Computational Modeling in Tissue Engineering pp 85-105 | Cite as
Computational Modeling of Mass Transport and Its Relation to Cell Behavior in Tissue Engineering Constructs
Abstract
Effective recapitulation of extracellular matrix properties into a Tissue Engineering strategy is strongly involved with the need for a proper transport environment. Consumption and production of soluble medium components gives rise to gradients which influence cell behavior in various ways. Understanding how transport related phenomena can shape these gradients is targeted in this chapter by the combined use of experiments and mathematical modeling. An overview of different models is given that describe solute transport and its relation to specific cell behavior. From the simulation results important information can be extracted which help to unravel mechanisms that drive solute transport. Finally we describe the genuine efforts that have been taken to translate this information into real tissue engineering setups (e.g., optimization of culture conditions and controlled-release of growth factors).
Keywords
Vascular Endothelial Growth Factor Oxygen Tension Solute Transport Oxygen Uptake Rate Fluorescence Recovery After PhotobleachingReferences
- 1.Acosta, M.A., Ymele-Leki, P., Kostov, Y.V., Leach, J.B.: Fluorescent microparticles for sensing cell microenvironment oxygen levels within 3D scaffolds. Biomaterials 30(17), 3068–3074 (2009). doi: 10.1016/j.biomaterials.2009.02.021 CrossRefGoogle Scholar
- 2.Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P.: Molecular Biology of the Cell, 4th edn. Garland Science, New York (2007)Google Scholar
- 3.Ando, R., Mizuno, H., Miyawaki, A.: Regulated fast nucleocytoplasmic shuttling observed by reversible protein highlighting. Science 306(5700), 1370–1373 (2004). doi: 10.1126/science.1102506. 306/5700/1370[pii]CrossRefGoogle Scholar
- 4.Arany, Z., Foo, S.Y., Ma, Y.H., Ruas, J.L., Bommi-Reddy, A., Girnun, G., Cooper, M., Laznik, D., Chinsomboon, J., Rangwala, S.M., Baek, K.H., Rosenzweig, A., Spiegelman, B.M.: HIF-independent regulation of VEGF and angiogenesis by the transcriptional coactivator PGC-1 alpha. Nature 451(7181), 1008–U1008 (2008). doi: 10.1038/Nature06613 CrossRefGoogle Scholar
- 5.Armstrong, C.G., Lai, W.M., Mow, V.C.: An analysis of the unconfined compression of articular-cartilage. J. Biomech. Eng. Trans. ASME 106(2), 165–173 (1984)CrossRefGoogle Scholar
- 6.Beard, D.A., Qian, H.: Chemical Biophysics: Quantitative Analysis of Cellular Systems. Cambridge Texts in Biomedical Engineering. Cambridge University Press, New York (2008)CrossRefGoogle Scholar
- 7.Bibby, S.R.S., Jones, D.A., Ripley, R.M., Urban, J.P.G.: Metabolism of the intervertebral disc: effects of low levels of oxygen, glucose, and pH on rates of energy metabolism of bovine nucleus pulposus cells. Spine 30(5), 487–496 (2005)CrossRefGoogle Scholar
- 8.Bishop, J.R., Schuksz, M., Esko, J.D.: Heparan sulphate proteoglycans fine-tune mammalian physiology. Nature 446(7139), 1030–1037 (2007). doi: 10.1038/Nature05817 CrossRefGoogle Scholar
- 9.Blum, J.J., Lawler, G., Reed, M., Shin, I.: Effect of cytoskeletal geometry on intracellular diffusion. Biophys. J. 56(5), 995–1005 (1989). doi: 10.1016/S0006-3495(89)82744-4. S0006-3495(89)82744-4[pii]CrossRefGoogle Scholar
- 10.Brown, D.A., MacLellan, W.R., Laks, H., Dunn, J.C., Wu, B.M., Beygui, R.E.: Analysis of oxygen transport in a diffusion-limited model of engineered heart tissue. Biotechnol. Bioeng. 97(4), 962–975 (2007). doi: 10.1002/bit.21295 CrossRefGoogle Scholar
- 11.Brown, E.B., Wu, E.S., Zipfel, W., Webb, W.W.: Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery. Biophys. J. 77(5), 2837–2849 (1999)CrossRefGoogle Scholar
