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March  2022, 27(3): 1323-1343. 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 2022 Early access  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 and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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