Biomechanics and Modeling in Mechanobiology

, Volume 10, Issue 4, pp 577–589 | Cite as

A multiphysics/multiscale 2D numerical simulation of scaffold-based cartilage regeneration under interstitial perfusion in a bioreactor

  • Riccardo Sacco
  • Paola Causin
  • Paolo Zunino
  • Manuela T. RaimondiEmail author
Original Paper


In vitro tissue engineering is investigated as a potential source of functional tissue constructs for cartilage repair, as well as a model system for controlled studies of cartilage development and function. Among the different kinds of devices for the cultivation of 3D cartilage cell colonies, we consider here polymeric scaffold-based perfusion bioreactors, where an interstitial fluid supplies nutrients and oxygen to the growing biomass. At the same time, the fluid-induced shear acts as a physiologically relevant stimulus for the metabolic activity of cells, provided that the shear stress level is appropriately tuned. In this complex environment, mathematical and computational modeling can help in the optimal design of the bioreactor configuration. In this perspective, we propose a computational model for the simulation of the biomass growth, under given inlet and geometrical conditions, where nutrient concentration, fluid dynamic field and cell growth are consistently coupled. The biomass growth model is calibrated with respect to the shear stress dependence on experimental data using a simplified short-time analysis in which the nutrient concentration and the fluid-induced shear stress are assumed constant in time and uniform in space. Volume averaging techniques are used to derive effective parameters that allow to upscale the microscopic structural properties to the macroscopic level. The biomass growth predictions obtained in this way are significant for long times of culture.


