doi: 10.3934/dcdsb.2021092

A free boundary problem of the cancer invasion

School of Mathematical Science, Harbin Engineering University, Harbin, 150001, Heilongjiang, China

Received  November 2020 Revised  January 2021 Published  March 2021

Fund Project: The first author is supported by NSFC grant 11626072

This paper deals with a free boundary problem for the cancer invasion model over a one dimensional habitat in the micro-environment, in which the free boundary represents the spreading front and is caused by tumour cells and acid-mediated. In this problem it is assumed that the tumour cells spread from the given initial region, and the spreading front expands at a speed that is proportional to the tumour cell and acids' population gradient at the front. The main objective is to realize the dynamics/variations of the healthy cells, tumour cells, acid-mediated and the free boundary. We prove a spreading-vanishing dichotomy for this model, namely the tumour cells either successfully spreads to infinity as time tends to infinite at the front, or it fails to establish and dies out in long run while the healthy cells stabilizes at a positive steady-state. The long time behavior of solution and criteria for spreading and vanishing are obtained.

Citation: Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021092
References:
[1]

L. Bianchini and A. Fasano, A model combining acid-mediated tumour invasion and nutrient dynamics, Nonlinear Anal.: RWA, 10 (2009), 1955-1975.  doi: 10.1016/j.nonrwa.2008.03.001.  Google Scholar

[2]

A. BertuzziA. FasanoA. Gandolfi and C. Sinisgalli, Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit, Journal of Theoretical Biology, 262 (2010), 142-150.  doi: 10.1016/j.jtbi.2009.09.024.  Google Scholar

[3]

Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar

[4]

Y. H. Du and Z. M. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.  Google Scholar

[5]

Y. H. DuZ. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.  Google Scholar

[6]

Y. H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Annales De Linstitut Henri Poincare Non Linear Analysis, 32 (2015), 279-305.  doi: 10.1016/j.anihpc.2013.11.004.  Google Scholar

[7]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal, 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

[8]

Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete contin. Dyn. Syst. Ser., 19 (2014), 3105-3132.   Google Scholar

[9]

Y. H. DuH. Matsuzawa and M. L. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal, 46 (2014), 375-396.  doi: 10.1137/130908063.  Google Scholar

[10]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer research, 56 (1996), 5745-5753.   Google Scholar

[11]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equat, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

[12]

J. B. McGillenE. A. GaffneyN. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, Journal of Mathematical Biology, 68 (2014), 1199-1224.  doi: 10.1007/s00285-013-0665-7.  Google Scholar

[13]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Cont. Dyn. Syst. A, 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[14]

K. SmallboneD. J. GavaghanR. A. Gatenby and P. K. Maini, The role of acidity in solid tumour growth and invasion, Journal of Theoretical Biology, 235 (2005), 476-484.  doi: 10.1016/j.jtbi.2005.02.001.  Google Scholar

[15]

R. VenkatasubramanianM. A. Henson and N. S. Forbes, Incorporating energy metabolism into a growth model of multicellular tumor spheroids, Journal of Theoretical Biology, 242 (2006), 440-453.  doi: 10.1016/j.jtbi.2006.03.011.  Google Scholar

[16]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[17]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar

[18]

M. X. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.   Google Scholar

[19]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[20]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[21]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal. TMA., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[22]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model,, Nonlinear Anal.: RWA., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.  Google Scholar

[23]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

show all references

References:
[1]

L. Bianchini and A. Fasano, A model combining acid-mediated tumour invasion and nutrient dynamics, Nonlinear Anal.: RWA, 10 (2009), 1955-1975.  doi: 10.1016/j.nonrwa.2008.03.001.  Google Scholar

[2]

A. BertuzziA. FasanoA. Gandolfi and C. Sinisgalli, Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit, Journal of Theoretical Biology, 262 (2010), 142-150.  doi: 10.1016/j.jtbi.2009.09.024.  Google Scholar

[3]

Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar

[4]

Y. H. Du and Z. M. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.  Google Scholar

[5]

Y. H. DuZ. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.  Google Scholar

[6]

Y. H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Annales De Linstitut Henri Poincare Non Linear Analysis, 32 (2015), 279-305.  doi: 10.1016/j.anihpc.2013.11.004.  Google Scholar

[7]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal, 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

[8]

Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete contin. Dyn. Syst. Ser., 19 (2014), 3105-3132.   Google Scholar

[9]

Y. H. DuH. Matsuzawa and M. L. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal, 46 (2014), 375-396.  doi: 10.1137/130908063.  Google Scholar

[10]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer research, 56 (1996), 5745-5753.   Google Scholar

[11]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equat, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

[12]

J. B. McGillenE. A. GaffneyN. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, Journal of Mathematical Biology, 68 (2014), 1199-1224.  doi: 10.1007/s00285-013-0665-7.  Google Scholar

[13]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Cont. Dyn. Syst. A, 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[14]

K. SmallboneD. J. GavaghanR. A. Gatenby and P. K. Maini, The role of acidity in solid tumour growth and invasion, Journal of Theoretical Biology, 235 (2005), 476-484.  doi: 10.1016/j.jtbi.2005.02.001.  Google Scholar

[15]

R. VenkatasubramanianM. A. Henson and N. S. Forbes, Incorporating energy metabolism into a growth model of multicellular tumor spheroids, Journal of Theoretical Biology, 242 (2006), 440-453.  doi: 10.1016/j.jtbi.2006.03.011.  Google Scholar

[16]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[17]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar

[18]

M. X. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.   Google Scholar

[19]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[20]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[21]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal. TMA., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[22]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model,, Nonlinear Anal.: RWA., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.  Google Scholar

[23]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[1]

Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493

[2]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[3]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403

[4]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256

[5]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[6]

Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373

[7]

Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195

[8]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[9]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021129

[10]

Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021066

[11]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017

[12]

Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021049

[13]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392

[14]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[15]

Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270

[16]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[17]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109

[18]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067

[19]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[20]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (34)
  • HTML views (53)
  • Cited by (0)

Other articles
by authors

[Back to Top]