0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Hoeks, A. P. G., Reneman, R. S., and Peronneau, P. A., 1981, “A Multi-Gate Pulsed Doppler System With Serial Data Processing,” IEEE Trans. Sonics Ultrason., 28, pp. 242–247. [CrossRef]
Tortoli, P., Guidi, F., Guidi, G., and Atzeni, C., 1996, “Spectral Velocity Profiles for Detailed Ultrasound Flow Analysis,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., 43(4), pp. 654–659. [CrossRef]
Evans, D. H., and McDicken, W. N., 2000, Doppler Ultrasound—Physics, Instrumentation and Signal Processing, John Wiley & Sons Inc., Chichester, UK.
Cobbold, R. S. C., 2007, Foundations of Biomedical Ultrasound, Oxford University Press, New York.
Bambi, G., Morganti, T., Ricci, S., Boni, E., Guidi, F., Palombo, C., and TortoliP., 2004, “A Novel Ultrasound Instrument for Investigation of Arterial Mechanics,” Ultrasonics, 42, pp. 731–737. [CrossRef] [PubMed]
Womersley, J. R., 1955, “Method for the Calculation of Velocity, Rate of Flow, and Viscous Drag in Arteries When the Pressure Gradient is Known,” J. Physiol., 127, pp. 553–563. [PubMed]
Womersley, J. R., 1955, “Oscillatory Motion of a Viscous Liquid in a Thin-Walled Elastic Tube: I The Linear Approximation for Long Waves,” Phil. Mag., 46, pp. 199–221.
Womersley, J. R., 1957, “Oscillatory Flow in Arteries: The Constrained Elastic Tube as a Model of Arterial Flow and Pulse Transmission,” Phys. Med. Biol., 2, pp. 178–187. [CrossRef] [PubMed]
Nichols, W. N., and O'Rourke, M. F., 1990, McDonald's Blood Flow in Arteries, Arnold, London.
Warriner, R. K., Johnston, K. W., and Cobbold, R. S. C., 2008, “A Viscoelastic Model of Arterial Wall Motion in Pulsatile Flow: Implications for Doppler Ultrasound Clutter Assessment,” Physiol. Meas., 29, pp. 157–179. [CrossRef] [PubMed]
Balocco, S., Basset, O., Courbebaisse, G., Boni, E., Frangi, A. F., Tortoli, P., and Cachard, C., 2010. “Estimation of the Viscoelastic Properties of Vessel Walls Using a Computational Model and Doppler,” Phys. Med. Biol., 55, pp. 3557–3575. [CrossRef] [PubMed]
Ku, D. N., 1997, “Blood Flow in Arteries,” Ann. Rev. Fluid Mech., 29, pp. 399–434. [CrossRef]
Struijk, P. C., Stewart, P. A., Fernando, K. L., Mathews, V. J., Loupas, T., Steegers, E. A. P., and Wladimiroff, J. W., 2005, “Wall Shear Stress and Related Hemodynamic Parameters in the Fetal Descending Aorta Derived From Color Doppler Velocity Profiles,” Ultrasound Med. Biol., 31, pp. 1441–1450. [CrossRef] [PubMed]
Leguy, C. A. D., Bosboom, E. M. H., Hoeks, A. P. G., and van de Vosse, F. N., 2009, “Assessment of Blood Volume Flow in Slightly Curved Arteries From Velocity Profile,” J. Biomech., 42, pp. 1664–1672. [CrossRef] [PubMed]
Leguy, C. A. D., Bosboom, E. M. H., Hoeks, A. P. G., and van de Vosse, F. N., 2009, “Model Based Assessment of Dynamic Blood Volume Flow From Ultrasound Measurements,” Med. Biol. Eng. Comput., 47, pp. 641–648. [CrossRef] [PubMed]
Ponzini, R., Vergara.C., Redaelli, A., and Veneziani, A., 2006, “Reliable CFD Estimation of Flow Rate in Haemodynamics Measures,” Ultrasound Med. Biol., 32(10), pp. 1545–1555. [CrossRef] [PubMed]
Vergara, C., Ponzini, R., Veneziani, A., Redaelli, A., Neglia, D., and Parodi, O., 2010, “Womersley Number-Based Estimation of Flow Rate With Doppler Ultrasound: Sensitivity Analysis and First Clinical Applications,” Comput. Meth. Prog. Biomed., 98, pp. 151–160. [CrossRef]