- 12.Brown, G.C.: Control of respiration and Atp synthesis in mammalian mitochondria and cells. Biochem. J. 284, 1–13 (1992)Google Scholar
- 13.Bursac, P.M., Freed, L.E., Biron, R.J., Vunjak-Novakovic, G.: Mass transfer studies of tissue engineered cartilage. Tissue Eng. 2(2), 141–150 (1996). doi: 10.1089/ten.1996.2.141 CrossRefGoogle Scholar
- 14.Cartmell, S.H., Porter, B.D., Garcia, A.J., Guldberg, R.E.: Effects of medium perfusion rate on cell-seeded three-dimensional bone constructs in vitro. Tissue Eng. 9(6), 1197–1203 (2003)CrossRefGoogle Scholar
- 15.Cassell, O.C., Morrison, W.A., Messina, A., Penington, A.J., Thompson, E.W., Stevens, G.W., Perera, J.M., Kleinman, H.K., Hurley, J.V., Romeo, R., Knight, K.R.: The influence of extracellular matrix on the generation of vascularized, engineered, transplantable tissue. Ann. N. Y. Acad. Sci. 944, 429–442 (2001)CrossRefGoogle Scholar
- 16.Chen, R.R., Silva, E.A., Yuen, W.W., Brock, A.A., Fischbach, C., Lin, A.S., Guldberg, R.E., Mooney, D.J.: Integrated approach to designing growth factor delivery systems. FASEB J. 21(14), 3896–3903 (2007). doi: 10.1096/fj.06-7873com CrossRefGoogle Scholar
- 17.Chen, Y., Whetstone, H.C., Lin, A.C., Nadesan, P., Wei, Q.X., Poon, R., Alman, B.A.: Beta-catenin signaling plays a disparate role in different phases of fracture repair: implications for therapy to improve bone healing. PLoS Med 4(7), 1216–1229 (2007). doi: 10.1371/journal.pmed.0040249. ARTN e249CrossRefGoogle Scholar
- 18.Clague, D.S., Phillips, R.J.: Hindered diffusion of spherical macromolecules through dilute fibrous media. Phys. Fluids 8(7), 1720–1731 (1996)MATHCrossRefGoogle Scholar
- 19.Contois, D.E.: Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures. J. Gen. Microbiol. 21, 40–50 (1959)CrossRefGoogle Scholar
- 20.Crabtree, H.G.: Observations on the carbohydrate metabolism of tumours. Biochem. J. 23(3), 536–545 (1929)Google Scholar
- 21.Crampin, E.J., Hackborn, W.W., Maini, P.K.: Pattern formation in reaction-diffusion models with nonuniform domain growth. Bull. Math. Biol. 64(4), 747–769 (2002). doi: 10.1006/bulm.2002.0295 CrossRefGoogle Scholar
- 22.Croll, T.I., Gentz, S., Mueller, K., Davidson, M., O’Connor, A.J., Stevens, G.W., Cooper-White, J.J.: Modelling oxygen diffusion and cell growth in a porous, vascularising scaffold for soft tissue engineering applications. Chem. Eng. Sci. 60(17), 4924–4934 (2005). doi: 10.1016/j.cea.2005.03.051 CrossRefGoogle Scholar
- 23.Dallon, J.C.: Multiscale modeling of cellular systems in biology. Curr. Opin. Colloid Interface Sci. 15(1–2), 24–31 (2010). doi: 10.1016/j.cocis.2009.05.007 CrossRefGoogle Scholar
- 24.Dehmelt, L., Bastiaens, P.I.H.: Spatial organization of intracellular communication: insights from imaging. Nat. Rev. Mol. Cell Biol. 11(6), 440–452 (2010). doi: 10.1038/Nrm2903 CrossRefGoogle Scholar
- 25.Demol, J., Lambrechts, D., Geris, L., Schrooten, J., Van Oosterwyck, H.: Towards a quantitative understanding of oxygen tension and cell density evolution in fibrin hydrogels. Biomaterials 32(1), 107–118 (2011). doi: 10.1016/j.biomaterials.2010.08.093 CrossRefGoogle Scholar
- 26.Dor, Y., Porat, R., Keshet, E.: Vascular endothelial growth factor and vascular adjustments to perturbations in oxygen homeostasis. Am. J. Physiol. Cell Physiol. 280(6), C1367–C1374 (2001)Google Scholar
- 27.Durlofsky, L., Brady, J.F.: Analysis of the Brinkman equation as a model for flow in porous-media. Phys. Fluids 30(11), 3329–3341 (1987)MATHCrossRefGoogle Scholar
- 28.Efrat, S., Linde, S., Kofod, H., Spector, D., Delannoy, M., Grant, S., Hanahan, D., Baekkeskov, S.: Beta-cell lines derived from transgenic mice expressing a hybrid insulin gene-oncogene. Proc. Natl. Acad. Sci. U. S. A. 85(23), 9037–9041 (1988)CrossRefGoogle Scholar
