An analytical theory for the unconfined creep behavior of a cylindrical inclusion (simulating a soft tissue tumor) embedded in a cylindrical background sample (simulating normal tissue) is presented and analyzed in this paper. Both the inclusion and the background are considered as fluid-filled, porous materials, each of them being characterized by a set of mechanical properties. Specifically, in this paper, the inclusion is considered to be less permeable than the background. The cylindrical sample is compressed using a constant pressure within two frictionless plates and is allowed to expand in an unconfined way along the radial direction. Analytical expressions for the effective Poisson's ratio (EPR) and fluid pressure inside and outside the inclusion are derived and analyzed. The theoretical results are validated using finite element models (FEMs). Statistical analysis shows excellent agreement between the results obtained from the developed model and the results from FEM. Thus, the developed theoretical model can be used in medical imaging modalities such as ultrasound poroelastography to extract the mechanical parameters of tissues and/or to better understand the impact of different mechanical parameters on the estimated displacements, strains, stresses, and fluid pressure inside a tumor and in the surrounding tissue.
References
1.
Biot
, M. A.
, 1941
, “General Theory of Three-Dimensional Consolidation
,” J. Appl. Phys.
, 12
(2
), pp. 155
–164
.2.
Biot
, M. A.
, 1962
, “Mechanics of Deformation and Acoustic Propagation in Porous Media
,” J. Appl. Phys.
, 33
(4
), pp. 1482
–1498
.3.
Cheng
, A. H.-D.
, 2016
, Poroelasticity
, Vol. 27
, Springer
, Berlin, Germany
.4.
Mow
, V. C.
, and Lai
, W. M.
, 1980
, “Recent Developments in Synovial Joint Biomechanics
,” SIAM Rev.
, 22
(3
), pp. 275
–317
.5.
Mow
, V. C.
, Ratcliffe
, A.
, and Woo
, S. L.
, 2012
, Biomechanics of Diarthrodial Joints
, Vol. 1
, Springer Science & Business Media
, Berlin, Germany
.6.
Mow
, V. C.
, Kuei
, S.
, Lai
, W. M.
, and Armstrong
, C. G.
, 1980
, “Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments
,” ASME J. Biomech. Eng.
, 102
(1
), pp. 73
–84
.7.
Mow
, V. C.
, Bachrach
, N. M.
, Setton
, L. A.
, and Guilak
, F.
, 1994
, “Stress, Strain, Pressure and Flow Fields in Articular Cartilage and Chondrocytes
,” In: Mow
, V. C.
, Tran-Son-Tay
, R.
, Guilak
, F.
, and Hochmuth
, R. M.
, eds., Cell Mechanics and Cellular Engineering
, Springer, New York.8.
Ehlers
, W.
, and Markert
, B.
, 2001
, “A Linear Viscoelastic Biphasic Model for Soft Tissues Based on the Theory of Porous Media
,” ASME J. Biomech. Eng.
, 123
(5
), pp. 418
–424
.9.
Suh
, J.
, and DiSilvestro
, M.
, 1999
, “Biphasic Poroviscoelastic Behavior of Hydrated Biological Soft Tissue
,” ASME J. Appl. Mech.
, 66
(2
), pp. 528
–535
.10.
Cowin
, S. C.
, and Doty
, S. B.
, 2007
, Tissue Mechanics
, Springer Science & Business Media
, New York
.11.
Pflaster
, D.
, Yuan
, Y.
, and Krag
, I.
, 1996
, “A Poroelastic Finite Element Formulation Including Transport and Swelling in Soft Tissue Structures
,” ASME J. Biomech. Eng.
, 118
(1
) pp. 1
–9
.12.
Armstrong
, C.
, Lai
, W.
, and Mow
, V.
, 1984
, “An Analysis of the Unconfined Compression of Articular Cartilage
,” ASME J. Biomech. Eng.
, 106
(2
), pp. 165
–173
.13.
Eshelby
, J. D.
, 1957
, “The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,” Proc. R. Soc. London, Ser. A
, 241
(1226
), pp. 376
–396
.14.
Rice
, J.
, Rudnicki
, J.
, and Simons
, D. A.
, 1978
, “Deformation of Spherical Cavities and Inclusions in Fluid-Infiltrated Elastic Materials
,” Int. J. Solids Struct.
, 14
(4
), pp. 289
–303
.15.
Song
, Y.
, Hu
, H.
, and Rudnicki
, J. W.
, 2016
, “Shear Properties of Heterogeneous Fluid-Filled Porous Media With Spherical Inclusions
,” Int. J. Solids Struct.
, 83
, pp. 154
–168
.16.
Song
, Y.
, Hu
, H.
, Rudnicki
, J. W.
