March  2020, 15(1): 57-85. doi: 10.3934/nhm.2020003

Incompressible limit of a continuum model of tissue growth for two cell populations

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

2. 

Francis Crick Institute, 1 Midland Rd, London NW1 1AT, UK

3. 

LAGA, Universite Paris 13, 99 avenue Jean-Baptiste Clement, 93430 Villetaneuse, France

* Corresponding author: Pierre Degond

Received  January 2019 Revised  September 2019 Published  December 2019

This paper investigates the incompressible limit of a system modelling the growth of two cells population. The model describes the dynamics of cell densities, driven by pressure exclusion and cell proliferation. It has been shown that solutions to this system of partial differential equations have the segregation property, meaning that two population initially segregated remain segregated. This work is devoted to the incompressible limit of such system towards a free boundary Hele Shaw type model for two cell populations.

Citation: 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
References:
[1]

R. Araujo and D. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling,, D.L.S. Bull. Math. Biol., 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.

[2]

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

[3]

M. BertschM. E. GurtinD. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: the effect of a sedentary colony,, Q. Appl. Math., 19 (1984), 1-12.  doi: 10.1007/BF00275928.

[4]

M. BertschM. Gurtin and D. Hilhorst, On a degenerate diffusion equation of the form c(z)t = $\phi$(zx)x with application to population dynamics, J. Differ. Equ., 67 (1987), 56-89.  doi: 10.1016/0022-0396(87)90139-2.

[5]

M. BertschM. Gurtin and D. Hilhorst, On interacting populations that disperse to avoid crowding: The case of equal dispersal velocities,, Nonlinear Anal. Theory Methods Appl., 11 (1987), 493-499.  doi: 10.1016/0362-546X(87)90067-8.

[6]

M. BertschD. HilhorstH. Izuhara and M. Mimura, A non linear parabolic-hyperbolic system for contact inhibition of cell growth,, Differ. Equ. Appl., 4 (2010), 137-157.  doi: 10.7153/dea-04-09.

[7]

D. BreschT. ColinE. GrenierB. Ribba and O. Saut, Computational modeling of solid tumor growth: The avascular stage,, SIAM J. Sci. Comput., 32 (2010), 2321-2344.  doi: 10.1137/070708895.

[8]

F. Bubba, B. Perthame, C. Pouchol and M. Schmidtchen, Hele-shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues, preprint, arXiv: 1901.01692.

[9]

S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration,, J. Math. Biol., 16 (1983), 181-198.  doi: 10.1007/BF00276056.

[10]

H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison,, J. Math. Biol., 58 (2008), 657-687.  doi: 10.1007/s00285-008-0212-0.

[11]

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

[12]

A. J. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes and segregation in reaction-(cross-) diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695–5718, arXiv: 1711.05434. doi: 10.1137/17M1158379.

[13]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure,, Commun. Comput. Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.

[14]

M. ChaplainL. Graziano and L. Preziozi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math Med Biol., 23 (2006), 197-229.  doi: 10.1093/imammb/dql009.

[15]

A. ChertockP. DegondS. Hecht and J.-P. Vincent, Incompressible limit of a continuum model of tissue growth with segregation for two cell populations, Math. Biosci. Eng., 16 (2019), 5804-5835. 

[16]

P. Ciarletta, L. Foret and M. Ben Amar, The radial growth phase of malignant melanoma: Multi-phase modelling, numerical simulations and linear stability analysis, J. R. Soc. Interface, 8 (2011), 345–368, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3030817/. doi: 10.1098/rsif.2010.0285.

[17]

S. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655.  doi: 10.1080/03605300701743848.

[18]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Am. Math. Soc., 360 (2008), 5291-5342.  doi: 10.1090/S0002-9947-08-04468-1.

[19]

G. Galiano, On a cross-diffusion population model deduced from mutation and splitting of a single species, Comput. Math. Appl., 64 (2012), 1927–1936, URL http://www.sciencedirect.com/science/article/pii/S0898122112002507. doi: 10.1016/j.camwa.2012.03.045.

[20]

G. Galiano, S. Shmarev and J. Velasco, Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem, Discrete Contin. Dyn. Syst., 35 (2015), 1479–1501, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10564. doi: 10.3934/dcds.2015.35.1479.

[21]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.

[22]

P. GwiazdaB. Perthame and A. Swierczewska-Gwiazda, A two species hyperbolic-parabolic model of tissue growth, Comm. Partial Differential Equations, 44 (2019), 1605-1618.  doi: 10.1080/03605302.2019.1650064.

[23]

S. Hecht and N. Vauchelet, Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint, Commun. Math. Sci., 15 (2017), 1913–1932, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5669502/. doi: 10.4310/CMS.2017.v15.n7.a6.

[24]

I. Kim and N. Požár, Porous medium equation to Hele-Shaw flow with general initial density, Trans. Amer. Math. Soc., 370 (2018), 873-909.  doi: 10.1090/tran/6969.

[25]

A. J. Lotka, Contribution to the theory of periodic reactions,, J. Chem. Biol. Phys., 14 (1909), 271-274.  doi: 10.1021/j150111a004.

[26]

A. MelletB. Perthame and F. Quirós, A Hele-Shaw problem for tumor growth,, J. Funct. Anal., 273 (2017), 3061-3093.  doi: 10.1016/j.jfa.2017.08.009.

[27]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.  doi: 10.1007/BF00276035.

[28]

B. PerthameF. QuiròsM. Tang and N. Vauchelet, Derivation of a Hele-Shaw type system from a cell model with active motion, Interfaces Free Bound., 16 (2014), 489-508.  doi: 10.4171/IFB/327.

[29]

B. PerthameF. Quirós and J. L. Vázquez, The Hele–Shaw asymptotics for mechanical models of tumor growth, Arch. Ration. Mech. Anal., 212 (2014), 93-127.  doi: 10.1007/s00205-013-0704-y.

[30]

B. Perthame and N. Vauchelet, Incompressible limit of a mechanical model of tumour growth with viscosity, Philos. Trans. Roy. Soc. A, Math. Phys. Eng. Sci., 373 (2015), 20140283, 16pp, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4535270/. doi: 10.1098/rsta.2014.0283.

[31]

J. RanftM. BasanJ. ElgetiJ.-F. JoannyJ. Prost and F. Jülicher, Fluidization of tissues by cell division and apoptosis, Proc. Natl. Acad. Sci., 107 (2010), 20863-20868.  doi: 10.1073/pnas.1011086107.

[32]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. of Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[33]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

show all references

References:
[1]

R. Araujo and D. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling,, D.L.S. Bull. Math. Biol., 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.

[2]

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

[3]

M. BertschM. E. GurtinD. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: the effect of a sedentary colony,, Q. Appl. Math., 19 (1984), 1-12.  doi: 10.1007/BF00275928.

[4]

M. BertschM. Gurtin and D. Hilhorst, On a degenerate diffusion equation of the form c(z)t = $\phi$(zx)x with application to population dynamics, J. Differ. Equ., 67 (1987), 56-89.  doi: 10.1016/0022-0396(87)90139-2.

[5]

M. BertschM. Gurtin and D. Hilhorst, On interacting populations that disperse to avoid crowding: The case of equal dispersal velocities,, Nonlinear Anal. Theory Methods Appl., 11 (1987), 493-499.  doi: 10.1016/0362-546X(87)90067-8.

[6]

M. BertschD. HilhorstH. Izuhara and M. Mimura, A non linear parabolic-hyperbolic system for contact inhibition of cell growth,, Differ. Equ. Appl., 4 (2010), 137-157.  doi: 10.7153/dea-04-09.

[7]

D. BreschT. ColinE. GrenierB. Ribba and O. Saut, Computational modeling of solid tumor growth: The avascular stage,, SIAM J. Sci. Comput., 32 (2010), 2321-2344.  doi: 10.1137/070708895.

[8]

F. Bubba, B. Perthame, C. Pouchol and M. Schmidtchen, Hele-shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues, preprint, arXiv: 1901.01692.

[9]

S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration,, J. Math. Biol., 16 (1983), 181-198.  doi: 10.1007/BF00276056.

[10]

H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison,, J. Math. Biol., 58 (2008), 657-687.  doi: 10.1007/s00285-008-0212-0.

[11]

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

[12]

A. J. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen, Splitting schemes and segregation in reaction-(cross-) diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695–5718, arXiv: 1711.05434. doi: 10.1137/17M1158379.

[13]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure,, Commun. Comput. Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.

[14]

M. ChaplainL. Graziano and L. Preziozi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math Med Biol., 23 (2006), 197-229.  doi: 10.1093/imammb/dql009.

[15]

A. ChertockP. DegondS. Hecht and J.-P. Vincent, Incompressible limit of a continuum model of tissue growth with segregation for two cell populations, Math. Biosci. Eng., 16 (2019), 5804-5835. 

