Skip to main content

The effect of velocity filtering in pressure estimation

Abstract

Velocity field measurements allow, in principle, the evaluation of the pressure field by integrating the equations of fluid motion. Unavoidable experimental uncertainty, however, may result in unreliable estimates. In this study, we use the Poisson pressure equation to estimate the relative pressure from experimental velocities, and investigate how pre-processing with smoothing and solenoidal filters affects this estimate. For diffusion dominated laminar flow or for turbulent flow modeled through an eddy viscosity, measurement noise significantly affects the results. In this case, solenoidal filtering provides superior performance over other smoothing approaches, as it preserves the second spatial derivatives of the velocity field. For laminar flows dominated by advection or acceleration components of the pressure gradient, the choice of the filter appears to have little effect under limited noise, while smoothing produces improved relative pressure estimates for higher noise intensities. The above statements are verified using idealized flow conditions, numerical fluid dynamics simulations, and velocity fields from in-vivo and in-vitro magnetic resonance velocimetry.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

References

  1. Adrian RJ (2005) Twenty years of particle image velocimetry. Exp Fluids 39(2):159–169

    Article  Google Scholar 

  2. Andersen AH, Kirsch JE (1996) Analysis of noise in phase contrast MR imaging. Med Phys 23(6):857–869

    Article  Google Scholar 

  3. Azijli I, Dwight RP (2015) Solenoidal filtering of volumetric velocity measurements using Gaussian process regression. Exp Fluids 56(11):1–18

    Article  Google Scholar 

  4. Banko AJ, Coletti F, Schiavazzi D, Elkins CJ, Eaton JK (2015) Three-dimensional inspiratory flow in the upper and central human airways. Exp Fluids 56(6):1–12

    Article  Google Scholar 

  5. Bermejo J, Martínez-Legazpi P, del Álamo JC (2015) The clinical assessment of intraventricular flows. Annu Rev Fluid Mech 47:315–342

    Article  Google Scholar 

  6. Blinde P, Michaelis D, Van Oudheusden B, Weiss PE, de Kat R, Laskari A, Jeon JY, David L, Schanz D, Huhn F, Gesemann S, Novara M, McPhaden C, Neeteson N, Rival D, Schneiders JFG, Schrijer F (2016) Comparative assessment of PIV-based pressure evaluation techniques applied to a transonic base flow. In: 18th international symposium on applications of laser techniques to fluid mechanics

  7. Briant JK, Cohen BS (1989) Flow distribution through human and canine airways during inhalation and exhalation. J Appl Physiol 67(4):1649–1654

    Google Scholar 

  8. Charonko JJ, King CV, Smith BL, Vlachos PP (2010) Assessment of pressure field calculations from particle image velocimetry measurements. Meas Sci Technol 21(10):105401

    Article  Google Scholar 

  9. Cohen BS, Sussman RG, Lippmann M (1993) Factors affecting distribution of airflow in a human tracheobronchial cast. Respir Physiol 93(3):261–278

    Article  Google Scholar 

  10. Coletti F, Muramatsu K, Schiavazzi D, Elkins CJ, Eaton JK (2014) Fluid flow and scalar transport through porous fins. Phys Fluids 26(5):055104

    Article  Google Scholar 

  11. Comer JK, Kleinstreuer C, Zhang Z (2001) Flow structures and particle deposition patterns in double-bifurcation airway models. Part 1. Air flow fields. J Fluid Mech 435:25–54

    MATH  Google Scholar 

  12. Dabiri JO, Bose S, Gemmell BJ, Colin SP, Costello JH (2013) An algorithm to estimate unsteady and quasi-steady pressure fields from velocity field measurements. J Exp Biol 217:331–336

  13. Deardorff JW (1970) A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J Fluid Mech 41(02):453–480

    Article  MATH  Google Scholar 

  14. Donati F, Figueroa CA, Smith NP, Lamata P, Nordsletten DA (2015) Non-invasive pressure difference estimation from PC-MRI using the work-energy equation. Med Image Anal 26(1):159–172

    Article  Google Scholar 

  15. Ebbers T, Farnebäck G (2009) Improving computation of cardiovascular relative pressure fields from velocity MRI. J Magn Reson Imaging 30(1):54–61

    Article  Google Scholar 

  16. Ebbers T, Wigström L, Bolger AF, Engvall J, Karlsson M (2001) Estimation of relative cardiovascular pressures using time-resolved three-dimensional phase contrast MRI. Magn Reson Med 45(5):872–879

    Article  Google Scholar 

  17. Elkins CJ, Alley MT (2007) Magnetic resonance velocimetry: applications of magnetic resonance imaging in the measurement of fluid motion. Exp Fluids 43(6):823–858

