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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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