Tensile strip experiments of bovine corneas have shown that the tissue exhibits a nonlinear rate-dependent stress-strain response and a highly nonlinear creep response that depends on the applied hold stress. In this paper, we present a constitutive model for the finite deformation, anisotropic, nonlinear viscoelastic behavior of the corneal stroma. The model formulates the elastic and viscous response of the stroma as the average of the elastic and viscous response of the individual lamellae weighted by a probability density function of the preferred in-plane lamellar orientations. The result is a microstructure-based model that incorporates the viscoelastic properties of the matrix and lamellae and the lamellar architecture in the response of the stroma. In addition, the model includes a fully nonlinear description of the viscoelastic response of the lamellar(fiber) level. This is in contrast to previous microstructure-based models of fibrous soft tissues, which relied on quasilinear viscoelastic formulations of the fiber viscoelasticity. Simulations of recent tensile strip experiments show that the model is able to predict, well within the bounds of experimental error and natural variations, the cyclic stress-strain behavior and nonlinear creep behavior observed in uniaxial tensile experiments of excised strips of bovine cornea.
Issue Section:
Research Papers
Keywords:
cornea,
viscoelasticity,
anisotropy,
creep,
mechanical behavior,
tissue mechanics,
biomechanics,
creep,
elastic deformation,
eye,
physiological models,
probability,
proteins,
stress-strain relations,
tensile strength,
viscoelasticity
Topics:
Anisotropy,
Cornea,
Creep,
Deformation,
Density,
Stress,
Viscoelasticity,
Biological tissues,
Simulation,
Probability,
Strips,
Errors
1.
Maurice
, D. M.
, 1957, “The Structure and Transparency of the Cornea
,” J. Physiol. (London)
0022-3751, 136
, pp. 263
–286
.2.
Komai
, Y.
, and Ushiki
, T.
, 1991, “The Three-Dimensional Organisation of Collagen Fibrils in the Human Cornea and Sclera
,” Invest. Ophthalmol. Visual Sci.
0146-0404, 32
, pp. 2244
–2258
.3.
Boote
, C.
, Dennis
, S.
, Newton
, R. H.
, Puri
, H.
, and Meek
, K. M.
, 2003, “Collagen Fibrils Appear More Closely Packed in the Prepupillary Cornea: Optical and Biomechanical Implications
,” Invest. Ophthalmol. Visual Sci.
0146-0404, 44
, pp. 2941
–2948
.4.
Meek
, K. M.
, Blamiers
, T.
, Elliott
, G. F.
, Gyi
, T. J.
, and Nave
, C.
, 1987, “The Organization of Collagen Fibrils in the Human Corneal Stroma: A Synchrotron X-Ray Diffraction Study
,” Curr. Eye Res.
0271-3683, 6
, pp. 841
–846
.5.
Daxer
, A.
, and Fratzl
, P.
, 1997, “Collagen Fibril Orientation in the Corneal Stroma and Its Implications in Keratoconous
,” Invest. Ophthalmol. Visual Sci.
0146-0404, 38
, pp. 121
–129
.6.
Aghamohammadzadeh
, H.
, Newton
, R. H.
, and Meek
, K. M.
, 2005, “X-Ray Scattering Used to Map the Preferred Collagen Orientation in the Human Cornea and Limbus
,” Structure (London)
0969-2126, 12
, pp. 249
–256
.7.
Boote
, C.
, Dennis
, S.
, Huang
, Y.
, Quantock
, A. J.
, and Meek
, K. M.
, 2005, “Lamellar Orientation in Human Cornea in Relation to Mechanical Properties
,” J. Struct. Biol.
1047-8477, 149
, pp. 1
–6
.8.
Nyquist
, G. W.
, 1968, “Rheology of the Cornea: Experimental Techniques and Results
,” Exp. Eye Res.
0014-4835, 7
, pp. 183
–188
.9.
Woo
, S. L.-Y.
, Kobayashi
, A. S.
, Schlegel
, W. A.
, and Lawrence
, C.
, 1972, “Nonlinear Material Properties of Intact Cornea and Sclera
,” Exp. Eye Res.
