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 parameters. Specifically, in this derivation, the inclusion is assumed to have significantly higher interstitial permeability than the background. The formulations of the effective Poisson's ratio (EPR) and fluid pressure in the inclusion and in the background are derived for the case of a sample subjected to a creep compression. The developed analytical expressions are validated using finite element models (FEM). Statistical comparison between the results obtained from the developed model and the results from FEM demonstrates accuracy of the proposed theoretical model higher than 99.4%. The model presented in this paper complements the one reported in the companion paper (Part I), which refers to the case of an inclusion having less interstitial permeability than the background.

References

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.
4.
Khoshghalb
,
A.
,
2013
, “
On Creep Laboratory Tests in Soil Mechanics
,”
Multiphysical Testing of Soils and Shales
,
Springer
, New York, pp.
255
260
.
5.
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
.
6.
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
.
7.
Eshelby, J. D.
, 1957, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. London
,
241
(
1226
), pp.
376
396
.
8.
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
.
9.
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
.
10.
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
.
11.
Berryman
,
J. G.
,
1985
, “
Scattering by a Spherical Inhomogeneity in a Fluid-Saturated Porous Medium
,”
J. Math. Phys.
,
26
(
6
), pp.
1408
1419
.
12.
Kanj
,
M.
, and
Abousleiman
,
Y.
,
2005
, “
Porothermoelastic Analyses of Anisotropic Hollow Cylinders With Applications
,”
Int. J. Numer. Anal. Methods Geomech.
,
29
(
2
), pp.
103
126
.
13.
Cui
,
L.
, and
Abousleiman
,
Y.
,
2001
, “
Time-Dependent Poromechanical Responses of Saturated Cylinders
,”
J. Eng. Mech.
,
127
(
4
), pp.
391
398
.
14.
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
.
15.
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.short
16.
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/55/22/5451.short
17.
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
.
18.
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.long
19.
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
.
20.
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
.
21.
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
.
22.
Jain
,
R. K.
,
1998
, “
Delivery of Molecular and Cellular Medicine to Solid Tumors
,”
J. Controlled Release
,
53
(
1–3
), pp.
49
67
.
23.
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
.
24.
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
.
25.
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
.
26.
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 Pt. 1
), pp.
7022
7032
.http://cancerres.aacrjournals.org/content/48/24_Part_1/7022.short
27.
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. Oncol. Biol. Phys.
,
43
(
5
), pp.
1111
1123
.
28.
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
.
29.
Byrne
,
H.
, and
Chaplain
,
M. A.
,
1996
, “
Modelling the Role of Cell-Cell Adhesion in the Growth and Development of Carcinomas
,”
Math. Comput. Modell.
,
24
(
12
), pp.
1
17
.
30.
Jones
,
A.
,
Byrne
,
H.
,
Gibson
,
J.
, and
Dold
,
J.
,
2000
, “
A Mathematical Model of the Stress Induced During Avascular Tumour Growth
,”
J. Math. Biol.
,
40
(
6
), pp.
473
499
.
31.
Konofagou
,
E. E.
,
Harrigan
,
T. P.
,
Ophir
,
J.
, and
Krouskop
,
T. A.
,
2001
, “
Poroelastography: Imaging the Poroelastic Properties of Tissues
,”
Ultrasound Med. Biol.
,
27
(
10
), pp.
1387
1397
.
32.
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
.
33.
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
), pp.
6525
6541
.
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
), pp.
6291
6313
.
35.
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
.
36.
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
.http://medicaldiagnostics.asmedigitalcollection.asme.org/article.aspx?articleid=2681016
37.
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 I
,”
ASME J. Biomech. Eng.
(in press).
38.
Swartz
,
M. A.
, and
Fleury
,
M. E.
,
2007
, “
Interstitial Flow and Its Effects in Soft Tissues
,”
Annu. Rev. Biomed. Eng.
,
9
, pp.
229
256
.
39.
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
.
40.
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
.
41.
Verruijt
,
A.
,
2013
, “
Theory and Problems of Poroelasticity
,” Delft University of Technology, Delft, The Netherlands.
42.
Grodzinsky
,
A.
,
Roth
,
V.
,
Myers
,
E.
