Biomechanics and Modeling in Mechanobiology

, Volume 12, Issue 4, pp 763–780 | Cite as

A multiscale approach in the computational modeling of the biophysical environment in artificial cartilage tissue regeneration

  • Paola CausinEmail author
  • Riccardo Sacco
  • Maurizio Verri
Original Paper


Tissue Engineering is a strongly interdisciplinary scientific area aimed at understanding the principles of tissue growth to produce biologically functional replacements for clinical use. To achieve such an ambitious goal, complex biophysical phenomena must be understood in order to provide the appropriate environment to cells (nutrient delivery, fluid-mechanical loading and structural support) in the bioengineered device. Such a problem has an inherent multiphysics/multiscale nature, as it is characterized by material heterogeneities and interplaying processes occurring within a wide range of temporal and spatial scales. In this context, computational models are useful to gain a quantitative and comprehensive understanding of phenomena often difficult to be accessed experimentally. In this paper, we propose a mathematical and computational model that represents, to our knowledge, the first example of a self-consistent multiscale description of coupled nutrient mass transport, fluid-dynamics and biomass production in bioengineered constructs. We specifically focus on articular cartilage regeneration based on dynamically perfused bioreactors, and we investigate by numerical simulations three issues critical in this application. First, we study oxygen distribution in the construct, since achieving an optimal level throughout the construct is a main control variable to improve tissue quality. Second, we provide a quantitative evaluation of how interstitial perfusion can enhance nutrient delivery and, ultimately, biomass production, compared with static culture. Third, we perform a sensitivity analysis with respect to biophysical parameters related to matrix production, assessing their role in tissue regeneration.


Tissue Engineering Multiscale model Mass transfer in heterogeneous media Model of biomass synthesis Interstitial perfusion bioreactor 



