Mathematical models of malignant tumour

I. V. Zhukova ; E. P. Kolpak

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2014), pp. 5-18

Résumé

The mathematical model of a malignant tumor, representing an initial-boundary problem for the system of three differential equations in partial derivatives, has been developed. In models there are three types of cells that communicate among themselves, dividing, normal and dead. The inhibiting effect of cells on each other, that the growth of continuously dividing cells is accompanied by destruction of normal cells and by formation the dead cells, is taken into account. Kinetic functions are constructed on the principle of pair interactions. In the developed model the growth of tumor cells is accompanied by displacement of normal by dead cells. In the model of immune response the estimation of the critical masses, in excess of which the decision of problem can become an increasing unrestrictedly, has been given. Mathematical analysis of stability of stationary solutions is based on the first method of Lyapunov. In the linear approximation, the estimation of the growth rate of the tumor, growing in the form of thread, has been given. It is proved that the solution of a nonlinear boundary problem can be represented as a spreading wave, the minimum velocity of wave propagation was found. The algorithm of solving a nonlinear boundary value problem for a system of differential equations based on the finite-difference approximation of differential operators has been developed. The numerical experiments were put and a comparison of their results with the results of analytical research was done. Bibliogr. 33. Il. 1.

Détail

Keywords: mathematical modeling, differential equations, boundary value problems, numerical methods, tumor.

Bibliographie

[1] Moiseenko V. M., Blinov N. N., Hanson K. P., “Biotherapy in malignant tumors”, Rus. Oncology. J., 1997, no. 5, 57–59 | MR

[2] Kolbin A. S., Kurylev A. A., Pavlysh A. V., Proskurin M. A., Balykina Yu. E., “Scientific analysis of outcomes in Oncology. Features pharmacoeconomic expertise”, Medical equipment. Evaluation and selection, 2:8 (2012), 87–93 | Zbl

[3] V. M. Moiseenko, A. F. Urmancheev, K. P. Hanson (eds.), Lectures on basic and clinical oncology, “Publishing house H-L”, St. Petersburg, 2004, 704 pp.

[4] Stepanova N. V., Chernavsky D. S., “A mathematical model of immunosuppresion by malignant growth”, Mathematical modeling, 1983, 379–386 | Zbl

[5] Astanin S. A., Kolobov A. V., Lobanov A. I., Pimenova T. P., Polezhaev A. A., Solyanik G. I., “The Influence of spatial heterogeneity of the environment on the growth and invasion of the tumor. Analysis of methods of mathematical modeling”, Medicine in the mirror information, eds. O. M. Belotserkovsky, A. S. Kholodov, Science, M., 2008, 188–223

[6] Astanin S. A., Lobanov A. I., “Three-dimensional model of growth revascularisations tumors in the tissue”, Math. Computer. Education, v. 1, 2005, 759–769

[7] Kolobov A. V., Anashkina A. A., Gubernov V. V., Polezhaev A. A., “Mathematical model of tumor growth given the dichotomy migration and proliferation”, Computer research and modeling, 1:4 (2009), 415–422 | Zbl

[8] Kolobov A. V., Gubernov V. V., Polezhaev A. A., “Sustained waves in the growth model of invasive tumors”, Biophysics, 54:2 (2009), 334–342

[9] Kolobov A. V., Anashkina A. A., Gubernov V. V., Polezhaev A. A., “The effect of random mobility of malignant cells on the sustainability front tumors”, Computer research and modeling, 1:2 (2009), 225–232 | Zbl

[10] Kolobov A. V., Gubernov V. V., Polezhaev A. A., Solyanik G. I., “The role of transport processes in the simulation of tumor growth”, Dynamic models of processes in cells and subcellular structures, eds. G. Yu. Riznichenko, A. B. Rubin, Scientific. issled. centre «Regular and chaotic dynamics». Izhevsk Institute of computer science, M.–Izhevsk, 2010, 355–374

[11] Byrne H. M., Breward C. J. W., Lewis C. E., “The role of cell-cell interactions in a two-phase model for avascular tumour growth”, J. of Mathem. Biology, 45:2 (2001), 125–131 | MR

[12] Chaplain M. A. J., Sherratt J. A., “A new mathematical model for avascular tumor growth”, J. of Mathem. Biology, 43:4 (2000), 291–312 | MR

[13] Mohammed Shuker Mahmood, Silvia Mahmood, Dusan Dobrota, “Formulation and numerical simulations of a continuum model of avascular tumor growth”, Mathematical Biosciences, 231:2 (2011), 159–171 | DOI | MR | Zbl

[14] Akoev I. G., Biophysics knows cancer, Nauka, M., 1988, 160 pp.

