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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

* Corresponding author: Zhennan Zhou

Received  February 2018 Revised  July 2018 Published  October 2018

Fund Project: J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135

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.

Citation: Jian-Guo Liu, Min Tang, Li Wang, Zhennan Zhou. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018297
##### References:
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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. [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. [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. [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. [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. [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. [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. [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.
Example 1: expanding disk with constant nutrient and initial data (5.55). Left: plot of solution at time $t = 0.5$ with different $\gamma = 20,\ 40, \ 80$. Here $\Delta r = 0.05$, and $\Delta t = 5\times 10^{-5}$ for $\gamma = 20, \ 40$ and $\Delta t = 2.5\times 10^{-5}$ for $\gamma = 80$. Right: comparison of the numerical solution with $\gamma = 80$ with the analytical solution (5.56) at different times $t = 0.0975$, $t = 0.2975$, $t = 0.4975$. Here the black solid curve is the numerical solution and the red dashed curve is the analytical solution
Example 2: a single annulus with constant nutrient and initial data (5.57). Left: plot of solution at time $t = 0.6$ with different $\gamma = 20,\ 40, \ 80$. Here $\Delta r = 0.05$, and $\Delta t = 2.5\times 10^{-5}$. Right: comparison of the numerical solution with $\gamma = 80$ with the analytical solution (5.58) at different times $t = 0.2494$, $t = 0.4994$, $t = 0.8$. Here we use $\Delta r = 0.025$ and $\Delta t = 6.25 \times 10^{-6}$. The black solid curve is the numerical solution and the red dashed curve is the analytical solution
Example 3: a double annulus with constant nutrient and initial data (5.59). Here we compare the numerical solution (black solid curve) and analytical solution (red dashed curve) at time $t = 0.2495$ (left) and $t = 0.6$ (right). Here we use $\Delta r = 0.025$ and $\Delta t = 5\times 10^{-6}$
Example 4: a 1D in vitro model with linear growing function. Left: plots of $n$ at time $t = 0.5$ with various $\gamma = 20, \ 40, \ 80$. The red curve is the analytical solution (5.61). Here $\Delta x = 0.05$ and $\Delta t = 2.5e-5$
Example 5: a 1D in vivo model with linear growing function. Left: plots of $n$ at time $t = 0.5$ with various $\gamma = 20, \ 40, \ 80$. The red curve is the analytical solution (5.61). Here $\Delta x = 0.05$ and $\Delta t = 2.5\times 10^{-5}$
A comparison of the front propagation speed for 1D in vitro model and in vivo model. The dots represent the position of the right boundary at each time, and the curves are computed via (3.37) and (3.39). Here $\Delta x = 0.05$, $\Delta t = 2.5 \times 10^{-5}$, $\gamma = 80$
A comparison of the front propagation speed in the 2D radial symmetric in vitro model and in vivo model. The dots represent the the position of the right boundary at each time, and the curve are computed via (3.43) and (3.44). Here $\Delta x = 0.05$, $\Delta t = 2.5\times 10^{-5}$, $\gamma = 80$
Plot of $n$ at four different times with initial data (5.63). From left to right, up to down, $t = 0$, $t = 0.0177$, $t = 0.0311$, $t = 0.05$
Plot of $n$ at four different times with initial data (5.64). From left to right, up to down, $t = 0$, $t = 0.0177$, $t = 0.0311$, $t = 0.05$
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