Skip to main content
SpringerLink
Log in

Porosity and Diffusion in Biological Tissues. Recent Advances and Further Perspectives

  • Chapter
  • First Online:
Constitutive Modelling of Solid Continua

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 262))

Abstract

We present a review of porosity and diffusion in biological tissues from different perspectives. We first introduce the topic by illustrating experimental evidence related to diffusion in porous media and review a number of state of the art experimental techniques. We then proceed by providing a revisited derivation of the equations of poroelasticity from the microstructure (via asymptotic homogenization), which is especially aimed at giving a first insight on the topic to both students and scientists who are not familiar with the subject. Results based on this kind of models have only recently been presented in the literature and could possibly complement the experiments by getting a more thorough understanding on the complex interplay between porosity and diffusion. We investigate further the matter by exploring the role of diffusion in driving growth and stresses in the context of linear elastic modeling for tumors and cellular automata. We finally conclude the chapter by (a) discussing diffusion in nonlinear, “active” materials, i.e., those which are possibly characterized by growth and remodeling, and (b) offering an overview on cutting edge research problems on diffusion for this class of complex materials.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Note that here \(\mathbb {R}_+^*\) stands for \(\mathbb {R}_+\) excluding 0.

  2. 2.

    The literature on growth and remodeling—especially on the mechanical aspects of these phenomena—has been proliferating in the last few years, and even attempting to provide an adequate list of authors is not an easy task.

  3. 3.

    This means that \(\mathrm{{Grad}}\,\mathbf {F}_{\mathrm {g}}\) is not rated among the variables determining the kinematic picture of the theory. Of course, it can be computed a posteriori.

References

  1. Alexandrakis G, Brown EB, Tong RT, McKee TD, Campbell RB, Boucher Y, Jain RK (2004) Two-photon fluorescence correlation microscopy reveals the two-phase nature of transport in tumors. Nat Med 10:203–207

    Article  Google Scholar 

  2. Ambrosi D, Ateshian GA, Arruda EM, Cowin SC, Dumais J, Goriely A, Holzapfel GA, Humphrey JD, Kemkemer R, Kuhl E, Olbering JE, Taber LA, Garikipati K (2011) Perspectives on biological growth and remodeling. J Mech Phys Solids 59:863–883

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosi D, Preziosi L, Vitale G (2010) The insight of mixtures theory for growth and remodeling. Z Angew Math Phys 61:177–191

    Article  MathSciNet  MATH  Google Scholar 

  4. Antoine EE, Vlachos PP, Rylander MN (2015) Tunable collagen I hydrogels for engineered physiological tissue micro-environments. PLoS ONE 10:e0122500

    Article  Google Scholar 

  5. Atanacković TM, Pilipović S, Stanković B, Zorica D (2014) Fractional calculus with applications in mechanics – vibrations and diffusion processes. Wiley, GB

    Book  MATH  Google Scholar 

  6. Atanacković TM, Pilipović S, Stanković B, Zorica D (2014) Fractional calculus with applications in mechanics – wave propagation, impact and variational principles. Wiley, London

    Book  MATH  Google Scholar 

  7. Atanacković TM, Stanković B (2009) Generalized wave equation in nonlocal elasticity. Acta Mech 208:1–10

    Article  MATH  Google Scholar 

  8. Ateshian GA (2007) On the theory of reactive mixtures for modeling biological growth. Biomech. Model. Mechanobiol. 6:423–445

    Article  Google Scholar 

  9. Ateshian GA, Humphrey J (2012) Continuum mixture models of biological growth and remodeling: past successes and future opportunities. Ann Rev Biomed Eng 14:97–111

    Article  Google Scholar 

  10. Ateshian GA, Weiss JA (2010) Anisotropic hydraulic permeability under finite deformation. J Biomech Eng 132, 111004-1—111004-7