[15] Dolgih V. T., Tumor growth, textbook, Phoenix, Rostov on/D., 2007, 160 pp.

[16] Zhimulev I. F., General and molecular genetics, Publishing house of Novosibirsk University, Novosibirsk, 2002, 459 pp.

[17] Vasilyev Yu. M., Biology of malignant growth, Nauka, M., 1965, 180 pp.

[18] Emmanuel N. M., Kinetics of experimental tumor processes, Nauka, M., 1977, 419 pp.

[19] Balykin Yu. E. Kolpak E. P., “Mathematical models of thyroid follicle functioning”, Vestnik St. Petersburg University, ser. 10: Applied mathematics, computer science, control processes, 2013, no. 3, 20–31

[20] Gorbunova E. A., Kolpak E. P., “Mathematical model of solitary population”, Vestnik St. Petersburg University, ser. 10: Applied mathematics, computer science, control processes, 2012, no. 4, 18–30

[21] Zhukova I. V., Kolpak E. P., “Mathematical model of solid tumor”, Natural and mathematical science in the modern world, 2013, no. 13, 18–25

[22] Slepkov V. A., Suhovolskiy V. G., Hhlebopros Z. G., “Population dynamics in the modelling of the growth of cancerous tumors”, Biophysics, 52:4 (2007), 733–740

[23] Ambrosi D., Mollica F., “On the mechanics of a growing tumor”, Intern. journal of Engineering Science, 40:12 (2002), 1297–1316 | DOI | MR | Zbl

[24] Gerlee P., Anderson A. R. A., “Evolution of cell motility in an individual-based model of tumour growth”, J. of Theoretical Biology, 259:1 (2009), 67–83 | DOI | MR

[25] Martin N. K., Gaffney E. A., Gatenby R. A., Maini P. K., “Tumour-stromal interactions in acid-mediated invasion: a mathematical model”, J. of Theoretical Biology, 267:3 (2010), 461–470 | DOI | MR

[26] Wise S. M., Lowengrub J. S., Cristini V., “An adaptive multigrid algorithm for simulating solid growth using mixture models”, Mathematical and computer modeling, 53:1 (2011), 1–20 | DOI | MR | Zbl

[27] Kim Yangjin, Lawler Sean, Nowicki M. O., Chiocca E. A., Friedman A., “A mathematical model for formation of glioma cells outside the tumor spheroid core”, J. of Theoretical Biology, 260:3 (2009), 359–371 | DOI | MR

[28] Matrosov A. V., “Computational instability of the algorithm of the method of initial functions”, Vestnik St. Petersburg University, ser. 10: Applied mathematics, computer science, control processes, 2010, no. 4, 30–39

[29] Matrosov A. V., “Convergence of power series in the method of initial functions”, Vestnik St. Petersburg University, ser. 10: Applied mathematics, computer science, control processes, 2012, no. 1, 41–51 | MR

[30] Tikhonov A. N., Samarsky A. A., Equations of mathematical physics, Nauka, M., 1977, 735 pp. | MR

[31] Olemskoy I. V., “Structural approach to the design of explicit one-stage methods”, Computational Mathematics and Mathematical Physics, 43:7 (2003), 961–974 | MR

[32] Kolpak E. P., Zhukova I. V. Stepanova D. S., Kritsky A. V., “On numerical methods for solving evolutionary equations on the example of the mathematical model of the «prey-predator»”, Young Scientist, 2014, no. 4(63), 20–30

[33] Kolpak E. P., Gorbunova E. A., Zhukova I. V., “Mathematical model of population waves”, Natural and mathematical science in the modern world, 2014, no. 3(15), 25–30