0014-4835, 14
, pp. 29
–39
.10.
Andreassen
, T. T.
, Simonsen
, A. H.
, and Oxlund
, H.
, 1980, “Biomechanical Properties of Keratoconous and Normal Corneas
,” Exp. Eye Res.
0014-4835, 31
, pp. 435
–441
.11.
Jue
, W.
, and Maurice
, D. M.
, 1991, “The Mechanical Properties of the Rabbit and Human Cornea
,” J. Biomech.
0021-9290, 24
, pp. 907
–922
.12.
Hoeltzel
, D. A.
, Altman
, P.
, Buzard
, K.
, and Choe
, K. I.
, 1992, “Strip Extensiometry for Comparison of the Mechanical Response of Bovine, Rabbit, and Human Corneas
,” ASME J. Biomech. Eng.
0148-0731, 114
, pp. 202
–215
.13.
Hjortdal
, J. O.
, 1996, “Regional Elastic Performance of the Human Cornea
,” J. Biomech.
0021-9290, 29
, pp. 931
–942
.14.
Shin
, T. J.
, Vito
, R. P.
, Johnson
, L.
, and McCarey
, B. E.
, 1997, “The Distribution of Strain in the Human Cornea
,” J. Biomech.
0021-9290, 30
, pp. 497
–503
.15.
Kampmeier
, J.
, Radt
, B.
, Birngruber
, R.
, and Brinkmann
, R.
, 2000, “Thermal and Biomechanical Parameters of Porcine Cornea
,” Cornea
0277-3740, 19
, pp. 355
–363
.16.
Wollensak
, G.
, Spoerl
, E.
, and Seiler
, T.
, 2003, “Stress-Strain Measurements of Human and Porcine Corneas After Riboflavin-Ultraviolet-A-Induced Cross-Linking
,” J. Cataract Refractive Surg.
0886-3350, 29
, pp. 1780
–1785
.17.
Boyce
, B. L.
, Jones
, R. E.
, Nguyen
, T. D.
, and Grazier
, J. M.
, 2007, “Stress-Controlled Viscoelastic Tensile Response of Bovine Cornea
,” J. Biomech.
0021-9290, 40
, pp. 2367
–2376
.18.
Hukins
, D. W. L.
, 1984, “Collagen Orientation
,” The Connective Tissue Matrix
, D. W. L.
Hukins
, ed., Macmillan
, London
, Vol. 5
, pp. 212
–240
.19.
Huang
, C. Y.
, Mow
, V. C.
, and Ateshian
, G. A.
, 2001, “The Role of Flow-Independent Viscoelasticity in the Biphasic Tensile and Compressive Response of Articular Cartilage
,” ASME J. Biomech. Eng.
0148-0731, 123
, pp. 410
–417
.20.
Liao
, J.
, and Vesely
, I.
, 2004, “Relationship Between Collagen Fibrils, Glycosaminoglycans, and Stress Relaxation in Mitral Valve Chordae Tendineae
,” J. Biomed. Eng.
0141-5425, 2004
, pp. 977
–983
.21.
Elliott
, D. M.
, Robinson
, P. S.
, Gimbel
, J. A.
, Sarver
, J. J.
, Abboud
, J. A.
, Iozzo
, R. V.
, and Soslowsky
, L. J.
, 2003, “Effect of Altered Matrix Proteins on Quasilinear Viscoelastic Properties in Transgenic Mouse Tail Tendons
,” Ann. Biomed. Eng.
0090-6964, 31
, pp. 599
–605
.22.
Puxkandl
, R.
, Zizak
, I.
, Paris
, O.
, Keckes
, J.
, Tesch
, W.
, Bernstorff
, S.
, Purslow
, P.
, and Fratzl
, P.
, 2002, “Viscoelastic Properties of Collagen: Synchrotron Radiation Investigations and Structural Model
,” Philos. Trans. R. Soc. London, Ser. B
0962-8436, 357
, pp. 191
–197
.23.
Pinsky
, P. M.
, and Datye
, D. V.
, 1991, “A Microstructurally-Based Finite Element Model of the Incised Human Cornea
,” J. Biomech.
