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Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics
1. | Department of Mathematics and Department of Physics, Duke University, Durham, NC 27708-0320, USA |
2. | Institute of Natural Sciences and Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, China |
3. | School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA |
4. | Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China |
In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density $ n$ is governed by the Darcy law via the pressure $ p(n) = n^{γ}$. For finite $ γ$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $ γ \to ∞$, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.
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L. N. de Almeida, F. Bubba, B. Perthame and C. Pouchol, Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations, arXiv preprint, 2018, arXiv: 1803.10629. Google Scholar |
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P. Fast and M. J. Shelley,
A moving overset grid method for interface dynamics applied to non-Newtonian Hele-Shaw flow, J. Comput. Phys., 195 (2004), 117-142.
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A. Friedman and B. Hu,
Stability and instability of Lyapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.
doi: 10.1090/S0002-9947-08-04468-1. |
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[18] |
S. Jin and B. Yan,
A class of asymmptotic-preserving schemes for the Fokker-Planck-Landau equation, J. Compt. Phys., 230 (2011), 6420-6437.
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[19] |
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Porous medium equation to Hele-Shaw flow with general initial density, Tran. AMS., 370 (2018), 873-909.
doi: 10.1090/tran/6969. |
[20] |
J.-G. Liu, L. Wang and Z. Zhou,
Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations, Math. Comp., 87 (2018), 1165-1189.
doi: 10.1090/mcom/3250. |
[21] |
J.-G. Liu, M. Tang, L. Wang and Z. Zhou,
An accurate front capturing scheme for tumor growth models with a free boundary limit, J. Compt. Phys., 364 (2018), 73-94.
doi: 10.1016/j.jcp.2018.03.013. |
[22] |
J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini,
Nonlinear modelling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91.
doi: 10.1088/0951-7715/23/1/R01. |
[23] |
B. Merriman, J. K. Bence and S. J. Osher,
Motion of Multiple Junctions: A level set approach, J. Compt. Phys., 112 (1994), 334-363.
doi: 10.1006/jcph.1994.1105. |
[24] |
B. Perthame, Some mathematical models of tumor growth, https://www.ljll.math.upmc.fr/perthame/cours_M2.pdf. Google Scholar |
[25] |
B. Perthame, F. Quirós and J. L. Vázquez,
The Hele-Shaw asymptotics for mechanical models of tumor growth, Arch. Rational Mech. Anal., 212 (2014), 93-127.
doi: 10.1007/s00205-013-0704-y. |
[26] |
B. Perthame, M. Tang and N. Vauchelet,
Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient, Math. Model. Methods Appl. Sci., 24 (2014), 2601-2626.
doi: 10.1142/S0218202514500316. |
[27] |
L. Preziosi and A. Tosin,
Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications, J. Math. Biol., 58 (2009), 625-656.
doi: 10.1007/s00285-008-0218-7. |
[28] |
J. Ranft, M. Basan, J. Elgeti, J.F. Joanny, J. Prost and F. Jülicher, Fluidization of tissues by cell division and apaptosis, Proc. Natl. Acad. Sci., 107 (2010), 20863-20868. Google Scholar |
[29] |
T. Roose, S. J. Chapman and P. K. Maini,
Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007), 179-208.
doi: 10.1137/S0036144504446291. |
[30] |
S. J. Ruuth,
Efficient algorithms for diffusion-generated motion by mean curvature, J. Comput. Phy., 144 (1998), 603-625.
doi: 10.1006/jcph.1998.6025. |
[31] |
T. L. Stepien, E. M. Rutter and Y. Kuang,
A data-motivated density-dependent diffusion model of in vitro glioblastoma growth, Math. Biosc. Eng., 12 (2015), 1157-1172.
doi: 10.3934/mbe.2015.12.1157. |
[32] |
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, Chin. Ann. Math. Ser. B, 34 (2013), 295-318.
doi: 10.1007/s11401-013-0761-4. |
[33] |
X. Xu, D. Wang and X. Wang,
An efficient threshold dynamics method for wetting on rough surfaces, J. Comput. Phy., 330 (2017), 510-528.
doi: 10.1016/j.jcp.2016.11.008. |
show all references
References:
[1] |
D. G. Aronson, L. A. Caffarelli and S. Kamin,
How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal., 14 (1983), 639-658.
doi: 10.1137/0514049. |
[2] |
N. Bellomo, N. K. Li and P. K. Maini,
On the foundations of cancer modeling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 4 (2008), 593-646.
doi: 10.1142/S0218202508002796. |
[3] |
M. Bertsch, R. Dal Passo and M. Mimura,
A free boundary problem arising in a simplied tumor growth model of contact inhibition, Interfaces Free Bound., 12 (2010), 235-250.
doi: 10.4171/IFB/233. |
[4] |
H. Byrne and D. Drasdo,
Individual based and continuum models of growing cell populations: A comparison, J. Math. Biol., 58 (2009), 657-687.
doi: 10.1007/s00285-008-0212-0. |
[5] |
C. Chatelain, T. Balois, P. Ciarletta and M. Ben Amar, Emergence of microstructural patterns in skin cancer: A phase separation analysis in a binary mixture, New J. Phys., 13 (2011), 115013. Google Scholar |
[6] |
K. Craig, I. Kim and Y. Yao,
Congested aggregation via newtonian interaction, Arch. Rational Mech. Anal., 227 (2018), 1-67.
