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Research Papers

Constitutive Relations for Pressure-Driven Stiffening in Poroelastic Tissues

[+] Author and Article Information
Adam M. Reeve

Auckland Bioengineering Institute and
Department of Engineering Science,
University of Auckland,
Auckland 1010, New Zealand
e-mail: aree035@aucklanduni.ac.nz

Martyn P. Nash

Faculty of Engineering,
Auckland Bioengineering Institute and
Department of Engineering Science,
University of Auckland,
Auckland 1010, New Zealand
e-mail: martyn.nash@auckland.ac.nz

Andrew J. Taberner

Faculty of Engineering,
Auckland Bioengineering Institute and
Department of Engineering Science,
University of Auckland,
Auckland 1010, New Zealand
e-mail: a.taberner@auckland.ac.nz

Poul M. F. Nielsen

Faculty of Engineering,
Auckland Bioengineering Institute and
Department of Engineering Science,
University of Auckland,
Auckland 1010, New Zealand
e-mail: p.nielsen@auckland.ac.nz

Manuscript received November 18, 2013; final manuscript received April 2, 2014; accepted manuscript posted May 14, 2014; published online June 13, 2014. Assoc. Editor: Guy M. Genin.

J Biomech Eng 136(8), 081011 (Jun 13, 2014) (9 pages) Paper No: BIO-13-1545; doi: 10.1115/1.4027666 History: Received November 18, 2013; Revised April 02, 2014; Accepted May 14, 2014

Vascularized biological tissue has been shown to increase in stiffness with increased perfusion pressure. The interaction between blood in the vasculature and other tissue components can be modeled with a poroelastic, biphasic approach. The ability of this model to reproduce the pressure-driven stiffening behavior exhibited by some tissues depends on the choice of the mechanical constitutive relation, defined by the Helmholtz free energy density of the skeleton. We analyzed the behavior of a number of isotropic poroelastic constitutive relations by applying a swelling pressure, followed by homogeneous uniaxial or simple-shear deformation. Our results demonstrate that a strain-stiffening constitutive relation is required for a material to show pressure-driven stiffening, and that the strain-stiffening terms must be volume-dependent.

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Figures

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Fig. 1

Uniaxial deformation of an isotropic, pressurized cube. The dotted lines indicate the initial, unperfused geometry; dashed lines indicate the perfused, swollen geometry; and solid lines indicate the deformed geometry under a homogeneous uniaxial extension along the x-axis.

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Fig. 2

First principal Cauchy stress against extension ratio for uniaxial extension at a range of perfusion pressures with (a) the Mooney–Rivlin constitutive relation (Eq. (14)) and (b) the exponential constitutive relation (Eq. (15))

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Fig. 3

(a) Volume ratio against perfusion pressure and (b) initial uniaxial stiffness against perfusion pressure for the Mooney–Rivlin (Eq. (14)) and exponential (Eq. (15)) constitutive relations

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Fig. 4

Uniaxial stiffness against perfusion pressure at a range of extension (λ) ratios when using the exponential constitutive relation (Eq. (15))

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Fig. 5

Initial shear stiffness against perfusion pressure for the Mooney–Rivlin (Eq. (14)) and exponential (Eq. (15)) constitutive relations

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Fig. 6

Initial uniaxial stiffness against perfusion pressure for the Mooney–Rivlin relation (Eq. (14)) with two sets of parameters. In the first, c2 = (c1/4), and in the second, c2 = 2c1.

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Fig. 7

Initial uniaxial stiffness against perfusion pressure for the exponential relation (Eq. (15)) with two sets of parameters. In the first, c2 = 2.0, and in the second, c2 = 0.1. For both simulations, c1 and K were set to 8.93 kPa and 17.1 kPa, respectively.

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Fig. 8

Initial uniaxial stiffness against perfusion pressure for the power-law relation with (a) a range of c3 values and (b) a range of c2 values

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Fig. 9

(a) First principal Cauchy stress against extension ratio at a range of perfusion pressures with the modified form of the exponential constitutive relation (Eq. (19)). (b) Initial uniaxial stiffness against perfusion pressure for the exponential (Eq. (15)) and modified exponential constitutive relations.

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