Tissue Engineering


Extracellular matrix




Computational fluid-dynamics


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  1. Atala A, Lanza RP, Thomson JA, Nerem R (2011) Principles of regenerative medicine. Academic Press, New YorkGoogle Scholar
  2. Bernard S, Pujo-Menjouet L, Mackey MC (2003) Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data. Biophys J 84(5): 3414–3424CrossRefGoogle Scholar
  3. Boschetti F, Raimondi MT, Migliavacca F, Dubini G (2006) Prediction of the micro-fluid dynamic environment imposed to three-dimensional engineered cell systems in bioreactors. J Biomech 39(3): 418–425CrossRefGoogle Scholar
  4. Brezzi F, Marini LD, Pietra P (1989) Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J Numer Anal 26: 1342–1355MathSciNetzbMATHCrossRefGoogle Scholar
  5. Buschmann MD, Gluzband YA, Grodzinsky AJ, Kimura JH, Hunziker EB (1992) Chondrocytes in agarose culture synthesize a mechanically functional extracellular matrix. J Orthop Res 10(6): 745–758CrossRefGoogle Scholar
  6. Chung CA, Chen CW, Chen CP, Tseng CS (2007) Enhancement of cell growth in tissue-engineering constructs under direct perfusion: modeling and simulation. Biotech Bioeng 97(6): 1603–1616CrossRefGoogle Scholar
  7. Cioffi M, Kuffer J, Strobel S, Dubini G, Martin I, Wendt D (2008) Computational evaluation of oxygen and shear stress distributions in 3d perfusion culture systems: macro-scale and micro-structured models. J Biomech 41(14): 2918–2925CrossRefGoogle Scholar
  8. Cowin SC (2004) Tissue growth and remodeling. Ann Rev Biomed Eng 6: 77–107CrossRefGoogle Scholar
  9. Das RH, van Osch GJ, Kreukniet M, Oostra J, Weinans H, Jahr H (2010) Effects of individual control of ph and hypoxia in chondrocyte culture. J Orthop Res 28(4): 537–545Google Scholar
  10. Devarapalli M, Lawrence BJ, Sundararajan VM (2009) Modeling nutrient consumptions in large flow–through bioreactors for tissue engineering. Biotechnol Bioeng 103(5): 1003–1015CrossRefGoogle Scholar
  11. DiMicco MA, Sah RL (2003) Dependence of cartilage matrix composition on biosynthesis, diffusion, and reaction. Transp Porous Med 50: 57–73CrossRefGoogle Scholar
  12. Freed LE, Vunjak-Novakovic G (2001) Tissue engineering bioreactors. In: Lanza RP, Langer RS, Vacanti JP (eds) Principles of tissue engineering. Academic Press, San DiegoGoogle Scholar
  13. Freyria AM, Yang Y, Chajra H, Rousseau CF, Ronziere MC, Herbage D, El Haj AJ (2005) Optimization of dynamic culture conditions: effects on biosynthetic activities of chondrocytes grown in collagen sponges. Tissue Eng 11(5–6): 674–684CrossRefGoogle Scholar
  14. Fu X, Viskanta R, Gore JP (1998) Prediction of effective thermal conductivity of cellular ceramics. Int Commun Heat Mass Transf 25(2): 151–160CrossRefGoogle Scholar
  15. Galban CJ, Locke BR (1999) Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold. Biotechnol Bioeng 65(2): 121–132CrossRefGoogle Scholar
  16. Galban CJ, Locke BR (1999) Effects of spatial variation of cells and nutrient product concentrations coupled with product inhibition on cell growth in a polymer scaffold. Biotechnol Bioeng 64(6): 633–643CrossRefGoogle Scholar
  17. Galbusera F, Cioffi M, Raimondi MT, Pietrabissa R (2007) Computational modelling of combined cell population dynamics and oxygen transport in engineered tissue subject to interstitial perfusion. Comput Methods Biomech Biomed Eng 10(4): 279–287CrossRefGoogle Scholar
  18. Gatti E, Micheletti S, Sacco R (1998) A new Galerkin framework for the drift-diffusion equation in semiconductors. East-West J Numer Math 6(2): 101–135MathSciNetzbMATHGoogle Scholar
  19. Grimshaw MJ, Mason RM (2001) Modulation of bovine articular chondrocyte gene expression in vitro by oxygen tension. Osteoarthritis Cartilage 9: 357–364CrossRefGoogle Scholar
  20. Hsu CT, Cheng P (1990) Thermal dispersion in a porous medium. Int J Heat Mass Transf 33(8): 1587–1597zbMATHCrossRefGoogle Scholar
  21. Klein TJ, Sah RL (2007) Modulation of depth-dependent properties in tissue-engineered cartilage with a semi–permeable membrane and perfusion: a continuum model of matrix metabolism and transport. Biomech Model Mechanobiol 6: 21–32CrossRefGoogle Scholar
  22. Klisch S, Chen SS, Sah RL, Hoger A (2003) A growth mixture theory for cartilage with application to growth-related experiments on cartilage explants. J Biomech Eng 125: 169–179CrossRefGoogle Scholar
  23. Knudson W, Knudson CB (1991) Assembly of a chondrocyte-like pericellular matrix on non-chondrogenic cells. role of the cell surface hyaluronan receptors in the assembly of a pericellular matrix. J Cell Sci 99(2): 227–235Google Scholar
  24. Laganà M, Raimondi MT (2011) A miniaturized, optically accessible bioreactor for systematic 3D tissue engineering research. Biomed Microdev doi: 10.1007/s10544-011-9600-0
  25. Landau L, Lifshtiz E (1959) Fluid mechanics. Pergamon Press, OxfordGoogle Scholar
  26. Langer RS, Vacanti JP (1993) Tissue engineering. Science 260(5110): 920–926CrossRefGoogle Scholar
  27. Lee S, Sundararaghavan V (2011) Multi-scale modeling of moving interface problems with flux and field jumps: application to oxidative degradation of ceramic matrix composites. Int J Numer Methods Eng 85(6): 784–804zbMATHCrossRefGoogle Scholar
  28. 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: 571–594MathSciNetzbMATHCrossRefGoogle Scholar
