https://orcid.org/0000-0002-0265-2753
Abstract
Soft biological tissues often function as highly deformable membranes in vivo and exhibit impressive mechanical behavior effectively characterized by planar biaxial testing. The Generalized Anisotropic Inverse Mechanics (GAIM) method links full-field deformations and boundary forces from mechanical testing to quantify material properties of soft, anisotropic, heterogeneous tissues. In this study, we introduced an orthotropic constraint to GAIM to improve the quality and physical significance of its mechanical characterizations. We evaluated the updated GAIM method using simulated and experimental biaxial testing datasets obtained from soft tissue analogs (PDMS and TissueMend) with well-defined mechanical properties. GAIM produced stiffnesses (first Kelvin moduli, K1) that agreed well with previously published Young's moduli of PDMS samples. It also matched the stiffness moduli determined via uniaxial testing for TissueMend, a collagen-rich patch intended for tendon repair. We then conducted the first biaxial testing of TissueMend and confirmed that the sample was mechanically anisotropic via a relative anisotropy metric produced by GAIM. Next, we demonstrated the benefits of full-field laser micrometry in distinguishing between spatial variations in thickness and stiffness. Finally, we conducted an analysis to verify that results were independent of partitioning scheme. The success of the newly implemented constraints on GAIM suggests notable potential for applying this tool to soft tissues, particularly following the onset of pathologies that induce mechanical and structural heterogeneities.
References
1.
Billiar
, K. L.
, and Sacks
, M. S.
, 2000
, “Biaxial Mechanical Properties of the Natural and Glutaraldehyde Treated Aortic Valve Cusp—Part I: Experimental Results
,” ASME J. Biomech. Eng.
, 122
(1
), pp. 23
–30
.10.1115/1.4296242.
Ross
, C.
, Laurence
, D.
, Wu
, Y.
, and Lee
, C. H.
, 2019
, “Biaxial Mechanical Characterizations of Atrioventricular Heart Valves
,” J. Vis. Exp.
, (146), p. e59170.10.3791/591703.
Sacks
, M. S.
, and Chuong
, C. J.
, 1993
, “Biaxial Mechanical Properties of Passive Right Ventricular Free Wall Myocardium
,” ASME J. Biomech. Eng.
, 115
(2
), pp. 202
–205
.10.1115/1.28941224.
Witzenburg
, C.
, Raghupathy
, R.
, Kren
, S. M.
, Taylor
, D. A.
, and Barocas
, V. H.
, 2012
, “Mechanical Changes in the Rat Right Ventricle With Decellularization
,” J. Biomech.
, 45
(5
), pp. 842
–849
.10.1016/j.jbiomech.2011.11.0255.
Nemavhola
, F.
, 2021
, “Study of Biaxial Mechanical Properties of the Passive Pig Heart: Material Characterisation and Categorisation of Regional Differences
,” Int. J. Mech. Mater. Eng.
, 16
(1
), pp. 1
–14
.10.1186/s40712-021-00128-46.
Lanir
, Y.
, and Fung
, Y. C.
, 1974
, “Two-Dimensional Mechanical Properties of Rabbit Skin—I. Experimental System
,” J. Biomech.
, 7
(1
), pp. 29
–34
.10.1016/0021-9290(74)90067-07.
Lanir
, Y.
, and Fung
, Y. C.
, 1974
, “Two-Dimensional Mechanical Properties of Rabbit Skin—II. Experimental Results
,” J. Biomech.
, 7
(2
), pp. 171
–182
.10.1016/0021-9290(74)90058-X8.
Schneider
, D. C.
, Davidson
, T. M.
, and Nahum
, A. M.
, 1984
, “In Vitro Biaxial Stress-Strain Response of Human Skin
,” Arch. Otolaryngol.
, 110
(5
), pp. 329
–333
.10.1001/archotol.1984.008003100530129.
Jiang
, M.
, Sridhar
, R. L.
, Robbins
, A. B.
, Freed
, A. D.
, and Moreno
, M. R.
, 2021
, “A Versatile Biaxial Testing Platform for Soft Tissues
,” J. Mech. Behav. Biomed. Mater.
, 114
, p. 104144
.10.1016/j.jmbbm.2020.10414410.
Eilaghi
, A.
, Flanagan
, J. G.
, Tertinegg
, I.
, Simmons
, C. A.
, Wayne Brodland
, G.
, and Ross Ethier
, C.
, 2010
, “Biaxial Mechanical Testing of Human Sclera
,” J. Biomech.
, 43
(9
), pp. 1696
–1701
.10.1016/j.jbiomech.2010.02.03111.
Cruz Perez
, B.
, Tang
, J.
, Morris
, H.
, Palko
, J.
, Pan
, X.
, Hart
, R.
, and Liu
, J.
, 2014
, “Biaxial Mechanical Testing of Posterior Sclera Using High-Resolution Ultrasound Speckle Tracking for Strain Measurements
,” J. Biomech.
, 47
(5
), pp. 1151
–1156
.10.1016/j.jbiomech.2013.12.00912.
Ndlovu
, Z.
, Desai
, D.
, Pandelani
, T.
, Ngwangwa
, H.
, and Nemavhola
, F.
, 2022
, “Biaxial Estimation of Biomechanical Constitutive Parameters of Passive Porcine Sclera Soft Tissue
,” Appl. Bionics Biomech.
, 2022
, pp. 1
–11
.10.1155/2022/477559513.
Humphrey
, J. D.
, 2002
, “Cardiovascular Solid Mechanics—Cells, Tissues, and Organs
,” Springer, New York.14.
Humphrey
, J. D.
, 1998
, “Computer Methods in Membrane Biomechanics
,” Comput. Methods Biomech. Biomed. Eng.
, 1
(3
), pp. 171
–210
.10.1080/0149573980893670115.
Ventsel
, E.
, and Krauthammer
, T.
, 2001
, Thin Plates and Shells
, CRC Press
, Boca Raton, FL.16.
Demer
, L. L.
, and Yin
, F. C.
, 1983
, “Passive Biaxial Mechanical Properties of Isolated Canine Myocardium
,” J. Physiol.
, 339
(1
), pp. 615
–630
.10.1113/jphysiol.1983.sp01473817.
Humphrey
, J. D.
, Strumpf
, R. K.
, and Yin
, F. C. P.
, 1990
, “Biaxial Mechanical Behavior of Excised Ventricular Epicardium
,” Am. J. Physiol.
, 259
(1
), pp. H101
–H108
.10.1152/ajpheart.1990.259.1.H10118.
Vande Geest
, J. P.
, Sacks
, M. S.
, and Vorp
, D. A.
, 2006
, “The Effects of Aneurysm on the Biaxial Mechanical Behavior of Human Abdominal Aorta
,” J. Biomech.
, 39
(7
), pp. 1324
–1334
.10.1016/j.jbiomech.2005.03.00319.
Nemavhola
, F.
, 2017
, “Biaxial Quantification of Passive Porcine Myocardium Elastic Properties by Region
,” Eng. Solid Mech.
, 5
(3
), pp. 155
–166
.20.
Sacks
, M. S.
, 2000
, “Biaxial Mechanical Evaluation of Planar Biological Materials
,” J. Elast. Phys. Sci. Solids
, 61
(1
), pp. 199
–246
.10.1023/A:101091702867121.
Rivlin
, R. S.
, 1948
, “Large Elastic Deformations of Isotropic Materials IV. further Developments of the General Theory
,” Philos. Trans. R. Soc., A
, 241
(835
), pp. 379
–397
.10.1098/rsta.1948.002422.
Rivlin
, R. S.
, and Saunders
, D. W.
, 1951
, “Large Elastic Deformations of Isotropic Materials VII. Experiments on the Deformation of Rubber
,” Philos. Trans. R. Soc., A
, 243
(865
), pp. 251
–288
.10.1098/rsta.1951.000423.
