Stability analysis and simulations of tumor growth model based on system of reaction-diffusion equation in two-dimensions

Ali Sadiq Alabdrabalnabi; Ishtiaq Ali; Ali Sadiq Alabdrabalnabi; Ishtiaq Ali

doi:10.3934/math.2024567

AIMS Mathematics

2024, Volume 9, Issue 5: 11560-11579. doi: 10.3934/math.2024567

Previous Article Next Article

Research article Special Issues

Stability analysis and simulations of tumor growth model based on system of reaction-diffusion equation in two-dimensions

Ali Sadiq Alabdrabalnabi ^,,
Ishtiaq Ali ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa, 31982, Saudi Arabia

Received: 08 January 2024 Revised: 28 February 2024 Accepted: 11 March 2024 Published: 25 March 2024
MSC : 35B35, 35K57, 65N35

In this study, we introduce a novel framework for exploring the dynamics of tumor growth and an evolution model for two-stage carcinogenic mutations in two-dimensions based on a system of reaction-diffusion equations. It is shown theoretically that the system is globally stable in the absence of both delay and diffusion. The inclusion of diffusion does not destabilize the system, while including delay does capture the key elements of how normal cells convert into cancer cells. To further validate these results, several numerical experiments are performed for different parameter values involved in the model equation. These parameter values are chosen in the sense that they have some biological meanings using the steady states of the equilibrium points. For the purpose of simulation, a stable Euler scheme is used for temporal discretization, while a Fourier spectral method is used for space variables, which is a natural choice due to the periodic boundary conditions in the model equation. The numerical simulation results further confirm our theoretical justification.
- tumor growth model,
- reaction-diffusion system with delay,
- equilibrium points,
- stability analysis,
- numerical simulations
Citation: Ali Sadiq Alabdrabalnabi, Ishtiaq Ali. Stability analysis and simulations of tumor growth model based on system of reaction-diffusion equation in two-dimensions[J]. AIMS Mathematics, 2024, 9(5): 11560-11579. doi: 10.3934/math.2024567

Related Papers:

Abstract

In this study, we introduce a novel framework for exploring the dynamics of tumor growth and an evolution model for two-stage carcinogenic mutations in two-dimensions based on a system of reaction-diffusion equations. It is shown theoretically that the system is globally stable in the absence of both delay and diffusion. The inclusion of diffusion does not destabilize the system, while including delay does capture the key elements of how normal cells convert into cancer cells. To further validate these results, several numerical experiments are performed for different parameter values involved in the model equation. These parameter values are chosen in the sense that they have some biological meanings using the steady states of the equilibrium points. For the purpose of simulation, a stable Euler scheme is used for temporal discretization, while a Fourier spectral method is used for space variables, which is a natural choice due to the periodic boundary conditions in the model equation. The numerical simulation results further confirm our theoretical justification.

References

[1]	J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3 Eds., New York: Springer, 2001. http://doi.org/10.1007/b98869
[2]	J. D. Murray, Mathematical Biology: I. An Introduction, Berlin/Heidelberg: Springer, 2002. http://doi.org/10.1007/b98868
[3]	C. M. Beauséjour, A. Krtolica, F. Galimi, M. Narita, S. W. Lowe, P. Yaswen, et al., Reversal of human cellular senescence: Roles of the p53 and p16 pathways, EMBO J., 22 (2003), 4212–4222. http://doi.org/10.1093/emboj/cdg417 doi: 10.1093/emboj/cdg417
[4]	K. Camphausen, M. A. Moses, C. Ménard, M. Sproull, W. Beecken, J. Folkman, et al., Radiation abscopal antitumor effect is mediated through p53, Cancer Res., 63 (2003), 1990–1993. http://doi.org/10.1016/S0360-3016(02)03449-1 doi: 10.1016/S0360-3016(02)03449-1
[5]	Z. Chen, L. C. Trotman, D. Shaffer, H. K. Lin, Z. A. Dotan, M. Niki, et al., Crucial role of p53-dependent cellular senescence in suppression of Pten-deficient tumorigenesis, Nature, 436 (2005), 725–730. http://doi.org/10.1038/nature03918 doi: 10.1038/nature03918
[6]	J. S. Fridman, S. W. Lowe, Control of apoptosis by p53, Oncogene, 22 (2003), 9030–9040. http://doi.org/10.1038/sj.onc.1207116
[7]	M. S. Greenblatt, W. P. Bennett, M. Hollstein, C. C. Harris, Mutations in the p53 tumor suppressor gene: clues to cancer etiology and molecular pathogenesis, Cancer Res., 54 (1994), 4855–4878.
[8]	J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, New York: Springer, 1993. http://doi.org/10.1007/978-1-4612-4342-7
[9]	B. Hat, K. Puszynski, T. Lipniacki, Exploring mechanisms of oscillations in p53 and nuclear factor-$\kappa$B systems, IET Syst. Biol., 3 (2009), 342–355. http://doi.org/10.1049/iet-syb.2008.0156 doi: 10.1049/iet-syb.2008.0156
[10]	E. Michalak, A. Villunger, M. Erlacher, A. Strasser, Death squads enlisted by the tumor suppressor p53, Biochem. Bioph. Res. Co., 331 (2005), 786–798. http://doi.org/10.1016/j.bbrc.2005.03.183 doi: 10.1016/j.bbrc.2005.03.183
[11]	J. C. Arciero, T. L. Jackson, D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, DCDS-B, 4 (2004), 39–58. http://doi.org/10.3934/dcdsb.2004.4.39 doi: 10.3934/dcdsb.2004.4.39
[12]	S. Kruś, Pathological Anatomy (In Polish), Warsaw: PZWL, 2001.
[13]	M. J. Piotrowska, U. Foryś, M. Bodnar, J. Poleszczuk, A simple model of carcinogenic mutations with time delay and diffusion, Math. Biosci. Eng., 10 (2013), 861–872. http://doi.org/10.3934/mbe.2013.10.861 doi: 10.3934/mbe.2013.10.861
[14]	U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations, J. Appl. Anal., 11 (2005), 200–281. http://doi.org/10.1515/JAA.2005.283 doi: 10.1515/JAA.2005.283
[15]	K. R. Swanson, E. C. Alvord, J. D. Murray, A quantitative model for differential motility of gliomas in grey and white matter, Cell Prolif., 33 (2000), 317–329. http://doi.org/10.1046/j.1365-2184.2000.00177.x doi: 10.1046/j.1365-2184.2000.00177.x
[16]	S. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. R. Swanson, M. Pélégrini-Issac, et al., Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Magn. Reson. Med., 54 (2005), 616–624. http://doi.org/10.1002/mrm.20625 doi: 10.1002/mrm.20625
[17]	A. Swan, T. Hillen, J. C. Bowman, A. D. Murtha, A Patient-Specific Anisotropic Diffusion Model for Brain Tumour Spread, Bull. Math. Biol., 80 (2018), 1259–1291. http://doi.org/ 10.1007/s11538-017-0271-8 doi: 10.1007/s11538-017-0271-8
[18]	E. Konukoglu, O. Clatz, B. H. Menze, B. Stieltjes, M. A. Weber, E. Mandonnet, et al., Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations, IEEE Trans. Med. Imaging, 29 (2010), 77–95. http://doi.org/10.1109/TMI.2009.2026413 doi: 10.1109/TMI.2009.2026413
[19]	H. Canuto, Z. Quaterolli, Spectral Methods: Fundamentals in Single Domains, Berlin/Heidelberg: Springer, 2006. http://doi.org/10.1007/978-3-540-30726-6
[20]	D. Gottlieb, S. A. Orszag, Numerical analysis of spectral methods: Theory and applications, Philadelphia: SIAM, 1977. http://doi.org/10.1137/1.9781611970425
[21]	L. N. Trefethen, Spectral methods in MATLAB, Philadelphia: SIAM, 2000. http://doi.org/10.1137/1.9780898719598
[22]	I. Ali, S. U. Khan, A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method, Mathematics, 11 (2023), 1328. http://doi.org/10.3390/math11061328 doi: 10.3390/math11061328
[23]	I. Ali, M. T. Saleem, Spatiotemporal Dynamics of Reaction–Diffusion System and Its Application to Turing Pattern Formation in a Gray–Scott Model, Mathematics, 11 (2023), 1459. http://doi.org/10.3390/math11061459 doi: 10.3390/math11061459
[24]	S. U. Khan, M. Ali, I. Ali, A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis, Adv. Differ. Equ., 1 (2019), 161. http://doi.org/10.1186/s13662-019-2096-2 doi: 10.1186/s13662-019-2096-2
[25]	X. Ma, Y. Wang, X. Zhu, W. Liu, Q. Lan, W. Xiao, A Spectral Method for Two-Dimensional Ocean Acoustic Propagation, J. Mar. Sci. Eng., 9 (2021), 892. http://doi.org/10.3390/jmse9080892 doi: 10.3390/jmse9080892
[26]	X. Ma, Y. Wang, X. Zhu, W. Liu, W. Xiao, Q. Lan, A High-Efficiency Spectral Method for Two-Dimensional Ocean Acoustic Propagation Calculations, Entropy, 23 (2021), 1227. http://doi.org/10.3390/e23091227 doi: 10.3390/e23091227
[27]	H. Tu, Y. Wang, Q. Lan, W. Liu, W. Xiao, S. Ma, A Chebyshev-Tau spectral method for normal modes of underwater sound propagation with a layered marine environment, J. Sound Vib., 492 (2021), 115784. http://doi.org/10.1016/j.jsv.2020.115784 doi: 10.1016/j.jsv.2020.115784
[28]	H. Tu, Y. Wang, Q. Lan, W. Liu, W. Xiao, S. Ma, Applying a Legendre collocation method based on domain decomposition to calculate underwater sound propagation in a horizontally stratified environment, J. Sound Vib., 511 (2021), 116364. http://doi.org/10.1016/j.jsv.2021.116364 doi: 10.1016/j.jsv.2021.116364
[29]	H. Tu, Y. Wang, C. Yang, X. Wang, S. Ma, W. Xiao, et al., A novel algorithm to solve for an underwater line source sound field based on coupled modes and a spectral method, J. Comput. Phys., 468 (2022), 111478. http://doi.org/10.1016/j.jcp.2022.111478 doi: 10.1016/j.jcp.2022.111478
[30]	H. Tu, Y. Wang, W. Liu, X. Ma, W. Xiao, Q. Lan, A Chebyshev spectral method for normal mode and parabolic equation models in underwater acoustics, Math. Probl. Eng., 2020 (2020), 7461314. http://doi.org/10.1155/2020/7461314 doi: 10.1155/2020/7461314
[31]	A. Ali, S. U. Khan, I. Ali, F. U. Khan, On dynamics of stochastic avian influenza model with asymptomatic carrier using spectral method, Math. Methods Appl. Sci., 45 (2022), 8230–8246. http://doi.org/10.1002/mma.8183 doi: 10.1002/mma.8183
[32]	S. U. Khan, I. Ali, Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind, Comput. Appl. Math., 38 (2019), 125. http://doi.org/10.1007/s40314-019-0897-2 doi: 10.1007/s40314-019-0897-2
[33]	I. Ali, M. T. Saleem, Applications of Orthogonal Polynomials in Simulations of Mass Transfer Diffusion Equation Arising in Food Engineering, Symmetry, 15 (2023), 527. http://doi.org/10.3390/sym15020527 doi: 10.3390/sym15020527
[34]	I. Ali, S. U. Khan, Asymptotic Behavior of Three Connected Stochastic Delay Neoclassical Growth Systems Using Spectral Technique, Mathematics, 10 (2022), 3639. http://doi.org/ 10.3390/math10193639 doi: 10.3390/math10193639
[35]	I. Ali, S. U. Khan, Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method, Symmetry, 14 (2022), 1838. http://doi.org/10.3390/sym14091838 doi: 10.3390/sym14091838
[36]	I. Ali, S. U. Khan, Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method, AIMS Mathematics, 8 (2023), 4220–4236. http://doi.org/ 10.3934/math.2023210 doi: 10.3934/math.2023210
[37]	J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, New York: Chapman and Hall/CRC, 2003. http://doi.org/10.1201/9781420036114
[38]	D. Gottlieb, M. Y. Hussaini, S. A. Orszag, Theory and Applications of Spectral Methods, Symposium of Spectral Methods for Partial Differential Equations, 1984.
[39]	K. Dehingia, Y. Alharbi, V. Pandey, A Mathematical Tumor Growth Model for Exploring Saturated Response of M2 Macrophages, Healthcare Anal., 5 (2024), 100306. http://doi.org/10.1016/j.health.2024.100306 doi: 10.1016/j.health.2024.100306
[40]	A. Das, K. Dehingia, H. K. Sarmah, K. Hosseini, K. Sadri, S. Salahshour, Analysis of a Delay-Induced Mathematical Model of Cancer, Adv. Cont. Discr. Mod., 15 (2022), 2022. http://doi.org/10.1186/s13662-022-03688-7 doi: 10.1186/s13662-022-03688-7
[41]	K. Dehingia, K. Hosseini, S. Salahshour, D. Baleanu, A Detailed Study on a Tumor Model with Delayed Growth of Pro-Tumor Macrophages, Int. J. Appl. Comput. Math., 8 (2022), 245. http://doi.org/10.1007/s40819-022-01433-y doi: 10.1007/s40819-022-01433-y
[42]	S. H. Xu, J. Wu, Qualitative Analysis of a Time-Delayed Free Boundary Problem for Tumor Growth with Angiogenesis and Gibbs-Thomson Relation, Math. Biosci. Eng., 16 (2019), 7433–7446. http://doi.org/ 10.3934/mbe.2019372 doi: 10.3934/mbe.2019372
[43]	P. R. Nyarko, M. Anokye, Mathematical Modeling and Numerical Simulation of a Multiscale Cancer Invasion of Host Tissue, AIMS Mathematics, 5 (2020), 3111–3124. http://doi.org/10.3934/math.2020200 doi: 10.3934/math.2020200
[44]	K. Dehingia, S.-W. Yao, K. Sadri, A. Das, H. K. Sarmah, A. Zeb, et al., A Study on Cancer-Obesity-Treatment Model with Quadratic Optimal Control Approach for Better Outcomes, Results Phys., 42 (2022), 105963. http://doi.org/10.1016/j.rinp.2022.105963 doi: 10.1016/j.rinp.2022.105963
[45]	K. Dehingia, H. Sarmah, A. Das, C. Park, K. Hosseini, A Study on a Gene Therapy Model for the Combined Treatment of Cancer, Eurasian J. Math. Comput. Appl., 10 (2022), 15–36. http://doi.org/ 10.3121/cmr.4.3.218 doi: 10.3121/cmr.4.3.218
[46]	F. A. Rihan, H. J. Alsakaji, S. Kundu, O. Mohamed, Dynamics of a Time-Delay Differential Model for Tumour-Immune Interactions with Random Noise, Alexandria Eng. J., 61 (2022), 11913–11923. http://doi.org/10.1016/j.aej.2022.05.027 doi: 10.1016/j.aej.2022.05.027
[47]	H. J. Alsakaji, F. A. Rihan, K. Udhayakumar, F. El Ktaibi, Stochastic Tumor-Immune Interaction Model with External Treatments and Time Delays: An Optimal Control Problem, Math. Biosci. Eng., 20 (2023), 19270–19299. http://doi.org/10.3934/mbe.2023852 doi: 10.3934/mbe.2023852
[48]	O. Bavi, M. Hosseininia, M. Hajishamsaei, M. H. Heydari, Glioblastoma Multiforme Growth Prediction Using a Proliferation-Invasion Model Based on Nonlinear Time-Fractional 2D Diffusion Equation, Chaos, Solitons Fractals, 170 (2023), 113393. http://doi.org/10.1016/j.chaos.2023.113393 doi: 10.1016/j.chaos.2023.113393
[49]	O. Bavi, M. Hosseininia, M. H. Heydari, A Mathematical Model for Precise Predicting Microbial Propagation Based on Solving Variable-Order Fractional Diffusion Equation, Math. Methods Appl. Sci., 46 (2023), 17313–17327. http://doi.org/10.1002/mma.9501 doi: 10.1002/mma.9501
[50]	R. Ahangar, X. B. Lin, Multistage evolutionary model for carcinogenesis mutations, EJDE, 10 (2003), 33–53.
[51]	U. Foryś, Multi-dimensional Lotka-Volterra system for carcinogenesis mutations, Math. Methods Appl. Sci., 32 (2009), 2287–2308. http://doi.org/10.1002/mma.1137 doi: 10.1002/mma.1137
[52]	U. Foryś, B. Zduniak, Two-stage model of carcinogenic mutations with the influence of delays, DCDS-B, 19 (2014), 2501–2519. http://doi.org/10.3934/dcdsb.2014.19.2501 doi: 10.3934/dcdsb.2014.19.2501

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)