Advertisement

Journal of Mathematical Biology

, Volume 72, Issue 7, pp 1775–1809 | Cite as

An investigation of the influence of extracellular matrix anisotropy and cell–matrix interactions on tissue architecture

  • R. J. Dyson
  • J. E. F. GreenEmail author
  • J. P. Whiteley
  • H. M. Byrne
Article

Abstract

Mechanical interactions between cells and the fibrous extracellular matrix (ECM) in which they reside play a key role in tissue development. Mechanical cues from the environment (such as stress, strain and fibre orientation) regulate a range of cell behaviours, including proliferation, differentiation and motility. In turn, the ECM structure is affected by cells exerting forces on the matrix which result in deformation and fibre realignment. In this paper we develop a mathematical model to investigate this mechanical feedback between cells and the ECM. We consider a three-phase mixture of collagen, culture medium and cells, and formulate a system of partial differential equations which represents conservation of mass and momentum for each phase. This modelling framework takes into account the anisotropic mechanical properties of the collagen gel arising from its fibrous microstructure. We also propose a cell–collagen interaction force which depends upon fibre orientation and collagen density. We use a combination of numerical and analytical techniques to study the influence of cell–ECM interactions on pattern formation in tissues. Our results illustrate the wide range of structures which may be formed, and how those that emerge depend upon the importance of cell–ECM interactions.

Keywords

Multiphase model Collagen fibres Cell aggregation Mechanics 

Mathematics Subject Classification

92C10 92C15 92C17 76T30 76Z99 35Q92 

Notes

Acknowledgments

We thank A.M. Soto and C. Sonnenschein (Tufts University) for the initial discussions which led to the development of the model, and D.J. Smith (University of Birmingham) for assistance with aspects of the numerics. RJD gratefully acknowledges the support of the University of Birmingham’s System Science for Health initiative and the hospitality of the School of Mathematical Sciences at the University of Adelaide. JEFG is supported by a Discovery Early Career Researcher Award (DE130100031) from the Australian Research Council. The work of HMB was supported in part by award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).

Supplementary material

Supplementary material 1 (avi 18260 KB)

Supplementary material 2 (avi 19655 KB)

Supplementary material 3 (avi 18876 KB)