Treloar
, L. R. G.
, 1948
, “Stresses and Birefringence in Rubber Subjected to General Homogeneous Strain
,” Proc. Phys. Soc.
, 60
(2
), pp. 135
–144
.10.1088/0959-5309/60/2/30324.
Pearce
, D.
, Nemcek
, M.
, and Witzenburg
, C.
, 2022
, “Combining Unique Planar Biaxial Testing With Full-Field Thickness and Displacement Measurement for Spatial Characterization of Soft Tissues
,” Curr. Protoc.
, 2
(7
), p. e493
.10.1002/cpz1.49325.
Kroon
, M.
, and Holzapfel
, G. A.
, 2008
, “Estimation of the Distributions of Anisotropic, Elastic Properties and Wall Stresses of Saccular Cerebral Aneurysms by Inverse Analysis
,” Proc. R. Soc. A
, 464
(2092
), pp. 807
–825
.10.1098/rspa.2007.033226.
Genovese
, K.
, Casaletto
, L.
, Humphrey
, J. D.
, and Lu
, J.
, 2014
, “Digital Image Correlation-Based Point-Wise Inverse Characterization of Heterogeneous Material Properties of Gallbladder In Vitro
,” Proc. R. Soc. A
, 470
(2167
), p. 20140152
.10.1098/rspa.2014.015227.
Zhou
, X.
, and Lu
, J.
, 2008
, “Inverse Formulation for Geometrically Exact Stress Resultant Shells
,” Int. J. Numer. Methods Eng.
, 74
(8
), pp. 1278
–1302
.10.1002/nme.221528.
Lu
, J.
, Zhou
, X.
, and Raghavan
, M. L.
, 2008
, “Inverse Method of Stress Analysis for Cerebral Aneurysms
,” Biomech. Model. Mechanobiol.
, 7
(6
), pp. 477
–486
.10.1007/s10237-007-0110-129.
Davis
, F.
, Luo
, Y.
, Avril
, S.
, Duprey
, A.
, and Lu
, J.
, 2015
, “Pointwise Characterization of the Elastic Properties of Planar Soft Tissues: Application to Ascending Thoracic Aneurysms
,” Biomech. Model. Mechanobiol.
, 14
(5
), pp. 967
–978
.10.1007/s10237-014-0646-930.
Raghupathy
, R.
, and Barocas
, V. H.
, 2013
, “Robust Image Correlation Based Strain Calculator for Tissue Systems (20130022, Dr. Victor Barocas)
,” accessed Sept. 12, 2024, https://license.umn.edu/product/robust-image-correlation-based-strain-calculator-for-tissue-systems31.
Zhou
, B.
, Ravindran
, S.
, Ferdous
, J.
, Kidane
, A.
, Sutton
, M. A.
, and Shazly
, T.
, 2016
, “Using Digital Image Correlation to Characterize Local Strains on Vascular Tissue Specimens
,” J. Vis. Exp.
, 2016
(107
), p. 53625
.10.3791/5362532.
Yang
, J.
, and Bhattacharya
, K.
, 2019
, “Augmented Lagrangian Digital Image Correlation
,” Exp. Mech.
59
(2
), pp. 187
–205
.10.1007/s11340-018-00457-033.
Raghupathy
, R.
, and Barocas
, V. H.
, 2010
, “Generalized Anisotropic Inverse Mechanics for Soft Tissues
,” ASME J. Biomech. Eng.
, 132
(8
), p. 081006
.10.1115/1.400125734.
Raghupathy
, R.
, Witzenburg
, C.
, Lake
, S. P.
, Sander
, E. A.
, and Barocas
, V. H.
, 2011
, “Identification of Regional Mechanical Anisotropy in Soft Tissue Analogs
,” ASME J. Biomech. Eng.
, 133
(9
), p. 091011
.10.1115/1.400517035.
Shih
, E. D.
, Provenzano
, P. P.
, Witzenburg
, C. M.
, Barocas
, V. H.
, Grande
, A. W.
, and Alford
, P. W.
, 2022
, “Characterizing Tissue Remodeling and Mechanical Heterogeneity in Cerebral Aneurysms
,” J. Vasc. Res.
, 59
(1
), pp. 34
–42
.10.1159/00051969436.
Johnston
, I. D.
, McCluskey
, D. K.
, Tan
, C. K. L.
, and Tracey
, M. C.
, 2014
, “Mechanical Characterization of Bulk Sylgard 184 for Microfluidics and Microengineering
,” J. Micromech. Microeng.
, 24
(3
), p. 035017
.10.1088/0960-1317/24/3/03501737.
Aurora
, A.
, McCarron
, J.
, Iannotti
, J. P.
, and Derwin
, K.
, 2007
, “Commercially Available Extracellular Matrix Materials for Rotator Cuff Repairs: State of the Art and Future Trends
,” J. Shoulder Elb. Surg.
, 16
(5
), pp. S171
–S178
.10.1016/j.jse.2007.03.00838.
Derwin
, K. A.
, Baker
, A. R.
, Spragg
, R. K.
, Leigh
, D. R.
, and Iannotti
, J. P.
, 2006
, “Commercial Extracellular Matrix Scaffolds for Rotator Cuff Tendon Repair: Biomechanical, Biochemical, and Cellular Properties
,” J. Bone Jt. Surg.
, 88
(12
), pp. 2665
–2672
.10.2106/JBJS.E.0130739.
Witzenburg
, C.
, Dhume
, R. Y.
, Lake
, S. P.
, and Barocas
, V. H.
, 2016
, “Automatic Segmentation of Mechanically Inhomogeneous Tissues Based on Deformation Gradient Jump
,” IEEE Trans. Med. Imaging
, 35
(1
), pp. 29
–41
.10.1109/TMI.2015.245331640.
Jayne
, B. A.
, and Hunt
, M. O.
, 1969
, “Plane Stress and Plane Strain in Orthotropic and Anisotropic Media
,” Wood Fiber Sci.
, 1
(3
), pp. 236
–247
.https://wfs.swst.org/index.php/wfs/article/view/1539/153941.
Krenk
, S.
, 1979
, “On the Elastic Constants of Plane Orthotropic Elasticity
,” J. Compos. Mater
, 13
(2
), pp. 108
–116
.10.1177/00219983790130020242.
Li
, Y.
, and Barbič
, J.
, 2014
, “Stable Orthotropic Materials
,” SCA 2014 - Proceedings of ACM SIGGRAPH/Eurographics Symposium on Compututer Animation
, Copenhagen, Denmark, July 21–23, pp. 41
–46
.https://viterbiweb.usc.edu/~jbarbic/orthotropic/LiBarbicStableOrthotropicMaterials-SCA2014.pdf43.
Cowin
, S. C.
, and Mehrabadi
, M. M.
, 1995
, “Anisotropic Symmetries of Linear Elasticity
,” ASME Appl. Mech. Rev
, 48
(5
), pp. 247
–285
.10.1115/1.300510244.
Thomson
, W.
, 1856
, “XXI. Elements of a Mathematical Theory of Elasticity
,” Philos. Trans. R. Soc. London
, 146
, pp. 481
–498
.10.1098/rstl.1856.002245.
Makinde
, A.
, Thibodeau
, L.
, and Neale
, K. W.
, 1992
, “Development of an Apparatus for Biaxial Testing Using Cruciform Specimens
,” Exp. Mech.
32
(2
), pp. 138
–144
.10.1007/BF0232472546.
Boehler
, J. P.
, Demmerle
, S.
, and Koss
, S.
, 1994
, “A New Direct Biaxial Testing Machine for Anisotropic Materials
,” Exp. Mech.
, 34
(1
), pp. 1
–9
.10.1007/BF0232843547.
Waldman
, S. D.
, and Lee
, J. M.
, 2005
, “Effect of Sample Geometry on the Apparent Biaxial Mechanical Behaviour of Planar Connective Tissues
,” Biomaterials
, 26
(35
), pp. 7504
–7513
.10.1016/j.biomaterials.2005.05.05648.
Lecompte
, D.
, Smits
, A.
, Sol
, H.
, Vantomme
, J.
, and Van Hemelrijck
, D.
, 2007
, “Mixed Numerical–Experimental Technique for Orthotropic Parameter Identification Using Biaxial Tensile Tests on Cruciform Specimens
,” Int. J. Solids Struct.
, 44
(5
), pp. 1643
–1656
.10.1016/j.ijsolstr.2006.06.05049.
Jacobs
, N. T.
, Cortes
, D. H.
, Vresilovic
, E. J.
, and Elliott
, D. M.
, 2013
, “Biaxial Tension of Fibrous Tissue: Using Finite Element Methods to Address Experimental Challenges Arising From Boundary Conditions and Anisotropy
,” ASME J. Biomech. Eng.
, 135
(2
), p. 021004
.10.1115/1.402350350.
Estrada
, J. B.
, and Franck
, C.
, 2015
, “Intuitive Interface for the Quantitative Evaluation of Speckle Patterns for Use in Digital Image and Volume Correlation Techniques
,” ASME J. Appl. Mech.
, 82
(9
), p. 095001
.10.1115/1.403082151.
Debes
, J.
, and Fung
, Y.
, 1995
, “Biaxial Mechanics of Excised Canine Pulmonary Arteries
,” Am. J. Physiol.
, 269
(2
), pp. H433
–H442
.10.1152/ajpheart.1995.269.2.H43352.
Lee
, J. M.
, and Langdon
, S. E.
, 1996
, “Thickness Measurement of Soft Tissue Biomaterials: A Comparison of Five Methods
,” J. Biomech.
, 29
(6
), pp. 829
–832
.10.1016/0021-9290(95)00121-253.
Murdock
, K.
, Martin
, C.
, and Sun
, W.
, 2018
, “Characterization of Mechanical Properties of Pericardium Tissue Using Planar Biaxial Tension and Flexural Deformation
,” J. Mech. Behav. Biomed. Mater.
, 77
, pp. 148
–156
.10.1016/j.jmbbm.2017.08.03954.
Avril
, S.
, and Evans
, S.
, 2017
, Material Parameter Identification and Inverse Problems in Soft Tissue Biomechanics
, Vol. 573
, Springer International Publishing
, Cham
.55.
Ariati
, R.
, Sales
, F.
, Souza
, A.
, Lima
, R. A.
, and Ribeiro
, J.
, 2021
, “Polydimethylsiloxane Composites Characterization and Its Applications: A Review
,” Polymers (Basel)
, 13
(23
), pp. 4258
–21
.10.3390/polym1323425856.
Long
, R.
, Hall
, M. S.
, Wu
, M.
, and Hui
, C. Y.
, 2011
, “Effects of Gel Thickness on Microscopic Indentation Measurements of Gel Modulus
,” Biophys. J
, 101
(3
), pp. 643
–650
.10.1016/j.bpj.2011.06.04957.
Toyjanova
, J.
, Bar-Kochba
, E.
, López-Fagundo
, C.
, Reichner
, J.
, Hoffman-Kim
, D.
, and Franck
, C.
, 2014
, “High Resolution, Large Deformation 3D Traction Force Microscopy
,” PLoS One
, 9
(4
), p. e90976
.10.1371/journal.pone.009097658.
Beussman
, K. M.
, Mollica
, M.
, Leonard
, A.
, Miles
, J.
, Hocter
, J.
, Song
, Z.
, Stolla
, M.
, Han
, S.
, Emery
, A.
, Thomas
, W.
, and Sniadecki
, N.