    Article  Google Scholar 

  18. Foucaut JM, Stanislas M (1058) Some considerations on the accuracy and frequency response of some derivative filters applied to particle image velocimetry vector fields. Meas Sci Technol 13(7):2002

    Google Scholar 

  19. Freudenhammer D, Baum E, Peterson B, Böhm B, Jung B, Grundmann S (2014) Volumetric intake flow measurements of an IC engine using magnetic resonance velocimetry. Exp Fluids 55(5):1–18

    Article  Google Scholar 

  20. Gresho PM, Sani RL (1987) On pressure boundary conditions for the incompressible Navier-Stokes equations. Int J Numer Methods Fluids 7(10):1111–1145

    Article  MATH  Google Scholar 

  21. Grundmann S, Wassermann F, Lorenz R, Jung B, Tropea C (2012) Experimental investigation of helical structures in swirling flows. Int J Heat Fluid Flow 37:51–63

    Article  Google Scholar 

  22. Gudbjartsson H, Patz S (1995) The Rician distribution of noisy MRI data. Magn Reson Med 34(6):910–914

    Article  Google Scholar 

  23. Ham F, Iaccarino G (2004) Energy conservation in collocated discretization schemes on unstructured meshes. Annu Res Briefs 3–14:2004

    Google Scholar 

  24. Ham F, Mattsson K, Iaccarino G (2006) Accurate and stable finite volume operators for unstructured flow solvers. In: Annual Research Briefs, Center for Turbulence Research, Stanford University, pp 243–261

  25. Hearst RJ, Buxton ORH, Ganapathisubramani B, Lavoie P (2012) Experimental estimation of fluctuating velocity and scalar gradients in turbulence. Exp fluids 53(4):925–942

    Article  Google Scholar 

  26. Heroux M, Bartlett R, Howle V, Hoekstra R, Hu J, Kolda T, Lehoucq R, Long K, Pawlowski R, Phipps E, Salinger A, Thornquist H, Tuminaro R, Willenbring J, Williams A (2003) An overview of trilinos. Technical Report SAND2003-2927, Sandia National Laboratories

  27. Jalal S, Nemes A, Van de Moortele T, Schmitter S, Coletti F (2016) Three-dimensional inspiratory flow in a double bifurcation airway model. Exp Fluids 57(9):148

    Article  Google Scholar 

  28. Kassinos S, Langer C, Iaccarino G, Moin P (2007) Complex effects in large eddy simulations. Springer Science and Business Media, Berlin

    Book  MATH  Google Scholar 

  29. Krittian SBS, Lamata P, Michler C, Nordsletten DA, Bock J, Bradley CP, Pitcher A, Kilner PJ, Markl M, Smith NP (2012) A finite-element approach to the direct computation of relative cardiovascular pressure from time-resolved MR velocity data. Med Image Anal 16(5):1029–1037

    Article  Google Scholar 

  30. Lamata P, Pitcher A, Krittian S, Nordsletten D, Bissell MM, Cassar T, Barker AJ, Markl M, Neubauer S, Smith NP (2014) Aortic relative pressure components derived from four-dimensional flow cardiovascular magnetic resonance. Magn Reson Med 72(4):1162–1169

    Article  Google Scholar 

  31. Laskari A, de Kat R, Ganapathisubramani B (2016) Full-field pressure from snapshot and time-resolved volumetric PIV. Exp Fluids 57(3):1–14

    Article  Google Scholar 

  32. Lilly DK (1967) The representation of small scale turbulence in numerical simulation experiments. In: IBM scientific computing symposium on environmental sciences, IBM, Data processing Division, pp 195–210

  33. Liu X, Katz J (2006) Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. Exp Fluids 41(2):227–240

    Article  Google Scholar 

  34. Longest PW, Vinchurkar S (2007) Effects of mesh style and grid convergence on particle deposition in bifurcating airway models with comparisons to experimental data. Med Eng Phys 29(3):350–366

    Article  Google Scholar 

  35. Markl M, Chan FP, Alley MT, Wedding KL, Draney MT, Elkins CJ, Parker DW, Wicker R, Taylor CA, Herfkens RJ et al (2003) Time-resolved three-dimensional phase-contrast MRI. J Magn Reson Imaging 17(4):499–506

    Article  Google Scholar 

  36. Markl M, Kilner PJ, Ebbers T (2011) Comprehensive 4D velocity mapping of the heart and great vessels by cardiovascular magnetic resonance. J Cardiovasc Magn Reson 13(7):10–1186

    Google Scholar 

  37. Pelc NJ, Herfkens RJ, Shimakawa A, Enzmann DR (1991) Phase contrast cine magnetic resonance imaging. Magn Reson Q 7(4):229–254

