The role of structural viscoelasticity in deformable porous media with incompressible
constituents: Applications in biomechanics

Maurizio Verri; Giovanna Guidoboni; Lorena Bociu; Riccardo Sacco; Maurizio Verri; Giovanna Guidoboni; Lorena Bociu; Riccardo Sacco

doi:10.3934/mbe.2018042

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 4: 933-959. doi: 10.3934/mbe.2018042

Previous Article Next Article

The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics

1.
Politecnico di Milano, Dipartimento di Matematica, Italy
2.
University of Missouri, Department of Electrical Engineering and Computer Science, USA
3.
NC State University, Department of Mathematics, USA
4.
Politecnico di Milano, Dipartimento di Matematica, Italy

Received: 24 July 2017 Accepted: 27 December 2017 Published: 01 August 2018
MSC : Primary: 35Q74, 35B44; Secondary: 35C10

The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimer’s disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.
- Deformable porous media flow,
- incompressible constituents,
- viscoelasticity,
- explicit solution,
- velocity blow-up,
- confined compression
Citation: Maurizio Verri, Giovanna Guidoboni, Lorena Bociu, Riccardo Sacco. The role of structural viscoelasticity in deformable porous media with incompressibleconstituents: Applications in biomechanics[J]. Mathematical Biosciences and Engineering, 2018, 15(4): 933-959. doi: 10.3934/mbe.2018042

Related Papers:

Abstract

The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimer’s disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.

References

[1]	[ P. Augat,S. Schorlemmer, The role of cortical bone and its microstructure in bone strength, Age and Ageing, 35 (2006): ii27-ii31.
[2]	[ H. T. Banks,K. Bekele-Maxwell,L. Bociu,M. Noorman,G. Guidoboni, Sensitivity analysis in poro-elastic and poro-visco-elastic models with respect to boundary data, Quart. Appl. Math. In press, 75 (2017): 697-735.
[3]	[ H. Barucq,M. Madaune-Tort,P. Saint-Macary, On nonlinear Biot's consolidation models, Nonlinear Anal Theory Methods Appl., 63 (2005): e985-e995, Invited Talks from the Fourth World Congress of Nonlinear Analysts (WCNA 2004).
[4]	[ H. Barucq,M. Madaune-Tort,P. Saint-Macary, Some existence-uniqueness results for a class of one-dimensional nonlinear Biot models, Nonlinear Anal Theory Methods Appl., 61 (2005): 591-612.
[5]	[ M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941): 155-164.
[6]	[ L. Bociu,G. Guidoboni,R. Sacco,J. Webster, Analysis of nonlinear poro-elastic and poro-viscoelastic models, Arch. Rational Mech. Anal., 222 (2016): 1445-1519.
[7]	[ S. Canic,J. Tambaca,G. Guidoboni,A. Mikelic,C. J. Hartley,D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math., 67 (2006): 164-193.
[8]	[ Y. Cao,S. Chen,A. J. Meir, Analysis and numerical approximations of equations of nonlinear poroelasticity, Discrete Continuous Dyn Syst Ser B, 18 (2013): 1253-1273.
[9]	[ P. Causin,G. Guidoboni,A. Harris,D. Prada,R. Sacco,S. Terragni, A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head, Math. Biosci., 257 (2014): 33-41.
[10]	[ P. Causin,R. Sacco,M. Verri, A multiscale approach in the computational modeling of the biophysical environment in artificial cartilage tissue regeneration, Biomech. Model. Mechanobiol, 12 (2013): 763-780.
[11]	[ D. Chapelle,J. Sainte-Marie,J.-F. Gerbeau,I. Vignon-Clementel, A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech., 46 (2010): 91-101.
[12]	[ O. Coussy, Poromechanics, John Wiley & Sons Ltd, 2004.
[13]	[ S. C. Cowin, Bone poroelasticity, J. Biomech., 32 (1999): 217-238.
[14]	[ R. de Boer, Theory of Porous Media. Highlights in the Historical Development and Current State, Springer, Berlin/New York, 2000.
[15]	[ E. Detournay,A. H.-D. Cheng, Poroelastic response of a borehole in a non-hydrostatic stress field, Int J Rock Mech Min Sci Geomech Abstr, 25 (1988): 171-182.
[16]	[ E. Detournay,A. H.-D. Cheng, Fundamentals of poroelasticity, Comprehensive rock engineering, 2 (1993): 113-171.
[17]	[ J. C. Downs,J. K. Suh,K. A. Thomas,A. J. Bellezza,R. T. Hart,C. F. Burgoyne, Viscoelastic material properties of the peripapillary sclera in normal and early-glaucoma monkey eyes, Invest. Ophthalmol. Vis. Sci., 46 (2005): 540-546.
[18]	[ J. W. Freeman,M. D. Woods,D. A. Cromer,L. D. Wright,C. T. Laurencin, Tissue engineering of the anterior cruciate ligament: The viscoelastic behavior and cell viability of a novel braid-twist scaffold, J Biomater Sci Polym Ed, 20 (2009): 1709-1728.
[19]	[ I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Elsevier/Academic Press, Amsterdam, 2015.
[20]	[ A. P. Guérin,B. Pannier,S. J. Marchais,G. M. London, Arterial structure and function in end-stage renal disease, Curr. Hypertens. Rep., 10 (2008): 107-111.
[21]	[ W. M. Lai,J. S. Hou,V. C. Mow, A triphasic theory for the swelling and deformation behaviors of articular cartilage, ASME J. Biomech. Eng., 113 (1991): 245-258.
[22]	[ V. C. Mow,S. C. Kuei,W. M. Lai,C. G. Armstrong, Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiments, ASME J. Biomech. Eng., 102 (1980): 73-84.
[23]	[ H. T. Nia,L. Han,Y. Li,C. Ortiz,A. Grodzinsky, Poroelasticity of cartilage at the nanoscale, Biophys. J., 101 (2011): 2304-2313.
[24]	[ M. S. Osidak,E. O. Osidak,M. A. Akhmanova,S. P. Domogatsky,A. S. Domogatskaya, Fibrillar, fibril-associated and basement membrane collagens of the arterial wall: Architecture, elasticity and remodeling under stress, Curr. Pharm. Des., 21 (2015): 1124-1133.
[25]	[ S. Owczarek, A Galerkin method for Biot consolidation model, Math. Mech. Solids, 15 (2010): 42-56.
[26]	[ N. Özkaya, M. Nordin, D. Goldsheyder and D. Leger, Fundamentals of Biomechanics. Equilibrium, Motion, and Deformation, Springer, New York, 1999.
[27]	[ P. J. Phillips,M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity Ⅰ: The continuous in time case, Comput Geosci, 11 (2007): 131-144.
[28]	[ P. J. Phillips,M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity Ⅱ: The continuous in time case, Comput Geosci, 11 (2007): 145-158.
[29]	[ P. J. Phillips,M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity, Comput Geosci, 12 (2008): 417-435.
[30]	[ L. Recha-Sancho, F. T. Moutos, J. Abell, F. Guilak and C. E. Semino, Dedifferentiated human articular chondrocytes redifferentiate to a cartilage-like tissue phenotype in a poly($\varepsilon$-caprolactone)/self-assembling peptide composite scaffold, Materials, 9 (2016), p472.
[31]	[ R. Sacco,P. Causin,P. Zunino,M. T. Raimondi, A multiphysics/multiscale 2D numerical simulation of scaffold-based cartilage regeneration under interstitial perfusion in a bioreactor, Biomech. Model. Mechanobiol., 10 (2011): 577-589.
[32]	[ I. Sack,B. Beierbach,J. Wuerfel,D. Klatt,U. Hamhaber,S. Papazoglou,P. Martus,J. Braun, The impact of aging and gender on brain viscoelasticity, NeuroImage, 46 (2009): 652-657.
[33]	[ A. P. S. Selvadurai, On the mechanics of damage-susceptible poroelastic media, Key Engineering Materials, 251/252 (2003): 363-374.
[34]	[ A. Settari and D. A. Walters, Advances in Coupled Geomechanical and Reservoir Modeling with Applications to Reservoir Compaction, Technical report, SPE Reservoir Simulation Symposium, 1999.
[35]	[ R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251 (2000): 310-340.
[36]	[ R. E. Showalter,N. Su, Partially saturated flow in a poroelastic medium, Discrete Continuous Dyn Syst Ser B, 1 (2001): 403-420.
[37]	[ M. A. Soltz,G. A. Ateshian, Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization at an impermeable contact interface in confined compression, J. Biomech., 31 (1998): 927-934.
[38]	[ K. Terzaghi, Erdbaumechanik auf Bodenphysikalischer Grundlage, Deuticke, Wien, 1925.
[39]	[ A. A. Tofangchi Mahyari, Computational Modelling of Fracture and Damage in Poroelastic Media, PhD thesis, Department of Civil Engineering and Applied Mechanics, McGill University, 1997.
[40]	[ A. Zenisek, The existence and uniqueness theorem in Biot's consolidation theory, Apl Mat, 29 (1984): 194-211.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)