, 2023
, “Black Dots: High-Yield Traction Force Microscopy Reveals Structural Factors Contributing to Platelet Forces
,” Acta Biomater.
, 163
, pp. 302
–311
.10.1016/j.actbio.2021.11.01359.
Li
, H.
, Matsunaga
, D.
, Matsui
, T.
, Aosaki
, H.
, Kinoshita
, G.
, Inoue
, K.
, Doostmohammadi
, A.
, and Deguchi
, S.
, 2022
, “Wrinkle Force Microscopy: A Machine Learning Based Approach to Predict Cell Mechanics From Images
,” Commun. Biol.
, 5
(1
), pp. 1
–3
.10.1038/s42003-022-03288-x60.
Notbohm
, J.
, Napiwocki
, B. N.
, de Lange
, W. J.
, Stempien
, A.
, Saraswathibhatla
, A.
, Craven
, R. J.
, Salick
, M. R.
, Ralphe
, J. C.
, and Crone
, W. C.
, 2019
, “Two-Dimensional Culture Systems to Enable Mechanics-Based Assays for Stem Cell-Derived Cardiomyocytes
,” Exp. Mech.
, 59
(9
), pp. 1235
–1248
.10.1007/s11340-019-00473-861.
Franck
, C.
, Hong
, S.
, Maskarinec
, S. A.
, Tirrell
, D. A.
, and Ravichandran
, G.
, 2007
, “Three-Dimensional Full-Field Measurements of Large Deformations in Soft Materials Using Confocal Microscopy and Digital Volume Correlation
,” Exp. Mech.
, 47
(3
), pp. 427
–438
.10.1007/s11340-007-9037-962.
Holzapfel
, G. A.
, Gasser
, T. C.
, and Ogden
, R. W.
, 2000
, “A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models
,” J. Elasticity
, 61
(1–3
), pp. 1
–48
.10.1023/A:101083531656463.
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
, 3
(6
), pp. 15
–35
.10.1098/rsif.2005.007364.
Gasparotti
, E.
, Vignali
, E.
, Quartieri
, S.
, Lazzeri
, R.
, and Celi
, S.
, 2023
, “Numerical Investigation on Circular and Elliptical Bulge Tests for Inverse Soft Tissue Characterization
,” Biomech. Model. Mechanobiol.
, 22
(5
), pp. 1697
–1707
.10.1007/s10237-023-01730-565.
Vignali
, E.
, Gasparotti
, E.
, Capellini
, K.
, Fanni
, B.
, Landini
, L.
, Positano
, V.
, and Celi
, S.
, 2021
, “Modeling Biomechanical Interaction Between Soft Tissue and Soft Robotic Instruments: Importance of Constitutive Anisotropic Hyperelastic Formulations
,” Int. J. Rob. Res.
, 40
(1
), pp. 224
–235
.10.1177/027836492092747666.
Wertheim
, M. G.
, 1847
, “Mémoire Sur L'élasticité et la Cohésion Des Principaux Tissus du Corps Humain
,” Ann. Chim. Phys.
, 21
, pp. 385
–414
.https://archive.org/details/mmoiresurlespro00chevgoog/page/n68/mode/2up67.
Roy
, C. S.
, 1881
, “The Elastic Properties of the Arterial Wall
,” J. Physiol.
, 3
(2
), pp. 125
–159
.10.1113/jphysiol.1881.sp00008868.
Humphrey
, J.
, 2003
, “Continuum Biomechanics of Soft Biological Tissues
,” Proc. Math. Phys. Eng. Sci.
, 459
(2029
), pp. 3
–46
.10.1098/rspa.2002.106069.
Witzenburg
, C.
, and Barocas
, V. H.
, 2016
, “A Nonlinear Anisotropic Inverse Method for Computational Dissection of Inhomogeneous Planar Tissues
,” Comput. Methods Biomech. Biomed. Eng.
, 19
(15
), pp. 1630
–1646
.10.1080/10255842.2016.1176154
