| 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 |
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.
| [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. |
show all 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. |
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).
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).
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
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)
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
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] |
Pierre Degond, Sophie Hecht, Nicolas Vauchelet. Incompressible limit of a continuum model of tissue growth for two cell populations. Networks and Heterogeneous Media, 2020, 15 (1) : 57-85. doi: 10.3934/nhm.2020003 |
| [2] |
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 and Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339 |
| [3] |
M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks and Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399 |
| [4] |
Matthew H. Chan, Peter S. Kim, Robert Marangell. Stability of travelling waves in a Wolbachia invasion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 609-628. doi: 10.3934/dcdsb.2018036 |
| [5] |
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 |
| [6] |
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 |
| [7] |
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 |
| [8] |
Caleb Mayer, Eric Stachura. Traveling wave solutions for a cancer stem cell invasion model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5067-5093. doi: 10.3934/dcdsb.2020333 |
| [9] |
Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056 |
| [10] |
Feng Dai, Bin Liu. Global boundedness for a $ \mathit{\boldsymbol{N}} $-dimensional two species cancer invasion haptotaxis model with tissue remodeling. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 311-341. doi: 10.3934/dcdsb.2021044 |
| [11] |
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 and Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527 |
| [12] |
Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092 |
| [13] |
Yuchi Qiu, Weitao Chen, Qing Nie. Stochastic dynamics of cell lineage in tissue homeostasis. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3971-3994. doi: 10.3934/dcdsb.2018339 |
| [14] |
Dmitry Treschev. Travelling waves in FPU lattices. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867 |
| [15] |
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 |
| [16] |
Tiffany A. Jones, Lou Caccetta, Volker Rehbock. Optimisation modelling of cancer growth. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 115-123. doi: 10.3934/dcdsb.2017006 |
| [17] |
Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks and Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021 |
| [18] |
Julien Cividini. Pattern formation in 2D traffic flows. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395 |
| [19] |
Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 |
| [20] |
Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555 |
2021 Impact Factor: 1.398
[Back to Top]