    Google Scholar 

  11. Auriault J-L, Boutin C, Geindreau C (2010) Homogenization of coupled phenomena in heterogeneous media. Wiley, London

    Google Scholar 

  12. Baaijens F, Bouten C, Driessen N (2010) Modeling cartilage remodeling. J Biomech 43:166–175

    Article  Google Scholar 

  13. Bakhvalov NS, Panasenko G (2012) Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials. Kluwer, Dordrecht

    MATH  Google Scholar 

  14. Bear J, Fel LG, Zimmels Y (2010) Effects of material symmetry on the coefficients of transport in anisotropic porous media. Transp Porous Med 82:347–361

    Article  Google Scholar 

  15. Bell E, Ivarsson B, Merrill C (1979) Production of a tissue-like structure by contraction of collagen lattices by human fibroblasts of different proliferative potential in vitro. Proc Natl Acad Sci 76:1274–1278

    Article  Google Scholar 

  16. Bennethum LS, Murad MA, Cushman JH (2000) Macroscale thermodynamics and the chemical potential for swelling porous media. Transp Porous Med 39:187–225

    Article  MathSciNet  Google Scholar 

  17. Bensoussan A, Lions J-L, Papanicolaou G (2011) Asymptotic analysis for periodic structures. American Mathematical Society Chelsea Publishing, Providence RI

    MATH  Google Scholar 

  18. Bi X, Li G, Doty SB, Camacho NP (2005) A novel method for determination of collagen orientation in cartilage by Fourier transform infrared imaging spectroscopy (FT-IRIS). Osteoarthr Cartil 13:1050–1058

    Article  Google Scholar 

  19. Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26:182–185

    Article  MathSciNet  MATH  Google Scholar 

  20. Biot MA (1956) General solutions of the equations of elasticity and consolidation for a porous material. J Appl Mech 23:91–96

    MathSciNet  MATH  Google Scholar 

  21. Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. ii. higher frequency range. J Acoust Soc Am 28, 179–191

    Article  MathSciNet  Google Scholar 

  22. Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33:1482–1498

    Article  MathSciNet  MATH  Google Scholar 

  23. Bottaro A, Ansaldi T (2012) On the infusion of a therapeutic agent into a solid tumor modeled as a poroelastic medium. J Biomech Eng 134:084501

    Article  Google Scholar 

  24. Burridge R, Keller JB (1981) Poroelasticity equations derived from microstructure. J Acoust Soc Am 70:1140–1146

    Article  MATH  Google Scholar 

  25. Byrne HM, Chaplain MA (1995) Growth of nonnecrotic tumors in the presence of absence of inhibitors. Math Biosci 130:151–181

    Article  MATH  Google Scholar 

  26. Bynre HM, Drasdo D (2009) Individual-based and continuum models of growing cell populations. J Math Biol 58:657–687

    Article  MathSciNet  MATH  Google Scholar 

  27. Carfagna M, Grillo A (2017) The spherical design algorithm in the numerical simulation of biological tissues with statistical fiber-reinforcement. Comput Visual Sci 18:157–184

    Article  MATH  Google Scholar 

  28. Carpinteri A, Cornetti P, Sapora A (2011) A fractional calculus approach to nonlocal elasticity. Eur Phys J Spec Top 193:193–204

    Article  MATH  Google Scholar 

  29. Casciari JJ, Sotirchos SV, Sutherland RM (1992) Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH. J Cell Physio 151:386–394

    Article  Google Scholar 

  30. Cermelli P, Fried E, Sellers S (2001) Configurational stress, yield and flow in rate-independent plasticity. Proc R Soc A 457:1447–1467

    Article  MathSciNet  MATH  Google Scholar 

  31. Chalasani R, Poole-Warren L, Conway RM, Ben-Nissan B (2007) Porous orbital implants in enucleation: a systematic review. Surv. Ophthalmol. 52:145–155