Voltairas, P. A., Fotiadis, D. I., Massalas, C. V., and Michalis, L. K., 2005, “Anharmonic Analysis of Arterial Blood Pressure and Flow Pulses,” J. Biomech., 38, pp. 1423–1431. [CrossRef] [PubMed]
Cinthio, M., Ahlgren, Å. R., Bergkvist, J., Jansson, T., Persson, H. W., and Lindström, K., 2006, “Longitudinal Movements and Resulting Shear Strain of the Arterial Wall,” Am. J. Physiol. Heart Circ. Physiol., 291, pp. 394–402. [CrossRef]
Daidzic, N. E., and Hossain, M. S., 2010, “The Model of Micro-Fluidic Pump With Vibrating Boundaries,” Proc. 14th AMME Conference, Cairo, Egypt, May 25–27, AM-054 (MP-9).
Hossain, M. S., and Daidzic, N. E., 2012, “The Shear-Driven Fluid Motion Using Oscillating Boundaries,” ASME J. Fluids Eng., 134(5), pp. 111–122.
Vennemann, P., Lindken, R., and Westerweel, J., 2007, “In vivo Whole-Field Blood Velocity Measurement Techniques,” Exp. Fluids, 42, pp. 495–511. [CrossRef]
Daidzic, N. E., Schmidt, E., Hasan, M. M., and Altobelli, S., 2005, “Gas-Liquid Phase Distribution and Void Fraction Measurements Using MRI,” Nucl. Eng. Des., 235, pp. 1163–1178. [CrossRef]
Lemmin, U., and Rolland, T., 1997, “Acoustic Velocity Profiler for Laboratory and Field Studies,” J. Hydraul. Eng., 123(12), pp. 1089–1097. [CrossRef]
Takeda, Y., 1999, “Ultrasonic Doppler Method or Velocity Profile Measurement in Fluid Dynamics and Fluid Engineering,” Exp. Fluids, 26, pp. 177–178. [CrossRef]
Takeda, Y. (ed.), 2012, Ultrasonic Doppler Velocity Profiler for Fluid Flow, Springer, Berlin.
Longo, S., 2006, “The Effects of Air Bubbles on Ultrasound Velocity Measurements,” Exp. Fluids, 41, pp. 593–602. [CrossRef]
Morgan, G. W., and Kiely, J. P., 1954, “Wave Propagation in a Viscous Liquid Contained in a Flexible Tube,” J. Acoust. Soc. Amer., 26(3), pp. 323–328. [CrossRef]
White, F. M., 1991, Viscous Fluid Flow, McGraw-Hill, New York.
Dwight, H. B., 1961, Tables of Integrals and other Mathematical Data, Macmillan, New York.
Abramowitz, M., and Stegun, I. A., 1972, Handbook of Mathematical Functions, Dover, New York.
Schlichting, H., and Gersten, K., 2001, Boundary-Layer Theory, Springer, Berlin.
Ramnarine, K. V., Nassiri, D. K., Hoskins, P. R., and Lubbers, J., 1998, “Validation of a New Blood-Mimicking Fluid for Use in Doppler Flow Test Objects,” Ultrasound Med. Biol., 24(3), pp. 451–459. [CrossRef] [PubMed]
Press, W. H., Vetterling, W. T., Teukolsky, S. A., and Flannery, B. P., 1992, Numerical Recipes in FORTRAN: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK.
Iserles, A., and Nørsett, S. P., 2004, “On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation,” BIT Num. Math., 44, pp. 755–772. [CrossRef]
McDonald, D. A., 1955, “The Relation of Pulsatile Pressure to Flow in Arteries,” J. Physiol., 127, pp. 533–552. [PubMed]
Hale, J. F., McDonald, D. A., and Womersley, J. R., 1955, “Velocity Profiles of Oscillating Arterial Flow, With Some Calculations of Viscous Drag and the Reynolds Number,” J. Physiol., 128, pp. 629–640. [PubMed]

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

The experimental hydraulic test rig utilizing multigate PW Doppler US

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 12

Reconstructed flow data for triangular flow excitation using first ten harmonics

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In