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March  2017, 10(1): 299-311. doi: 10.3934/krm.2017012

On interfaces between cell populations with different mobilities

Tommaso Lorenzi 1,2,3, , Alexander Lorz 3,4,, and  Benoît Perthame 3,

1. 

School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK
2. 

Centre de Mathématiques et de Leurs Applications, ENS Cachan, CNRS, Cachan 94230 Cedex, France
3. 

Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, UMR 7598, Laboratoire Jacques-Louis Lions, Équipe MAMBA, 4, place Jussieu 75005, Paris, France
4. 

CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

* Corresponding author: Alexander Lorz

Received  January 2016 Revised  April 2016 Published  November 2016

Partial differential equations describing the dynamics of cell population densities from a fluid mechanical perspective can model the growth of avascular tumours. In this framework, we consider a system of equations that describes the interaction between a population of dividing cells and a population of non-dividing cells. The two cell populations are characterised by different mobilities. We present the results of numerical simulations displaying two-dimensional spherical waves with sharp interfaces between dividing and non-dividing cells. Furthermore, we numerically observe how different ratios between the mobilities change the morphology of the interfaces, and lead to the emergence of finger-like patterns of invasion above a threshold. Motivated by these simulations, we study the existence of one-dimensional travelling wave solutions.

Keywords: Cell populations, tissue growth, cancer invasion, interfaces, travelling waves, Pattern formation.
Mathematics Subject Classification: 35C07, 35B35, 35B36, 92C10.
Citation: Tommaso Lorenzi, Alexander Lorz, Benoît Perthame. On interfaces between cell populations with different mobilities. Kinetic & Related Models, 2017, 10 (1) : 299-311. doi: 10.3934/krm.2017012
References:
[1]

D. Ambrosi and F. Mollica, On the mechanics of a growing tumor, International Journal of Engineering Science, 40 (2002), 1297-1316.  doi: 10.1016/S0020-7225(02)00014-9.  Google Scholar

[2]

R. Araujo and D. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bulletin of Mathematical Biology, 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.  Google Scholar

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E. Baratchart, S. Benzekry, A. Bikfalvi, T. Colin, L. S. Cooley, R. Pineau, E. J. Ribot, O. Saut and W. Souleyreau, Computational modelling of metastasis development in renal cell carcinoma, PLoS Computational Biology, 11 (2015), e1004626. doi: 10.1371/journal.pcbi.1004626.  Google Scholar

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H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison, Journal of Mathematical Biology, 58 (2009), 657-687.  doi: 10.1007/s00285-008-0212-0.  Google Scholar

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H. Byrne and M.A. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Mathematical and Computer Modelling, 24 (1996), 1-17.  doi: 10.1016/S0895-7177(96)00174-4.  Google Scholar

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P. Ciarletta, L. Foret and M. Ben~Amar, The radial growth phase of malignant melanoma: Muti-phase modelling, numerical simulation and linear stability, J. R. Soc. Interface, 8 (2011), 345-368.  doi: 10.1098/rsif.2010.0285.  Google Scholar

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E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97–115, URL http://dx.doi.org/10.1017/S0956792598003660. doi: 10.1017/S0956792598003660.  Google Scholar

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D. Drasdo and S. Hoehme, Modeling the impact of granular embedding media, and pulling versus pushing cells on growing cell clones, New Journal of Physics, 14 (2012), 055025. doi: 10.1088/1367-2630/14/5/055025.  Google Scholar

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A. Friedman, A hierarchy of cancer models and their mathematical challenges, Discrete and Continuous Dynamical Systems Series B, 4 (2004), 147-159.  doi: 10.3934/dcdsb.2004.4.147.  Google Scholar

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H. Greenspan, On the growth and stability of cell cultures and solid tumors, Journal of Theoretical Biology, 56 (1976), 229-242.  doi: 10.1016/S0022-5193(76)80054-9.  Google Scholar

[15]

M. Kowalczyk, B. Perthame and N. Vauchelet, Transversal instability for the thermodiffusive reaction-diffusion system, Chinese Annals of Mathematics, Series B, 36 (2015), 871-882.  doi: 10.1007/s11401-015-0981-x.  Google Scholar

[16]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[17]

M. Mimura, H. Sakaguchi and M. Matsushita, Reaction diffusion modelling of bacterial colony patterns, Physica A, 282 (2000), 283-303.  doi: 10.1016/S0378-4371(00)00085-6.  Google Scholar

[18]

B. Perthame, F. Quirós, M. Tang and N. Vauchelet, Derivation of a hele-shaw type system from a cell model with active motion, Interfaces and Free Boundaries, 16 (2014), 489-508.  doi: 10.4171/IFB/327.  Google Scholar

[19]

B. Perthame, F. Quirós and J. L. Vázquez, The hele-shaw asymptotics for mechanical models of tumor growth, ARMA, 212 (2014), 93–127, URL http://hal.upmc.fr/docs/00/83/19/32/PDF/Hele_Shaw.pdf. doi: 10.1007/s00205-013-0704-y.  Google Scholar

[20]

L. Preziosi, Cancer Modelling and Simulation, CRC Press, 2003. doi: 10.1201/9780203494899.  Google Scholar

[21]

I. Ramis-Conde, D. Drasdo, A.R. Anderson and M.A. Chaplain, Modeling the influence of the e-cadherin-$β$-catenin pathway in cancer cell invasion: A multiscale approach, Biophysical Journal, 95 (2008), 155-165.   Google Scholar

[22]

J. Ranft, M. Basan, J. Elgeti, J.-F. Joanny, J. Prost and F. Jülicher, Fluidization of tissues by cell division and apoptosis, Proceedings of the National Academy of Sciences, 107 (2010), 20863-20868.  doi: 10.1073/pnas.1011086107.  Google Scholar

[23]

T. Roose, S.J. Chapman and P.K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.  doi: 10.1137/S0036144504446291.  Google Scholar

[24]

P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or heleshaw cell containing a more viscous liquid, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 245, The Royal Society, 1958,312–329 doi: 10.1098/rspa.1958.0085.  Google Scholar

[25]

J.A. Sherratt and M.A. Chaplain, A new mathematical model for avascular tumour growth, Journal of Mathematical Biology, 43 (2001), 291-312.  doi: 10.1007/s002850100088.  Google Scholar

[26]

M. Tang, N. Vauchelet, I. Cheddadi, I. Vignon-Clementel, D. Drasdo and B. Perthame, Composite waves for a cell population system modeling tumor growth and invasion, Chinese Annals of Mathematics, Series B, 34 (2013), 295-318.  doi: 10.1007/s11401-013-0761-4.  Google Scholar

show all references

References:
[1]

D. Ambrosi and F. Mollica, On the mechanics of a growing tumor, International Journal of Engineering Science, 40 (2002), 1297-1316.  doi: 10.1016/S0020-7225(02)00014-9.  Google Scholar

[2]

R. Araujo and D. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bulletin of Mathematical Biology, 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.  Google Scholar

[3]

E. Baratchart, S. Benzekry, A. Bikfalvi, T. Colin, L. S. Cooley, R. Pineau, E. J. Ribot, O. Saut and W. Souleyreau, Computational modelling of metastasis development in renal cell carcinoma, PLoS Computational Biology, 11 (2015), e1004626. doi: 10.1371/journal.pcbi.1004626.  Google Scholar

[4]

H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM Journal on Mathematical Analysis, 16 (1985), 1207-1242.  doi: 10.1137/0516088.  Google Scholar

[5]

A. Brú, S. Albertos, J.L. Subiza, J.L. García-Asenjo and I. Brú, The universal dynamics of tumor growth, Biophysical Journal, 85 (2003), 2948-2961.   Google Scholar

[6]

H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Mathematical Biosciences, 135 (1996), 187-216.  doi: 10.1016/0025-5564(96)00023-5.  Google Scholar

[7]

H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison, Journal of Mathematical Biology, 58 (2009), 657-687.  doi: 10.1007/s00285-008-0212-0.  Google Scholar

[8]

H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Mathematical Medicine and Biology, 20 (2003), 341-366.  doi: 10.1093/imammb/20.4.341.  Google Scholar

[9]

H. Byrne and M.A. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Mathematical and Computer Modelling, 24 (1996), 1-17.  doi: 10.1016/S0895-7177(96)00174-4.  Google Scholar

[10]

P. Ciarletta, L. Foret and M. Ben~Amar, The radial growth phase of malignant melanoma: Muti-phase modelling, numerical simulation and linear stability, J. R. Soc. Interface, 8 (2011), 345-368.  doi: 10.1098/rsif.2010.0285.  Google Scholar

[11]

E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97–115, URL http://dx.doi.org/10.1017/S0956792598003660. doi: 10.1017/S0956792598003660.  Google Scholar

[12]

D. Drasdo and S. Hoehme, Modeling the impact of granular embedding media, and pulling versus pushing cells on growing cell clones, New Journal of Physics, 14 (2012), 055025. doi: 10.1088/1367-2630/14/5/055025.  Google Scholar

[13]

A. Friedman, A hierarchy of cancer models and their mathematical challenges, Discrete and Continuous Dynamical Systems Series B, 4 (2004), 147-159.  doi: 10.3934/dcdsb.2004.4.147.  Google Scholar

[14]

H. Greenspan, On the growth and stability of cell cultures and solid tumors, Journal of Theoretical Biology, 56 (1976), 229-242.  doi: 10.1016/S0022-5193(76)80054-9.  Google Scholar

[15]

M. Kowalczyk, B. Perthame and N. Vauchelet, Transversal instability for the thermodiffusive reaction-diffusion system, Chinese Annals of Mathematics, Series B, 36 (2015), 871-882.  doi: 10.1007/s11401-015-0981-x.  Google Scholar

