2013, 10(4): 997-1015. doi: 10.3934/mbe.2013.10.997

Modeling of the migration of endothelial cells on bioactive micropatterned polymers

1. 

Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France

2. 

INSERM, IECB, UMR 5248, F-33607 Pessac, France

3. 

Univ. Bordeaux, IECB, UMR 5248, F-33607 Pessac, France

4. 

INRIA, F-33400 Talence, France, France

5. 

CNRS, IMB, UMR 5251, F-33400 Talence, France

Received  June 2012 Revised  April 2013 Published  June 2013

In this paper, a macroscopic model describing endothelial cell migration on bioactive micropatterned polymers is presented. It is based on a system of partial differential equations of Patlak-Keller-Segel type that describes the evolution of the cell densities. The model is studied mathematically and numerically. We prove existence and uniqueness results of the solution to the differential system. We also show that fundamental physical properties such as mass conservation, positivity and boundedness of the solution are satisfied. The numerical study allows us to show that the modeling results are in good agreement with the experiments.
Citation: Thierry Colin, Marie-Christine Durrieu, Julie Joie, Yifeng Lei, Youcef Mammeri, Clair Poignard, Olivier Saut. Modeling of the migration of endothelial cells on bioactive micropatterned polymers. Mathematical Biosciences & Engineering, 2013, 10 (4) : 997-1015. doi: 10.3934/mbe.2013.10.997
References:
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A. Anderson and M. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857.  doi: 10.1006/bulm.1998.0042.  Google Scholar

[2]

K. Anselme, P. Davidson, A. M. Popa, M. Giazzon, M. Liley and L. Ploux, The interaction of cells and bacteria with surfaces structured at the nanometre scale,, Acta Biomaterialia, 6 (2010), 3824.  doi: 10.1016/j.actbio.2010.04.001.  Google Scholar

[3]

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[4]

A. Blanchet, J. Dolbeault and B. Perthame, Two dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solution,, Electron. J. Differential Equations, 2006 ().   Google Scholar

[5]

P. Carmeliet and M. Tessier-Lavigne, Common mechanisms of nerve and blood vessel wiring,, Nature, 436 (2005), 193.  doi: 10.1038/nature03875.  Google Scholar

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C. S Chen, M. Mrksich, S. Huang, G. M. Whitesides and D. E. Ingber, Geometric control of cell life and death,, Science, 276 (1997), 1425.  doi: 10.1126/science.276.5317.1425.  Google Scholar

[7]

L. E. Dike, C. S. Chen, M. Mrksich, J. Tien, G. M. Whitesides and D. E. Ingber, Geometric control of switching between growth, apoptosis, and differentiation during angiogenesis using micropatterned substr'ates,, in Vitro Cell. Dev. Biol., 35 (1999), 441.   Google Scholar

[8]

J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in $\mathbbR^2$,, C. R. Math. Acad. Sci. Paris, 339 (2004), 611.  doi: 10.1016/j.crma.2004.08.011.  Google Scholar

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R. Eymard, T. Gallouet and R. Herbin, "Finite Volume Methods,", Handbook of Numerical Analysis, (2007).   Google Scholar

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A. Folch and M. Toner, Microengineering of cellular interactions,, Annu. Rev. Biomed. Eng., 2 (2000), 227.   Google Scholar

[11]

J. Folkman and C. Haudenschild, Angiogenesis in vitro,, Nature, 288 (1980), 551.   Google Scholar

[12]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

[13]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2008), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[14]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results,, Nonlinear Differ. Equ. Appl., 8 (2001), 399.  doi: 10.1007/PL00001455.  Google Scholar

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W. Hundsdorfer and J. G. Verwer, "Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations,", Springer Series in Comput. Math., 33 (2003).   Google Scholar

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Y. Ito, Surface micropatterning to regulate cell functions,, Biomaterials, 20 (1999), 2333.  doi: 10.1016/S0142-9612(99)00162-3.  Google Scholar

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R. K. Jain, Molecular regulation of vessel maturation,, Nat. Med., 9 (2003), 685.  doi: 10.1038/nm0603-685.  Google Scholar

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R. K. Jain, P. Au, J. Tam, D. G. Duda and D. Fukumura, Engineering vascularized tissue,, Nat Biotechnol, 23 (2005), 821.  doi: 10.1038/nbt0705-821.  Google Scholar

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M. Kamei, W. B. Saunders, K. J. Bayless, L. Dye, G. E. Davis and B. M. Weinstein, Endothelial tubes assemble from intracellular vacuoles,, in vivo, 442 (2006), 453.  doi: 10.1038/nature04923.  Google Scholar

[21]

E. F. Keller and L. A. Segel, Traveling band of chemotactic bacteria: A theoretical analysis,, Journal of Theo. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[22]

