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:
[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.

[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.

[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.

[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 ().

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[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.

[33]

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

[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.

[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.

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.

[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.

[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.

[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 ().

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[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.

[33]

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

[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.

[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.

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