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Research Papers

One-Dimensional Model for Propagation of a Pressure Wave in a Model of the Human Arterial Network: Comparison of Theoretical and Experimental Results

[+] Author and Article Information
Masashi Saito, Yuki Ikenaga, Yoshiaki Watanabe, Takaaki Asada, Pierre-Yves Lagrée

Laboratory of Ultrasonic Electronics,  Doshisha University, 1-3 Tatara-Miyakodani, Kyotanabeshi, Kyoto, 610-0321, JapanMurata Manufacturing Co., Ltd. 10-1, Higashi Kotari 1-chome, Nagaokakyoshi, Kyoto 617-8555, Japan e-mail: asada@murata.co.jp CNRS and Université Pierre et Marie Curie Paris 06, Institut Jean le Rond d’Alembert, Boîte 162, 4 place Jussieu, 75252 Paris, France e-mail: pierre-yves.lagree@upmc.fr

Mami Matsukawa1

Laboratory of Ultrasonic Electronics,  Doshisha University, 1-3 Tatara-Miyakodani, Kyotanabeshi, Kyoto, 610-0321, Japanmmatsuka@mail.doshisha.ac.jpMurata Manufacturing Co., Ltd. 10-1, Higashi Kotari 1-chome, Nagaokakyoshi, Kyoto 617-8555, Japan e-mail: asada@murata.co.jpmmatsuka@mail.doshisha.ac.jp CNRS and Université Pierre et Marie Curie Paris 06, Institut Jean le Rond d’Alembert, Boîte 162, 4 place Jussieu, 75252 Paris, France e-mail: pierre-yves.lagree@upmc.frmmatsuka@mail.doshisha.ac.jp

1

Corresponding author.

J Biomech Eng 133(12), 121005 (Dec 23, 2011) (9 pages) doi:10.1115/1.4005472 History: Received July 11, 2011; Revised November 16, 2011; Published December 23, 2011; Online December 23, 2011

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Diagram and details of the viscoelastic tubes and the flow input. The pulse flow was input from the left and pressure waves were measured at points 1, 2, and 3. The distance between the input point and the measurement points was 27.5 cm (point 1), 55.0 cm (point 2), and 83.0 cm (point 3).

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Figure 2

Time series of pressure waves in viscoelastic tubes measured at three points for tubes (a) A, (b) B, (c) C, and (d ) D

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Figure 3

Discrete values of the radius (R) and flux (Q) in mother and daughter tubes. Superscripts A and B denote daughter tubes A and B. Subscripts denote the position in the tube.

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Figure 4

Example of cost functions calculated from elasticity E0 /(1 − σ2 ) = 240, 250, and 260 kPa, attenuation ɛ¯ν=0.01-0.08, and nonlinearity ɛ¯p=0.01-0.08. The optimum parameters for pressure waves in tube A were estimated as E0 /(1 − σ2 ) = 250 kPa, ɛ¯ν=0.034, and ɛ¯p=0.046.

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Figure 5

Experimental and theoretical pressure waves as a function of time (tube A). The optimum parameters used were estimated as E0 /(1 – σ2 ) = 250 kPa, ɛ¯ν=0.034, and ɛ¯p = 0.046.

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Figure 6

Comparison of the wave measured at point 1 and theoretical waves calculated for four conditions: (a) E0 /(1 − σ2 ) = 250 kPa, ɛ¯ν = 0.0, ɛ¯p = 0.0, and no fluid convection effect; (b) E0 /(1 − σ2 ) = 250 kPa, ɛ¯ν = 0.034, ɛ¯p = 0.0, and no fluid convection effect; (c) E0 /(1 − σ2 ) = 250 kPa, ɛ¯ν = 0.034, ɛ¯p = 0.0, and a fluid convection effect; and (d) E0 /(1 − σ2 ) = 250 kPa, ɛ¯ν = 0.034, ɛ¯p = 0.046, an no fluid convection effect.

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Figure 7

Experimental and theoretical pressure waves in tube B. The optimum parameters used were estimated as E0 /(1 − σ2 )  = 75 kPa, ɛ¯ν = 0.070, and ɛ¯p = 0.070.

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Figure 8

Experimental and theoretical pressure waves in tube C. The optimum parameters used were estimated as E0 /(1 − σ2 ) = 250 kPa, ɛ¯ν = 0.034, and ɛ¯p = 0.046.

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Figure 9

Experimental and theoretical pressure waves in tube D. The optimum parameters used were estimated as E0 /(1 − σ2 )  = 250 kPa, ɛ¯ν = 0.034, and ɛ¯p = 0.046.

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Figure 10

Effect of fluid viscosity on the pressure wave at point 1. An increase in flow viscosity involving a coefficient of (a) 1.0, (b) 2.0, or (c) 3.0 leads to marked augmentation of the offset level.

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Figure 11

Comparison of velocity profiles for a Womersley flow with α = 12.94 and a Hagen-Poiseuille flow of the same flux

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Figure 12

Structure of the simple arterial network and part details. Silicone tubes were connected to the flexible tubes to model reflection points. The measurement position was at the left carotid artery.

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Figure 13

Time series of measured and simulated pressure waves in the simple human artery model. The optimum parameters used were estimated as E0 /(1 − σ2 ) = 170 kPa, ɛ¯ν = 0.036, and ɛ¯p = 0.010.

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Figure 14

Time series of measured and simulated flow velocity waves in the simple human artery model. The optimum parameters used were estimated as E0 /(1 − σ2 ) = 170 kPa, ɛ¯ν = 0.036, and ɛ¯p = 0.010.

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