An analytical poroelastic model of a non-homogeneous medium under creep compression for ultrasound poroelastography applications - Part I.

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 (FEM). 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 the different mechanical parameters on the estimated displacements, strains, stresses and fluid pressure inside a tumor and in the surrounding tissue.

[1]  H M Byrne,et al.  A mathematical model of the stress induced during avascular tumour growth , 2000, Journal of mathematical biology.

[2]  Jeffrey C Bamber,et al.  Towards an acoustic model-based poroelastic imaging method: I. Theoretical foundation. , 2006, Ultrasound in medicine & biology.

[3]  Jonathan Ophir,et al.  The feasibility of using poroelastographic techniques for distinguishing between normal and lymphedematous tissues in vivo , 2007, Physics in medicine and biology.

[4]  M. Biot.  MECHANICS OF DEFORMATION AND ACOUSTIC PROPAGATION IN POROUS MEDIA , 1962 .

[5]  R K Jain,et al.  Transport of fluid and macromolecules in tumors. III. Role of binding and metabolism. , 1991 .

[6]  Cass T. Miller,et al.  A multiphase model for three-dimensional tumor growth , 2013, New journal of physics.

[7]  D Pflaster,et al.  A poroelastic finite element formulation including transport and swelling in soft tissue structures. , 1996, Journal of biomechanical engineering.

[8]  Hui Zhi,et al.  Comparison of Ultrasound Elastography, Mammography, and Sonography in the Diagnosis of Solid Breast Lesions , 2007, Journal of ultrasound in medicine : official journal of the American Institute of Ultrasound in Medicine.

[9]  Dean G. Duffy,et al.  Advanced Engineering Mathematics with MATLAB , 2016 .

[10]  A. Khoshghalb.  On Creep Laboratory Tests in Soil Mechanics , 2013 .

[11]  Ricky T. Tong,et al.  Effect of vascular normalization by antiangiogenic therapy on interstitial hypertension, peritumor edema, and lymphatic metastasis: insights from a mathematical model. , 2007, Cancer research.

[12]  A. Bertuzzi,et al.  A MATHEMATICAL MODEL FOR TUMOR CORDS INCORPORATING THE FLOW OF INTERSTITIAL FLUID , 2005 .

[13]  R K Jain,et al.  Mechanisms of heterogeneous distribution of monoclonal antibodies and other macromolecules in tumors: significance of elevated interstitial pressure. , 1988, Cancer research.

[14]  Van C. Mow,et al.  Recent Developments in Synovial Joint Biomechanics , 1980 .

[15]  Deformation of spherical cavities and inclusions in fluid-infiltrated elastic materials , 1978 .

[16]  Xueqin Zhao,et al.  Discrimination Between Cervical Cancer Cells and Normal Cervical Cells Based on Longitudinal Elasticity Using Atomic Force Microscopy , 2015, Nanoscale Research Letters.

[17]  R K Jain,et al.  Transport of molecules in the tumor interstitium: a review. , 1987, Cancer research.

[18]  J. N. Reddy,et al.  Effect of Permeability on the Performance of Elastographic Imaging Techniques , 2013, IEEE Transactions on Medical Imaging.

[19]  T. Krouskop,et al.  Elastography: Ultrasonic estimation and imaging of the elastic properties of tissues , 1999, Proceedings of the Institution of Mechanical Engineers. Part H, Journal of engineering in medicine.

[20]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[21]  Dewey H. Hodges,et al.  Theory and problems , 1996 .

[22]  R. Skalak,et al.  Time-dependent behavior of interstitial fluid pressure in solid tumors: implications for drug delivery. , 1995, Cancer research.

[23]  Mostafa Fatemi,et al.  Automated Compression Device for Viscoelasticity Imaging , 2017, IEEE Transactions on Biomedical Engineering.

[24]  A. Segall,et al.  Thermoelastic Analysis of Thick-Walled Vessels Subjected to Transient Thermal Loading , 2001 .