Tissue engineering Artificial cartilage Computational model Multiphysics Multiscale Interstitial perfusion Bioreactor Fluid dynamics Transport Shear stress 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arnold V (1973) Ordinary differential equations. The MIT Press, CambridgezbMATHGoogle Scholar
  2. Bear J (1972) Dynamics of fluids in porous materials. American Elsevier, New YorkGoogle 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): 418–425Google Scholar
  4. Chung C, Chen C, Chen C, Tseng C (2007) Enhancement of cell growth in tissue-engineering constructs under direct perfusion: modeling and simulation. Biotechnol Bioeng 97(6): 1603–1616CrossRefGoogle Scholar
  5. Cioffi M, Boschetti F, Raimondi MT, Dubini G (2006) Modeling evaluation of the fluid-dynamic microenvironment in tissue-engineered constructs: a micro-CT based model. Biotechnol Bioeng 93(3): 500–510CrossRefGoogle Scholar
  6. Cioffi M, Kueffer J, Stroebel 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
  7. Contois DE (1959) Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures. J Gen Microbiol 21: 40–50Google Scholar
  8. Dahlquist G, Bjorck A (2003) Numerical methods. Dover, New YorkzbMATHGoogle Scholar
  9. Davisson T, Sah R, Ratcliffe A (2002) Perfusion increases cell content and matrix synthesis in chondrocyte three-dimensional cultures. Tissue Eng 8(5): 807–816CrossRefGoogle Scholar
  10. Dunkelman N, Zimber M, Lebaron R, Pavelec R, Kwan M, Purchio A (1995) Cartilage production by rabbit articular chondrocytes on polyglycolic acid scaffolds in a closed bioreactor system. Biotech Bioeng 46: 299–305CrossRefGoogle Scholar
  11. Freed L, Vunjak-Novakovic G (2000) Tissue engineering bioreactors. In: Lanza RP, Langer R, Vacanti J (eds) Principles of tissue engineering. Academic Press, San DiegoGoogle Scholar
  12. Freyria AM, Yang Y, Chajra H, Rousseau C, Ronzire MC, Herbage D, Haj AE (2005) Optimization of dynamic culture conditions: effects on biosynthetic activities of chondrocytes grown in collagen sponges. Tissue Eng 11(5–6): 674–684CrossRefGoogle Scholar
  13. Galban CJ, Locke BR (1999) Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold. Biotechnol Bioeng 65(2): 121–132CrossRefGoogle Scholar
  14. 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
  15. Galbusera F, Cioffi M, Raimondi MT (2008) An in silico bioreactor for simulating laboratory experiments in tissue engineering. Biomed Microdevices 10(4): 547–554CrossRefGoogle Scholar
  16. 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 Engin 10(4): 279–287CrossRefGoogle Scholar
  17. Grodzinsky A, Levenston M, Jin M, Frank E (2000) Cartilage tissue remodeling in response to mechanical forces. Ann Rev Biomed Engin 2(1): 691–713CrossRefGoogle Scholar
  18. Guilak F, Hung C (2005) Basic orthopaedic biomechanics and mechano-biology chap physical regulation of cartilage metabolism. Lippincott Williams and Wilkins, Baltimore, pp 259–300Google Scholar
  19. Haj AJE, Wood MA, Thomas P, Yang Y (2005) Controlling cell biomechanics in orthopaedic tissue engineering and repair. Pathologie Biologie 53(10): 581–589CrossRefGoogle Scholar
  20. Hsu C, Cheng P (1990) Thermal dispersion in a porous medium. Int J Heat Mass Transfer 33(8): 1587–1597zbMATHCrossRefGoogle Scholar
  21. Lemon G, King JR (2007) Multiphase modelling of cell behaviour on artificial scaffolds: effects of nutrient depletion and spatially nonuniform porosity. Math Med Biol 24: 57–83zbMATHCrossRefGoogle Scholar
  22. 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
  23. Mahmoudifar N, Doran P (2005) Tissue engineering of human cartilage and osteochondral composites using recirculation bioreactors. Biomaterials 26: 7012–7024CrossRefGoogle Scholar
  24. Masaro L, Zhu XX (1999) Physical models of diffusion for polymer solutions, gels and solids. Prog Polym Sci 24: 731–775CrossRefGoogle Scholar
  25. Maxwell J (1881) Treatise on electricity and magnetism, vol. I. Clarendon Press, Oxford. Reprinted by Dover, New York, 1954Google Scholar
  26. Palsson B, Bhatia S (2004) Tissue engineering, chap. Scaling up for ex vivo cultivation. Pearson Education, London, pp 223–243Google Scholar
  27. Palsson E (2001) A three-dimensional model of cell movement in multicellular systems. Future Gener Comput Syst 17(7): 835–852. doi: 10.1016/S0167-739X(00)00062-5 zbMATHCrossRefGoogle Scholar
  28. Pazzano D, Mercier K, Moran J, Fong S, DiBiasio D, Rulfs J, Kohles S, Bonassar L (2000) Comparison of chondrogensis in static and perfused bioreactor culture. Biotechnol Prog 16(5): 893–896CrossRefGoogle Scholar
  29. Quarteroni A, Valli A (1997) Numerical approximation of partial differential equations, 2nd edn. Springer-Verlag, New YorkGoogle Scholar
  30. Raimondi M (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
  31. Raimondi M, Boschetti F, Falcone L, Fiore G, Remuzzi A, Marinoni E, Marazzi M, Pietrabissa R (2002) Mechanobiology of engineered cartilage cultured under a quantified fluid-dynamic environment. In: Biomechanics and modeling in mechanobiology, vol 1. Springer-Verlag, Berlin, pp 69–82Google Scholar
  32. Raimondi M, Boschetti F, Falcone L, Migliavacca F, Remuzzi A, Dubini G (2004) The effect of media perfusion on three-dimensional cultures of human chondrocytes: integration of experimental and computational approaches. Biorheology 41(3–4): 401–410Google Scholar
  33. Raimondi M, 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
  34. Raimondi M, 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
  35. 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. 9Google Scholar
  36. Schulz R, Bader A (2007) Cartilage tissue engineering and bioreactor systems for the cultivation and stimulation of chondrocytes. Eur Biophys J 36: 539–568CrossRefGoogle Scholar
  37. Silver F (2006) Mechanosensing and mechanochemical transduction in extracellular matrix, chap mechanochemical sensing and transduction. Springer, US, pp 211–261Google Scholar
  38. Wang S, Tarbell J (2000) Effect of fluid flow on smooth muscle cells in a 3-dimensional collagen gel model. Arterioscler Thromb Vasc Biol 20(10): 2220–2225CrossRefGoogle Scholar
  39. Whitaker S (1999) The method of volume averaging. Theory and application of transport in porous media. Kluwer Academic Publishers, DordrechtGoogle Scholar
  40. Wood BD, Quintard M, Whitaker S (2002) Calculation of effective diffusivities for biofilms and tissues. Biotech Bioeng 77(5): 495–514CrossRefGoogle Scholar
  41. Wood BD, Whitaker S (1998) Diffusion and reaction in biofilms. Chem Eng Sci 53: 397–425CrossRefGoogle Scholar
  42. Zhou P-H, Liu S-Q, Peng H (2008) The effect of hyaluronic acid on IL-1b-induced chondrocyte apoptosis in a rat model of osteoarthritis. Orthop Res 26(12): 1643–1648CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Riccardo Sacco
    • 1
  • Paola Causin
    • 2
  • Paolo Zunino
    • 3
  • Manuela T. Raimondi
    • 4
    • 5
    Email author
  1. 1.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanoItaly
  2. 2.Dipartimento di Matematica “F. Enriques”Università degli Studi di MilanoMilanoItaly
  3. 3.MOX, Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanoItaly
  4. 4.LaBS, Dipartimento di Ingegneria StrutturalePolitecnico di MilanoMilanoItaly
  5. 5.IRCCS Galeazzi Orthopaedic InstituteMilanoItaly

Personalised recommendations