0021-9290, 24
, pp. 907
–922
.24.
Pinsky
, P. M.
, van der Heide
, D.
, and Chernyak
, D.
, 2005, “Computational Modeling of Mechanical Anisotropy in the Cornea and Sclera
,” J. Cataract Refractive Surg.
0886-3350, 31
, pp. 136
–145
.25.
Guarnieri
, F. A.
, 1999, “Modelo Biomechanico del ojo Para Diseno Asistido por Computadora de la Cirurgua Refractive
,” Ph.D. thesis, Universidad Nacional del Litoral, Santa Fe, Argentina.26.
Pandolfi
, A.
, and Manganiello
, F.
, 2006, “A Model for the Human Cornea: Constitutive Formulation and Numerical Analysis
,” Biomech. Model. Mechanobiol.
, 5
, pp. 237
–246
.27.
Lanir
, Y.
, 1983, “Constitutive Equations for Fibrous Connective Tissues
,” J. Biomech.
0021-9290, 16
, pp. 1
–12
.28.
Gasser
, T. C.
, Ogden
, R. W.
, and Holzapfel
, G. A.
, 2006, “Hyperelastic Modelling of Arterial Layers With Distributed Collagen Fibre Orientations
,” J. R. Soc., Interface
1742-5689, 3
, pp. 15
–35
.29.
Pioletti
, D. P.
, Rakotomanana
, L. R.
, Benvenuti
, J. F.
, and Leyvraz
, P. F.
, 1998, “Viscoelastic Constitutive Law in Large Deformations: Application to Human Knee Ligaments and Tendons
,” J. Biomech.
0021-9290, 31
, pp. 753
–757
.30.
Holzapfel
, G. A.
, and Gasser
, T. C.
, 2000, “A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models
,” J. Elast.
0374-3535, 61
, pp. 1
–48
.31.
Bischoff
, J. E.
, Arruda
, E. M.
, and Grosh
, K.
, 2004, “A Rheological Network Model for the Continuum Anisotropic and Viscoelastic Behavior of Soft Tissue
,” Biomech. Model. Mechanobiol.
, 3
, pp. 56
–65
.32.
Limbert
, G.
, and Middleton
, J.
, 2004, “A Transversely Isotropic Viscohyperelastic Material: Application to the Modeling of Biological Soft Connective Tissues
,” Int. J. Solids Struct.
0020-7683, 2004
, pp. 4237
–4260
.33.
Merodio
, J.
, and Rajagopal
, K. R.
, 2007, “On Constitutive Equations for Anisotropic Nonlinearly Viscoelastic Solids
,” Math. Mech. Solids
1081-2865, 12
, pp. 131
–147
.34.
Lanir
, Y.
, 1980, “A Microstructure Model for the Rheology of Mammalian Tendon
,” ASME J. Biomech. Eng.
0148-0731, 102
, pp. 332
–339
.35.
Thornton
, G. M.
, Frank
, C. B.
, and Shrive
, N. G.
, 2001, “Ligament Creep Behavior Can be Predicted From Stress Relaxation by Incorporating Fiber Recruitment
,” J. Rheol.
0148-6055, 45
, pp. 493
–507
.36.
Sverdlik
, A.
, and Lanir
, Y.
, 2002, “Time-Dependent Mechanical Behavior of Sheep Digitial Tendons, Including the Effects of Preconditioning
,” ASME J. Biomech. Eng.
0148-0731, 124
, pp. 78
–84
.37.
Bischoff
, J. E.
, 2006, “Reduced Parameter Formulation for Incorporating Viscoelasticity Into Tissue Level Biomechanics
,” Ann. Biomed. Eng.
0090-6964, 34
, pp. 1164
–1172
.38.
Thornton
, G. M.
, Oliynyk
, A.
, Frank
, C. B.
, and Shrive
, N. G.
, 1997, “Ligament Creep Behavior Cannot be Predicted From Stress Relaxation at Low Stress: A Biomechanical Study of the Rabbit Medial Collateral Ligament
,” J. Orthop. Res.
0736-0266, 15
, pp. 652
–656
.39.
Provenzano
, P.
, Lakes
, R. S.