[25]  Malisa Sarntinoranont,et al.  Interstitial Stress and Fluid Pressure Within a Growing Tumor , 2004, Annals of Biomedical Engineering.

[26]  R. Jain,et al.  Delivery of molecular and cellular medicine to solid tumors. , 1998, Journal of controlled release : official journal of the Controlled Release Society.

[27]  V C Mow,et al.  The significance of electromechanical and osmotic forces in the nonequilibrium swelling behavior of articular cartilage in tension. , 1981, Journal of biomechanical engineering.

[28]  M Muskat,et al.  THE FLOW OF HOMOGENEOUS FLUIDS THROUGH POROUS MEDIA: ANALOGIES WITH OTHER PHYSICAL PROBLEMS , 1937 .

[29]  Farshid Guilak,et al.  Stress, Strain, Pressure and Flow Fields in Articular Cartilage and Chondrocytes , 1994 .

[30]  A. Arneodo,et al.  From elasticity to inelasticity in cancer cell mechanics: A loss of scale-invariance , 2016 .

[31]  J. Adam.  A mathematical model of tumor growth. II. effects of geometry and spatial nonuniformity on stability , 1987 .

[32]  R K Jain,et al.  Transport of fluid and macromolecules in tumors. IV. A microscopic model of the perivascular distribution. , 1991, Microvascular research.

[33]  Matija Snuderl,et al.  Coevolution of solid stress and interstitial fluid pressure in tumors during progression: implications for vascular collapse. , 2013, Cancer research.

[34]  Jeffrey C Bamber,et al.  Coupling between elastic strain and interstitial fluid flow: ramifications for poroelastic imaging , 2006, Physics in medicine and biology.

[35]  J. Suh,et al.  Biphasic Poroviscoelastic Behavior of Hydrated Biological Soft Tissue , 1999 .

[36]  R K Jain,et al.  Transport of fluid and macromolecules in tumors. II. Role of heterogeneous perfusion and lymphatics. , 1990, Microvascular research.

[37]  W Ehlers,et al.  A linear viscoelastic biphasic model for soft tissues based on the Theory of Porous Media. , 2001, Journal of biomechanical engineering.

[38]  Todd H. Skaggs,et al.  Analytical solutions of the one-dimensional advection–dispersion solute transport equation subject to time-dependent boundary conditions , 2013 .

[39]  R. Skalak,et al.  Macro- and Microscopic Fluid Transport in Living Tissues: Application to Solid Tumors , 1997 .

[40]  Byungkyu Kim,et al.  Cell Stiffness Is a Biomarker of the Metastatic Potential of Ovarian Cancer Cells , 2012, PloS one.

[41]  Rizwan-Uddin,et al.  Structure and Growth of Tumors: The Effect of Cartesian, Cylindrical, and Spherical Geometries , 1998, Annals of the New York Academy of Sciences.

[42]  Pu Chen,et al.  Numerical Modeling of Fluid Flow in Solid Tumors , 2011, PloS one.

[43]  Paolo A. Netti,et al.  A poroelastic model for interstitial pressure in tumors , 1995 .

[44]  P. Rzymski,et al.  Changes in ultrasound shear wave elastography properties of normal breast during menstrual cycle. , 2011, Clinical and experimental obstetrics & gynecology.

[45]  Francis A. Duck,et al.  Physical properties of tissue : a comprehensive reference book , 1990 .

[46]  Y. Fung,et al.  Biomechanics: Mechanical Properties of Living Tissues , 1981 .

[47]  A. Fyles,et al.  The relationship between elevated interstitial fluid pressure and blood flow in tumors: a bioengineering analysis. , 1999, International journal of radiation oncology, biology, physics.

[48]  Mickael Tanter,et al.  Viscoelastic shear properties of in vivo breast lesions measured by MR elastography. , 2005, Magnetic resonance imaging.

