Um segundo estudo de invasão populacional dinâmica a partir da equação do telégrafo reativo e formulação de elementos de contorno - Um ensaio sobre o crescimento tumoral in vitro

Autores

DOI:

https://doi.org/10.20873/jbb.uft.cemaf.v10n2.pettres

Palavras-chave:

método dos elementos de fronteira, equação reativa-telegráfica com geração, início e reprodução de células cancerosas, condição suficiente da positividade da solução, crescimento do tumor

Resumo

Este artigo é a continuação de um estudo já realizado sobre o uso da equação do telégrafo reativo para analisar problemas de dinâmica populacional a partir de uma formulação do método dos elementos de contorno (BEM). Neste artigo, o modelo numérico simula a evolução de um tumor como um problema de densidade populacional de células cancerosas a partir de diferentes termos reativos acoplados à equação do telégrafo reativo para descrever o crescimento e distribuição da população, semelhante ao experimento de crescimento do tumor in vitro. O modelo matemático desenvolvido é denominado D-BEM, usa uma solução fundamental independente do tempo e o método das diferenças finitas é combinado com o BEM para aproximar os termos de tempo derivativos e a quadratura Gaussiana é usada para calcular as integrais de domínio. A solução do sistema de equações é baseada no método de eliminação gaussiana e foi verificada a estabilidade da formulação proposta. Como a equação do telégrafo possui comportamento ondulatório, pode ocorrer um fenômeno de mudança de fase que pode levar ao aparecimento de densidade populacional negativa, para tanto, foi desenvolvido um algoritmo que garantir a positividade da solução. Resultados importantes foram obtidos e demonstram o efeito do parâmetro de atraso no crescimento do tumor. Em um dos casos testados, os resultados indicaram um comportamento oscilatório no crescimento tumoral quando o parâmetro de retardo assumiu valores crescentes. O importante resultado dessa antítese para o crescimento do tumor, bem como toda a formulação dos elementos de contorno são apresentados a seguir.

Referências

Alharbi W, Petrovskii S. Critical domain problem for the reaction–telegraph equation model of population dynamics. Mathematics, v.6, n.4, p.6–59, 2018. https://doi.org/10.3390/math6040059

Aviv-Sharon E, Aharoni A. Generalized logistic growth mod-eling of the COVID-19 pandemic in Asia. Infectious Dis-ease Modelling, v.5, p.502-509, 2020. https://doi.org/10.1016/j.idm.2020.07.003

Bacaer N. Verhulst and the logistic equation (1838), A Short History of Mathematical Population Dynamics. Springer, London, p.35-39, 2011. https://doi.org/10.1007/978-0-85729-115-8

Burkowski FJ. A computer simulation of the growth of a tumour in vitro. Computer Programs in Biomedicine, v.7, n.3, p.203-210, 1977. https://doi.org/10.1016/0010-468X(77)90028-9

Byrne HM, Chaplain MAJ. Growth of nonnecrotic tumors in the presence and absence of inhibitors. Mathematical Biosci-ences, v.130, n.2, p.151-182, 1995. https://doi.org/10.1016/0025-5564(94)00117-3

Cattaneo C. Sur uneforme de l’equation de la chaleur eliminant le paradoxed’une propagation instantanee. Comptes Rendus de l'Académiedes Sciences, v.247, p.431-433, 1958.

Chandrasekharaiah DS. Hyperbolic thermoelasticity: a review of recent literature. Applied Mechanics Reviews, v.51, n.12, p.705-729, 1998. https://doi.org/10.1115/1.3098984

Chiu SN, Stoyan D, Kendall WS, Mecke J. Stochastic geome-try and its applications. John Wiley & Sons, 2013. https://doi.org/10.1002/bimj.4710390510

Cirilo ER, Petrovskii S, Romeiro N, Natti P. Investigation into the Critical Domain Problemfor the Reaction-telegraph equa-tion Using Advanced Numerical Algorithms. International Journal of Applied and Computational Mathematics, v.5, n.54, 2019. https://doi.org/10.1007/s40819-019-0633-z

Fort J, Méndez V. Wavefronts in time-delayed reaction-diffusion systems. Theory and Comparison to Experiment. Reports on Progress in Physics, v. 65, p. 895, 2002.

Forys U, Marciniak-Czochra A. Logistic equations in tumour growth modelling. International Journal of Applied Mathe-matics and Computer Science, v.13, n.3, p.317-325, 2003.

Friedl P, Wolf, A. Tumor cell invasion and migration: diversi-ty and escape mechanisms. Nature Reviews Cancer, v.3, n.5, p.362–374, 2003. https://doi.org/10.1038/nrc1075

Gatenby RA, Vincent TL. Application of quantitative models from population biology and evolutionary game theory to tumour therapeutic strategies. American Association for Cancer Research, v.2, n.9, p.919-927, 2003.

Godoi PHV. Mathematical modeling of two-dimensional biological invasion via telegraph equation. Master's Thesis in Applied and Computational Mathematics, State Universi-ty of Londrina, Londrina, 2021.

