American Institute of Mathematical Sciences

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March  2013, 8(1): 131-147. doi: 10.3934/nhm.2013.8.131

Modeling contact inhibition of growth: Traveling waves

Michiel Bertsch 1, , Masayasu Mimura 2, and  Tohru Wakasa 3,

1. 

Istituto per le Applicazioni del Calcolo Mauro Picone, CNR, University of Rome Tor Vergata, Via dei Taurini 19, 00185 Rome, Italy
2. 

Meiji Institute of Advanced Mathematical Sciences, Meiji University, 1-1-1, Higashi-mita, Tama-ku, Kawasaki, 214-8571
3. 

Department of Basic Sciences, Kyushu Institute of Technology, 1-1, Sensui-cho, Tobata, Kitakyuchu, 804-8550, Japan

Received  January 2012 Revised  October 2012 Published  April 2013

We consider a simplified 1-dimensional PDE-model describing the effect of contact inhibition in growth processes of normal and abnormal cells. Varying the value of a significant parameter, numerical tests suggest two different types of contact inhibition between the cell populations: the two populations move with constant velocity and exhibit spatial segregation, or they stop to move and regions of coexistence are formed. In order to understand the different mechanisms, we prove that there exists a segregated traveling wave solution for a unique wave speed, and we present numerical results on the ``stability" of the segregated waves. We conjecture the existence of a non-segregated standing wave for certain parameter values.
Keywords: Degenerate parabolic PDE, tumor growth model, population dynamics of cells., free boundary, traveling wave solutions.
Mathematics Subject Classification: Primary: 35R35; Secondary: 35Q92, 92C15, 92C1.
Citation: Michiel Bertsch, Masayasu Mimura, Tohru Wakasa. Modeling contact inhibition of growth: Traveling waves. Networks and Heterogeneous Media, 2013, 8 (1) : 131-147. doi: 10.3934/nhm.2013.8.131
References:
[1]

D. G. Aronson, Density-dependent interaction systems, in "Dynamics and Modelling of Reactive Systems" (eds. W. H. Steward, W. R .Ray and C. C Conley), Academic Press, New York-London, (1980), 161-176.

[2]

M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010), 235-250. doi: 10.4171/IFB/233.

[3]

M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differ. Equ. Appl., 4 (2012), 137-157. doi: 10.7153/dea-04-09.

[4]

M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Singular limit problem of a nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013). doi: 10.7153/dea-04-09.

[5]

M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Traveling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013).

[6]

M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, in preparation (2013).

[7]

Z. Biró, Stability of traveling waves for degenerate reaction-diffusion equations of KPP-type, Adv. Nonlinear Stud., 2 (2002), 357-371.

[8]

M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour deveropment, Math. Med. Bio., 23 (2006), 197-229. doi: 10.1093/imammb/dql009.

[9]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, New York-Toronto-London, 1955.

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[11]

G. Garcia-Ramos, F. Sanches-Garduño and P. K. Maini, Dispersal can sharpen parapatric boundaries on a spatially varying environment, Ecology, 81 (2000), 749-760.

[12]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753.

[13]

B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[14]

D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion, J. Differential Equations, 244 (2008), 2870-2889. doi: 10.1016/j.jde.2008.02.018.

[15]

S. Kamin and P. Rosenau, Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Ser. 9 Mat. Appl., 15 (2004), 271-280.

[16]

N. S. Kolmogorov, N. Petrovsky and I. G. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière e son application a un problème biologique, (Russian), Bull. Univ. État Moscou, Série internationale A, 1 (1937), 1-26.

[17]

G. S. Medvedev, K. Ono and P. Holmes, Traveling wave solutions of the degenerate Kolmogorov-Petrovsky-Piskunov equation, European J. Appl. Math., 14 (2003), 343-367. doi: 10.1017/S0956792503005102.

[18]

J. A. Sherratt, Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations, Proc. R. Soc. Lond. A, 456 (2000), 2365-2386. doi: 10.1098/rspa.2000.0616.

[19]

F. Sanches-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.

[20]

Y. Tsukatani, K. Suzuki and K. Takahashi, Loss of density-dependent growth inhibition and dissociation of $\alpha$-catenin from E-cadherin, J. Cell. Physiol., 173 (1997), 54-63.

show all references

References:
[1]

D. G. Aronson, Density-dependent interaction systems, in "Dynamics and Modelling of Reactive Systems" (eds. W. H. Steward, W. R .Ray and C. C Conley), Academic Press, New York-London, (1980), 161-176.

[2]

M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010), 235-250. doi: 10.4171/IFB/233.

[3]

M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differ. Equ. Appl., 4 (2012), 137-157. doi: 10.7153/dea-04-09.

[4]

M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Singular limit problem of a nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013). doi: 10.7153/dea-04-09.

[5]

M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Traveling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013).

[6]

M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, in preparation (2013).

[7]

Z. Biró, Stability of traveling waves for degenerate reaction-diffusion equations of KPP-type, Adv. Nonlinear Stud., 2 (2002), 357-371.

[8]

M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour deveropment, Math. Med. Bio., 23 (2006), 197-229. doi: 10.1093/imammb/dql009.

[9]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, New York-Toronto-London, 1955.

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[11]

G. Garcia-Ramos, F. Sanches-Garduño and P. K. Maini, Dispersal can sharpen parapatric boundaries on a spatially varying environment, Ecology, 81 (2000), 749-760.

[12]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753.

[13]

B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[14]

D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion, J. Differential Equations, 244 (2008), 2870-2889. doi: 10.1016/j.jde.2008.02.018.

[15]

S. Kamin and P. Rosenau, Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Ser. 9 Mat. Appl., 15 (2004), 271-280.

[16]

N. S. Kolmogorov, N. Petrovsky and I. G. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière e son application a un problème biologique, (Russian), Bull. Univ. État Moscou, Série internationale A, 1 (1937), 1-26.

[17]

G. S. Medvedev, K. Ono and P. Holmes, Traveling wave solutions of the degenerate Kolmogorov-Petrovsky-Piskunov equation, European J. Appl. Math., 14 (2003), 343-367. doi: 10.1017/S0956792503005102.

[18]

J. A. Sherratt, Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations, Proc. R. Soc. Lond. A, 456 (2000), 2365-2386. doi: 10.1098/rspa.2000.0616.

[19]

F. Sanches-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.

[20]

Y. Tsukatani, K. Suzuki and K. Takahashi, Loss of density-dependent growth inhibition and dissociation of $\alpha$-catenin from E-cadherin, J. Cell. Physiol., 173 (1997), 54-63.

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