- 29.Elisseeff, J., McIntosh, W., Fu, K., Blunk, T., Langer, R.: Controlled-release of IGF-I and TGF-beta 1 in a photopolymerizing hydrogel for cartilage tissue engineering. J. Orthop. Res. 19(6), 1098–1104 (2001)CrossRefGoogle Scholar
- 30.Engler, A.J., Sen, S., Sweeney, H.L., Discher, D.E.: Matrix elasticity directs stem cell lineage specification. Cell 126(4), 677–689 (2006). doi: 10.1016/j.cell.2006.06.044. S0092-8674(06)00961-5[pii]CrossRefGoogle Scholar
- 31.Esko, J.D., Selleck, S.B.: Order out of chaos: assembly of ligand binding sites in heparan sulfate. Annu. Rev. Biochem. 71, 435–471 (2002). doi: 10.1146/annurev.biochem.71.110601.13545 CrossRefGoogle Scholar
- 32.Evans, R.C., Quinn, T.M.: Dynamic compression augments interstitial transport of a glucose-like solute in articular cartilage. Biophys. J. 91(4), 1541–1547 (2006). doi: 10.1529/biophysj.105.080366 CrossRefGoogle Scholar
- 33.Evans, R.C., Quinn, T.M.: Solute convection in dynamically compressed cartilage. J. Biomech. 39(6), 1048–1055 (2006). doi: 10.1016/j.jbiomech.2005.02.017. S0021-9290(05)00127-2[pii]CrossRefGoogle Scholar
- 34.Fassnacht, D., Portner, R.: Experimental and theoretical considerations on oxygen supply for animal cell growth in fixed-bed reactors. J. Biotechnol. 72(3), 169–184 (1999). S0168165699001297[pii]CrossRefGoogle Scholar
- 35.Fleury, M.E., Boardman, K.C., Swartz, M.A.: Autologous morphogen gradients by subtle interstitial flow and matrix interactions. Biophys. J. 91(1), 113–121 (2006). doi: 10.1529/biophysj.105.080192 CrossRefGoogle Scholar
- 36.Galban, C.J., Locke, B.R.: Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold. Biotechnol. Bioeng. 65(2), 121–132 (1999)CrossRefGoogle Scholar
- 37.Garzon-Alvarado, D.A., Garcia-Aznar, J.M., Doblare, M.: A reaction-diffusion model for long bones growth. Biomech. Model. Mechan. 8(5), 381–395 (2009). doi: 10.1007/s10237-008-0144-z CrossRefGoogle Scholar
- 38.Gefen, A., Cornelissen, L.H., Gawlitta, D., Bader, D.L., Oomens, C.W.: The free diffusion of macromolecules in tissue-engineered skeletal muscle subjected to large compression strains. J. Biomech. 41(4), 845–853 (2008). doi: 10.1016/j.jbiomech.2007.10.023. S0021-9290(07)00471-X[pii]CrossRefGoogle Scholar
- 39.Gerber, H.P., Vu, T.H., Ryan, A.M., Kowalski, J., Werb, Z., Ferrara, N.: VEGF couples hypertrophic cartilage remodeling, ossification and angiogenesis during endochondral bone formation. Nat. Med. 5(6), 623–628 (1999)CrossRefGoogle Scholar
- 40.Geris, L., Gerisch, A., Maes, C., Carmeliet, G., Weiner, R., Vander Sloten, J., Van Oosterwyck, H.: Mathematical modeling of fracture healing in mice: comparison between experimental data and numerical simulation results. Med. Biol. Eng. Comput. 44(4), 280–289 (2006). doi: 10.1007/s11517-006-0040-6 CrossRefGoogle Scholar
- 41.Geris, L., Schugart, R., Van Oosterwyck, H.: In silico design of treatment strategies in wound healing and bone fracture healing. Philos. Trans. A Math. Phys. Eng. Sci. 368(1920), 2683–2706 (2010). doi: 10.1098/rsta.2010.0056. 368/1920/2683[pii]CrossRefGoogle Scholar
- 42.Griffith, L.G., Swartz, M.A.: Capturing complex 3D tissue physiology in vitro. Nat. Rev. Mol. Cell Biol. 7(3), 211–224 (2006). doi: 10.1038/Nrm1858 CrossRefGoogle Scholar
- 43.Grimshaw, M.J., Mason, R.M.: Bovine articular chondrocyte function in vitro depends upon oxygen tension. Osteoarthr. Cartil. 8(5), 386–392 (2000). doi: 10.1053/joca.1999.0314. S1063-4584(99)90314-X[pii]CrossRefGoogle Scholar
- 44.Gross, J.D., Constantinidis, I., Sambanis, A.: Modeling of encapsulated cell systems. J. Theor. Biol. 244(3), 500–510 (2007). doi: 10.1016/j.jtbi.2006.08.012. S0022-5193(06)00365-1[pii]MathSciNetCrossRefGoogle Scholar
- 45.Guaccio, A., Borselli, C., Oliviero, O., Netti, P.A.: Oxygen consumption of chondrocytes in agarose and collagen gels: a comparative analysis. Biomaterials 29(10), 1484–1493 (2008). doi: 10.1016/j.biomaterials.2007.12.020 CrossRefGoogle Scholar