[16]

P. Ciarletta, L. Foret and M. Ben Amar, The radial growth phase of malignant melanoma: Multi-phase modelling, numerical simulations and linear stability analysis, J. R. Soc. Interface, 8 (2011), 345–368, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3030817/. doi: 10.1098/rsif.2010.0285.

[17]

S. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655.  doi: 10.1080/03605300701743848.

[18]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Am. Math. Soc., 360 (2008), 5291-5342.  doi: 10.1090/S0002-9947-08-04468-1.

[19]

G. Galiano, On a cross-diffusion population model deduced from mutation and splitting of a single species, Comput. Math. Appl., 64 (2012), 1927–1936, URL http://www.sciencedirect.com/science/article/pii/S0898122112002507. doi: 10.1016/j.camwa.2012.03.045.

[20]

G. Galiano, S. Shmarev and J. Velasco, Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem, Discrete Contin. Dyn. Syst., 35 (2015), 1479–1501, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10564. doi: 10.3934/dcds.2015.35.1479.

[21]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.

[22]

P. GwiazdaB. Perthame and A. Swierczewska-Gwiazda, A two species hyperbolic-parabolic model of tissue growth, Comm. Partial Differential Equations, 44 (2019), 1605-1618.  doi: 10.1080/03605302.2019.1650064.

[23]

S. Hecht and N. Vauchelet, Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint, Commun. Math. Sci., 15 (2017), 1913–1932, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5669502/. doi: 10.4310/CMS.2017.v15.n7.a6.

[24]

I. Kim and N. Požár, Porous medium equation to Hele-Shaw flow with general initial density, Trans. Amer. Math. Soc., 370 (2018), 873-909.  doi: 10.1090/tran/6969.

[25]

A. J. Lotka, Contribution to the theory of periodic reactions,, J. Chem. Biol. Phys., 14 (1909), 271-274.  doi: 10.1021/j150111a004.

[26]

A. MelletB. Perthame and F. Quirós, A Hele-Shaw problem for tumor growth,, J. Funct. Anal., 273 (2017), 3061-3093.  doi: 10.1016/j.jfa.2017.08.009.

[27]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.  doi: 10.1007/BF00276035.

[28]

B. PerthameF. QuiròsM. Tang and N. Vauchelet, Derivation of a Hele-Shaw type system from a cell model with active motion, Interfaces Free Bound., 16 (2014), 489-508.  doi: 10.4171/IFB/327.

[29]

B. PerthameF. Quirós and J. L. Vázquez, The Hele–Shaw asymptotics for mechanical models of tumor growth, Arch. Ration. Mech. Anal., 212 (2014), 93-127.  doi: 10.1007/s00205-013-0704-y.

[30]

B. Perthame and N. Vauchelet, Incompressible limit of a mechanical model of tumour growth with viscosity, Philos. Trans. Roy. Soc. A, Math. Phys. Eng. Sci., 373 (2015), 20140283, 16pp, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4535270/. doi: 10.1098/rsta.2014.0283.

[31]

J. RanftM. BasanJ. ElgetiJ.-F. JoannyJ. Prost and F. Jülicher, Fluidization of tissues by cell division and apoptosis, Proc. Natl. Acad. Sci., 107 (2010), 20863-20868.  doi: 10.1073/pnas.1011086107.

[32]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. of Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[33]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

Figure 1.  Densities $ n_1 $ (blue), $ n_2 $ (red) and pressure $ p $ as functions of position $ x $ at different times: a) $ t = 0 $, (b) $ t = 0.1 $, (c) $ t = 0.3 $, (d) $ t = 0.6 $, (e) $ t = 1 $ and (f) $ t = 2 $; in the case $ \epsilon = 1 $ with the initial densities and growth rate defined by (44)-(45)
Figure 2.  Densities $ n_1 $ (blue), $ n_2 $ (red) and pressure $ p $ as functions of position $ x $ at different times: (ⅰ) $ t = 0.5 $, (ⅱ) $ t = 1 $, (ⅲ) $ t = 1.5 $; and for different values of $ \epsilon $: (a) $ \epsilon = 1 $, (b) $ \epsilon = 0.1 $, (c) $ \epsilon = 0.01 $, (d) $ \epsilon = 0.001 $, (e) Hele-Shaw system
Figure 3.  Densities $ n_1 $ (blue), $ n_2 $ (red) and $ p $ (black) as functions of position $ x $ for different growth function at different times: (ⅰ) $ t = 0.3 $, (ⅱ) $ t = 0.6 $, (ⅲ) $ t = 1 $, (ⅳ) $ t = 1.5 $
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