    Google Scholar 

  38. Pelc NJ, Sommer FG, Li KC, Brosnan TJ, Herfkens RJ, Enzmann DR (1994) Quantitative magnetic resonance flow imaging. Magn Reson Q 10(3):125–147

    Google Scholar 

  39. Pozrikidis C (2001) A note on the regularization of the discrete Poisson-Neumann problem. J Comput Phys 172(2):917–923

    Article  MATH  MathSciNet  Google Scholar 

  40. Schiavazzi D, Coletti F, Iaccarino G, Eaton JK (2014) A matching pursuit approach to solenoidal filtering of three-dimensional velocity measurements. J Comput Phys 263:206–221

    Article  MATH  MathSciNet  Google Scholar 

  41. Schneiders JFG, Pröbsting S, Dwight RP, van Oudheusden BW, Scarano F (2016) Pressure estimation from single-snapshot tomographic PIV in a turbulent boundary layer. Exp Fluids 57(4):1–14

    Article  Google Scholar 

  42. Schroeder WJ, Lorensen B, Martin K (2004) The visualization toolkit. Kitware, New York

    Google Scholar 

  43. Sherman TF (1981) On connecting large vessels to small. The meaning of Murray’s law. J Gen Physiol 78(4):431–453

    Article  Google Scholar 

  44. Smagorinsky J (1963) General circulation experiments with the primitive equations: I. The basic experiment. Mon Weather Rev 91(3):99–164

    Article  Google Scholar 

  45. Song SM, Leahy RM, Boyd DP, Brundage BH, Napel S (1994) Determining cardiac velocity fields and intraventricular pressure distribution from a sequence of ultrafast CT cardiac images. Med Imaging IEEE Trans 13(2):386–397

    Article  Google Scholar 

  46. Vahanian A, Baumgartner H, Bax J, Butchart E, Dion R, Filippatos G, Flachskampf F, Hall R, Iung B, Kasprzak J et al (2007) Guidelines on the management of valvular heart disease. Eur Heart J 28(2):230–268

    Google Scholar 

  47. Van Oudheusden BW (2013) PIV-based pressure measurement. Meas Sci Technol 24(3):032001

    Article  Google Scholar 

  48. Weibel ER (1963) Geometry and dimensions of airways of conductive and transitory zones. Springer, New York

    Google Scholar 

  49. Whiting CH, Jansen KE (2001) A stabilized finite element method for the incompressible Navier-Stokes equations using a hierarchical basis. Int J Numer Methods Fluids 35(1):93–116

    Article  MATH  Google Scholar 

  50. Wilcox DC (1998) Turbulence modeling for CFD. DCW industries, La Canada Flintridge

    Google Scholar 

  51. Wilson N, Wang K, Dutton RW, Taylor C (2001) A software framework for creating patient specific geometric models from medical imaging data for simulation based medical planning of vascular surgery. In: Medical image computing and computer-assisted intervention-MICCAI 2001, Springer, pp 449–456

  52. Worth P, Longest, Vinchurkar S (2007) Validating CFD predictions of respiratory aerosol deposition: effects of upstream transition and turbulence. J Biomech 40(2):305–316

    Article  Google Scholar 

  53. Yang GZ, Kilner PJ, Wood NB, Underwood SR, Firmin DN (1996) Computation of flow pressure fields from magnetic resonance velocity mapping. Magn Reson Med 36(4):520–526

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the two anonymous reviewers for their comments and feedback that greatly contributed to improve the consistency and quality of the present contribution. This work was supported by the American Heart Association Grant #15POST23010012 (Daniele Schiavazzi), the Leducq Foundation as part of a Transatlantic Network of Excellence for Cardiovascular Research, and a National Science Foundation (Chemical, Bioengineering, Environmental, and Transport Systems) CAREER Grant #1453538 (Filippo Coletti). The authors would like to thank Gianluca Iaccarino, Javier Urzay Lobo, Gianluca Geraci, and Dante De Santis for the interesting and motivating discussions, and Jorge Bernate for providing the large eddy simulation results on the human airways model. We also acknowledge the open source SimVascular project at http://www.simvascular.org.

Author information

Affiliations

Authors

Corresponding author

Correspondence to F. Coletti.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Schiavazzi, D.E., Nemes, A., Schmitter, S. et al. The effect of velocity filtering in pressure estimation. Exp Fluids 58, 50 (2017). https://doi.org/10.1007/s00348-017-2314-1

Download citation

Keywords

  • Velocity Field
  • Large Eddy Simulation
  • Relative Pressure
  • Neumann Boundary Condition
  • Computational Fluid Dynamic Simulation