,
Grossman
,
W.
, and
Mow
,
V.
,
1981
, “
The Significance of Electromechanical and Osmotic Forces in the Nonequilibrium Swelling Behavior of Articular Cartilage in Tension
,”
ASME J. Biomech. Eng.
,
103
(
4
), pp.
221
231
.
43.
Muskat
,
M.
, and
Wyckoff
,
R. D.
,
1937
,
Flow of Homogeneous Fluids Through Porous Media
,
McGraw-Hill Book
,
New York
.
44.
Schmidt
,
J. E.
, and
Sonnemann
,
G.
,
1960
, “
Transient Temperatures and Thermal Stresses in Hollow Cylinders Due to Heat Generation
,”
ASME J. Heat Transfer
,
82
(
4
), pp.
273
278
.
45.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
,
1959
,
Conduction of Heat in Solids
,
2nd ed.
,
Clarendon Press
,
Oxford, UK
.
46.
Duffy
,
D. G.
,
2010
,
Advanced Engineering Mathematics With MATLAB
,
CRC Press
, Boca Raton, FL.
47.
Hibbitt
,
K.
,
2005
,
Abaqus/Explicit User's Manual, Version 6.5
, Vol.
1
,
Sorensen
, Providence, RI.
48.
Chaudhry
,
A.
,
Unnikrishnan
,
G.
,
Reddy
,
J.
,
Krouskop
,
T. A.
, and
Righetti
,
R.
,
2013
, “
Effect of Permeability on the Performance of Elastographic Imaging Techniques
,”
IEEE Trans. Med. Imaging
,
32
(
2
), pp.
189
199
.
49.
Adam
,
J. A.
,
1987
, “
A Mathematical Model of Tumor Growth. ii. effects of Geometry and Spatial Nonuniformity on Stability
,”
Math. Biosci.
,
86
(
2
), pp.
183
211
.
50.
Bertuzzi
,
A.
,
Fasano
,
A.
, and
Gandolfi
,
A.
,
2005
, “
A Mathematical Model for Tumor Cords Incorporating the Flow of Interstitial Fluid
,”
Math. Models Methods Appl. Sci.
,
15
(
11
), pp.
1735
1777
.
51.
Rizwan-Uddin
, and
Saeed
,
I. M.
,
1998
, “
Structure and Growth of Tumors: The effect of Cartesian, Cylindrical, and Spherical Geometries
,”
Ann. N. Y. Acad. Sci.
,
858
(
1
), pp.
127
136
.http://verl.npre.illinois.edu/Documents/J-98-01.pdf
52.
Sciumè
,
G.
,
Shelton
,
S.
,
Gray
,
W. G.
,
Miller
,
C. T.
,
Hussain
,
F.
,
Ferrari
,
M.
,
Decuzzi
,
P.
, and
Schrefler
,
B.
,
2013
, “
A Multiphase Model for Three-Dimensional Tumor Growth
,”
New J. Phys.
,
15(1
), p.
015005
.
53.
Zhi
,
H.
,
Ou
,
B.
,
Luo
,
B.-M.
,
Feng
,
X.
,
Wen
,
Y.-L.
, and
Yang
,
H.-Y.
,
2007
, “
Comparison of Ultrasound Elastography, Mammography, and Sonography in the Diagnosis of Solid Breast Lesions
,”
J. Ultrasound Med.
,
26
(
6
), pp.
807
815
.
54.
Sinkus
,
R.
,
Tanter
,
M.
,
Xydeas
,
T.
,
Catheline
,
S.
,
Bercoff
,
J.
, and
Fink
,
M.
,
2005
, “
Viscoelastic Shear Properties of In Vivo Breast Lesions Measured by Mr Elastography
,”
Magn. Reson. Imaging
,
23
(
2
), pp.
159
165
.
55.
Rzymski
,
P.
,
Skórzewska
,
A.
, and
Opala
,
T.
,
2011
, “
Changes in Ultrasound Shear Wave Elastography Properties of Normal Breast During Menstrual Cycle
,”
Clin. Exp. Obstet. Gynecol.
,
38
(
2
), pp.
137
142
.https://www.ncbi.nlm.nih.gov/pubmed/21793274
You do not currently have access to this content.