doi: 10.1007/s00205-017-1156-6. |
[7] |
L. N. de Almeida, F. Bubba, B. Perthame and C. Pouchol, Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations, arXiv preprint, 2018, arXiv: 1803.10629. Google Scholar |
[8] |
A. J. DeGregoria and L. W. Schwartz,
A boundary-integral method for two-phase displacement in Hele-Shaw cells, J. Fluid Mech., 164 (1986), 383-400.
doi: 10.1017/S0022112086002604. |
[9] |
S. Esedoglu, S. Ruuth and R. Tsai,
Threshold dynamics for high order geometric motions, Interfaces Free Bound., 10 (2008), 263-282.
doi: 10.4171/IFB/189. |
[10] |
P. Fast and M. J. Shelley,
A moving overset grid method for interface dynamics applied to non-Newtonian Hele-Shaw flow, J. Comput. Phys., 195 (2004), 117-142.
doi: 10.1016/j.jcp.2003.08.034. |
[11] |
R. P. Fedkiw, B. Merriman and S. Osher,
Simplified discretization of systems of hyperbolic conservation laws containing advection equations, J. Comput. Phys., 157 (2000), 302-326.
doi: 10.1006/jcph.1999.6379. |
[12] |
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. |
[13] |
A. Friedman,
Mathematical analysis and challenges arising from models of tumor growth, Math. Model. Methods Appl. Sci., 17 (2007), 1751-1772.
doi: 10.1142/S0218202507002467. |
[14] |
A. Friedman and B. Hu,
Stability and instability of Lyapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.
doi: 10.1090/S0002-9947-08-04468-1. |
[15] |
H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. Google Scholar |
[16] |
T. Y. Hou, Z. Li, S. Osher and H. Zhao,
A hybrid method for moving interface problems with application to the Hele-Shaw flow, J. Comput. Phys., 134 (1997), 236-252.
doi: 10.1006/jcph.1997.5689. |
[17] |
S. Jin and L. Wang,
An asymptotic-preserving scheme for the Vlasov-Poisson-Fokker-Planck system in the high field regime, Acta Math. Sci., 31 (2011), 2219-2232.
doi: 10.1016/S0252-9602(11)60395-0. |
[18] |
S. Jin and B. Yan,
A class of asymmptotic-preserving schemes for the Fokker-Planck-Landau equation, J. Compt. Phys., 230 (2011), 6420-6437.
doi: 10.1016/j.jcp.2011.04.002. |
[19] |
I. Kim and N. Požár,
Porous medium equation to Hele-Shaw flow with general initial density, Tran. AMS., 370 (2018), 873-909.
doi: 10.1090/tran/6969. |
[20] |
J.-G. Liu, L. Wang and Z. Zhou,
Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations, Math. Comp., 87 (2018), 1165-1189.
doi: 10.1090/mcom/3250. |
[21] |
J.-G. Liu, M. Tang, L. Wang and Z. Zhou,
An accurate front capturing scheme for tumor growth models with a free boundary limit, J. Compt. Phys., 364 (2018), 73-94.
doi: 10.1016/j.jcp.2018.03.013. |
[22] |
J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini,
Nonlinear modelling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91.
doi: 10.1088/0951-7715/23/1/R01. |
[23] |
B. Merriman, J. K. Bence and S. J. Osher,
Motion of Multiple Junctions: A level set approach, J. Compt. Phys., 112 (1994), 334-363.
doi: 10.1006/jcph.1994.1105. |
[24] |
B. Perthame, Some mathematical models of tumor growth, https://www.ljll.math.upmc.fr/perthame/cours_M2.pdf. Google Scholar |
[25] |
B. Perthame, F. Quirós and J. L. Vázquez,
The Hele-Shaw asymptotics for mechanical models of tumor growth, Arch. Rational Mech. Anal., 212 (2014), 93-127.
doi: 10.1007/s00205-013-0704-y. |
[26] |
B. Perthame, M. Tang and N. Vauchelet,
Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient, Math. Model. Methods Appl. Sci., 24 (2014), 2601-2626.
doi: 10.1142/S0218202514500316. |
[27] |
L. Preziosi and A. Tosin,
Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications, J. Math. Biol., 58 (2009), 625-656.
doi: 10.1007/s00285-008-0218-7. |
[28] |
J. Ranft, M. Basan, J. Elgeti, J.F. Joanny, J. Prost and F. Jülicher, Fluidization of tissues by cell division and apaptosis, Proc. Natl. Acad. Sci., 107 (2010), 20863-20868. Google Scholar |
[29] |
T. Roose, S. J. Chapman and P. K. Maini,
Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007), 179-208.
doi: 10.1137/S0036144504446291. |
[30] |
S. J. Ruuth,
Efficient algorithms for diffusion-generated motion by mean curvature, J. Comput. Phy., 144 (1998), 603-625.
doi: 10.1006/jcph.1998.6025. |
[31] |
T. L. Stepien, E. M. Rutter and Y. Kuang,
A data-motivated density-dependent diffusion model of in vitro glioblastoma growth, Math. Biosc. Eng., 12 (2015), 1157-1172.
doi: 10.3934/mbe.2015.12.1157. |
[32] |
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, Chin. Ann. Math. Ser. B, 34 (2013), 295-318.
doi: 10.1007/s11401-013-0761-4. |
[33] |
X. Xu, D. Wang and X. Wang,
An efficient threshold dynamics method for wetting on rough surfaces, J. Comput. Phy., 330 (2017), 510-528.
doi: 10.1016/j.jcp.2016.11.008. |









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