    Article  Google Scholar 

  32. Challamel N, Zorica D, Atanacković TM, Spasić DT (2013) On the fractional generalization of Eringen’s nonlocal elasticity for wave propagation. C R Mec 341:298–303

    Article  Google Scholar 

  33. Cheng AH-D (2016) Poroelasticity. Springer, Switzerland

    Book  MATH  Google Scholar 

  34. Cowin SC (1999) Bone poroelasticity. J Biomech 32:217–238

    Article  Google Scholar 

  35. Cowin SC (2000) How is a tissue built? J Biomech Eng 122:553–569

    Article  Google Scholar 

  36. Crevacore E, Di Stefano S, Grillo A (2019) Coupling among deformation, fluid flow, structural reorganisation and fiber reorientation in fiber-reinforced, transversely isotropic biological tissues. Int J Non-Linear Mech 111:1–13

    Article  Google Scholar 

  37. Cyron C, Humphrey J (2017) Growth and remodeling of load-bearing biological soft tissues. Meccanica 52:645–664

    Article  MathSciNet  Google Scholar 

  38. Dalwadi MP, Griffiths IM, Bruna M (2015) Understanding how porosity gradients can make a better filter using homogenization theory. Proc R Soc A 471:20150464

    Article  MathSciNet  MATH  Google Scholar 

  39. Davit Y, Bell CG, Byrne HM, Chapman LA, Kimpton LS, Lang GE, Leonard KH, Oliver JM, Pearson NC, Shipley RJ et al (2013) Homogenization via formal multiscale asymptotics and volume averaging: how do the two techniques compare? Adv Water Resour 62:178–206

    Article  Google Scholar 

  40. Dehghani H, Penta R, Merodio J (2018) The role of porosity and solid matrix compressibility on the mechanical behavior of poroelastic tissues. Mater Res Exp 6:035404

    Article  Google Scholar 

  41. Deutsch A, Dormann S (2005) Cellular automaton modeling of biological pattern formation: characterization, applications, and analysis. Birkh’auser, Basel

    Google Scholar 

  42. DiCarlo A, Quiligotti S (2002) Growth and balance. Mech Res Commun 29:449–456

    Article  MathSciNet  MATH  Google Scholar 

  43. Di Domenico CD, Lintz M, Bonassar LJ (2018) Molecular transport in articular cartilage - what have we learned from the past 50 years? Nat Rev Rheumatol 14:393–403

    Article  Google Scholar 

  44. Di Paola M, Zingales M (2008) Long-range cohesive interactions of non-local continuum faced by fractional calculus. Int J Solids Struct 45:5642–5659

    Article  MATH  Google Scholar 

  45. Di Stefano S, Ramírez-Torres A, Penta R, Grillo A (2018) Self-influenced growth through evolving material inhomogeneities. Int J Non-Linear Mech 106:174–187

    Article  Google Scholar 

  46. Di Stefano S, Carfagna M, Knodel MM, Kotaybah H, Federico S, Grillo A (2019) Anelastic reorganisation of fiber-reinforced biological tissues. Comput Visual Sci in press. https://doi.org/10.1007/s00791-019-00313-1

    Article  MathSciNet  Google Scholar 

  47. Driessen N, Wilson W, Bouten C, Baaijens F (2004) A computational model for collagen fibre remodelling in the arterial wall. J Theoret Biol 226:53–64

    Article  Google Scholar 

  48. Elson EL, Magde D (1974) Fluorescence correlation spectroscopy. I. Conceptual basis and theory. Biopolymers 13:1–27

    Article  Google Scholar 

  49. Epstein M, Maugin GA (1996) On the geometrical material structure of anelasticity. Acta Mech 115:119–131

    Article  MathSciNet  MATH  Google Scholar 

  50. Epstein M, Maugin GA (2000) Thermomechanics of volumetric growth in uniform bodies. Int J Plasticity 16:951–978

    Article  MATH  Google Scholar 

  51. Erikson A, Andersen HN, Naess SN, Sikorski P, De Lange Davies C (2008) Physical and chemical modifications of collagen gels: impact on diffusion. Biopolymers 89:135–143

    Article  Google Scholar 

  52. Estrada-Rodriguez G, Gimperlein H, Painter KJ (2017) Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion. SIAM J Appl Math 78:1155–1173

    Article  MathSciNet  MATH  Google Scholar 

  53. Evans RC, Quinn TM (2005) Solute diffusivity correlates with mechanical properties and matrix density of compressed articular cartilage. Arch Biochem Biophys 442:1–10

    Article  Google Scholar 

  54. Federico S (2012) Covariant formulation of the tensor algebra of non-linear elasticity. Int J Non-Linear Mech 47:273–284

    Article  Google Scholar 

  55. Federico S, Gasser TC (2010) Nonlinear elasticity of biological tissues with statistical fiber orientation. J R Soc Interface 7:955–966

    Article  Google Scholar 

  56. Federico S, Grillo A (2012) Elasticity and permeability of porous fiber-reinforced materials under large deformations. Mech Mater 44:58–71

    Article  Google Scholar 

  57. Federico S, Herzog W (2008) On the anisotropy and inhomogeneity of permeability in articular cartilage. Biomech Model Mechanobiol 77:367–378

    Article  Google Scholar 

  58. Federico S, Herzog W (2008) On the permeability of fiber-reinforced porous materials. Int J Solids Struct 45:2160–2172

    Article  MATH  Google Scholar 

  59. Fel L, Bear J (2010) Dispersion and dispersivity tensors in saturated porous media with uniaxial symmetry. Transp Porous Med 85:259–268

    Article  MathSciNet  Google Scholar 

  60. Fung YC (1990) Biomechanics: motion, flow, stress, and growth. Springer, New York

    Book  MATH  Google Scholar 

  61. Garikipati K, Arruda E, Grosh K, Narayanan H, Calve S (2004) A continuum treatment of growth in biological tissues: the coupling of mass transport and mechanics. J Mech Phys Solids 52:1595–1625

    Article  MathSciNet  MATH  Google Scholar 

  62. Giverso C, Scianna M, Grillo A (2015) Growing avascular tumours as elasto-plastic bodies by the theory of evolving natural configurations. Mech Res Commun 68:31–39

    Article  Google Scholar 

  63. Goriely A (2016) The mathematics and mechanics of biological growth. Springer, New York

    MATH  Google Scholar 

  64. Grillo A, Carfagna M, Federico S (2017) Non-darcian flow in fiber-reinforced biological tissues. Meccanica 52:3299–3320

    Article  MathSciNet  MATH  Google Scholar 

  65. Grillo A, Carfagna M, Federico S (2018) An Allen-Cahn approach to the remodelling of fiber-reinforced anisotropic materials. J Eng Math 109:139–172

    Article  MATH  Google Scholar 

  66. Grillo A, Federico S, Wittum G (2012) Growth, mass transfer, and remodeling in fiber-reinforced, multi-constituent materials. Int J Non-Linear Mech 47:388–401

    Article  Google Scholar 

  67. Guinot V (2002) Modelling using stochastic, finite state cellular automata: rule inference from continuum models. Appl Math Model 26:701–714

    Article  MATH  Google Scholar 

  68. Hak S, Reitan NK, Haraldseth O, De Lange Davies C (2010) Intravital microscopy in window chambers: a unique tool to study tumor angiogenesis and delivery of nanoparticles. Angiogenesis 13:113–130

    Article  Google Scholar 

  69. Hardin RH, Sloane NJH (1996) McLaren’s improved Snub cube and other new spherical designs in three dimensions. Discrete Comput Geom 15:429–441

    Article  MathSciNet  MATH  Google Scholar 

  70. Hariton I, deBotton G, Gasser TC, Holzapfel GA (2007) Stress-driven collagen fiber remodeling in arterial walls. Biomech Model Mechanobiol 6:163–175

    Article  Google Scholar 

  71. Hassanizadeh SM (1986) Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy’s and Fick’s laws. Adv Water Resour 9, 208–222

    Google Scholar 

  72. Hassanizadeh SM, Leijnse A (1995) A non-linear theory of high-concentration-gradient dispersion in porous media. Adv Water Resour 18:203–215

    Article  Google Scholar 

  73. Holmes MH, Mow VC (1990) The nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration. J Biomech 23:1145–1156

    Article  Google Scholar 

  74. Holmes MH (2012) Introduction to perturbation methods. Springer, New York

    Google Scholar 

  75. Hori M, Nemat-Nasser S (1999) On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mech Mater 31:667–682

    Article  Google Scholar 

  76. Interian R, Rodríguez-Ramos R, Valdés-Ravelo F, Ramírez-Torres A, Ribeiro CC, Conci A (2017) Tumor growth modeling by cellular automata. Math Mech Complex Syst 5:239–259

    Article  MathSciNet  MATH  Google Scholar 

  77. Jain RK (1987) Transport of molecules in the tumor interstitium: A review. Cancer Res. 47:3039–3051

    Google Scholar 

  78. Jain RK (1990) Physiological barriers to delivery of monoclonal antibodies and other macromolecules in tumors. Cancer Res 50:814s–819s

    Google Scholar 

  79. Jain RK, Stylianopoulos T (2010) Delivering nanomedicine to solid tumors. Nat Rev Clin Oncol 7:653–664

    Article  Google Scholar 

  80. Jacob JT, Burgoyne CF, McKinnon SJ, Tanji TM, LaFleur PK, Duzman E (1998) Biocompatibility response to modified baerveldt glaucoma drains. J Biomed Mater Res 43:99–107

    Article  Google Scholar 

  81. Karageorgiou V, Kaplan D (2005) Porosity of 3D biomaterial scaffolds and osteogenesis. Biomaterials 26:5474–5491

    Article  Google Scholar 

  82. Kihara T, Ito J, Miyake J (2013) Measurement of biomolecular diffusion in extracellular matrix condensed by fibroblasts using fluorescence correlation spectroscopy. PLoS ONE 8:e82382

    Article  Google Scholar 

  83. Kröner E (1959) Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch Ration Mech Anal 4:273–334

    Article  MathSciNet  MATH  Google Scholar 

  84. Landau LD, Lifshitz EM (1987) Fluid mechanics. In: Revised—Translated by Sykes, JB, Reid, WH (eds) Course of theoretical physics, second english edition, vol 6. Pergamon Press, Oxford, Frankfurt

    Google Scholar 

  85. Lanza R, Langer R, Vacanti JP (2011) Principles of tissue engineering. Academic, London

    Google Scholar 

  86. Loh QL, Choong C (2013) Three-dimensional scaffolds for tissue engineering applications: role of porosity and pore size. Tissue Eng Part B Rev 19:485–502

    Article  Google Scholar 

  87. Loret B, Simões FMF (2005) A framework for deformation, generalized diffusion, mass transfer and growth in multi-species multi-phase biological tissues. Eur J Mech A/Solids 24:757–781

    Article  MathSciNet  MATH  Google Scholar 

  88. Loret B, Simões FMF (2017) Biomechanical aspects of soft tissues. CRC Press, Boca Raton

    Book  Google Scholar 

  89. Maroudas A (1970) Distribution and diffusion of solutes in articular cartilage. Biophys J 10:365–379

    Article  Google Scholar 

  90. Mascheroni P, Penta R (2017) The role of the microvascular network structure on diffusion and consumption of anticancer drugs. Int J Num Meth Biomed Eng 33:e2857

