# American Institute of Mathematical Sciences

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

## 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

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.

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 [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

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
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).

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).

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

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)

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

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

 [1] R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339 [2] M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks & Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399 [3] Matthew H. Chan, Peter S. Kim, Robert Marangell. Stability of travelling waves in a Wolbachia invasion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 609-628. doi: 10.3934/dcdsb.2018036 [4] Yangjin Kim, Hans G. Othmer. Hybrid models of cell and tissue dynamics in tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1141-1156. doi: 10.3934/mbe.2015.12.1141 [5] Jan Kelkel, Christina Surulescu. On some models for cancer cell migration through tissue networks. Mathematical Biosciences & Engineering, 2011, 8 (2) : 575-589. doi: 10.3934/mbe.2011.8.575 [6] Kazuhisa Ichikawa. Synergistic effect of blocking cancer cell invasion revealed by computer simulations. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1189-1202. doi: 10.3934/mbe.2015.12.1189 [7] Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056 [8] Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure & Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527 [9] Yuchi Qiu, Weitao Chen, Qing Nie. Stochastic dynamics of cell lineage in tissue homeostasis. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3971-3994. doi: 10.3934/dcdsb.2018339 [10] Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761 [11] Tiffany A. Jones, Lou Caccetta, Volker Rehbock. Optimisation modelling of cancer growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 115-123. doi: 10.3934/dcdsb.2017006 [12] Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867 [13] Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395 [14] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 [15] Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555 [16] Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217 [17] Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809 [18] Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609 [19] Christian Engwer, Markus Knappitsch, Christina Surulescu. A multiscale model for glioma spread including cell-tissue interactions and proliferation. Mathematical Biosciences & Engineering, 2016, 13 (2) : 443-460. doi: 10.3934/mbe.2015011 [20] Paolo Ubezio. Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 323-335. doi: 10.3934/dcdsb.2004.4.323

2018 Impact Factor: 1.38