- 46.Gurdon, J.B., Bourillot, P.Y.: Morphogen gradient interpretation. Nature 413(6858), 797–803 (2001)CrossRefGoogle Scholar
- 47.Gurskaya, N.G., Verkhusha, V.V., Shcheglov, A.S., Staroverov, D.B., Chepurnykh, T.V., Fradkov, A.F., Lukyanov, S., Lukyanov, K.A.: Engineering of a monomeric green-to-red photoactivatable fluorescent protein induced by blue light. Nat. Biotechnol. 24(4), 461–465 (2006). doi: 10.1038/Nbt1191 CrossRefGoogle Scholar
- 48.Hacker, U., Nybakken, K., Perrimon, N.: Heparan sulphate proteoglycans: the sweet side of development. Nat. Rev. Mol. Cell Biol. 6(7), 530–541 (2005). 10.1038/nrm1681]nrm1681[pii]CrossRefGoogle Scholar
- 49.Happel, J.: Viscous flow relative to arrays of cylinders. AIChE J. 5(2), 174–177 (1959)CrossRefGoogle Scholar
- 50.Helm, C.L.E., Fleury, M.E., Zisch, A.H., Boschetti, F., Swartz, M.A.: Synergy between interstitial flow and VEGF directs capillary morphogenesis in vitro through a gradient amplification mechanism. Proc. Natl. Acad. Sci. U. S. A. 102(44), 15779–15784 (2005). doi: 10.1073/pnas.0503681102 CrossRefGoogle Scholar
- 51.Heywood, H.K., Bader, D.L., Lee, D.A.: Rate of oxygen consumption by isolated articular chondrocytes is sensitive to medium glucose concentration. J. Cell. Physiol. 206(2), 402–410 (2006). doi: 10.1002/Jcp.20491 CrossRefGoogle Scholar
- 52.Heywood, H.K., Knight, M.M., Lee, D.A.: Both superficial and deep zone articular chondrocyte subpopulations exhibit the crabtree effect but have different basal oxygen consumption rates. J. Cell. Physiol. 223(3), 630–639 (2010). doi: 10.1002/Jcp.22061 Google Scholar
- 53.Higdon, J.J.L., Ford, G.D.: Permeability of three-dimensional models of fibrous porous media. J. Fluid Mech. 308, 341–361 (1996)MATHCrossRefGoogle Scholar
- 54.Hilton, M.J., Gutierrez, L., Martinez, D.A., Wells, D.E.: EXT1 regulates chondrocyte proliferation and differentiation during endochondral bone development. Bone 36(3), 379–386 (2005). doi: 10.1016/j.bone.2004.09.025 CrossRefGoogle Scholar
- 55.Hrabe, J., Hrabetova, S., Segeth, K.: A model of effective diffusion and tortuosity in the extracellular space of the brain. Biophys. J. 87(3), 1606–1617 (2004). doi: 10.1529/biophysj.103.039495 CrossRefGoogle Scholar
- 56.Hunziker, E.B., Driesang, I.M.: Functional barrier principle for growth-factor-based articular cartilage repair. Osteoarthr. Cartil. 11(5), 320–327 (2003). S1063458403000311[pii]CrossRefGoogle Scholar
- 57.Johansson, L., Lofroth, J.E.: Diffusion and interaction in gels and solutions. 4. Hard-sphere Brownian dynamics simulations. J. Chem. Phys. 98(9), 7471–7479 (1993)CrossRefGoogle Scholar
- 58.Karsenty, G., Wagner, E.F.: Reaching a genetic and molecular understanding of skeletal development. Dev. Cell 2(4), 389–406 (2002)CrossRefGoogle Scholar
- 59.Kholodenko, B.N.: Cell-signalling dynamics in time and space. Nat. Rev. Mol. Cell Biol. 7(3), 165–176 (2006). doi: 10.1038/Nrm1838 CrossRefGoogle Scholar
- 60.Kirkpatrick, C.J., Fuchs, S., Unger, R.E.: Co-culture systems for vascularization—learning from nature. Adv. Drug Deliv. Rev. 63(4–5), 291–299 (2011). doi: 10.1016/j.addr.2011.01.009 CrossRefGoogle Scholar
- 61.Kleinman, H.K., Philp, D., Hoffman, M.P.: Role of the extracellular matrix in morphogenesis. Curr. Opin. Biotechnol. 14(5), 526–532 (2003). S0958166903001186[pii]CrossRefGoogle Scholar
- 62.Kosto, K.B., Deen, W.M.: Diffusivities of macromolecules in composite hydrogels. AIChE J. 50(11), 2648–2658 (2004). doi: 10.1002/Aic.10216 CrossRefGoogle Scholar
- 63.Koziel, L., Kunath, M., Kelly, O.G., Vortkamp, A.: Ext1-dependent heparan sulfate regulates the range of Ihh signaling during endochondral ossification. Dev. Cell 6(6), 801–813 (2004)CrossRefGoogle Scholar
