Localization of Solutions to Equations of Tumor Dynamics
УДК 517.9
Abstract
This article discusses a mathematical model of tumor dynamics. The tissue is considered as a multiphase three-component medium consisting of extracellular matrix, tumor cells, and extracellular fluid. The extracellular matrix is generally deformable. In the case of the predominant extracellular fluid — tumor cell interaction, the original system of equations is reduced to the one parabolic equation degenerating on the solution with a special right-hand side. The property of a finite perturbation propagation velocity for tumor cell saturation is revealed. The introduction describes the essence of the problem. The second part presents the derivation of a mathematical model of tumor dynamics as a three-phase medium. The third part describes a mathematical model for the case when mechanical interaction with extracellular fluid is neglected. The fourth part considers the case of predominant fluid-cell interaction. The fifth part provides a proof of the theorem on the localization of the solution to the equation for the saturation of tumor cell.
Downloads
Metrics
References
Mollica F., Preziosi L., Rajagopal K.R. Modeling of Biological Materials. Boston: Birkhauser, 2007. 357 p. DOI: 10.1007/ b138320
Astanin S., Tosin A. Mathematical Model of Tumor Cord Growth Along the Source of Nutrient. Mathematical Modeling of Natural Phenomena. 2007. Vol. 2. P. 153-177. DOI: 10.1051/ mmnp:2007007
Adman J.A., Bellomo N. A Survey of Models on Tumor Immune Systems Dynamics. Boston: Birkhauser, 1996. 344 p. DOI: 10.1007/978-0-8176-8119-7
Ambrosi D., Preziosi L. On the Closure of Mass Balance Models for Tumor Growth. Mathematical Models and Methods in Applied Sciences. 2002. Vol. 12. P 737-754. DOI: 10.1142/ S0218202502001878
Preziosi L. Cancer Modeling and Simulation. Boca Raton: CRCPress.Chapman Hall, 2003. 452 p. DOI: 10.1201/9780203494899
Preziosi L., Farina A. On Darcy’s Law for Growing Porous Media. International Journal of Non-Linear Mechanics. 2001. Vol. 37. P. 485-491. DOI: 10.1016/S0020-7462(01)00022-1
Kowalczyk R. Preventing Blow-up in a Chemotaxis Model. Journal of Mathematical Analysis and Applications. 2005. Vol. 305. P. 566-588. DOI: 10.1016/j.jmaa.2004.12.009
Papin A.A., Sibin A.N. Simulation of the Motion of a Mixture of Liquid and Solid Particles in Porous Media with Regard to Internal Suffosion. Fluid Dynamics. 2019. Vol. 54. No 4. P. 520-534. DOI: 10.1134/S0015462819030108
Papin A.A., Sibin A.N. Heat and Mass Transfer in Melting Snow. Journal of Applied Mechanics and Technical Physics. 2021. Vol. 62. No 1. P. 96-104. DOI: 10.1134/S0021894421010120
Koleva M., Vulkov L. Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals. Computation. 2023. Vol. 11. No 204. P. 1-16. DOI: 10.3390/ computation11100204
Antontsev S.N., Diaz J.I., Shmarev S. Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics. Washington D.C.: Progress in Nonlinear Differential Equations and Their Applications, 2002. 332 p. DOI: 10.1115/1.1483358
Favin A., Marinoschi G. Degenerate Nonlinear Diffusion Equations. Berlin: Springer, 2012. 143 p. DOI: 10.1007/978-3642-28285-0
Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N. Linear and Quasilinear Equations of Parabolic Type. M.: Nauka, 1967. 736 p. (in Russ.).
Antontsev S.N., Papin A.A. Localization of Solutions of Equations of a Viscous Gas, with Viscosity Depending on the Density. Dinamika Sploshnoy Sredy, Novosibirsk: Institut Gidrodinamiki. 1988. Vol. 82. P. 24-40. (in Russ.).
Antontsev S.N. Localization of Solutions of Degenerate Equations of Continuum Mechanics. Novosibirsk: Akademiya Nauk SSSR Sibirskoe Otdelenie Instituta Gidrodinamiki, 1986. 109 p. (in Russ.).
Copyright (c) 2024 Вардан Баландурович Погосян, Маргарита Андреевна Токарева, Александр Алексеевич Папин
This work is licensed under a Creative Commons Attribution 4.0 International License.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
