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Research Papers

A Theoretical Study of Mechanical Stability of Arteries

[+] Author and Article Information
Alexander Rachev

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, 315 Ferst Drive, Atlanta, GA 30332alexander.rachv@me.gatech.edu

J Biomech Eng 131(5), 051006 (Mar 27, 2009) (10 pages) doi:10.1115/1.3078188 History: Received April 15, 2008; Revised December 23, 2008; Published March 27, 2009

This study proposes a mathematical model for studying stability of arteries subjected to a longitudinal extension and a periodic pressure. An artery was considered as a straight composite beam comprised of an external thick-walled tube and a fluid core. The dynamic criterion for stability was used, based on analyzing the small transverse vibrations superposed on the finite deformation of the vessel under static load. In contrast to the case of a static pressurization, in which buckling is only possible if the load produces a critical axial compressive force, a loss of stability of arteries under periodic pressure occurs under many combinations of load parameters. Instability occurs as a parametric resonance characterized by an exponential increase in the amplitude of transverse vibrations over several bands of pressure frequencies. The effects of load parameters were analyzed on the basis of the results for a dynamic and static stability of a rabbit thoracic aorta. Under normal physiological loads the artery is in a stable configuration. Static instability occurs under high distending pressures and low longitudinal stretch ratios. When the artery is subjected to periodic pressure, an independent increase in the mean pressure, amplitude of the periodic pressure, or frequency, most often, but not always, increases the risk of stability loss. In contrary, an increase in longitudinal stretch ratio most likely, but not certain, stabilizes the vessel. It was shown that adaptive geometrical remodeling due to an increase in mean pressure and flow does not affect artery stability.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 2

Strutt’s diagram. Stable (S) and unstable (US) domains.

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Figure 1

Free body diagram of an arterial segment before stability loss considered as a composite beam (upper figure), and the deformed configuration of the beam due to a transverse deformation (lower figure)

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Figure 3

Critical static pressure versus initial length for a rabbit thoracic aorta at different longitudinal stretch ratios

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Figure 4

Critical static pressure versus longitudinal stretch ratio for a rabbit thoracic aorta. The initial length is 40 mm.

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Figure 5

Time course of displacement f(t̃) (a) and phase portrait f(t̃)−df(t̃)/dt̃ (b) for a rabbit thoracic aorta. Mean pressure P¯=13.33 kPa, longitudinal stretch ratio λ=1.7, periodic pressure amplitude Pa=2.67 kPa, and frequency f=3.5 Hz. Initial conditions f(0)=0.01 mm and df/dt̃(0)=0.

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Figure 6

Trajectories of the representative point for a rabbit thoracic aorta due to variation in a single pressure parameter. Curve M1-M2 corresponds to mean pressure P¯ from 8.00 kPa to 26.67 kPa, curve M3-M4 corresponds to pressure amplitude Pa from 1.00 kPa to 4.00 kPa, and curve M5-M6 corresponds to frequency f from 2 Hz to 7 Hz. Point M0 corresponds to P¯=13.33 kPa, Pa=2.67 kPa, and f=3.5 Hz. The initial length is 40 mm and the longitudinal stretch ratio λ=1.7, n=1.

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Figure 7

Time course of displacement f(t̃) (a) and phase portrait f(t̃)−df(t̃)/dt̃ (b) for a rabbit thoracic aorta with an initial length of 40 mm. Mean pressure P¯=13.33 kPa, longitudinal stretch ratio λ=1.7, periodic pressure amplitude Pa=2.67 kPa, and frequency f=7.0 Hz. Initial conditions f(0)=0.01 mm and df/dt̃(0)=0. The corresponding representative point is M6 in Fig. 6.

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Figure 8

Trajectories of the representative point for a rabbit thoracic aorta calculated for the adapted geometrical dimensions. The mean pressure and flow were independently varied from 13.33 kPa to 26.67 kPa (curve M0P), and from the baseline value of flow rate to its fivefold increase (curve M0Q). The trajectory of the representative point (curve L1L2) for longitudinal stretch ratio from 1.625 to 1.725 at baseline values of pressure parameters. Point M0 corresponds to P¯=13.33 kPa, Pa=2.67 kPa, f=3.5 Hz, and longitudinal stretch ratio λ=1.7 Points K1 and K2 correspond to P¯=13.33 kPa, Pa=2.67 kPa, f=2.7 Hz, λ=1.7, and initial length of 40 mm and 45 mm, respectively.

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Figure 9

Schematic representation of the arterial cross section (a) at the zero-stress state and (b) at the current deformed state

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