- 64.Kronenberg, H.M.: Developmental regulation of the growth plate. Nature 423(6937), 332–336 (2003). doi: 10.1038/Nature01657 CrossRefGoogle Scholar
- 65.Lanske, B., Karaplis, A.C., Lee, K., Luz, A., Vortkamp, A., Pirro, A., Karperien, M., Defize, L.H.K., Ho, C., Mulligan, R.C., AbouSamra, A.B., Juppner, H., Segre, G.V., Kronenberg, H.M.: PTH/PTHrP receptor in early development and Indian hedgehog-regulated bone growth. Science 273(5275), 663–666 (1996)CrossRefGoogle Scholar
- 66.Leddy, H.A., Guilak, F.: Site-specific effects of compression on macromolecular diffusion in articular cartilage. Biophys. J. 95(10), 4890–4895 (2008). doi: 10.1529/biophysj.108.137752 CrossRefGoogle Scholar
- 67.Lee, R.B., Urban, J.P.: Evidence for a negative Pasteur effect in articular cartilage. Biochem. J. 321(Pt 1), 95–102 (1997)Google Scholar
- 68.Lee, R.B., Urban, J.P.: Functional replacement of oxygen by other oxidants in articular cartilage. Arthritis Rheum. 46(12), 3190–3200 (2002). doi: 10.1002/art.10686 CrossRefGoogle Scholar
- 69.Levick, J.R.: Flow through interstitium and other fibrous matrices. Q. J. Exp. Physiol. CMS 72(4), 409–438 (1987)Google Scholar
- 70.Lewis, M.C., MacArthur, B.D., Malda, J., Pettet, G., Please, C.P.: Heterogeneous proliferation within engineered cartilaginous tissue: the role of oxygen tension. Biotechnol. Bioeng. 91(5), 607–615 (2005). doi: 10.1002/Bit.20508 CrossRefGoogle Scholar
- 71.Lutolf, M.P., Hubbell, J.A.: Synthetic biomaterials as instructive extracellular microenvironments for morphogenesis in tissue engineering. Nat. Biotechnol. 23(1), 47–55 (2005). doi: 10.1038/Nbt1055 CrossRefGoogle Scholar
- 72.Lutolf, M.R., Weber, F.E., Schmoekel, H.G., Schense, J.C., Kohler, T., Muller, R., Hubbell, J.A.: Repair of bone defects using synthetic mimetics of collagenous extracellular matrices. Nat. Biotechnol. 21(5), 513–518 (2003). doi: 10.1038/Nbt818 CrossRefGoogle Scholar
- 73.Mackie, J.S., Meares, P.: The diffusion of electrolytes in a cation-exchange resin membrane. 1. Theoretical. Proc. R. Soc. Lond. Ser. A 232(1191), 498–509 (1955a)Google Scholar
- 74.Mackie, J.S., Meares, P.: The diffusion of electrolytes in a cation-exchange resin membrane. 2. Experimental. Proc. R. Soc. Lond. Ser. A 232(1191), 510–518 (1955b)Google Scholar
- 75.Madzvamuse, A.: Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. J. Comput. Phys. 214(1), 239–263 (2006). doi: 10.1016/j.jcp.2005.09.012 MathSciNetMATHCrossRefGoogle Scholar
- 76.Maes, C., Carmeliet, G.: Vascular and nonvascular roles of VEGF in bone development. In: Ruhrberg, C. (ed.) VEGF in Development, pp. 79–90. Springer, New York (2008)CrossRefGoogle Scholar
- 77.Mahoney, M.J., Krewson, C., Miller, J., Saltzman, W.M.: Impact of cell type and density on nerve growth factor distribution and bioactivity in 3-dimensional collagen gel cultures. Tissue Eng. 12(7), 1915–1927 (2006)CrossRefGoogle Scholar
- 78.Maini, P.K.: Using mathematical models to help understand biological pattern formation. C. R. Biol. 327(3), 225–234 (2004). doi: 10.1016/j.crvi.2003.05.006 MathSciNetCrossRefGoogle Scholar
- 79.Makarenkova, H.P., Hoffman, M.P., Beenken, A., Eliseenkova, A.V., Meech, R., Tsau, C., Patel, V.N., Lang, R.A., Mohammadi, M.: Differential interactions of FGFs with heparan sulfate control gradient formation and branching morphogenesis. Sci. Signal 2(88), ra55 (2009). doi:2/88/ra55[pii]10.1126/scisignal.2000304Google Scholar
- 80.Malda, J., Rouwkema, J., Martens, D.E., le Comte, E.P., Kooy, F.K., Tramper, J., van Blitterswijk, C.A., Riesle, J.: Oxygen gradients in tissue-engineered PEGT/PBT cartilaginous constructs: measurement and modeling. Biotechnol. Bioeng. 86(1), 9–18 (2004). doi: 10.1002/Bit.20038 CrossRefGoogle Scholar
