American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence
  • DCDS-B Home
  • This Issue
  • Next Article
    Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises
July  2019, 24(7): 3011-3035. doi: 10.3934/dcdsb.2018297

Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics

Jian-Guo Liu 1, , Min Tang 2, , Li Wang 3, and  Zhennan Zhou 4,,

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.

Keywords: Tumor growth model, free boundary limit, Hele-Shaw flow model.
Mathematics Subject Classification: 35K55, 35B25, 76D27, 92C50.
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, 2019, 24 (7) : 3011-3035. doi: 10.3934/dcdsb.2018297
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

Figure 1.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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 $\Delta t = 5\times 10^{-6}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  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 $\Delta t = 2.5e-5$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 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 $\Delta t = 2.5\times 10^{-5}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  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 $\Delta x = 0.05$, $\Delta t = 2.5 \times 10^{-5}$, $\gamma = 80$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  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 $\Delta x = 0.05$, $\Delta t = 2.5\times 10^{-5}$, $\gamma = 80$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Plot of $n$ at four different times with initial data (5.63). From left to right, up to down, $t = 0$, $t = 0.0177$, $t = 0.0311$, $t = 0.05$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Plot of $n$ at four different times with initial data (5.64). From left to right, up to down, $t = 0$, $t = 0.0177$, $t = 0.0311$, $t = 0.05$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108
[2]

A. Bernardini, J. Bragard, H. Mancini. Synchronization between two Hele-Shaw Cells. Mathematical Biosciences & Engineering, 2004, 1 (2) : 339-346. doi: 10.3934/mbe.2004.1.339
[3]

Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1941-1974. doi: 10.3934/cpaa.2016023
[4]

Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Mesoscopic model for tumor growth. Mathematical Biosciences & Engineering, 2007, 4 (4) : 687-698. doi: 10.3934/mbe.2007.4.687
[5]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
[6]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
[7]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101
[8]

Shihe Xu, Yinhui Chen, Meng Bai. Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 997-1008. doi: 10.3934/dcdsb.2016.21.997
[9]

Zejia Wang, Suzhen Xu, Huijuan Song. Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2593-2605. doi: 10.3934/dcdsb.2018129
[10]

Elisabeth Logak, Chao Wang. The singular limit of a haptotaxis model with bistable growth. Communications on Pure & Applied Analysis, 2012, 11 (1) : 209-228. doi: 10.3934/cpaa.2012.11.209
[11]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[12]

Shihe Xu, Meng Bai, Fangwei Zhang. Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3535-3551. doi: 10.3934/dcdsb.2017213
[13]

T.L. Jackson. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 187-201. doi: 10.3934/dcdsb.2004.4.187
[14]

J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263
[15]

Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1173-1187. doi: 10.3934/mbe.2015.12.1173
[16]

Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925
[17]

Marcelo E. de Oliveira, Luiz M. G. Neto. Directional entropy based model for diffusivity-driven tumor growth. Mathematical Biosciences & Engineering, 2016, 13 (2) : 333-341. doi: 10.3934/mbe.2015005
[18]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 203-209. doi: 10.3934/dcdss.2020011
[19]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013
[20]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (140)
  • HTML views (550)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences