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Research Papers: Flows in Complex Systems

Application of Womersley Model to Reconstruct Pulsatile Flow From Doppler Ultrasound Measurements

[+] Author and Article Information
Nihad E. Daidzic

Mem. ASME
AAR Aerospace Consulting, LLC,
P.O. Box 208,
Saint Peter, MN 56082
e-mail: aaraerospace@cs.com

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 7, 2013; final manuscript received January 9, 2014; published online February 28, 2014. Assoc. Editor: Michael G. Olsen.

J. Fluids Eng 136(4), 041102 (Feb 28, 2014) (15 pages) Paper No: FE-13-1481; doi: 10.1115/1.4026481 History: Received August 07, 2013; Revised January 09, 2014

A Womersley model-based assessment of pulsatile rigid-tube flow is presented. Multigate Doppler ultrasound was used to measure axial velocities at many radial locations along a single interrogation beam going through the center of a stiff tube. However, a large impediment to Doppler ultrasound diagnostics and resolution close to the wall is considerable noise due to the presence of the wall-fluid interface as well as many other effects, such as spectral broadening, coherent scattering, time resolution, and Doppler angle uncertainty. Thus, our confidence in measured signals is questionable, especially in the wall vicinity where the important oscillatory shear stresses occur. In order to alleviate known biases and shortcomings of the pulsed Doppler ultrasound measurements we have applied Womersley's laminar axisymmetric rigid-tube approximation to reconstruct velocity profiles over the entire flow domain and specifically close to wall, enabling unambiguous determination of the shear stresses. We employ harmonic analysis of the measured velocity profiles at all or selected trusted tube radial locations over one or more periods. Each of estimated Fourier coefficients has a unique counterpart in the respective pressure gradient component. From ensemble-averaged cross-sectional pressure gradient components we compute velocity profiles, volume flow rate, wall shear stress, and other flow parameters. Estimation of the pressure gradients from spatially resolved pulsed Doppler ultrasound velocity measurements is an added benefit of our reconstruction method. Multigate pulsed Doppler ultrasound scanners offer powerful capabilities to noninvasively and nonintrusively measure velocity profiles for hemodynamic and other fluid flow applications. This flow reconstruction method can also be tailored for use with other flow diagnostic modalities, such as magnetic resonance imaging (MRI) and a wide class of optical methods.

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Figures

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Fig. 1

Normalized damping factor and phase as function of Womersley numbers at selected radial locations for velocity, VFR, and WSS

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Fig. 2

The experimental hydraulic test rig utilizing multigate PW Doppler US

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Fig. 3

Simplified space-time representation (LHS) and 1D sample volume longitudinal real and maximum resolution (RHS) of PW US range system

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Fig. 4

Flow characterization results recovered from the original McDonald's work [36] on dog femoral arteries

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Fig. 5

Harmonic analysis using 30-term Fourier expansion of sampled velocities at two radial locations for steady (Poiseuille) flow with measured Doppler US velocity profiles

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Fig. 6

Reconstructed flow data for steady-state (Poiseuille) flow using first ten harmonics

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Fig. 7

Harmonic analysis using 13-term Fourier expansion of sampled velocities at two radial locations for sinusoidal flow excitation and to the right the actual raw Doppler US velocity profiles with velocities close to wall discarded

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Fig. 8

Reconstructed flow data for sinusoidal flow excitation with velocity profiles plotted for each T/24-th of a period

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Fig. 9

Harmonic analysis using 22-term Fourier expansion of sampled velocities at two radial locations and measured axial velocities for sinusoidal flow excitation with very noisy US measurements and observed flow reversal close to wall

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Fig. 10

Reconstructed flow data for sinusoidal flow excitation with flow reversal with reconstructed data showing diminished flow reversal in wall vicinity compared to original

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Fig. 11

Harmonic analysis using 23-term Fourier expansion of sampled velocities at two radial locations for triangular flow excitation with raw US velocity profiles at various times within one period

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Fig. 12

Reconstructed flow data for triangular flow excitation using first ten harmonics

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