References

  1. Barocas VH, Tranquillo RT (1997) An anisotropic biphasic theory of tissue-equivalent mechanics: the interplay among cell traction, fibrillar network deformation, fibril alignment and cell contact guidance. J Biomech Eng 119:137–145CrossRefGoogle Scholar
  2. Barocas VH, Moon AG, Tranquillo RT (1995) The fibroblast-populated collagen microsphere assay of cell traction force—Part 2: measurement of the cell traction parameter. J Biomech Eng 117:161–170CrossRefGoogle Scholar
  3. Bissell MJ, Radisky D (2001) Putting tumours in context. Nat Rev Cancer 1:46–54CrossRefGoogle Scholar
  4. Breward CJW, Byrne HM, Lewis CE (2002) The role of cell–cell interactions in a two-phase model for avascular tumour growth. J Math Biol 45(2):125–152MathSciNetCrossRefzbMATHGoogle Scholar
  5. Byfield FJ, Reen RK, Shentu TP, Levitan I, Gooch KJ (2009) Endothelial actin and cell stiffness is modulated by substrate stiffness in 2D and 3D. J Biomech 42(8):1114–1119CrossRefGoogle Scholar
  6. Byrne HM, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20:341–366CrossRefzbMATHGoogle Scholar
  7. Byrne HM, King JR, McElwain DLS, Preziosi L (2003) A two-phase model of solid tumour growth. Appl Math Lett 16(4):567–573MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chauviere A, Hillen T, Preziosi L (2007) Modelling cell movement in anisotropic and heterogeneous network tissues. Netw Heterog Media 2(2):333–357MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cockburn B, Shu CW (1998) The Runge–Kutta discontinuous Galerkin method for conservation laws V. J Comput Phys 141:199–224MathSciNetCrossRefzbMATHGoogle Scholar
  10. Cook J (1995) Mathematical models for dermal wound healing: wound contraction and scar formation. PhD thesis, University of WashingtonGoogle Scholar
  11. Cukierman E, Bassi DE (2010) Physico-mechanical aspects of extracellular matrix influences on tumourigenic behaviors. Semin Cancer Biol 20(3):139–145Google Scholar
  12. Dhimolea E, Maffini MV, Soto AM, Sonnenschein C (2010) The role of collagen reorganization on mammary epithelial morphogenesis in a 3d culture model. Biomaterials 31:3622–3630CrossRefGoogle Scholar
  13. Drew DA (1983) Mathematical modelling of two-phase flow. Ann Rev Fluid Mech 15:261–291CrossRefGoogle Scholar
  14. Dyson RJ, Jensen OE (2010) A fibre-reinforced fluid model of anisotropic plant cell growth. J Fluid Mech 655:472–503MathSciNetCrossRefzbMATHGoogle Scholar
  15. Engler AJ, Sen S, Sweeney HL, Discher DE (2006) Matrix elasticity directs stem cell lineage specification. Cell 126:677–689CrossRefGoogle Scholar
  16. Ericksen JL (1960) Transversely isotropic fluids. Colloid Polym Sci 173(2):117–122Google Scholar
  17. Eriksson K, Estep D, Hansbo P, Johnson C (1996) Computational differential equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
  18. Gerisch A, Chaplain MAJ (2008) Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J Theor Biol 250(4):684–704MathSciNetCrossRefGoogle Scholar
  19. Green JEF, Friedman A (2008) The extensional flow of a thin sheet of incompressible, transversely isotropic fluid. Euro J Appl Math 19(3):225–257MathSciNetCrossRefzbMATHGoogle Scholar
  20. Green JEF, Waters SL, Shakesheff KM, Byrne HM (2009) A mathematical model of liver cell aggregation in vitro. Bull Math Biol 71:906–930MathSciNetCrossRefzbMATHGoogle Scholar
  21. Green JEF, Waters SL, Shakesheff KM, Edelstein-Keshet L, Byrne HM (2010) Non-local models for the interactions of hepatocytes and stellate cells during aggregation. J Theor Biol 267(1):106–120MathSciNetCrossRefGoogle Scholar
  22. Green JEF, Bassom AP, Friedman A (2013) A mathematical model for cell-induced gel compaction in vitro. Math Models Methods Appl Sci 23(1):127–163. doi: 10.1142/S0218202512500479 MathSciNetCrossRefzbMATHGoogle Scholar
  23. Häcker A (2012) A mathematical model for mesenchymal and chemosensitive cell dynamics. J Math Biol 64:361–401MathSciNetCrossRefzbMATHGoogle Scholar
  24. Hillen T (2006) M\(^5\) mesoscopic and macroscopic models for mesenchymal motion. J Math Biol 53(4):585–616MathSciNetCrossRefzbMATHGoogle Scholar
  25. Hinch EJ, Leal LG (1975) Constitutive equations in suspension mechanics. Part 1. General formulation. J Fluid Mech 71(3):481–495MathSciNetCrossRefzbMATHGoogle Scholar
  26. Hinch EJ, Leal LG (1976) Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations. J Fluid Mech 76(1):187–208CrossRefzbMATHGoogle Scholar
  27. Holloway C, Dyson R, Smith D (2015) Linear Taylor-Couette stability of a transversely isotropic fluid. Proc R Soc A 471:20150141. doi: 10.1098/rspa.2015.0141 MathSciNetCrossRefGoogle Scholar
  28. Ingber DE (2006) Mechanical control of tissue morphogenesis during embryological development. Int J Dev Biol 50:255–266CrossRefGoogle Scholar
  29. Ingber DE (2008) Can cancer be reversed by engineering the tumour microenvironment? Semin Cancer Biol 18(5):356–364CrossRefGoogle Scholar
  30. Jaalouk DE, Lammerding J (2009) Mechanotransduction gone awry. Nat Rev Mol Cell Biol 10:63–73CrossRefGoogle Scholar
  31. Kabla A, Mahadevan L (2007) Nonlinear mechanics of soft fibre networks. J R Soc Interface 4(12):99–106CrossRefGoogle Scholar
  32. Kirkpatrick ND, Andreou S, Hoying JB, Utzinger U (2007) Live imaging of collagen remodeling during angiogenesis. Am J Physiol Heart Circ Physiol 292(6):H3198–H3206CrossRefGoogle Scholar
  33. Knapp DM, Barocas VH, Moon AG, Yoo K, Petzold LR, Tranquillo RT (1997) Rheology of reconstituted type i collagen gel in confined compression. J Rheol 41:971–933CrossRefGoogle Scholar
  34. Korff T, Augustin HG (1999) Tensional forces in fibrillar extracellular matrices control directional capillary sprouting. J Cell Sci 112:3249–3258Google Scholar
  35. Krause S, Maffini MV, Soto AM, Sonnenschein C (2008) A novel 3d in vitro culture model to study stromal–epithelial interactions in the mammary gland. Tissue Eng 14:261–271CrossRefGoogle Scholar
  36. Kumar S, Weaver VM (2009) Mechanics, malignancy, and metastasis: the force journey of a tumour cell. Cancer Metastasis Rev 28:113–127Google Scholar
  37. Lee MEM (2001) Mathematical models of the carding process. PhD thesis, University of OxfordGoogle Scholar
  38. Lee MEM, Ockendon H (2005) A continuum model for entangled fibres. Euro J Appl Math 16:145–160MathSciNetCrossRefzbMATHGoogle Scholar
  39. Lemon G, King JR, Byrne HM, Jensen OE, Shakesheff KM (2006) Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. J Math Biol 52:571–594MathSciNetCrossRefzbMATHGoogle Scholar
  40. Lopez JI, Mouw JK, Weaver VM (2008) Biomechanical regulation of cell orientation and fate. Oncogene 27:6981–6993CrossRefGoogle Scholar
  41. Manoussaki D, Lubkin S, Vemon R, Murray J (1996) A mechanical model for the formation of vascular networks in vitro. Acta Biotheor 44(3–4):271–282CrossRefGoogle Scholar
  42. Martins-Green M, Bissell MJ (1995) Cell–ECM interactions in development. Semin Dev Biol 6:149–159CrossRefGoogle Scholar
  43. Murray JD (1993) Mathematical biology, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  44. Namy P, Ohayon J, Tracqui P (2004) Critical conditions for pattern formation and in vitro tubulogenesis driven by cellular traction fields. J Theor Biol 227:103–120MathSciNetCrossRefGoogle Scholar
  45. Nelson CM, Bissell MJ (2006) Of extracellular matrix, scaffolds, and signalling: tissue architecture regulates development, homeostasis and cancer. Ann Rev Cell Dev Biol 22:287–309CrossRefGoogle Scholar
  46. O’Dea RD, Waters SL, Byrne HM (2008) A two-fluid model for tissue growth within a dynamic flow environment. Euro J Appl Math 19(06):607–634MathSciNetCrossRefzbMATHGoogle Scholar
  47. O’Dea RD, Waters SL, Byrne HM (2010) A multiphase model for tissue construct growth in a perfusion bioreactor. Math Med Biol 27(2):95–127MathSciNetCrossRefzbMATHGoogle Scholar