Y. Lei, O. F. Zouani, M. Rémy, L. Ramy and M. C. Durrieu, Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery,, Small, ().  doi: 10.1002/smll.201202410.  Google Scholar

[23]

Y. Lei, O. F. Zouani, M. Rémy, C. Ayela and M. C. Durrieu, Geometrical microfeature cues for directing tubulogenesis of endothelial cells,, PLoS ONE, 7 (2012).  doi: 10.1371/journal.pone.0041163.  Google Scholar

[24]

X. D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes,, Journal of Computational Physics, 115 (1994), 200.  doi: 10.1006/jcph.1994.1187.  Google Scholar

[25]

B. Lubarsky and M. A. Krasnow., Tube morphogenesis: Making and shaping biological tubes,, Cell, 112 (2003), 19.   Google Scholar

[26]

R. M. Nerem, Tissue engineering: The hope, the hype, and the future,, Tissue Eng., 12 (2006), 1143.   Google Scholar

[27]

D. V. Nicolau, T. Taguchi, H. Taniguchi, H. Tanigawa and S. Yoshikawa, Patterning neuronal and glia cells on light-assisted functionalized photoresists,, Biosens. Bioelectron, 14 (1999), 317.   Google Scholar

[28]

Z. K. Otrock, R. A. Mahfouz, J. A. Makarem and A. I. Shamseddine, Understanding the biology of angiogenesis: Review of the most important molecular mechanisms,, Blood Cells Mol. Dis., 39 (2007), 212.  doi: 10.1016/j.bcmd.2007.04.001.  Google Scholar

[29]

E. M Ouhabaz, "Analysis of Heat Equations on Domains,", London Math. Soc. Monographs Series, 31 (2005).   Google Scholar

[30]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311.  doi: 10.1007/BF02476407.  Google Scholar

[31]

E. A. Phelps and A. J.Garcia, Engineering more than a cell: Vascularization strategies in tissue engineering,, Curr. Opin. Biotechnol, 21 (2010), 704.  doi: 10.1016/j.copbio.2010.06.005.  Google Scholar

[32]

M. I. Santos and R. L. Reis, Vascularization in bone tissue engineering: Physiology, current strategies, major hurdles and future challenges,, Macromol Biosci., 10 (2010), 12.  doi: 10.1002/mabi.200900107.  Google Scholar

[33]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology,, Adv. Differential Equations, 6 (2001), 21.   Google Scholar

[34]

Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients,, Arch. Rational Mech. Anal., 153 (2000), 91.  doi: 10.1007/s002050000082.  Google Scholar

[35]

F. Y Wang and L.Yan, Gradient estimate on convex domains and application,, To Appear in AMS. Proc., 141 (2013), 1067.  doi: 10.1090/S0002-9939-2012-11480-7.  Google Scholar

show all references

References:
[1]

A. Anderson and M. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857.  doi: 10.1006/bulm.1998.0042.  Google Scholar

[2]

K. Anselme, P. Davidson, A. M. Popa, M. Giazzon, M. Liley and L. Ploux, The interaction of cells and bacteria with surfaces structured at the nanometre scale,, Acta Biomaterialia, 6 (2010), 3824.  doi: 10.1016/j.actbio.2010.04.001.  Google Scholar

[3]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles,, I. Colloq. Math., 66 (1993), 319.   Google Scholar

[4]

A. Blanchet, J. Dolbeault and B. Perthame, Two dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solution,, Electron. J. Differential Equations, 2006 ().   Google Scholar

[5]

P. Carmeliet and M. Tessier-Lavigne, Common mechanisms of nerve and blood vessel wiring,, Nature, 436 (2005), 193.  doi: 10.1038/nature03875.  Google Scholar

[6]

C. S Chen, M. Mrksich, S. Huang, G. M. Whitesides and D. E. Ingber, Geometric control of cell life and death,, Science, 276 (1997), 1425.  doi: 10.1126/science.276.5317.1425.  Google Scholar

[7]

L. E. Dike, C. S. Chen, M. Mrksich, J. Tien, G. M. Whitesides and D. E. Ingber, Geometric control of switching between growth, apoptosis, and differentiation during angiogenesis using micropatterned substr'ates,, in Vitro Cell. Dev. Biol., 35 (1999), 441.   Google Scholar

[8]

J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in $\mathbbR^2$,, C. R. Math. Acad. Sci. Paris, 339 (2004), 611.  doi: 10.1016/j.crma.2004.08.011.  Google Scholar

[9]

R. Eymard, T. Gallouet and R. Herbin, "Finite Volume Methods,", Handbook of Numerical Analysis, (2007).   Google Scholar

[10]