  29. Lesman A, Blinder Y, Levenberg S (2010) Modeling of flow-induced shear stress applied on 3D cellular scaffolds: implications for vascular tissue engineering. Biotechnol Bioeng 105(3): 645–654CrossRefGoogle Scholar
  30. Malda J, Martens D, Tramper J, Blitterswijk C, van Riesle J (2003) Cartilage tissue engineering: controversy in the effect of oxygen. Crit Rev Biotechnol 23(3): 175–194Google Scholar
  31. Martin I, Wendt D, Heberer M (2004) The role of bioreactors in tissue engineering. Trends Biotechnol 22(2): 80–86CrossRefGoogle Scholar
  32. Nield DA, Bejan A (1998) Convection in porous media. Springer, New YorkGoogle Scholar
  33. Nikolaev NI, Obradovic B, Versteeg HK, Lemon G, Williams DJ (2010) A validated model of GAG deposition, cell distribution, and growth of tissue engineered cartilage cultured in a rotating bioreactor. Biotechnol Bioeng 105(4): 842–853Google Scholar
  34. Obradovic B, Carrier RL, Vunjak-Novakovic G, Freed LE (1999) Gas exchange is essential for bioreactor cultivation of tissue engineered cartilage. Biotechnol Bioeng 63(2): 197–205CrossRefGoogle Scholar
  35. Obradovic B, Meldon JH, Freed LE, Vunjak-Novakovic G (2000) Glycosaminoglycan deposition in engineered cartilage: experiments and mathematical model. AIChE J 46: 1547–5905CrossRefGoogle Scholar
  36. Potter K, Butler JJ, Adams C, Fishbein KW, McFarland EW, Horton WE, Spencer RGS (1998) Cartilage formation in a hollow fiber bioreactor studied by proton magnetic resonance microscopy. Matrix Biol 17(7): 513–523CrossRefGoogle Scholar
  37. Raimondi MT (2006) Engineered tissue as a model to study cell and tissue function from a biophysical perspective. Curr Drug Discov Technol 3(4): 245–268CrossRefGoogle Scholar
  38. Raimondi MT, Candiani G, Cabras M, Cioffi M, Laganà K, Moretti M, Pietrabissa R (2008) Engineered cartilage constructs subject to very low regimens of interstitial perfusion. Biorheology 45(3–4): 471–479Google Scholar
  39. Raimondi MT, Causin P, Laganà M, Zunino P, Sacco R (2012) Multiphysics computational modeling in cartilage tissue engineering. In: Baumann C (ed) Studies in mechanobiology, tissue engineering and biomaterials, vol 9. Springer, Berlin (in press)Google Scholar
  40. Raimondi MT, Causin P, Mara A, Nava M, Laganà M, Sacco R (2011) Breakthroughs in computational modeling of cartilage regeneration in perfused bioreactors. IEEE Trans Biomed Eng 58(12): 3496–3499CrossRefGoogle Scholar
  41. Raimondi MT, Moretti M, Cioffi M, Giordano C, Boschetti F, Laganà K, Pietrabissa R (2006) The effect of hydrodynamic shear on 3D engineered chondrocyte systems subject to direct perfusion. Biorheology 43(3–4): 215–222Google Scholar
  42. Raimondi MT, Boschetti F, Migliavacca F, Cioffi M, Dubini G (2005) Micro fluid dynamics in three-dimensional engineered cell systems in bioreactors. In: Ashammakhi N, Reis RL (eds) Topics in tissue engineering, vol 2, chap 9, pp 1–26Google Scholar
  43. Roos HG, Stynes M, Tobiska L (1996) Numerical methods for singularly perturbed differential equations. Springer, Berlin HeidelbergzbMATHCrossRefGoogle Scholar
  44. Sacco R, Causin P, Zunino P, Raimondi MT (2011) A multiphysics/multiscale 2D numerical simulation of scaffold-based cartilage regeneration under interstitial perfusion in a bioreactor. Biomech Model Mechanobiol 10: 577–589CrossRefGoogle Scholar
  45. Schulz RM, Bader A (2007) Cartilage tissue engineering and bioreactor systems for the cultivation and stimulation of chondrocytes. Eur Biophys J 36: 539–568CrossRefGoogle Scholar
  46. Secomb TW, Beard DA, Frisbee JC, Smith NP, Pries AR (2008) The role of theoretical modeling in microcirculation research. Microcirculation 15(8): 693–698CrossRefGoogle Scholar
  47. Singh H, Teoh SH, Low HT, Hutmacher DW (2005) Flow modelling within a scaffold under the influence of uni-axial and bi-axial bioreactor rotation. J Biotechol 119(2): 181–196CrossRefGoogle Scholar
  48. Vunjak-Novakovic G, Freshney RI (2006) Culture of cells for tissue engineering. Wiley-Liss, New YorkGoogle Scholar
  49. Wendt D, Stroebel S, Jacob M, John GT, Martin I (2006) Uniform tissues engineered by seeding and culturing cells in 3D scaffolds under perfusion at defined oxygen tensions. Biorheology 53: 481–488Google Scholar
  50. Whitaker S (1999) The method of volume averaging. Theory and application of transport in porous media. Kluwer, Dordrecht, The NetherlandsCrossRefGoogle Scholar
  51. Williams KA, Saini S, Wick TM (2002) Computational investigation of steady-state momentum and mass transfer in a bioreactor for the production of tissue-engineered cartilage. Biotechnol Prog 18: 951–963CrossRefGoogle Scholar
  52. Wilson C, Bonassar L, Kohles S (2002) Modelling the dynamic composition of engineered cartilage. Arch Biochem Biophys 408(2): 246–254CrossRefGoogle Scholar
  53. Wood BD, Quintard M, Whitaker S (2002) Calculation of effective diffusivities for biofilms and tissues. Biotechnol Bioeng 77(5): 495–514CrossRefGoogle Scholar
  54. Wyatt J, Mikulecky D, DeSimone J (1980) Network modelling of reaction-diffusion systems and their numerical solution using spice. Chem Eng Sci 35(10): 2115–2127CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica “F. Enriques”Università degli Studi di MilanoMilanoItaly
  2. 2.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanoItaly

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