- 81.Marcus, R.E.: The effect of low oxygen concentration on growth, glycolysis, and sulfate incorporation by articular chondrocytes in monolayer culture. Arthritis Rheum. 16(5), 646–656 (1973)CrossRefGoogle Scholar
- 82.Martin, I., Wendt, D., Heberer, M.: The role of bioreactors in tissue engineering. Trends Biotechnol. 22(2), 80–86 (2004). doi: 10.1016/j.tibtech.2003.12.001 CrossRefGoogle Scholar
- 83.Mauck, R.L., Hung, C.T., Ateshian, G.A.: Modeling of neutral solute transport in a dynamically loaded porous permeable gel: Implications for articular cartilage biosynthesis and tissue engineering (vol. 125, 602, 2003). J. Biomech. Eng. Trans. ASME 126(3), 392–392 (2004)Google Scholar
- 84.Mcelwain, D.L.S., Ponzo, P.J.: Model for growth of a solid tumor with nonuniform oxygen-consumption. Math. Biosci. 35(3–4), 267–279 (1977)MATHCrossRefGoogle Scholar
- 85.Mizuno, S., Allemann, F., Glowacki, J.: Effects of medium perfusion on matrix production by bovine chondrocytes in three-dimensional collagen sponges. J. Biomed. Mater. Res. 56(3), 368–375 (2001). doi: 10.1002/1097-4636(20010905). 56:3<368:AID-JBM1105>3.0.CO;2-V[pii]CrossRefGoogle Scholar
- 86.Mizutani, C.M., Nie, Q., Wan, F.Y.M., Zhang, Y.T., Vilmos, P., Sousa-Neves, R., Bier, E., Marsh, J.L., Lander, A.D.: Formation of the BMP activity gradient in the Drosophila embryo. Dev. Cell 8(6), 915–924 (2005). doi: 10.1016/j.devcel.2005.04.009 CrossRefGoogle Scholar
- 87.Moseley, J.B., Mayeux, A., Paoletti, A., Nurse, P.: A spatial gradient coordinates cell size and mitotic entry in fission yeast. Nature 459(7248), 857–U858 (2009). doi: 10.1038/Nature08074 CrossRefGoogle Scholar
- 88.Moser, H.: Structure and dynamics of bacterial populations maintained in the chemostat. Cold Spring Harb. Symp. Quant. Biol. 22, 121–137 (1957)CrossRefGoogle Scholar
- 89.Mow, V.C., Kuei, S.C., Lai, W.M., Armstrong, C.G.: Biphasic creep and stress relaxation of articular cartilage in compression? Theory and experiments. J. Biomech. Eng. 102(1), 73–84 (1980)CrossRefGoogle Scholar
- 90.Murry, C.E., Keller, G.: Differentiation of embryonic stem cells to clinically relevant populations: lessons from embryonic development. Cell 132(4), 661–680 (2008). doi: 10.1016/j.cell.2008.02.008 CrossRefGoogle Scholar
- 91.Ng, C.P., Swartz, M.A.: Fibroblast alignment under interstitial fluid flow using a novel 3-D tissue culture model. Am. J. Physiol. Heart Circ. Physiol. 284(5), H1771–H1777 (2003). doi: 10.1152/ajpheart.01008.2002 Google Scholar
- 92.Nimer, E., Schneiderman, R., Maroudas, A.: Diffusion and partition of solutes in cartilage under static load. Biophys. Chem. 106(2), 125–146 (2003). doi: 10.1016/S0301-4622(03)00157-1 CrossRefGoogle Scholar
- 93.Obradovic, B., Carrier, R.L., Vunjak-Novakovic, G., Freed, L.E.: Gas exchange is essential for bioreactor cultivation of tissue engineered cartilage. Biotechnol. Bioeng. 63(2), 197–205 (1999). doi: 10.1002/(SICI)1097-0290(19990420). 63:2<197:AID-BIT8>3.0.CO;2-2[pii]CrossRefGoogle Scholar
- 94.Obradovic, B., Meldon, J.H., Freed, L.E., Vunjak-Novakovic, G.: Glycosaminoglycan deposition in engineered cartilage: experiments and mathematical model. AIChE J. 46(9), 1860–1871 (2000)CrossRefGoogle Scholar
- 95.Ogston, A.G., Preston, B.N., Wells, J.D., Ogston, A.G., Preston, B.N., Snowden, J.M., Wells, J.D.: Transport of compact particles through solutions of chain-polymers. Proc. R. Soc. Lond. A Mat. 333(1594), 297–316 (1973)CrossRefGoogle Scholar
- 96.Otte, P.: Basic cell metabolism of articular cartilage. Manometric studies. Z. Rheumatol. 50(5), 304–312 (1991)Google Scholar
- 97.Page-McCaw, A., Ewald, A.J., Werb, Z.: Matrix metalloproteinases and the regulation of tissue remodelling. Nat. Rev. Mol. Cell Biol. 8(3), 221–233 (2007). doi: 10.1038/nrm2125. nrm2125 [pii]CrossRefGoogle Scholar
