Modeling of Tumor Occurrence and Growth – II

УДК 519.87:612

  • S.N. Antontsev Lavrentyev’s Institute of Hydrodynamics (Novosibirsk, Russia); Center for Mathematics, Fundamental Applications and Operations Research (CMAF-CIO), University of Lisbon (Lisbon, Portugal) Email: antontsevsn@mail.ru
  • А.А. Papin Altai State University (Barnaul, Russia) Email: papin@math.asu.ru
  • M.A. Tokareva Lavrentyev’s Institute of Hydrodynamics (Novosibirsk, Russia); Altai State University (Barnaul, Russia) Email: tma25@mail.ru
  • E.I. Leonova Altai State University (Barnaul, Russia) Email: leonova.eve@gmail.com
  • E.A. Gridushko Altai State University (Barnaul, Russia) Email: katya19992012@gmail.com
DOI_disc_logo https://doi.org/10.14258/izvasu(2021)1-12
Keywords: mathematical modeling, tumor modeling, hybrid models, multiphase systems, tumour cords, tumour — host interface, mixture theory

Abstract

This paper considers the mathematical model of tumor growth along a blood vessel. The model employs the mixture theory approach to describe a tissue that consists of cells, extracellular matrix, and liquid. The growing tumor tissue is supposed to be surrounded by the host tissue. Tumors, where complete oxidation of glucose prevails, are considered. Special attention is paid to consistent descriptions of oxygen consumption and growth processes based on the energy balance. The level set method is used to track an interface between the tissues. The simulations show localization of the tumor within a limited distance from the vessels and constant expansion speed along the vessels. Cancer disease manifests itself as abnormally excessive cell proliferation. This is the result of dysregulation of normal constraints on cellular proliferation. This fact has serious implications on the morphology of the growth. The intensive proliferation of tumor cells creates cell populations distant from blood vessels and deprived of nutrient and oxygen supply while most of the cells in the human body are within few cell diameters from a blood vessel. This leads to the formation of cylindrical structures around blood vessels.

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Author Biographies

S.N. Antontsev, Lavrentyev’s Institute of Hydrodynamics (Novosibirsk, Russia); Center for Mathematics, Fundamental Applications and Operations Research (CMAF-CIO), University of Lisbon (Lisbon, Portugal)

профессор, доктор физико-математических наук, главный научный сотрудник лаборатории механики неоднородных сред

А.А. Papin, Altai State University (Barnaul, Russia)

доктор физико-математических наук, заведующий кафедрой дифференциальных уравнений

M.A. Tokareva, Lavrentyev’s Institute of Hydrodynamics (Novosibirsk, Russia); Altai State University (Barnaul, Russia)

кандидат физико-математических наук, доцент кафедры дифференциальных уравнений

E.I. Leonova, Altai State University (Barnaul, Russia)

студентка

E.A. Gridushko, Altai State University (Barnaul, Russia)

студентка

References

Антонцев С.Н., Папин А.А., Токарева М.А., Леонова Э.И., Гридюшко Е.А. Моделирование возникновения и роста опухолей-I // Известия Алт. гос. ун-та. 2020. № 4(114), DOI:10.14258/izvasu(2020)4-11.

Astanin S., Tosin A. Mathematical model of tumour cord growth along the source of nutrient // Math. Model. Nat. Phenom., 2007. 2. 3. DOI:10.1051/mmnp:2007007.

Andreef M., Goodrich D., Pardee A.B. Cancer Medicine 6 / editors, Donald W Kufe [et al.], chapter Cell proliferation and differentiation. 2003.

Minchinton A.I., Tannock I.F. Drug penetration in solid tumours // Nature Reviews Cancer. 2006. 6. DOI:10.1038/ nrc1893.

Tannock I.F. The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour // Br. J. Cancer. 1968. 22. D0I:10.1038/bjc.1968.34.

Hirst G., Denekamp J. Tumour cell proliferation in relation to the vasculature // Cell Tissue Kinet. 1979. 12. D0I:10.1111/j.1365-2184.1979.tb00111.x.

Moore J.V, Hasleton P.S., Buckley C.H. Tumour cords in 52 human bronchial and cervical squamous cell carcinomas: Inferences for their cellular kinetics and radiobiology // Br. J. Cancer. 1985. 51. DOI:10.1038/bjc.1985.55.

Bertuzzi A., Gandolfi A. Cell kinetics in a tumor cord // J.Theor.Biol. 2000. 204. DOI:10.1006/jtbi.2000.1079.

Bertuzzi A., Fasano A., Gandolfi A., Marangi D. Cell kinetics in tumour cords studied by a model with variable cell cycle length // Math. Biosci. 2002. 177-178. D0I:10.1016/ s0025-5564(01)00114-6.

Bertuzzi A., Fasano A., Gandolfi A. A free boundary problem with unilateral constraints describing the evolution of a tumor cord under the influence of cell killing agents // SIAM J. Math. Anal. 2004. 36. № 3. DOI:10.1137/S003614002406060.

Bertuzzi A., Fasano A., Gandolfi A. A mathematical model for tumor cords incorporating the flow of interstitial fluid // Math. Models Methods Appl. Sci. 2005. 15. № 11. D0I:10.1142/S0218202505000959 .

Tosin A. Multiphase modeling and qualitative analysis of the problem of the growth of tumor cords // Networks and heterogeneous media. 2008. 3. № 1. DOI: 10.3934/nhm.2008.3.43.

Preziosi L., Tosin A. Multiphase and Multiscale Trends in Cancer Modelling // Math. Model. Nat. Phenom. 2009. 4. 3. DOI:10.1051/mmnp/20094301.

Astanin S., Preziosi L. Selected Topics on Cancer Modelling: Genesis - Evolution - Immune Competition -Therapy, chapter Multiphase Models of Tumour Growth. Birkhauser, 2008.

Byrne H.M., King J.R., McElwain D.L.S., Preziosi L. A two-phase model of solid tumor growth // Appl. Math. Lett. 2003. 16. DOI:10.1016/S0893-9659(03)00038-7.

Zheng X., Sweidan M. A mathematical model of angio-genesis and tumor growth: analysis and application in anti-angiogenesis therapy // Journal of Mathematical Biology. 2018. DOI:10.1007/s00285-018-1264-4.

Chaplain M.A.J., Graziano L., Preziosi L. Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development // Math. Med. Biol. 2006. 23. D0I:10.1093/imammb/dql009.

Ambrosi D., Preziosi L. On the closure of mass balance models for tumor growth // Math. Models Methods Appl. Sci. 2002. 12. DOI:10.1142/S0218202502001878.

Graziano L., Preziosi L. Modeling of Biological Materials / editors, F. Mollica ...[et al.], chapter Mechanics in tumour growth. Birkhauser. 2007. D0I:10.1007/b138320.

Navalitloha Y., Schwartz E.S., Groothuis E.N., Allen C.V., Levy R.M., Groothuis D.R. Therapeutic implications of tumor interstitial fluid pressure in subcutaneous RG-2 tumors // Neuro Oncol. 2006. 8. № 3. DOI:10.1215/15228517-2006-007.

Nelson D.L.,Cox M.M. I principi di biochimica di Lehninger. Zanichelli, 2002.

Smallbone K., Gatenby R.A., Gillies R.J., Maini P.K., Gavaghan D.J. Metabolic changes during carcinogenesis: Potential impact on invasiveness // J. Theor. Biol. 2007. 244. D0I:10.1016/j.jtbi.2006.09.010.

Gillies R.J., Gatenby R.A. Hypoxia and adaptive landscapes in the evolution of carcinogenesis // Cancer Metastasis Rev. 2007. D0I:10.1007/s10555-007-9065-z.

Gatenby R.A.,Gawlinski E.T., Gmitro A.F., Kaylor B., Gillies R.J. Acid-meditated tumour invasion: a multidisciplinary study // Cancer Res. 2006. 66. D0I:10.1158/0008-5472. CAN-05-4193.

Published
2021-03-17
How to Cite
Antontsev S., PapinА., Tokareva M., Leonova E., Gridushko E. Modeling of Tumor Occurrence and Growth – II // Izvestiya of Altai State University, 2021, № 1(117). P. 72-83 DOI: 10.14258/izvasu(2021)1-12. URL: http://izvestiya.asu.ru/article/view/%282021%291-12.
Issue
No 1(117) (2021): Izvestiya of Altai State University
Section
Математика и механика

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