[16]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[17]

M. Mimura, H. Sakaguchi and M. Matsushita, Reaction diffusion modelling of bacterial colony patterns, Physica A, 282 (2000), 283-303.  doi: 10.1016/S0378-4371(00)00085-6.  Google Scholar

[18]

B. Perthame, F. Quirós, M. Tang and N. Vauchelet, Derivation of a hele-shaw type system from a cell model with active motion, Interfaces and Free Boundaries, 16 (2014), 489-508.  doi: 10.4171/IFB/327.  Google Scholar

[19]

B. Perthame, F. Quirós and J. L. Vázquez, The hele-shaw asymptotics for mechanical models of tumor growth, ARMA, 212 (2014), 93–127, URL http://hal.upmc.fr/docs/00/83/19/32/PDF/Hele_Shaw.pdf. doi: 10.1007/s00205-013-0704-y.  Google Scholar

[20]

L. Preziosi, Cancer Modelling and Simulation, CRC Press, 2003. doi: 10.1201/9780203494899.  Google Scholar

[21]

I. Ramis-Conde, D. Drasdo, A.R. Anderson and M.A. Chaplain, Modeling the influence of the e-cadherin-$β$-catenin pathway in cancer cell invasion: A multiscale approach, Biophysical Journal, 95 (2008), 155-165.   Google Scholar

[22]

J. Ranft, M. Basan, J. Elgeti, J.-F. Joanny, J. Prost and F. Jülicher, Fluidization of tissues by cell division and apoptosis, Proceedings of the National Academy of Sciences, 107 (2010), 20863-20868.  doi: 10.1073/pnas.1011086107.  Google Scholar

[23]

T. Roose, S.J. Chapman and P.K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.  doi: 10.1137/S0036144504446291.  Google Scholar

[24]

P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or heleshaw cell containing a more viscous liquid, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 245, The Royal Society, 1958,312–329 doi: 10.1098/rspa.1958.0085.  Google Scholar

[25]

J.A. Sherratt and M.A. Chaplain, A new mathematical model for avascular tumour growth, Journal of Mathematical Biology, 43 (2001), 291-312.  doi: 10.1007/s002850100088.  Google Scholar

[26]

M. Tang, N. Vauchelet, I. Cheddadi, I. Vignon-Clementel, D. Drasdo and B. Perthame, Composite waves for a cell population system modeling tumor growth and invasion, Chinese Annals of Mathematics, Series B, 34 (2013), 295-318.  doi: 10.1007/s11401-013-0761-4.  Google Scholar

Figure 1.  Numerical observations in the case µ < ν

Plots of the computed m (left panel) and n (right panel) at time t = 1 for ν = 2 and µ = 1. We observe the emergence of a spherical wave of dividing cells pushing the surrounding non-dividing cells (left panel), and an invasive front made of non-dividing cells that are induced to move by the expansion of dividing cells (right panel).

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Figure 2.  Numerical observations in the case µ > ν

Plots of the computed m (left panel) and n (right panel) at time t = 1 for ν = 1 and µ = 2. We observe the appearance of numerical instabilities which result in finger-like patterns of dividing cells (left panel) that protrude through and dislocate the surrounding nondividing cells (right panel).

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Figure 4.  Travelling waves of Theorem 3.1 for µ < ν

Profiles of p (left panel), and m (right panel, red curve) and n (right panel, blue curve) for the travelling wave in the case where n has a compact support and µ < ν. The dashed line in the left panel highlights the value of PM, while the dashed line in the right panel highlights the value of (PM/Kγ)1/γ

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Figure 5.  Transient regime of Theorem 3.1 for µ > ν

Profiles of p (left panel), and m (right panel, red curve) and n (right panel, blue curve) in the case where n has a compact support and µ > ν. The dashed line in the left panel highlights the value of PM, while the dashed line in the right panel highlights the value of (PM/Kγ)1/γ. This figure shows a transient regime after which n is left behind and m propagates alone (see also Supplementary Movie S1)

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Figure 3.  The profile of p for the travelling wave when n has a finite support that coincides with [0, r]
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Figure 6.  Travelling waves of Theorem 4.1 for µ < ν

Profiles of p (left panel), and m (right panel, red curve) and n (right panel, blue curve) for the travelling wave in the case where n does not vanish at infinity and µ < ν. The dashed line in the left panel highlights the value of PM, while the dashed line in the right panel highlights the value of (PM/Kγ)1/γ

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Figure 7.  Transient regime of Theorem 4.1 for µ > ν

Profiles of p (left panel), and m (right panel, red curve) and n (right panel, blue curve) in the case where n does not vanish at infinity and µ > ν. The dashed line in the left panel highlights the value of PM, while the dashed line in the right panel highlights the value of (PM/Kγ)1/γ

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