- 98.Papas, K.K., Long Jr, R.C., Constantinidis, I., Sambanis, A.: Effects of oxygen on metabolic and secretory activities of beta TC3 cells. Biochim. Biophys. Acta 1291(2), 163–166 (1996). 0304-4165(96)00062-1[pii]CrossRefGoogle Scholar
- 99.Patterson, G.H., Lippincott-Schwartz, J.: A photoactivatable GFP for selective photolabeling of proteins and cells. Science 297(5588), 1873–1877 (2002)CrossRefGoogle Scholar
- 100.Patterson, J., Martino, M.M., Hubbell, J.A.: Biomimetic materials in tissue engineering. Mater. Today 13(1–2), 14–22 (2010)CrossRefGoogle Scholar
- 101.Phelps, E.A., Landazuri, N., Thule, P.M., Taylor, W.R., Garcia, A.J.: Bioartificial matrices for therapeutic vascularization. Proc. Natl. Acad. Sci. U. S. A. 107(8), 3323–3328 (2010). doi: 10.1073/pnas.0905447107 CrossRefGoogle Scholar
- 102.Phillips, J.E., Burns, K.L., Le Doux, J.M., Guldberg, R.E., Garcia, A.J.: Engineering graded tissue interfaces. Proc. Natl. Acad. Sci. U. S. A. 105(34), 12170–12175 (2008). doi: 10.1073/pnas.0801988105. 0801988105 [pii]CrossRefGoogle Scholar
- 103.Phillips, R.J., Deen, W.M., Brady, J.F.: Hindered transport of spherical macromolecules in fibrous membranes and gels. AIChE J. 35(11), 1761–1769 (1989)CrossRefGoogle Scholar
- 104.Picioreanu, C., van Loosdrecht, M.C., Heijnen, J.J.: A new combined differential-discrete cellular automaton approach for biofilm modeling: application for growth in gel beads. Biotechnol. Bioeng. 57(6), 718–731 (1998). doi: 10.1002/(SICI)1097-0290(19980320)57:6<718:AID-BIT9>. 3.0.CO;2-O [pii]CrossRefGoogle Scholar
- 105.Place, E.S., Evans, N.D., Stevens, M.M.: Complexity in biomaterials for tissue engineering. Nat. Mater. 8(6), 457–470 (2009). doi: 10.1038/Nmat2441 CrossRefGoogle Scholar
- 106.Pugh, C.W., Ratcliffe, P.J.: Regulation of angiogenesis by hypoxia: role of the HIF system. Na.t Med. 9(6), 677–684 (2003). doi:10.1038/nm0603-677nm0603-677 [pii]Google Scholar
- 107.Qutub, A.A., Mac Gabhann, F., Karagiannis, E.D., Vempati, P., Popel, A.S.: Multiscale models of angiogenesis. IEEE Eng. Med. Biol. Mag. 28(2), 14–31 (2009). doi: 10.1109/MEMB.2009.931791 CrossRefGoogle Scholar
- 108.Rajpurohit, R., Koch, C.J., Tao, Z., Teixeira, C.M., Shapiro, I.M.: Adaptation of chondrocytes to low oxygen tension: relationship between hypoxia and cellular metabolism. J. Cell. Physiol. 168(2), 424–432 (1996). doi: 10.1002/(SICI)1097-4652(199608). 168:2<424:AID-JCP21>3.0.CO;2-1[pii]10.1002/(SICI)1097-4652(199608)168:2<424:AID-JCP21>3.0.CO;2-1CrossRefGoogle Scholar
- 109.Rolfe, D.F., Brown, G.C.: Cellular energy utilization and molecular origin of standard metabolic rate in mammals. Physiol. Rev. 77(3), 731–758 (1997)Google Scholar
- 110.Sengers, B.G., Heywood, H.K., Lee, D.A., Oomens, C.W.J., Bader, D.L.: Nutrient utilization by bovine articular chondrocytes: A combined experimental and theoretical approach. J. Biomech. Eng. Trans. ASME 127(5), 758–766 (2005). doi: 10.1115/1.1993664 CrossRefGoogle Scholar
- 111.Sengers, B.G., van Donkelaar, C.C., Oomens, C.W.J., Baaijens, F.P.T.: Computational study of culture conditions and nutrient supply in cartilage tissue engineering. Biotechnol. Prog. 21(4), 1252–1261 (2005). doi: 10.1021/Bp0500157 CrossRefGoogle Scholar
- 112.Singh, A.B., Harris, R.C.: Autocrine, paracrine and juxtacrine signaling by EGFR ligands. Cell. Signal. 17(10), 1183–1193 (2005). doi: 10.1016/j.cellsig.2005.03.026 CrossRefGoogle Scholar
- 113.Soukane, D.M., Shirazi-Adl, A., Urban, J.P.G.: Computation of coupled diffusion of oxygen, glucose and lactic acid in an intervertebral disc. J. Biomech. 40(12), 2645–2654 (2007). doi: 10.1016/j.jbiomech.2007.01.003 CrossRefGoogle Scholar