  48. Olsen L, Maini PK, Sherratt JA, Dallon J (1999) Mathematical modelling of anisotropy in fibrous connective tissue. Math Biosci 158(2):145–170CrossRefzbMATHGoogle Scholar
  49. Osborne JM, Whiteley JP (2010) A numerical method for the multiphase viscous flow equations. Comput Methods Appl Mech Eng 199:3402–3417MathSciNetCrossRefzbMATHGoogle Scholar
  50. Oster GF, Murray JD, Harris AK (1983) Mechanical aspects of mesenchymal morphogenesis. J Embryol Exp Morphol 78:83–125zbMATHGoogle Scholar
  51. Painter KJ (2009) Modelling cell migration strategies in the extracellular matrix. J Math Biol 58:511–543MathSciNetCrossRefzbMATHGoogle Scholar
  52. Petersen OW, Ronnov-Jessen L, Howlett AR, Bissell MJ (1992) Interaction with basement membrane serves to rapidly distinguish growth and differentiation pattern of normal and malignant human breast epithelial cells. Proc Natl Acad Sci USA 89(19):9064–9068. doi: 10.1073/pnas.89.19.9064 CrossRefGoogle Scholar
  53. Petrie CJS (1999) The rheology of fibre suspensions. J Non-Newton Fluid Mech 87:369–402CrossRefzbMATHGoogle Scholar
  54. Peyton SR, Ghajar CM, Khatiwala CB, Putnam AJ (2007) The emergence of ECM mechanics and cytoskeletal tension as important regulators of cell function. Cell Biochem Biophys 47:300–320CrossRefGoogle Scholar
  55. Ronnov-Jessen L, Bissell MJ (2008) Breast cancer by proxy: can the microenvironment be both the cause and consequence? Trends Mol Med 15(1):5–13CrossRefGoogle Scholar
  56. Schreiber DI, Barocas VH, Tranquillo RT (2003) Temporal variations in cell migration and traction during fibroblast-mediated gel compaction. Biophys J 84:4102–4114CrossRefGoogle Scholar
  57. Soto AM, Sonnenschein C (2004) The somatic mutation theory of cancer: growing problems with the paradigm? BioEssays 26:1097–1107CrossRefGoogle Scholar
  58. Spain B (1953) Tensor calculus. Oliver and Boyd, EdinburghzbMATHGoogle Scholar
  59. Stevenson MD, Sieminski AL, McLeod CM, Byfield FJ, Barocas VH, Gooch KJ (2010) Pericellular conditions regulate extent of cell-mediated compaction of collagen gels. Biophys J 99:19–28CrossRefGoogle Scholar
  60. Strand DW, Franco OE, Basanta D, Anderson ARA, Hayward SW (2010) Perspectives on tissue interactions in development and disease. Curr Mol Med 10:95–112CrossRefGoogle Scholar
  61. Szymanska Z, Morales-Rodrigo C, Lachowicz M, Chaplain MAJ (2009) Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions. Math Models Methods Appl Sci 19(2):257–281MathSciNetCrossRefzbMATHGoogle Scholar
  62. Takakuda K, Miyairi H (1996) Tensile behaviour of fibroblasts cultured in collagen gel. Biomaterials 17(14):1393–1397CrossRefGoogle Scholar
  63. Thompson DW (1942) On growth and form, 2nd edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
  64. Tosin A, Ambrosi D, Preziosi L (2006) Mechanics and chemotaxis in the morphogenesis of vascular networks. Bull Math Biol 68(7):1819–1836MathSciNetCrossRefzbMATHGoogle Scholar
  65. Tranquillo RT, Murray JD (1993) Mechanistic model of wound contraction. J Surg Res 55:233–247CrossRefGoogle Scholar
  66. Vader D, Kabla A, Weitz D, Mahadevan L (2009) Strain-induced alignment in collagen gels. PLoS One 4(6):e5902. doi: 10.1371/journal.pone.0005902 CrossRefGoogle Scholar
  67. Weigelt B, Bissell MJ (2008) Unravelling the microenvironmental influences on the normal mammary gland and breast cancer. Semin Cancer Biol 18:311–321CrossRefGoogle Scholar
  68. Wipff PJ, Rifkin DB, Meister JJ, Hinz B (2007) Myofibroblast contraction activates latent TGF-\(\beta 1\) from the extracellular matrix. J Cell Biol 179(6):1311–1323CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • R. J. Dyson
    • 1
  • J. E. F. Green
    • 2
    Email author
  • J. P. Whiteley
    • 3
  • H. M. Byrne
    • 3
    • 4
  1. 1.School of MathematicsUniversity of BirminghamBirminghamUK
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  3. 3.Department of Computer ScienceUniversity of OxfordOxfordUK
  4. 4.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations