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Technical Briefs

# A Generalized Maxwell Model for Creep Behavior of Artery Opening Angle

[+] Author and Article Information
W. Zhang, X. Guo

Department of Biomedical Engineering, Indiana University Purdue University Indianapolis, 723 West Michigan Street, Indianapolis, IN 46202

G. S. Kassab1

Department of Biomedical Engineering, Department of Surgery, and Department of Cellular and Integrative Physiology, Indiana University Purdue University Indianapolis, 723 West Michigan Street, Indianapolis, IN 46202gkassab@iupui.edu

1

Corresponding author.

J Biomech Eng 130(5), 054502 (Sep 17, 2008) (5 pages) doi:10.1115/1.2979853 History: Received November 08, 2007; Revised March 28, 2008; Published September 17, 2008

## Abstract

An artery ring springs open into a sector after a radial cut. The opening angle characterizes the residual strain in the unloaded state, which is fundamental in understanding stress and strain in the vessel wall. A recent study revealed that the opening angle decreases with time if the artery is cut from the loaded state, while it increases if the cut is made from the no-load state due to viscoelasticity. In both cases, the opening angle approaches the same value in $3h$. This implies that the characteristic relaxation time is about $10,000s$. Here, the creep function of a generalized Maxwell model (a spring in series with six Voigt bodies) is used to predict the temporal change of opening angle in multiple time scales. It is demonstrated that the theoretical model captures the salient features of the experimental results. The proposed creep function may be extended to study the viscoelastic response of blood vessels under various loading conditions.

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## Figures

Figure 7

Temporal evolution of opening angles measured in experiments and predicted by the model

Figure 6

The strain recovery after the radial cut at (a) the inner surface with β=0.025 and (b) the outer surface with β=0.006. εr (Eq. 5) and εn (Eq. 7) are theoretical predictions with m=6, ρ=3.6, and τ=4.0s

Figure 5

Reduced creep function j(t) and reduced relaxation function g(t) for m=6, ρ=3.6, and τ=4.0s

Figure 4

The plots of tan(δ) versus log(ωτ0) of the generalized Maxwell model for τ=4.0s and β=0.025. Note that τ0=1.0s has been used to nondimensionalize the abscissa.

Figure 3

A generalized Maxwell viscoelastic model (a linear spring in serial with m Voigt elements)

Figure 2

Photographs of a porcine coronary artery at (a) the loaded state with hardened elastomer in the lumen and (b) the zero-stress state where the opening angle is larger than 180deg

Figure 1

An artery at the (a) no-load and (b) zero-stress states. ri and roRi and Ro are the inner and outer radii in the (r, θ, z) and (R, Θ, Z) coordinate systems, respectively, and Φ denotes the opening angle.

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