, and Duan
, Y.
, 2016
, “Dynamic Transverse Shear Modulus for a Heterogeneous Fluid-Filled Porous Solid Containing Cylindrical Inclusions
,” Geophys. J. Int.
, 206
(3
), pp. 1677
–1694
.17.
Sarntinoranont
, M.
, Rooney
, F.
, and Ferrari
, M.
, 2003
, “Interstitial Stress and Fluid Pressure Within a Growing Tumor
,” Ann. Biomed. Eng.
, 31
(3
), pp. 327
–335
.18.
Swabb
, E. A.
, Wei
, J.
, and Gullino
, P. M.
, 1974
, “Diffusion and Convection in Normal and Neoplastic Tissues
,” Cancer Res.
, 34
(10
), pp. 2814
–2822
.http://cancerres.aacrjournals.org/content/34/10/2814.short19.
Netti
, P. A.
, Baxter
, L. T.
, Boucher
, Y.
, Skalak
, R.
, and Jain
, R. K.
, 1995
, “Time-Dependent Behavior of Interstitial Fluid Pressure in Solid Tumors: Implications for Drug Delivery
,” Cancer Res.
, 55
(22
), pp. 5451
–5458
.http://cancerres.aacrjournals.org/content/55/22/545120.
Swartz
, M. A.
, Kaipainen
, A.
, Netti
, P. A.
, Brekken
, C.
, Boucher
, Y.
, Grodzinsky
, A. J.
, and Jain
, R. K.
, 1999
, “Mechanics of Interstitial-Lymphatic Fluid Transport: Theoretical Foundation and Experimental Validation
,” J. Biomech.
, 32
(12
), pp. 1297
–1307
.21.
Netti
, P. A.
, Berk
, D. A.
, Swartz
, M. A.
, Grodzinsky
, A. J.
, and Jain
, R. K.
, 2000
, “Role of Extracellular Matrix Assembly in Interstitial Transport in Solid Tumors
,” Cancer Res.
, 60
(9
), pp. 2497
–2503
.http://cancerres.aacrjournals.org/content/60/9/2497.short22.
Baxter
, L. T.
, and Jain
, R. K.
, 1989
, “Transport of Fluid and Macromolecules in Tumors—I: Role of Interstitial Pressure and Convection
,” Microvasc. Res.
, 37
(1
), pp. 77
–104
.23.
Jain
, R. K.
, Martin
, J. D.
, and Stylianopoulos
, T.
, 2014
, “The Role of Mechanical Forces in Tumor Growth and Therapy
,” Annu. Rev. Biomed. Eng.
, 16
, p. 321
.24.
Jain
, R. K.
, Tong
, R. T.
, and Munn
, L. L.
, 2007
, “Effect of Vascular Normalization by Antiangiogenic Therapy on Interstitial Hypertension, Peritumor Edema, and Lymphatic Metastasis: Insights From a Mathematical Model
,” Cancer Res.
, 67
(6
), pp. 2729
–2735
.25.
Jain
, R. K.
, 1998
, “Delivery of Molecular and Cellular Medicine to Solid Tumors
,” J. Controlled Release
, 53
(1–3
), pp. 49
–67
.26.
Netti
, P. A.
, Baxter
, L. T.
, Boucher
, Y.
, Skalak
, R.
, and Jain
, R. K.
, 1997
, “Macro-and Microscopic Fluid Transport in Living Tissues: Application to Solid Tumors
,” AIChE J.
, 43
(3
), pp. 818
–834
.27.
Baxter
, L. T.
, and Jain
, R. K.
, 1991
, “Transport of Fluid and Macromolecules in Tumors. iv. a Microscopic Model of the Perivascular Distribution
,” Microvasc. Res.
, 41
(2
), pp. 252
–272
.28.
Baxter
, L. T.
, and Jain
, R. K.
, 1991
, “Transport of Fluid and Macromolecules in Tumors—III: Role of Binding and Metabolism
,” Microvasc. Res.
, 41
(1
), pp. 5
–23
.29.
Baxter
, L. T.
, and Jain
, R. K.
, 1990
, “Transport of Fluid and Macromolecules in Tumors. ii. role of Heterogeneous Perfusion and Lymphatics
,” Microvasc. Res.
, 40
(2
), pp. 246
–263
.30.
Jain
, R. K.
, and Baxter
, L. T.
, 1988
, “Mechanisms of Heterogeneous Distribution of Monoclonal Antibodies and Other Macromolecules in Tumors: Significance of Elevated Interstitial Pressure
,” Cancer Res.
, 48
(24 Part 1
), pp. 7022
–7032
.http://cancerres.aacrjournals.org/content/48/24_Part_1/702231.