, Keenan
, T.
, and Vanderby
, R.
, Jr., 2001, “Nonlinear Ligament Viscoelasticity
,” Ann. Biomed. Eng.
0090-6964, 29
, pp. 908
–914
.40.
Hingorani
, R. V.
, Provenzano
, P.
, Lakes
, R. S.
, Escarcega
, A.
, and Vanderby
, R.
, Jr., 2004, “Nonlinear Viscoelasticity in Rabbit Medial Collateral Ligament
,” Ann. Biomed. Eng.
0090-6964, 32
, pp. 306
–312
.41.
Nguyen
, T. D.
, Jones
, R. E.
, and Boyce
, B. L.
, 2007, “Modeling the Anisotropic Finite-Deformation Viscoelastic Behavior of Soft Fiber-Reinforced Composites
,” Int. J. Solids Struct.
0020-7683, 44
, pp. 8366
–8389
.42.
Casey
, J.
, and Naghdi
, P. M.
, 1983, “On the Use of Invariance Requirements for Intermediate Configurations Associated With the Polar Decomposition of the Deformation Gradient
,” Q. Appl. Math.
0033-569X, 41
, pp. 339
–342
.43.
Imam
, A.
, and Johnson
, G. C.
, 1998, “Decomposition of the Deformation Gradient in Thermoelasticity
,” ASME J. Appl. Mech.
0021-8936, 65
, pp. 362
–366
.44.
Boote
, C.
, Dennis
, S.
, and Meek
, K. M.
, 2004, “Spatial Mapping of Collagen Fibril Organisation in Primate Cornea—An X-Ray Diffraction Investigation
,” J. Struct. Biol.
1047-8477, 146
, pp. 359
–267
.45.
Green
, M. S.
, and Tolbosky
, A. V.
, 1946, “A New Approach to the Theory of Relaxing Polymeric Media
,” J. Chem. Phys.
0021-9606, 14
, pp. 80
–92
.46.
Lubliner
, J.
, 1985, “A Model of Rubber Viscoelasticity
,” Mech. Res. Commun.
0093-6413, 12
, pp. 93
–99
.47.
Lion
, A.
, 1996, “A Constitutive Model for Carbon Black Filled Rubber. Experimental Investigations and Mathematical Representations
,” Continuum Mech. Thermodyn.
0935-1175, 8
, pp. 153
–169
.48.
Reese
, S.
, and Govindjee
, S.
, 1998, “A Theory of Finite Viscoelasticity and Numerical Aspects
,” Int. J. Solids Struct.
0020-7683, 35
, pp. 3455
–82
.49.
Bergstrom
, J. S.
, and Boyce
, M.
, 1998, “Constitutive Modeling of the Large Strain Time-Dependent Behavior of Elastomers
,” J. Mech. Phys. Solids
0022-5096, 46
, pp. 931
–954
.50.
Bergstrom
, J.
, and Boyce
, M.
, 2001, “Constitutive Modeling of the Time-Dependent and Cyclic Loading of Elastomers and Application to Soft Biological Tissues
,” Mech. Mater.
0167-6636, 3
, pp. 523
–530
.51.
Misof
, K.
, Rapp
, G.
, and Fratzl
, P.
, 1997, “A New Molecular Model for Collagen Elasticity Based on Synchrotron X-Ray Scattering Evidence
,” Biophys. J.
0006-3495, 72
, pp. 1376
–1381
.52.
Eyring
, H.
, 1936, “Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates
,” J. Chem. Phys.
0021-9606, 4
, pp. 283
–291
.53.
Ree
, T.
, and Eyring
, H.
, 1955, “A Theory of Non-Newtonian Flow. I Solution System of High Polymers
,” J. Appl. Phys.
0021-8979, 26
, pp. 800
–809
.54.
Boyce
, B. L.
, Grazier
, J. M.
, Jones
, R. E.
, and Nguyen
, T. D.
, 2008, “Full-Field Deformation of Bovine Cornea Under Constrained Inflation Conditions
,” Biomaterials
, 0142-9612doi: .Copyright © 2008
by American Society of Mechanical Engineers
You do not currently have access to this content.