    Article  MathSciNet  Google Scholar 

  91. Mascheroni P, Carfagna M, Grillo A, Boso D, Schrefler B (2018) An avascular tumor growth model based on porous media mechanics and evolving natural states. Math Mech Solids 23:686–712

    Article  MathSciNet  MATH  Google Scholar 

  92. Maugin GA, Epstein M (1998) Geometrical material structure of elastoplasticity. Int J Plast 14:109–115

    Article  MATH  Google Scholar 

  93. Mavko G, Mukerji T, Dvorkin J (2009) The rock physics handbook: tools for seismic analysis of porous media. Cambridge University Press, Cambridge

    Book  Google Scholar 

  94. Meerschaert MM, Mortensen J, Wheatcraft SW (2014) Fractional vector calculus for fractional advection-diffusion. Phys A 367:181–190

    Article  Google Scholar 

  95. Mei CC, Vernescu B (2010) Homogenization methods for multiscale mechanics. World Scientific, Singapore

    Book  MATH  Google Scholar 

  96. Menzel A (2005) Modelling of anisotropic growth in biological tissues – a new approach and computational aspects. Biomech Model Mechanobiol 3:147–171

    Article  Google Scholar 

  97. Meyvis TK, De Smedt SC, Van Oostveldt P, Demeester J (1999) Fluorescence recovery after photobleaching: a versatile tool for mobility and interaction measurements in pharmaceutical research. Pharm Res 16:1153–1162

    Article  Google Scholar 

  98. Mićunović MV (2009) Thermomechanics of viscoplasticity-fundamentals and applications. Springer, Heidelberg

    Book  MATH  Google Scholar 

  99. Minozzi M, Nardinocchi P, Teresi L, Varano V (2017) Growth-induced compatible strains. Math Mech Solids 22:62–71

    Article  MathSciNet  MATH  Google Scholar 

  100. Moreno-Arotzena O, Meier JG, Del Amo C, García-Aznar JM (2015) Characterization of fibrin and collagen gels for engineering wound healing models. Materials 8:1636–1651

    Article  Google Scholar 

  101. Mueller-Klieser WF, Freyer JP, Sutherland RM (1986) Influence of glucose and oxygen supply conditions on the oxygenation of multicellular spheroids. Br J Cancer 53:345–353

    Article  Google Scholar 

  102. Mueller-Klieser WF, Sutherland RM (1982) Oxygen tensions in multicell spheroids of two cell lines. Br J Cancer 45:256–264

    Article  Google Scholar 

  103. Netti PA, Berk DA, Swartz MA, Grodzinsky AJ, Jain RK (2000) Role of extracellular matrix assembly in interstitial transport in solid tumors. Cancer Res 60:2497–2503

    Google Scholar 

  104. Ngwa M, Agyingi E (2012) Effect of an external medium on tumor growth-induced stress. IAENG Int J Appl Math 42:229–236

    MathSciNet  Google Scholar 

  105. Nimer E, Schneiderman R, Maroudas A (2003) Diffusion and partition of solutes in cartilage under static load. Biophys Chem 106:125–146

    Article  Google Scholar 

  106. Olsson T, Klarbring A (2008) Residual stresses in soft tissue as a consequence of growth and remodeling: application to an arterial geometry. Eur J Mech A/Solids 27:959–974

    Article  MathSciNet  MATH  Google Scholar 

  107. Penta R, Ambrosi D, Quarteroni A (2015) Multiscale homogenization for fluid and drug transport in vascularized malignant tissues. Math Model Methods Appl Sci 25:79–108

    Article  MathSciNet  MATH  Google Scholar 

  108. Penta R, Ambrosi D, Shipley R (2014) Effective governing equations for poroelastic growing media. Q J Mech Appl Math 67:69–91

    Article  MathSciNet  MATH  Google Scholar 

  109. Penta R, Gerisch A (2017) The asymptotic homogenization elasticity tensor properties for composites with material discontinuities. Contin Mech Thermodyn 29:187–206

