Two Numerical Solutions for Solving a Mathematical Model of the Avascular Tumor Growth

Year 2021, Volume: 5 Issue: 3, 156 - 164, 20.09.2021

Sıla Övgü Korkut Uysal , Neslişah İmamoğlu Karabaş Yasemin Başbınar

https://doi.org/10.30621/jbachs.957601

Abstract

Objectives: Cancer which is one of the most challenging health problems overall the world is composed of various processes: tumorigenesis, angiogenesis, and metastasis. Attempting to understand the truth behind this complicated disease is one of the common objectives of many experts and researchers from different fields. To provide deeper insights any prognostic and/or diagnostic scientific contribution to this topic is so crucial. In this study, the avascular tumor growth model which is the earliest stage of tumor growth is taken into account from a mathematical point of view. The main aim is to solve the mathematical model of avascular tumor growth numerically.
Methods: This study has focused on the numerical solution of the continuum mathematical model of the avascular tumor growth described by Sharrett and Chaplin. Unlike the existing recent literature, the study has focused on the methods for the temporal domain. To obtain the numerical schemes the central difference method has been used in the spatial coordinates. This discretization technique has reduced the main partial differential equation into an ordinary differential equation which will be solved successively by two alternative techniques: the 4th order Runge-Kutta method (RK4) and the three-stage strongly-stability preserving Runge-Kutta method (SSP-RK3).
Results: The model has been solved by the proposed methods. The numerical results are discussed in both mathematical and biological angles. The biological compatibility of the methods is depicted in various figures. Besides biological outputs, the accuracies of the methods have been listed from a mathematical point of view. Furthermore, the rate of convergence of the proposed methods has also been discussed computationally.
Conclusion: All recorded results are evidence that the proposed schemes are applicable for solving such models. Moreover, all exhibited figures have proved the biological compatibility of the methods. It is observed that the quiescent cells which are one of the most mysterious cells in clinics tend to become proliferative for the selected parameters.

Keywords

Avascular Tumor Growth, Numerical Simulation, Mathematical Biology, Three-stage Strongly-stability Preserving Runge-Kutta method, 4th-order Runge-Kutta Method

References

• Byrne HM, Alarcon T, Owen MR, Webb SD, and Maini PK. Modelling aspects of cancer dynamics: a Review. Phil. Trans. R. Soc. A. 2006; 364;1563–1578. http://doi.org/10.1098/rsta.2006.1786
• Jiang Y, Pjesivac-Grbovic J, Cantrell C, Freyer JP. A multiscale model for avascular tumor growth. Biophysical Journal 2005; 89(6); 3884–3894. https://doi.org/10.1529/biophysj.105.060640
• Bagherpoorfard M, Soheili A. A numerical method based on the moving mesh for the solving of a mathematical model of the avascular tumor growth. Computational Methods for Differential Equations 2021; 9(2): 327-346. https://doi.org/10.22034/cmde.2020.31455.1472
• Kunz-Schughart LA, Kreutz M, Knuechel R. Multicellular spheroids: a three-dimensional in vitro culture system to study tumour biology. International Journal of Experimental Pathology, 1998; 79: 1-23. https://doi.org/10.1046/j.1365-2613.1998.00051.x
• Thomlinson RH, Gray LH. The histological structure of some human lung cancers and possible implications for radiotherapy. Br J Cancer 1955; 9: 539–549.
• Burton AC. Rate of growth of solid tumours as a problem of diffusion. Growth 1966; 30: 157–176.
• Greenspan HP: Models for the growth of a solid tumor by diffusion. Stud Appl Math 1972; 52: 317–340.
• Chaplin MA, Benson DL, Maini PK. Nonlinear diffusion of a growth inhibitory factor in multicell spheroids. Math. Biosci. 1994; 121(1): 1-13. https://doi.org/10.1016/0025-5564(94)90029-9
• Byrne HM, Chaplin MAJ. Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 1996; 135(2): 187-216. https://doi.org/10.1016/0025-5564(96)00023-5
• Ward JP, King JR. Mathematical modelling of avascular-tumour growth II: modelling growth saturation. Math. Med. Biol. 1997; 14(1): 39-69.
• Sherratt JA, Chaplin MA. A new mathematical model for avascular tumor growth. J. Math. Biol. 2001; 43(4): 291-312. https://doi.org/10.1007/s002850100088
• Unsal S, Acar A, Itik M, et al. Personalized Tumor Growth Prediction Using Multiscale Modeling. Journal of Basic and Clinical Health Sciences 2020; 4(3), 347-363. https://doi.org/10.30621/jbachs.2020.1245
• Roose T, Chapman S, Maini P. Mathematical Models of Avascular Tumor Growth. SIAM Review 2007; 49(2): 179- 208.
• Kuang Y, Nagy JD, Eikenberry SE. Introduction to Mathematical Oncology (Chapman & Hall/CRC Mathematical and Computational Biology), 1st ed. CRC Press; 2016.
• Mahmood MS, Mahmood S, Dobrota D. A numerical algorithm for avascular tumor growth model. Mathematics and Computers in Simulation 2010; 80(6): 1269-1277. https://doi.org/10.1016/j.matcom.2009.09.011.
• Said NM, Ibrahim A, Alias N, Numerical Simulation of Hypoxic Cell Regulation in Avascular Tumor Growth. AIP Conference Proceedings 1522, 400-404 (2013) https://doi.org/10.1063/1.4801153
• Darbyshire PM. A System of Coupled Nonlinear Partial Differential Equations Describing Avascular Tumour Growth Are Solved Numerically Using Parallel Programming to Assess Computational Speedup. Computational Biology and Bioinformatics. 2015; 3(5): 65-73. https://doi.org/10.11648/j.cbb.20150305.11
• Amoddeo A. Modeling Avascular Tumor Growth: Approach with an Adaptive Grid Numerical Technique. Journal of Multiscale Modelling 2018; 9(3): Article Id: 1840002. http://dx.doi.org/10.1142/S1756973718400024
• Bagherpoorfard M, Soheili AR. A numerical method based on the moving mesh for the solving of a mathematical model of the avascular tumor growth. Computational methods for Differential Equations 2021; 9(2): 327-346.
• Ali A, Hussain M, Ghaffar A et al. Numerical simulations and analysis for mathematical model of avascular tumor growth using Gompertz growth rate function. Alexandria Engineering Journal 2021; 60(4): 3731-3740. https://doi.org/10.1016/j.aej.2021.02.040
• Durran DR. Numerical Methods for Fluid Dynamics. 2nd ed. New York, NY: Springer; 2010.