuille's flow.

FIGURE 7

Hemodynamic metrics were evaluated with Poiseuille (dash‐dotted bars) and Womersley (solid bars) solutions for cardiovascular states of rest (R., black), fear (F., blue), and exercise (E., red). (a) Poiseuille's flow overestimated time‐averaged median wall shear stress (WSS) in conduit arteries. Underestimation (0.1%) in the femoral artery during a fear response was negligible. (b) in contrast, Poiseuille's flow consistently underestimated maximum WSS. Largest differences between methods occurred in the brachial artery. (c) evaluation of the oscillatory shear index (OSI) revealed physiologic levels of oscillatory shear stress occurred in conduit arteries despite fully monophasic, antegrade flows.