On interfaces between cell populations with different mobilities

Tommaso Lorenzi; Alexander Lorz; Benoît Perthame

doi:10.3934/krm.2017012

Article Contents

2017, Volume 10, Issue 1: 299-311. Doi: 10.3934/krm.2017012

This issue Previous Article On the classical limit of a time-dependent self-consistent field system: Analysis and computation Next Article Aggregated steady states of a kinetic model for chemotaxis

On interfaces between cell populations with different mobilities

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
* Corresponding author: Alexander Lorz

Received: January 2016

Revised: April 2016

Published: September 2017

Abstract / Introduction Full Text(HTML) Figure(7) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 35C07, 35B35, 35B36, 92C10.

Citation:

Full Text(HTML)

Figure 1. Numerical observations in the case µ < ν

Download: Full-size image PowerPoint slide

Figure 2. Numerical observations in the case µ > ν

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Figure 3. The profile of p for the travelling wave when n has a finite support that coincides with [0, r]

Download: Full-size image PowerPoint slide

Figure 6. Travelling waves of Theorem 4.1 for µ < ν

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253.
[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.
[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.
[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.
[20]	L. Preziosi, Cancer Modelling and Simulation, CRC Press, 2003. doi: 10.1201/9780203494899.
[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.
[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.
[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.
[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.
[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.
[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.