A. Folch and M. Toner, Microengineering of cellular interactions,, Annu. Rev. Biomed. Eng., 2 (2000), 227.   Google Scholar

[11]

J. Folkman and C. Haudenschild, Angiogenesis in vitro,, Nature, 288 (1980), 551.   Google Scholar

[12]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

[13]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2008), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[14]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results,, Nonlinear Differ. Equ. Appl., 8 (2001), 399.  doi: 10.1007/PL00001455.  Google Scholar

[15]

W. Hundsdorfer and J. G. Verwer, "Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations,", Springer Series in Comput. Math., 33 (2003).   Google Scholar

[16]

Y. Ito, Surface micropatterning to regulate cell functions,, Biomaterials, 20 (1999), 2333.  doi: 10.1016/S0142-9612(99)00162-3.  Google Scholar

[17]

R. K. Jain, Molecular regulation of vessel maturation,, Nat. Med., 9 (2003), 685.  doi: 10.1038/nm0603-685.  Google Scholar

[18]

R. K. Jain, P. Au, J. Tam, D. G. Duda and D. Fukumura, Engineering vascularized tissue,, Nat Biotechnol, 23 (2005), 821.  doi: 10.1038/nbt0705-821.  Google Scholar

[19]

G. S. Jiang and C. W Shu, Efficient implementation of weighted ENO schemes,, J. of Computational Physics, 126 (1996), 202.  doi: 10.1006/jcph.1996.0130.  Google Scholar

[20]

M. Kamei, W. B. Saunders, K. J. Bayless, L. Dye, G. E. Davis and B. M. Weinstein, Endothelial tubes assemble from intracellular vacuoles,, in vivo, 442 (2006), 453.  doi: 10.1038/nature04923.  Google Scholar

[21]

E. F. Keller and L. A. Segel, Traveling band of chemotactic bacteria: A theoretical analysis,, Journal of Theo. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[22]

Y. Lei, O. F. Zouani, M. Rémy, L. Ramy and M. C. Durrieu, Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery,, Small, ().  doi: 10.1002/smll.201202410.  Google Scholar

[23]

Y. Lei, O. F. Zouani, M. Rémy, C. Ayela and M. C. Durrieu, Geometrical microfeature cues for directing tubulogenesis of endothelial cells,, PLoS ONE, 7 (2012).  doi: 10.1371/journal.pone.0041163.  Google Scholar

[24]

X. D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes,, Journal of Computational Physics, 115 (1994), 200.  doi: 10.1006/jcph.1994.1187.  Google Scholar

[25]

B. Lubarsky and M. A. Krasnow., Tube morphogenesis: Making and shaping biological tubes,, Cell, 112 (2003), 19.   Google Scholar

[26]

R. M. Nerem, Tissue engineering: The hope, the hype, and the future,, Tissue Eng., 12 (2006), 1143.   Google Scholar

[27]

D. V. Nicolau, T. Taguchi, H. Taniguchi, H. Tanigawa and S. Yoshikawa, Patterning neuronal and glia cells on light-assisted functionalized photoresists,, Biosens. Bioelectron, 14 (1999), 317.   Google Scholar

[28]

Z. K. Otrock, R. A. Mahfouz, J. A. Makarem and A. I. Shamseddine, Understanding the biology of angiogenesis: Review of the most important molecular mechanisms,, Blood Cells Mol. Dis., 39 (2007), 212.  doi: 10.1016/j.bcmd.2007.04.001.  Google Scholar

[29]

E. M Ouhabaz, "Analysis of Heat Equations on Domains,", London Math. Soc. Monographs Series, 31 (2005).   Google Scholar

[30]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311.  doi: 10.1007/BF02476407.  Google Scholar

[31]

E. A. Phelps and A. J.Garcia, Engineering more than a cell: Vascularization strategies in tissue engineering,, Curr. Opin. Biotechnol, 21 (2010), 704.  doi: 10.1016/j.copbio.2010.06.005.  Google Scholar

[32]

M. I. Santos and R. L. Reis, Vascularization in bone tissue engineering: Physiology, current strategies, major hurdles and future challenges,, Macromol Biosci., 10 (2010), 12.  doi: 10.1002/mabi.200900107.  Google Scholar

[33]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology,, Adv. Differential Equations, 6 (2001), 21.   Google Scholar

[34]

Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients,, Arch. Rational Mech. Anal., 153 (2000), 91.  doi: 10.1007/s002050000082.  Google Scholar

[35]

F. Y Wang and L.Yan, Gradient estimate on convex domains and application,, To Appear in AMS. Proc., 141 (2013), 1067.  doi: 10.1090/S0002-9939-2012-11480-7.  Google Scholar

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