- 114.Southern, J., Pitt-Francis, J., Whiteley, J., Stokeley, D., Kobashi, H., Nobes, R., Kadooka, Y., Gavaghan, D.: Multi-scale computational modelling in biology and physiology. Prog. Biophys. Mol. Biol. 96(1–3), 60–89 (2008). doi: 10.1016/j.pbiomolbio.2007.07.019. S0079-6107(07)00067-3 [pii]CrossRefGoogle Scholar
- 115.Sternlicht, M.D., Werb, Z.: How matrix metalloproteinases regulate cell behavior. Annu. Rev. Cell Dev. Biol. 17, 463–516 (2001)CrossRefGoogle Scholar
- 116.Stevens, M.M., Marini, R.P., Schaefer, D., Aronson, J., Langer, R., Shastri, V.P.: In vivo engineering of organs: the bone bioreactor. Proc. Natl. Acad. Sci. U. S. A. 102(32), 11450–11455 (2005). doi: 10.1073/pnas.0504705102. 0504705102 [pii]CrossRefGoogle Scholar
- 117.Stylianopoulos, T., Barocas, V.H.: Volume-averaging theory for the study of the mechanics of collagen networks. Comput. Method Appl. Mech. Eng. 196(31–32), 2981–2990 (2007). doi: 10.1016/j.cma.2006.06.019 MathSciNetMATHCrossRefGoogle Scholar
- 118.Stylianopoulos, T., Diop-Frimpong, B., Munn, L.L., Jain, R.K.: Diffusion anisotropy in collagen gels and tumors: the Effect of fiber network orientation. Biophys. J. 99(10), 3119–3128 (2010). doi: 10.1016/j.bpj.2010.08.065 CrossRefGoogle Scholar
- 119.Sumi, T., Tsuneyoshi, N., Nakatsuji, N., Suemori, H.: Defining early lineage specification of human embryonic stem cells by the orchestrated balance of canonical Wnt/beta-catenin. Activin/Nodal BMP signal. Dev. 135(17), 2969–2979 (2008). doi: 10.1242/Dev.021121 Google Scholar
- 120.Swartz, M.A., Fleury, M.E.: Interstitial flow and its effects in soft tissues. Annu. Rev. Biomed. Eng. 9, 229–256 (2007). doi: 10.1146/annure/bioeng.9.060906.151850 CrossRefGoogle Scholar
- 121.Teleman, A.A., Strigini, M., Cohen, S.M.: Shaping morphogen gradients. Cell 105(5), 559–562 (2001). S0092-8674(01)00377-4 [pii]CrossRefGoogle Scholar
- 122.Tschumperlin, D.J., Dai, G.H., Maly, I.V., Kikuchi, T., Laiho, L.H., McVittie, A.K., Haley, K.J., Lilly, C.M., So, P.T.C., Lauffenburger, D.A., Kamm, R.D., Drazen, J.M.: Mechanotransduction through growth-factor shedding into the extracellular space. Nature 429(6987), 83–86 (2004)CrossRefGoogle Scholar
- 123.Tsuji, K., Bandyopadhyay, A., Harfe, B.D., Cox, K., Kakar, S., Gerstenfeld, L., Einhorn, T., Tabin, C.J., Rosen, V.: BMP2 activity, although dispensable for bone formation, is required for the initiation of fracture healing. Nat. Genet. 38(12), 1424–1429 (2006). doi: 10.1038/Ng1916 CrossRefGoogle Scholar
- 124.van der Eerden, B.C.J., Karperien, M., Wit, J.M.: Systemic and local regulation of the growth plate. Endocr. Rev. 24(6), 782–801 (2003). doi: 10.1210/Er.2002-0033 CrossRefGoogle Scholar
- 125.van Donkelaar, C.C., Huiskes, R.: The PTHrP-Ihh feedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation. Biomech. Model. Mechanbiol. 6(1–2), 55–62 (2007). doi: 10.1007/s10237-006-0035-0 CrossRefGoogle Scholar
- 126.Vortkamp, A., Lee, K., Lanske, B., Segre, G.V., Kronenberg, H.M., Tabin, C.J.: Regulation of rate of cartilage differentiation by Indian hedgehog and PTH-related protein. Science 273(5275), 613–622 (1996)CrossRefGoogle Scholar
- 127.Williams, R.M., Zipfel, W.R., Tinsley, M.L., Farnum, C.E.: Solute transport in growth plate cartilage: in vitro and in vivo. Biophys. J. 93(3), 1039–1050 (2007). doi: 10.1529/biophysj.106.097675 CrossRefGoogle Scholar
- 128.Wood, B.D., Quintard, M., Whitaker, S.: Calculation of effective diffusivities for biofilms and tissues. Biotechnol. Bioeng. 77(5), 495–516 (2002). 10.1002/bit.10075 [pii]CrossRefGoogle Scholar
- 129.Wood, B.D., Whitaker, S.: Diffusion and reaction in biofilms. Chem. Eng. Sci. 53(3), 397–425 (1998)CrossRefGoogle Scholar
- 130.Zhou, S., Cui, Z., Urban, J.P.G.: Nutrient gradients in engineered cartilage: metabolic kinetics measurement and mass transfer modeling. Biotechnol. Bioeng. 101(2), 408–421 (2008). doi: 10.1002/Bit.21887 CrossRefGoogle Scholar