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Research Papers

# A New Three-Dimensional Exponential Material Model of the Coronary Arterial Wall to Include Shear Stress Due to Torsion

[+] Author and Article Information
J. Scott Van Epps

Department of Surgery, Department of Bioengineering, the McGowan Institute for Regenerative Medicine, and the Center for Vascular Remodeling and Regeneration, University of Pittsburgh, Pittsburgh, PA 15219

David A. Vorp1

Department of Surgery, Department of Bioengineering, the McGowan Institute for Regenerative Medicine, and the Center for Vascular Remodeling and Regeneration, University of Pittsburgh, Pittsburgh, PA 15219vorpda@upmc.edu

1

Corresponding author.

J Biomech Eng 130(5), 051001 (Jul 10, 2008) (8 pages) doi:10.1115/1.2948396 History: Received June 28, 2007; Revised January 21, 2008; Published July 10, 2008

## Abstract

The biomechanical milieu of the coronary arteries is unique in that they experience mechanical deformations of twisting, bending, and stretching due to their tethering to the epicardial surface. Spatial variations in stresses caused by these deformations could account for the heterogeneity of atherosclerotic plaques within the coronary tree. The goal of this work was to utilize previously reported shear moduli to calculate a shear strain parameter for a Fung-type exponential model of the arterial wall and determine if this single constant can account for the observed behavior of arterial segments under torsion. A Fung-type exponential strain-energy function was adapted to include a torsional shear strain term. The material parameter for this term was determined from previously published data describing the relationship between shear modulus and circumferential stress and longitudinal stretch ratio. Values for the shear strain parameter were determined for three geometries representing the mean porcine left anterior descending coronary artery dimensions plus or minus one standard deviation. Finite element simulation of triaxial biomechanical testing was then used to validate the model. The mean value calculated for the shear strain parameter was $0.0759±0.0009$ ($N=3$ geometries). In silico triaxial experiments demonstrated that the shear modulus is directly proportional to the applied pressure at a constant longitudinal stretch ratio and to the stretch ratio at a constant pressure. Shear moduli determined from these simulations showed excellent agreement to shear moduli reported in literature. Previously published models describing the torsional shear behavior of porcine coronary arteries require a total of six independent constants. We have reduced that description into a single parameter in a Fung-type exponential strain-energy model. This model will aid in the estimation of wall stress distributions of vascular segments undergoing torsion, as such information could provide insight into the role of mechanical stimuli in the localization of atherosclerotic plaque formation.

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## Figures

Figure 1

(A) Schematic of an arterial segment of length L and inner and outer radii ri and ro, respectively, acted upon by a longitudinal stretch λ, an intraluminal pressure P, and a torque T, which generates a twist angle θ. (B) Schematic of a sector of an arterial ring with inner and outer radii Ri and Ro, respectively, and an opening angle Φ.

Figure 2

In silico triaxial experimental protocol demonstrated by a representative vessel geometry. Two views are shown of a hemivessel at each step with an angle of twist plotted as gray scale. An end view (top) and a side view into the open end of the hemivessel (bottom). (A) Vessel segment in the unloaded configuration; (B) stretched to the desired λ; (C) pressurized to the desired P; (D) twisted on the upper end by θ=−25deg; (E) and twisted in the opposite direction, θ=25deg.

Figure 3

Plots of computed inner (A) and outer (B) radii versus pressure for the three porcine LAD geometries (small, medium, and large) at stretch ratios of 1.2, 1.3, and 1.4

Figure 4

Calculated values for the parameter c7 for each porcine LAD geometry (small, medium, and large). The bars represent the average value from each combination of pressure (0kPa, 3kPa, 5kPa, 8kPa, 10.7kPa, 13.3kPa, and 16kPa) and stretch ratio (1.2, 1.3, and 1.4), a total of 21 combinations. The error bars represent the standard error of the mean.

Figure 5

Plots of torque versus polar moment of inertia (γ) for various stretch ratios at constant pressure (top) and various pressures at constant stretch ratio (bottom) for each of the three vessel geometries (small, medium, and large)

Figure 6

Shear modulus (G) as a function of pressure for various stretch ratios for each of the three vessel geometries (small, medium, and large). Individual points represent shear moduli determined by linear functions of mean circumferential wall stress from Lu (24). The error bars represent a 95% confidence interval. The lines represent shear moduli predicted from the in silico triaxial experiments using W defined by Eqs. 4,6; i.e., including the shear strain term (c7=0.0759).

Figure 7

Mesh independence analysis. The radial distribution of shear stress (σθz) for the nodes at the twist end of the vessel is shown for three different mesh densities (coarse, medium, and fine).

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