Milosevic
, M. F.
, Fyles
, A. W.
, and Hill
, R. P.
, 1999
, “The Relationship Between Elevated Interstitial Fluid Pressure and Blood Flow in Tumors: A Bioengineering Analysis
,” Int. J. Radiat. Oncology* Biol.* Phys.
, 43
(5
), pp. 1111
–1123
.32.
Netti
, P.
, Baxter
, L.
, Coucher
, Y.
, Skalak
, R.
, and Jain
, R.
, 1995
, “A Poroelastic Model for Interstitial Pressure in Tumors
,” Biorheology
, 32
(2–3
), pp. 346
–346
.33.
Stylianopoulos
, T.
, Martin
, J. D.
, Snuderl
, M.
, Mpekris
, F.
, Jain
, S. R.
, and Jain
, R. K.
, 2013
, “Coevolution of Solid Stress and Interstitial Fluid Pressure in Tumors During Progression: Implications for Vascular Collapse
,” Cancer Res.
, 73
(13
), pp. 3833
–3841
.34.
Leiderman
, R.
, Barbone
, P. E.
, Oberai
, A. A.
, and Bamber
, J. C.
, 2006
, “Coupling Between Elastic Strain and Interstitial Fluid Flow: Ramifications for Poroelastic Imaging
,” Phys. Med. Biol.
, 51
(24
), p. 6291
.35.
Ophir
, J.
, Cespedes
, I.
, Ponnekanti
, H.
, Yazdi
, Y.
, and Li
, X.
, 1991
, “Elastography: A Quantitative Method for Imaging the Elasticity of Biological Tissues
,” Ultrason. Imaging
, 13
(2
), pp. 111
–134
.36.
Ophir
, J.
, Alam
, S.
, Garra
, B.
, Kallel
, F.
, Konofagou
, E.
, Krouskop
, T.
, and Varghese
, T.
, 1999
, “Elastography: Ultrasonic Estimation and Imaging of the Elastic Properties of Tissues
,” Proc. Inst. Mech. Eng., Part H: J. Eng. Med.
, 213
(3
), pp. 203
–233
.37.
Righetti
, R.
, Garra
, B. S.
, Mobbs
, L. M.
, Kraemer-Chant
, C. M.
, Ophir
, J.
, and Krouskop
, T. A.
, 2007
, “The Feasibility of Using Poroelastographic Techniques for Distinguishing Between Normal and Lymphedematous Tissues In Vivo
,” Phys. Med. Biol.
, 52
(21
), p. 6525
.38.
Righetti
, R.
, Ophir
, J.
, Srinivasan
, S.
, and Krouskop
, T. A.
, 2004
, “The Feasibility of Using Elastography for Imaging the Poisson's Ratio in Porous Media
,” Ultrasound Med. Biol.
, 30
(2
), pp. 215
–228
.39.
Lakes
, R. S.
, 1998
, Viscoelastic Solids
, Vol. 9
, CRC Press
, Boca Raton, FL
.40.
Fung
, Y.-C.
, 2013
, Biomechanics: Mechanical Properties of Living Tissues
, Springer Science & Business Media
, New York
.41.
Duck
, F. A.
, 2013
, Physical Properties of Tissues: A Comprehensive Reference Book
, Academic Press
, San Diego, CA
.42.
Wellman
, P.
, Howe
, R. D.
, Dalton
, E.
, and Kern
, K. A.
, 1999
, Breast Tissue Stiffness in Compression Is Correlated to Histological Diagnosis
, Harvard Bio Robotics Laboratory
, Cambridge, MA
.43.
Wellman
, P. S.
, Dalton
, E. P.
, Krag
, D.
, Kern
, K. A.
, and Howe
, R. D.
, 2001
, “Tactile Imaging of Breast Masses: First Clinical Report
,” Arch. Surg.
, 136
(2
), pp. 204
–208
.44.
Plewes
, D. B.
, Bishop
, J.
, Samani
, A.
, and Sciarretta
, J.
, 2000
, “Visualization and Quantification of Breast Cancer Biomechanical Properties With Magnetic Resonance Elastography
,” Phys. Med. Biol.
, 45
(6
), p. 1591
.45.
Islam
, M. T.
, Chaudhry
, A.
, Unnikrishnan
, G.
, Reddy
, J.
, and Righetti
, R.
, 2018
, “An Analytical Poroelastic Model for Ultrasound Elastography Imaging of Tumors
,” Phys. Med. Biol.
, 63
(2
), p. 025031
.46.
Islam
, M. T.
, Chaudhry
, A.
, Unnikrishnan
, G.
, Reddy
, J.
, and Righetti
, R.