    Article  MathSciNet  MATH  Google Scholar 

  110. Penta R, Gerisch A (2017) An introduction to asymptotic homogenization. In: Gerisch A, Penta R, Lang J (eds) Multiscale models in mechano and tumor biology. Springer, Cham, pp 1–26

    MATH  Google Scholar 

  111. Penta R, Merodio J (2017) Homogenized modeling for vascularized poroelastic materials. Meccanica 52:3321–3343

    Article  MathSciNet  MATH  Google Scholar 

  112. Pluen A, Boucher Y, Ramanujan S, McKee TD, Gohongi T, Di Tomaso E, Brown EB, Izumi Y, Campbell RB, Berk DA, Jain RK (2001) Role of tumor-host interactions in interstitial diffusion of macromolecules: cranial versus subcutaneous tumors. Proc Natl Acad Sci 98, 4628–4633

    Article  Google Scholar 

  113. Pluen A, Netti PA, Jain RK, Berk DA (1999) Diffusion of macromolecules in agarose gels: comparison of linear and globular configurations. Biophys J 77:542–552

    Article  Google Scholar 

  114. Podlubny I (1999) Fractional differential equations. Academic, New York

    MATH  Google Scholar 

  115. Preziosi L, Vitale G (2011) A multiphase model of tumor and tissue growth including cell adhesion and plastic reorganization. Math Models Meth Appl Sci 21:1901–1932

    Article  MathSciNet  MATH  Google Scholar 

  116. Quiligotti S, Maugin GA, dell’Isola F (2003) An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures. Acta Mech 160:45–60

    Article  MATH  Google Scholar 

  117. Quinn TM, Kocian P, Meister JJ (2000) Static compression is associated with decreased diffusivity of dextrans in cartilage explants. Arch Biochem Biophys 384:327–334

    Article  Google Scholar 

  118. Ramanujan S, Pluen A, Mckee TD, Brown EB, Boucher Y, Jain RK (2002) Diffusion and convection in collagen gels: implications for transport in the tumor Interstitium. Biophys J 83:1650–1660

    Article  Google Scholar 

  119. Ramírez-Torres A, Di Stefano S, Grillo A, Rodríguez-Ramos R, Merodio J, Penta R (2018) An asymptotic homogenization approach to the microstructural evolution of heterogeneous media. Int J Non-Linear Mech 106:245–257

    Article  Google Scholar 

  120. Ramírez-Torres A, Valdés-Ravelo F, Rodríguez-Ramos R, Bravo-Castillero J, Guinovart-Díaz R, Sabina FJ (2016) Modeling avascular tumor growth via linear elasticity. In: Floryan JM (ed) Proceedings of the 24th international congress of theoretical and applied mechanics, ICTAM2016, Montreal

    Google Scholar 

  121. Rodriguez E, Hoger A, McCullogh A (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27:455–467

    Article  Google Scholar 

  122. Sadik S, Yavari A (2017) On the origins of the idea of the multiplicative decomposition of the deformation gradient. Math Mech Solids 22:771–772

    Article  MathSciNet  MATH  Google Scholar 

  123. Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, Switzerland

    MATH  Google Scholar 

  124. Sapora A, Cornetti P, Chiaia B, Lenzi EK, Evangelista LR (2017) Nonlocal diffusion in porous media: a spatial fractional approach. J Eng Mech 143, D4016007-1—D4016007-7

    Google Scholar 

  125. Shoga JS, Graham BT, Wang L, Price C (2017) Direct quantification of solute diffusivity in agarose and articular cartilage using correlation spectroscopy. Ann Biomed Eng 45:2461–2474

    Article  Google Scholar 

  126. Suhaimi H, Wang S, Thornton T, Das B (2015) On glucose diffusivity of tissue engineering membranes and scaffolds. Chem Eng Sci 126:244–256