[49]  G A Ateshian,et al.  A Theoretical Analysis of Water Transport Through Chondrocytes , 2007, Biomechanics and modeling in mechanobiology.

[50]  J. Rudnicki,et al.  Shear properties of heterogeneous fluid-filled porous media with spherical inclusions , 2016 .

[51]  J. Rudnicki,et al.  Dynamic transverse shear modulus for a heterogeneous fluid-filled porous solid containing cylindrical inclusions , 2016 .

[52]  Triantafyllos Stylianopoulos,et al.  The role of mechanical forces in tumor growth and therapy. , 2014, Annual review of biomedical engineering.

[53]  P. Gullino,et al.  Diffusion and convection in normal and neoplastic tissues. , 1974, Cancer research.

[54]  Md Tauhidul Islam,et al.  An analytical poroelastic model of a non-homogeneous medium under creep compression for ultrasound poroelastography applications - Part II. , 2018, Journal of biomechanical engineering.

[55]  Jianhua Deng,et al.  The enhanced anticoagulation for graphene induced by COOH+ ion implantation , 2015, Nanoscale Research Letters.

[56]  Savio Lau-Yuen Woo,et al.  Biomechanics of diarthrodial joints , 1990 .

[57]  M. Chaplain,et al.  Modelling the role of cell-cell adhesion in the growth and development of carcinomas , 1996 .

[58]  R. Jain,et al.  Role of extracellular matrix assembly in interstitial transport in solid tumors. , 2000, Cancer research.

[59]  Jonathan Ophir,et al.  The feasibility of using elastography for imaging the Poisson's ratio in porous media. , 2004, Ultrasound in medicine & biology.

[60]  W M Lai,et al.  An analysis of the unconfined compression of articular cartilage. , 1984, Journal of biomechanical engineering.

[61]  V. Mow,et al.  Biphasic creep and stress relaxation of articular cartilage in compression? Theory and experiments. , 1980, Journal of biomechanical engineering.

[62]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[63]  Md Tauhidul Islam,et al.  An Analytical Model of Tumors With Higher Permeability Than Surrounding Tissues for Ultrasound Elastography Imaging , 2018, Journal of Engineering and Science in Medical Diagnostics and Therapy.

[64]  M. Swartz,et al.  Interstitial flow and its effects in soft tissues. , 2007, Annual review of biomedical engineering.

[65]  James G. Berryman,et al.  Scattering by a spherical inhomogeneity in a fluid‐saturated porous medium , 1985 .

[66]  R K Jain,et al.  Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection. , 1989, Microvascular research.

[67]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[68]  Y. Abousleiman,et al.  Porothermoelastic analyses of anisotropic hollow cylinders with applications , 2005 .

[69]  M. Lekka.  Discrimination Between Normal and Cancerous Cells Using AFM , 2016, BioNanoScience.

[70]  Y. Abousleiman,et al.  TIME-DEPENDENT POROMECHANICAL RESPONSES OF SATURATED CYLINDERS , 2001 .

[71]  R. Howe,et al.  Tactile imaging of breast masses: first clinical report. , 2001, Archives of surgery.

[72]  J. Ophir,et al.  Elastography: A Quantitative Method for Imaging the Elasticity of Biological Tissues , 1991, Ultrasonic imaging.

[73]  J. Bishop,et al.  Visualization and quantification of breast cancer biomechanical properties with magnetic resonance elastography. , 2000, Physics in medicine and biology.

[74]  R K Jain,et al.  Mechanics of interstitial-lymphatic fluid transport: theoretical foundation and experimental validation. , 1999, Journal of biomechanics.

[75]  Md Tauhidul Islam,et al.  An analytical poroelastic model for ultrasound elastography imaging of tumors , 2018, Physics in medicine and biology.

[76]  T. Krouskop,et al.  Poroelastography: imaging the poroelastic properties of tissues. , 2001, Ultrasound in medicine & biology.