, 2018
, “An Analytical Model of Tumors With Higher Permeability Than Surrounding Tissues for Ultrasound Elastography Imaging
,” J. Eng. Sci. Med. Diagn. Ther.
, 1
(3
), p. 031006
.47.
Swartz
, M. A.
, and Fleury
, M. E.
, 2007
, “Interstitial Flow and Its Effects in Soft Tissues
,” Annu. Rev. Biomed. Eng.
, 9
, pp. 229
–256
.48.
Berry
, G. P.
, Bamber
, J. C.
, Armstrong
, C. G.
, Miller
, N. R.
, and Barbone
, P. E.
, 2006
, “Towards an Acoustic Model-Based Poroelastic Imaging Method—I: Theoretical Foundation
,” Ultrasound Med. Biol.
, 32
(12
), pp. 547
–567
.49.
Ateshian
, G. A.
, Costa
, K. D.
, and Hung
, C. T.
, 2007
, “A Theoretical Analysis of Water Transport Through Chondrocytes
,” Biomech. Model. Mechanobiol.
, 6
(1–2
), pp. 91
–101
.50.
Islam
, M.
, Reddy
, T.
, Righetti
, J.
, and Raffaella
, A.
, 2018
, “An Analytical Poroelastic Model of a Non-Homogeneous Medium Under Creep Compression for Ultrasound Poroelastography Applications—Part II
,” ASME J. Biomech. Eng.
, (Accepted manuscript).51.
Verruijt
, A.
, 2013
, “Theory and Problems of Poroelasticity
,” Delft University of Technology, Delft, The Netherlands.52.
Carslaw
, H. S.
, and Jaeger
, J. C.
, 1959
, Conduction of Heat in Solids
, 2nd ed., Clarendon Press
, Oxford, UK
.53.
Segall
, A.
, 2001
, “Thermoelastic Analysis of Thick-Walled Vessels Subjected to Transient Thermal Loading
,” ASME J. Pressure Vessel Technol.
, 123
(1
), pp. 146
–149
.54.
Guerrero
, J. P.
, Pontedeiro
, E.
, van Genuchten
, M. T.
, and Skaggs
, T.
, 2013
, “Analytical Solutions of the One-Dimensional Advection–Dispersion Solute Transport Equation Subject to Time-Dependent Boundary Conditions
,” Chem. Eng. J.
, 221
, pp. 487
–491
.55.
Hibbitt
, K.
, 2005
, “Sorensen, inc. abaqus/explicit user's manual, version 6.5. 1.
,” Providence, RI.56.
Nabavizadeh
, A.
, Kinnick
, R. R.
, Bayat
, M.
, Amador
, C.
, Urban
, M. W.
, Alizad
, A.
, and Fatemi
, M.
, 2017
, “Automated Compression Device for Viscoelasticity Imaging
,” IEEE Trans. Bio-Med. Eng.
, 64
(7
), pp. 1535
–1546
.57.
Lekka
, M.
, 2016
, “Discrimination Between Normal and Cancerous Cells Using Afm
,” Bionanoscience
, 6
(1
), pp. 65
–80
.58.
Soltani
, M.
, and Chen
, P.
, 2011
, “Numerical Modeling of Fluid Flow in Solid Tumors
,” PloS One
, 6
(6
), p. e20344
.59.
Laperrousaz
, B.
, Drillon
, G.
, Berguiga
, L.
, Nicolini
, F.
, Audit
, B.
, Satta
, V. M.
, Arneodo
, A.
, and Argoul
, F.
, 2016
, “From Elasticity to Inelasticity in Cancer Cell Mechanics: A Loss of Scale-Invariance
,” AIP Conf. Proc.
, 1760
, p. 020040
.60.
Zhao
, X.
, Zhong
, Y.
, Ye
, T.
, Wang
, D.
, and Mao
, B.
, 2015
, “Discrimination Between Cervical Cancer Cells and Normal Cervical Cells Based on Longitudinal Elasticity Using Atomic Force Microscopy
,” Nanoscale Res. Lett.
, 10
(1
), pp. 1
–8
.61.
Xu
, W.
, Mezencev
, R.
, Kim
, B.
, Wang
, L.
, McDonald
, J.
, and Sulchek
, T.
, 2012
, “Cell Stiffness Is a Biomarker of the Metastatic Potential of Ovarian Cancer Cells
,” PloS One
, 7
(10
), p. e46609
.62.
Jain
, R. K.
, 1987
, “Transport of Molecules in the Tumor Interstitium: A Review
,” Cancer Res.
, 47
(12
), pp. 3039
–3051
.http://cancerres.aacrjournals.org/content/47/12/3039.longCopyright © 2019 by ASME
You do not currently have access to this content.
