The article describes an invasive tumor with a large area of tissue damage and a small relative density of malignant cells. The dynamics of the inflammatory process is quantitatively described, following the change in cell concentration - the number of cells per unit volume of tissue. A mathematical model for the growth of a malignant tumor is proposed. In this tumor growth model, 3 types of cells are taken into account: malignant, healthy and white blood cells - lymphocytes. It is believed that the proliferation of malignant, healthy cells and lymphocytes in space occurs due to diffusion. Apoptosis (cell death after a certain number of divisions) in malignant cells is absent. That is, malignant cells are "immortal". Healthy cells can be "infected": because of the proliferation of tumor cells, the signaling molecules do not flow to healthy cells, which is why they begin to multiply uncontrollably. The constructed model was solved numerically by the predictor-corrector method. Several cases were considered: a model without malignant cells, a case with an insufficient number of lymphocytes, and when the immune system manages to defeat the disease. The model showed good convergence with experimental data.
chemotaxis, malignant tumor, immunotherapy, mathematical model
1. Babushkina N.A., Kuzina E.A., Loos A.A., Belyaeva E.V. Ocenka effektivnyh strategiy primeneniy protivoopuholevoy vakcinoterapii na osnove matematicheskogo modelirovaniya. Matematicheskaya biologiya i biofizika, 2019, t. 14, № 1, s. 34-54. [Babushkina N.A., Kuzina E.A., Loos A.A., Belyaeva E.V. Evaluation of effective strategies for the use of anticancer vaccine therapy based on mathematical modeling. Mathematical biology and Biophysics, 2019, vol. 14, no. 1, p. 34-54. (In Russ.)]
2. Beloteloe N.V., Lobanov A.I. Populyacionnye modeli s nelineynoy diffuziey. Matematicheskoe modelirovanie, 1997, t. 9, № 12, c. 43-56. [Belotelov N.V., Lobanov A.I. Population model with nonlinear diffusion. Mathematical modeling, 1997, vol. 9, no. 12, p. 43-56. (In Russ.)]
3. Bernet F. Kletochnaya immunologiya. M.: Mir, 1971, 542 c. [Burnett F. Cell immunology. M.: World, 1971, 542 p. (In Russ.)]
4. Vodop'yanov V.V., Vodop'yanova L.L. Matematicheskoe modelirovanie nefti v rizosfere rasteniy s ispol'zovaniem diffuzii. Vestnik UGATU, 2014, t. 18, № 4, vyp. 65, s. 178-182. [Vodopyanov V.V., Vodopyanova L.L. Mathematical modeling of oil in the rhizosphere of plants using diffusion. Bulletin of USATU, 2014, vol. 18, no. 4, iss. 65, pp. 178-182. (In Russ.)]
5. Kolobov A.V., Anashkin A.A., Gubernov V.V., Polezhaev A.A. Matematicheskaya model' rosta opuholi s uchetom dihotomii migracii i proliferacii. Komp'yuternye issledovaniya i modelirovanii, 2009, t. 1, № 4, s. 415-422. [Kolobov A.V., Anashkin A. A., Gubernov V.V., Polezhaev A.A. Mathematical model of tumor growth taking into account the dichotomy of migration and proliferation. Computer studies and modeling, 2009, vol. 1, no. 4, pp. 415-422. (In Russ.)]