Hillen T, Gatenby R, Hinow P. Partial Differential Equations in Cancer Modelling. Banff International Research Station for mathematiclal Innovation and Discovery. Workshopin February, p.2-6, 2015.

Holmes EE. Are Diffusion Models Too Simple? A Compari-son with Telegraph Models of Invasion. American Natural-ist, v.142, n.5, p.779-795, 1993.

Jiang C, Cui C, Li L, Shao Y. The anomalous diffusion of a tumor invading with different surrounding tissues. PLoS One, v.9, n.10, 2014. https://doi.org/10.1371/journal.pone.0109784

Johnson KE, Howard G, Mo W, Strasser MK, Lima EABF, Huang S, Brock A. Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect. PLoS Biology, v.17 n.8, e3000399, 2019.https://doi.org/10.1371/journal.pbio.3000399

Katt ME, Placone AL, Wong AD, Xu ZS, Searson PC. In Vitro Tumour Models: Advantages, Disadvantages, Varia-bles, and Selecting the Right Platform. Frontiers in bioengi-neering and biotechnology, v.4, n.12, 2016. https://doi.org/10.3389/fbioe.2016.00012

Kerbel RS. Tumour angiogenesis: past, present and the near future. Carcinogenesis, v.21, n.3, p.505-515, 2000. https://doi.org/10.1093/carcin/21.3.505

Konukoglu E, Clatz O, Bondiau PY, Delignette H, Ayache N. Extrapolation glioma invasion margin in brain magnetic res-onance images: Suggesting new irradiation margins. Medi-cal Image Analysis, v.14, n.2, p.111-125, 2010. https://doi.org/10.1016/j.media.2009.11.005

Lowengrub JS, Frieboes HB, Jin F, Chuang YL, Li X, Mack-lin P, Wise SM, Cristini V. Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity, v.23, n.1, 2010. https://doi:10.1088/0951-7715/23/1/r01

Ludwig CU, Durie BG, Salmon SE, Moon TE. Tumor growth stimulation in vitro by interferons. European Journal of Cancer and Clinical Oncology, v.19, n.11, p.1625-32, 1983. https://doi.org/10.1016/0277-5379(83)90095-0

Lurie SA, Belov PA. On the nature of the relaxation time, the Maxwell–Cattaneo and Fourier law in the thermodynamics of a continuous medium, and the scale effects in thermal conductivity. Continuum Mechanics and Thermodynamics, v.32, p.709-728, 2020. https://doi.org/10.1007/s00161-018-0718-7

Mahooti, M. Gaussian Elimina-tion(https://www.mathworks.com/matlabcentral/fileexchange/73485-gaussian-elimination), MATLAB Central File Ex-change. Retrieved July 5, 2020.

Malinzi J. A mathematical model for oncolytic virus spread using the telegraph equation. Communications in Nonlinear Science and Numerical Simulation, v.102, p.1-16, 2021. https://doi.org/10.1016/j.cnsns.2021.105944

Maxwell JC. On the Dynamical Theory of Gases. The Philo-sophical Transactions of the Royal Society, v.157, n.49, p.49-88, 1867.

Méndez V, Fedotov S, Horsthemke W. Reactions and transport: diffusion, inertia, and subdiffusion.In: Reaction-Transport Systems. Springer Series in Synergetics. Spring-er, Berlin, p.33-54, 2010. https://doi.org/10.1007/978-3-642-11443-4_2

Michelson S, Glicksman SA, Leith JT. Growth in solid heter-ogeneous human colon adenocarcinomas: comparison of simple logistical models. Cell Proliferation, v.20, p.343-355, 1987.

https://doi.org/10.1111/j.1365-2184.1987.tb01316.x

Mittal RC, Dahiya S. Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differen-tial quadrature methods. Computers & Mathematics with Applications, v.70, n.5, p.737-749, 2015. https://doi.org/10.1016/j.camwa.2015.04.022

Morton KW, Mayers DF. Numerical Solutions of Partial Differential Equations. Cambridge University Press, New York, 1994. https://doi.org/10.1002/zamm.19950750904

Oliveira TF. Mathematical modeling and numeric analysis of relaxation time for reactive-diffusive-telegraph equation. 2020. Master's Dissertation in Applied and Computational Mathematics - State University of Londrina, Londrina, Bra-zil.

Onsager L. Reciprocal relations in irreversible processes. Physical Review journals, v.37, 119p., 1931. https://doi.org/10.1103/PhysRev.37.405

Painter KJ, Hillen T. Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion. Journal of Theoretical Biology, v.323, n.21, p.25-39, 2013. https://doi.org/10.1016/j.jtbi.2013.01.014

Pettres R, Lacerda LA, Carrer JAM. A boundary element formulation for the heat equation with dissipative and heat generation terms. Engineering Analysis with Boundary El-ements, v.51, p.191-198, 2015. https://doi.org/10.1016/j.enganabound.2014.11.005

Pettres R. A first dynamic population invasion study from reactive-telegraph equation and boundary element formula-tion. Engineering Analysis with Boundary Elements, v.122, p.214-231, 2021. https://doi.org/10.1016/j.enganabound.2020.11.002

Pettres R. An introductory course on the boundary element method. v.1, p.150-153, 2020.

Preziosi L. Cancer Modelling and Simulation. Chapman Hall/CRC Press, 421 p. 2003.

Richards FJ. A flexible growth function for empirical use. Journal of Experimental Botany, v.10, n.2, p.290-301, 1959.

Sadhukhan S, Basu SK. Avascular tumour growth models based on anomalous diffusion. Journal of Biological Phys-ics, v.46, p.67-94, 2020. https://doi.org/10.1007/s10867-020-09541-w

Schwarzwälder MC. Non-Fourier Heat Conduction. The Maxwell-Cattaneo Equations. Department: Matemàtica Aplicada I Academic, Master's Degree Thesis in Science in Advanced Mathematics and Mathematical Engineering. 2015. 81p. Universitat Politècnica de Catalunya Facultat de Matemàtiquesi Estadística.

Spratt JS, Meyer JS, Spratt JA. Rates of growth of human neoplasms: part II. Journal of Surgical Oncology, v.61, p.68-73, 1996. https://doi.org/10.1002/1096-9098(199601)61:1<68::AID-JSO2930610102>3.0.CO;2-E

Sutherland RM, Durand RE. Hypoxic cells in an in vitro turnout model. International Journal of Radiation Biology and Related Studies in Physics, Chemistry and Medicine, v.23, n.3, p.235-246, 1973. https://doi.org/10.1080/09553007314550261

Swanson KR, Alvord Jr EC, Murray JD. A quantitative model for differential motility of gliomas in grey and white matter. Cell Proliferation, v.33, p.317-329, 2000. https://doi.org/10.1046/j.1365-2184.2000.00177.x

Swanson KR, Bridge C, Murray JD, JR Alvord EC. Virtual and real brain tumours: using mathematical modeling to quantify glioma growth and invasion. Journal of the Neuro-logical Sciences, v.216, n.1, p.1-10, 2003. https://doi.org/10.1016/j.jns.2003.06.001

Swanson KR. Mathematical modeling of the growth and con-trol of tumors. 1999. Ph.D thesis, University of Washing-ton.

Tao Y, Guo Q. Simulation of a model of tumors with virus-therapy. International Series of Numerical Mathematics, v.154, p.435-444, 2006.

Tilles PFC, Petrovskii SV. On the Consistency of the Reac-tion-Telegraph Process Within Finite Domains. Journal of Statistical Physics, v.177, p.569–587, 2019. https://doi.org/10.1007/s10955-019-02379-0

Vaghi C, Rodallec A, Fanciullino R, Ciccolini J, Mochel JPM, Mastri M, Poignard C, Ebos J, Benzekry S. A reduced Gompertz model for predicting tumor age using a population approach. BioRxiv, p.670869, 2019. https://doi.org/10.1101/670869

Varalta N, Gomes AV, Camargo RF. A prelude to the frac-tional calculus applied to tumour dynamic. Tema, v.15, n.2, p.211-221, 2014. https://doi.org/10.5540/tema.2014.015.02.0211

Vernotte P. Les paradoxes de lathéorie continue de l'équation de lachaleur. Comptes Rendus de l'Académiedes Sciences, v.246, n.22, p.3154-3155, 1958.

Ward JP, King JR. Mathematical modelling of avascular-tumour growth. IMA Journal of Mathematics Applied in Medicine and Biology, v.14, n.1, p.39-69, 1997. https://doi.org/10.1093/imammb/14.1.39

Wrobel LC. Potential and Viscous Flow Problems Using the Boundary Element Method. 1981. U.K. Ph.D. Thesis, Uni-versity of Southampton.

Yorke ED, Fuks Z, Norton L, Whitmore W, Ling CC. Model-ing the Development of Metastases from Primary and Local-ly Recurrent Tumours: Comparison with a Clinical Data Base for Prostatic Cancer. Cancer Research, v.53, n.13, p.2987-2993, 1993.

Zhukovsky KV. Exact solution of Guyer–Krumhansl type heat equation by operational method. International Journal of Heat and Mass Transfer, v.96, p.132-144, 2016. https://doi.org/10.1016/j.ijheatmasstransfer.2016.01.005

Journal of Biotechnology and Biodiversity (CeMAF / UFT)

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01-08-2022

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Pettres, R., Pettres, A. A., & Cirilo, E. R. (2022). Um segundo estudo de invasão populacional dinâmica a partir da equação do telégrafo reativo e formulação de elementos de contorno - Um ensaio sobre o crescimento tumoral in vitro. Journal of Biotechnology and Biodiversity, 10(2), 111–136. https://doi.org/10.20873/jbb.uft.cemaf.v10n2.pettres