    Article  Google Scholar 

  127. Suhaimi H, Das DB (2016) Glucose diffusion in tissue engineering membranes and scaffolds. Rev Chem Eng 32:629–650

    Article  Google Scholar 

  128. Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48(8):487–595

    Article  Google Scholar 

  129. Taffetani M, de Falco C, Penta R, Ambrosi D, Ciarletta P (2014) Biomechanical modelling in nanomedicine: multiscale approaches and future challenges. Arch. Appl. Mech. 84:1627–1645

    Article  Google Scholar 

  130. Tarasov VE (2008) Fractional vector calculus and fractional Maxwell’s equations. Ann Phys 323:2756–2778

    Article  MathSciNet  MATH  Google Scholar 

  131. Tomic A, Grillo A, Federico S (2014) Poroelastic materials reinforced by statistically oriented fibers–numerical implementation and application to articular cartilage. IMA J Appl Math 79:1027–1059

    Article  MathSciNet  MATH  Google Scholar 

  132. Valdés-Ravelo F, Ramírez-Torres A, Rodríguez-Ramos R, Bravo-Castillero J, Guinovart-Díaz R, Merodio J, Penta R, Conci A, Sabina FJ, García-Reimbert C (2018) Mathematical modeling of the interplay between stress and anisotropic growth of avascular tumors. J Mech Med Biol 18, 1850006–1–1850006–28

    Article  MATH  Google Scholar 

  133. Wang HF (2017) Theory of linear poroelasticity with applications to geomechanics and hydrogeology. University Press, Princeton

    Google Scholar 

  134. Wilson W, Driessen N, van Donkelaar CC, Ito K (2006) Prediction of collagen orientation in articular cartilage by a collagen remodeling algorithm. Osteoarthr Cartil 14:1196–1202

    Article  Google Scholar 

  135. Zingales M (2014) Fractional order theory of heat transport in rigid bodies. Commun Nonlinear Sci Numer Simul 19:3938–3953

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

LM is funded by EPSRC with project number EP/N509668/1. ART and AG acknowledge the Dipartimento di Scienze Matematiche (DISMA) “G.L. Lagrange” of the Politecnico di Torino, Dipartimento di Eccellenza 2018–2022 (Department of Excellence 2018–2022, Project code: E11G18000350001). PM is supported by MicMode-I2T (01ZX1710B) from the German Federal Ministry of Education and Research (BMBF).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow, G12 8QQ, UK

    Raimondo Penta & Laura Miller

  2. Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, 10129, Turin, Italy

    Alfio Grillo & Ariel Ramírez-Torres

  3. Braunschweig Integrated Centre of Systems Biology (BRICS), Helmholtz Centre for Infection Research (HZI), Rebenring 56, 38106, Braunschweig, Germany

    Pietro Mascheroni

  4. Departamento de Matemáticas, Facultad de Matemática y Computación, Universidad de La Habana, 10400, Havana, CP, Cuba

    Reinaldo Rodríguez-Ramos

Authors
  1. Raimondo Penta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laura Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alfio Grillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ariel Ramírez-Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Pietro Mascheroni
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Reinaldo Rodríguez-Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raimondo Penta .

Editor information

Editors and Affiliations

  1. Department of Continuum Mechanics and Structures, Escuela de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Madrid, Spain

    José Merodio

  2. School of Mathematics and Statistics, University of Glasgow, Glasgow, UK

    Raymond Ogden

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Penta, R., Miller, L., Grillo, A., Ramírez-Torres, A., Mascheroni, P., Rodríguez-Ramos, R. (2020). Porosity and Diffusion in Biological Tissues. Recent Advances and Further Perspectives. In: Merodio, J., Ogden, R. (eds) Constitutive Modelling of Solid Continua. Solid Mechanics and Its Applications, vol 262. Springer, Cham. https://doi.org/10.1007/978-3-030-31547-4_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-31547-4_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-31546-7

  • Online ISBN: 978-3-030-31547-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation