Точные решения типа бегущей волны одномерных моделей инвазии рака

Точные решения типа бегущей волны одномерных моделей инвазии рака

Шубина М. В.
Сибирский журнал индустриальной математики, 26, 3(95), 179-194 (2023)
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УДК 51–76 
DOI: 10.33048/SIBJIM.2023.26.314

Аннотация:

Получены точные аналитические решения уравнений непрерывных математических моделей роста и инвазии опухоли, основанных на модели Чаплейна и Лолас, для случая одного пространственного измерения. Модели представляются системой трёх нелинейных дифференциальных уравнений реакции-диффузии-таксиса в частных производных, описывающих взаимодействие раковых клеток, фермента, разрушающего внеклеточный матрикс и ткани. Построенные решения являются гладкими неотрицательными функциями, зависящими от переменной бегущей волны и определённых условий на модельные параметры.

Литература:
  1. Folkman J., Klagsbrun M. Angiogenic Factors, Science. 1987. V. 235, N 4787. P. 442–447. 
     
  2. Anderson A. R. A., Chaplain M. A. J. Continuous and discrete mathematical models of tumor-induced angiogenesis // Bull. Math. Biology. 1998. V. 60. P. 857–899. 
     
  3. Anderson A. R. A., Chaplain M. A. J., Newman E. L., Steele R. J. C., Thompson A. M. Mathematical modelling of tumour invasion and metastasis // J. Theor. Medicine. 2000. V. 2, N 2. P. 129-154. 
     
  4. Chaplain M. A. J., Lolas G. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity // Amer. Institute Math. Sci. 2006. V. 1, N 3. P. 399–439. 
     
  5. Enderling H., Chaplain M. A. J. Mathematical modeling of tumor growth and treatment // Curr. Pharm. Des. 2014. V. 20, N 30, P. 4934–4940. 
     
  6. Adam J. A., Bellomo N. A Survey Ofmodels for Tumour-Immune System Dynamics. Boston: Birkhüser, 1996. 
     
  7. Preziosi L. Cancer Modelling and Simulation. Boca Raton: Chapman Hall/CRC Press, 2003. 
     
  8. Bellomo N., Chaplain M. A. J., De Angelis E. Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy. Boston: Birkhüser, 2008. (Modeling and Simulation in Science, Engineering and Technology). 
     
  9. Araujo R. P., McElwain D. L. S. A history of the study of solid tumour growth: the contribution of mathematical modelling // Bull. Math. Biology. 2004. V. 66, N 5. P. 1039–1091. 
     
  10. Lowengrub J. S., Frieboes H. B., Jin F., Chuang Y.-L., Li X., Macklin P., Wise S. M., Cristini V. Nonlinear modelling of cancer: bridging the gap between cells and tumours // Nonlinearity. 2010. V. 23. P. R1–R91. 
     
  11. Gatenby R. A., Gawlinski E. T. A reaction-diffusion model of cancer invasion // Cancer Res. 1996. V. 56, N 24. P. 5745–5753. 
     
  12. Perumpanani A. J., Sherratt J. A., Norbury J., Byrne H. M. Biological inferences from a mathematical model for malignant invasion // Invasion Metastasis. 1996. V. 16, N 4–5. P. 209–221. 
     
  13. Patlak C. S. Random walk with persistence and external bias // Bull. Math. Biophys. 1953. V. 15, N 3. P. 311–338. 
     
  14. Keller E. F., Segel L. A. Initiation of slime mold aggregation viewed as an instability // J. Theor. Biology. 1970. V. 26, N 3. P. 399–415. 
     
  15. Keller E. F., Segel L. A. Model for Chemotaxis // J. Theor. Biology. 1971. V. 30, N 2. P. 225–234. 
     
  16. Keller E. F., Segel L. A. Traveling bands of chemotactic bacteria: a theoretical analysis // J. Theor. Biology. 1971. V. 30, N 2. P. 235–248. 
     
  17. Painter K. J. Mathematical models for chemotaxis and their applications in self-organisation phenomena // J. Theor. Biology. 2019. V. 481. P. 162–182. 
     
  18. Anderson A. R. A. A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion // Math. Medicine and Biology. 2005. V. 22, N 2. P. 163–186. 
     
  19. Chaplain M. A. J., Lolas G. Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system // Math. Models Methods Appl. Sci. 2005. V. 15. P. 1685– 1734. 
     
  20. Enderling H., Anderson A. R. A., Chaplain M. A. J., Munro A. J., Vaidya J. S. Mathematical Modelling of Radiotherapy Strategies for Early Breast Cancer // J. Theor. Biology. 2006. V. 241, N 1. P. 158–171. 
     
  21. Andasari V., Gerisch A., Lolas G., South A. P., Chaplain M. A. J. Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation // J. Math. Biology. 2010, V. 63, N 1. P. 141–171. 
     
  22. Gerisch A., Chaplain M. A. J. Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion // J. Theor. Biology. 2008. V. 250, N 4. P. 684–704. 
     
  23. Frieboes H. B., Zheng X., Sun C. H., Tromberg B., Gatenby R., Christini V. An integrated computational/experimental model of tumor invasion // Cancer Res. 2006. V. 66. P. 1597–1604. 
     
  24. Painter K. J. Modelling cell migration strategies in the extracellular matrix // J. Math. Biology. 2009. V. 58, N 4–5. P. 511–543. 
     
  25. Ramis-Conde I., Chaplain M. A. J., Anderson A. R. A. Mathematical modelling of cancer cell invasion of tissue // Math. Comput. Model. 2008. V. 47, N 5–6. P. 533–545. 
     
  26. Painter K. J., Armstrong N. A., Sherratt J. A. The impact of adhesion on cellular invasion processes in cancer and development // J. Theor. Biology. 2010. V. 264. P. 1057–1067. 
     
  27. Peng L., Trucu D., Lin P., Thompson A., Chaplain M. A. J. A multiscale mathematical model of tumour invasive growth // Bull. Math. Biology. 2017. V. 79, N 3. P. 389–429. 
     
  28. Domschke P., Trucu D., Gerisch A., Chaplain M. A. J. Structured models of cell migration incorporating molecular binding processes // J. Math. Biology. 2017. V. 75, N 5–6. P. 1517–1561. 
     
  29. Bitsouni V., Chaplain M. A. J., Eftimie R. Mathematical modelling of cancer invasion: the multiple roles of TGF-$\beta$ pathway on tumour proliferation and cell adhesion // Math. Models Methods Appl. Sci. 2017. V. 27, N 10. Article 1929.
     
  30. Bitsouni V., Trucu D., Chaplain M. A. J., Eftimie R. Aggregation and travelling wave dynamics in a two-population model of cancer cell growth and invasion // Math. Medicine and Biology. 2018. V. 35, N 4. P. 541–577.
     
  31. Szymanska Z., Cytowski M., Mitchell E., Macnamara C. K., Chaplain M. A. J. Computational modelling of cancer development and growth: modelling at multiple scales and multiscale modelling // Bull. Math. Biology. 2017. V. 80, N 5. P. 1366–1403.
     
  32. Pang P. Y. H., Wang Y. Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant // J. Diff. Equ. 2017. V. 263. P. 1269–1292.
     
  33. Ke Y., Zheng J. A note for global existence of a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant // Nonlinearity. 2018. V. 31, N 10. Article 4602.
     
  34. Bubba F., Pouchol C., Ferrand N., Vidal G., Almeida L., Perthame B., Sabbah M. A chemotaxis-based explanation of spheroid formation in 3d cultures of breast cancer cells // J. Theor. Biology. 2019. V. 479. P. 73–80.
     
  35. Xiang T., Zheng J. A new result for 2D boundedness of solutions to a chemotaxis–haptotaxis model with/without sub-logistic source // Nonlinearity. 2019. V. 32. Article 4890. 
     
  36. Tao Y., Winkler M. Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy // J. Diff. Equ. 2020. V. 268, N 9. Article 4973.
     
  37. Perumpanani A. J., Sherratt J. A., Norbury J., Byrne H. A two parameter family of travelling waves with a singular barrier arising from the modelling of matrix mediated malignant invasion // Physica. Ser. D: Nonlinear Phenomena. 1999. V. 126. P. 145–159.
     
  38. Marchant B. P., Norbury J., Sherratt J. A. Travelling wave solutions to a haptotaxis-dominated model of malignant invasion // Nonlinearity. 2001. V. 14, N 6. P. 1653–1671.
     
  39. Sherratt J. On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion // Math. Model. Nat. Phenom. 2010. V. 5, N 5. P. 64–79.
     
  40. Harley K., Van Heijster P., Marangell R., Pettet G. J., Wechselberger M. Existence of traveling wave solutions for a model of tumor invasion // J. Appl. Dynam. Syst. 2014. V. 13, N 1. P. 366–396.
     
  41. Olver P. J. Applications of Lie Groups to Differential Equations. Springer-Verl., 1986.
     
  42. Bateman H., Erdélyi A. Higher Transcendental Functions. V. 2. N. Y.; Toronto; London: McGraw-Hill Book Company, 1953.
М. В. Шубина
  1. Московский государственный университет им. М. В. Ломоносова, 
    НИИ ядерной физики им. Д. В. Скобельцына, 
    Ленинские горы, 1, стр. 2, г. Москва 119234, Россия

E-mail: yurova-m@rambler.ru

Статья поступила 13.02.2023 г. 
После доработки — 13.04.2023 г.
Принята к публикации 27.04.2023 г.

Abstract:

In this paper we obtain exact analytical solutions of equations of continuous mathematical models of tumour growth and invasion based on the model introduced by Chaplain and Lolas for the case of one space dimension. The models consist of a system of three nonlinear reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, the matrix degrading enzyme and the tissue. The obtained solutions are smooth nonnegative functions depending on the traveling wave variable and certain conditions on the model parameters.

References:
  1. Folkman J., Klagsbrun M. Angiogenic Factors. Science, 1987, Vol. 235, No. 4787, pp. 442–447. 
     
  2. Anderson A. R. A., Chaplain M. A. J. Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biology, 1998, Vol. 60, pp. 857–899. 
     
  3. Anderson A. R. A., Chaplain M. A. J., Newman E. L., Steele R. J. C., Thompson A. M. Mathematical modelling of tumour invasion and metastasis. J. Theor. Medicine, 2000, Vol. 2, No. 2, pp. 129-154. 
     
  4. Chaplain M. A. J., Lolas G. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Amer. Institute Math. Sci., 2006, Vol. 1, No. 3, pp. 399–439. 
     
  5. Enderling H., Chaplain M. A. J. Mathematical modeling of tumor growth and treatment. Curr. Pharm. Des., 2014, Vol. 20, No. 30, pp. 4934–4940. 
     
  6. Adam J. A., Bellomo N. A Survey Ofmodels for Tumour-Immune System Dynamics. Boston: Birkhüser, 1996. 
     
  7. Preziosi L. Cancer Modelling and Simulation. Boca Raton: Chapman Hall/CRC Press, 2003. 
     
  8. Bellomo N., Chaplain M. A. J., De Angelis E. Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy. Boston: Birkhüser, 2008. (Modeling and Simulation in Science, Engineering and Technology). 
     
  9. Araujo R. P., McElwain D. L. S. A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull. Math. Biology, 2004, Vol. 66, No. 5, pp. 1039–1091. 
     
  10. Lowengrub J. S., Frieboes H. B., Jin F., Chuang Y.-L., Li X., Macklin P., Wise S. M., Cristini V. Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity, 2010, Vol. 23, pp. R1–R91. 
     
  11. Gatenby R. A., Gawlinski E. T. A reaction-diffusion model of cancer invasion. Cancer Res., 1996, Vol. 56, No. 24, pp. 5745–5753. 
     
  12. Perumpanani A. J., Sherratt J. A., Norbury J., Byrne H. M. Biological inferences from a mathematical model for malignant invasion. Invasion Metastasis, 1996, Vol. 16, No. 4–5, pp. 209–221. 
     
  13. Patlak C. S. Random walk with persistence and external bias. Bull. Math. Biophys, 1953, Vol. 15, No. 3, pp. 311–338. 
     
  14. Keller E. F., Segel L. A. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biology, 1970, Vol. 26, No. 3, pp. 399–415. 
     
  15. Keller E. F., Segel L. A. Model for Chemotaxis. J. Theor. Biology, 1971, Vol. 30, No. 2, pp. 225–234. 
     
  16. Keller E. F., Segel L. A. Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biology, 1971, Vol. 30, No. 2, pp. 235–248. 
     
  17. Painter K. J. Mathematical models for chemotaxis and their applications in self-organisation phenomena. J. Theor. Biology, 2019, Vol. 481, pp. 162–182. 
     
  18. Anderson A. R. A. A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Medicine and Biology, 2005, Vol. 22, No. 2, pp. 163–186. 
     
  19. Chaplain M. A. J., Lolas G. Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci., 2005, Vol. 15, pp. 1685– 1734. 
     
  20. Enderling H., Anderson A. R. A., Chaplain M. A. J., Munro A. J., Vaidya J. S. Mathematical Modelling of Radiotherapy Strategies for Early Breast Cancer. J. Theor. Biology, 2006, Vol. 241, No. 1, pp. 158–171. 
     
  21. Andasari V., Gerisch A., Lolas G., South A. P., Chaplain M. A. J. Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation. J. Math. Biology, 2010, Vol. 63, No. 1, pp. 141–171. 
     
  22. Gerisch A., Chaplain M. A. J. Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion. J. Theor. Biology, 2008, Vol. 250, No. 4, pp. 684–704. 
     
  23. Frieboes H. B., Zheng X., Sun C. H., Tromberg B., Gatenby R., Christini V. An integrated computational/experimental model of tumor invasion. Cancer Res., 2006, Vol. 66, pp. 1597–1604. 
     
  24. Painter K. J. Modelling cell migration strategies in the extracellular matrix. J. Math. Biology, 2009, Vol. 58, No. 4–5, pp. 511–543. 
     
  25. Ramis-Conde I., Chaplain M. A. J., Anderson A. R. A. Mathematical modelling of cancer cell invasion of tissue. Math. Comput. Model., 2008, Vol. 47, No. 5–6, pp. 533–545. 
     
  26. Painter K. J., Armstrong N. A., Sherratt J. A. The impact of adhesion on cellular invasion processes in cancer and development. J. Theor. Biology, 2010, Vol. 264, pp. 1057–1067. 
     
  27. Peng L., Trucu D., Lin P., Thompson A., Chaplain M. A. J. A multiscale mathematical model of tumour invasive growth. Bull. Math. Biology, 2017. Vol. 79, No. 3, pp. 389–429. 
     
  28. Domschke P., Trucu D., Gerisch A., Chaplain M. A. J. Structured models of cell migration incorporating molecular binding processes. J. Math. Biology, 2017, Vol. 75, No. 5–6, pp. 1517–1561. 
     
  29. Bitsouni V., Chaplain M. A. J., Eftimie R. Mathematical modelling of cancer invasion: the multiple roles of TGF-$\beta$ pathway on tumour proliferation and cell adhesion. Math. Models Methods Appl. Sci., 2017, Vol. 27, No. 10, article 1929.
     
  30. Bitsouni V., Trucu D., Chaplain M. A. J., Eftimie R. Aggregation and travelling wave dynamics in a two-population model of cancer cell growth and invasion. Math. Medicine and Biology, 2018, Vol. 35, No. 4, pp. 541–577.
     
  31. Szymanska Z., Cytowski M., Mitchell E., Macnamara C. K., Chaplain M. A. J. Computational modelling of cancer development and growth: modelling at multiple scales and multiscale modelling. Bull. Math. Biology, 2017, Vol. 80, No. 5, pp. 1366–1403.
     
  32. Pang P. Y. H., Wang Y. Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant. J. Diff. Equ., 2017, Vol. 263, pp. 1269–1292.
     
  33. Ke Y., Zheng J. A note for global existence of a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant. Nonlinearity, 2018, Vol. 31, No. 10, article 4602.
     
  34. Bubba F., Pouchol C., Ferrand N., Vidal G., Almeida L., Perthame B., Sabbah M. A chemotaxis-based explanation of spheroid formation in 3d cultures of breast cancer cells. J. Theor. Biology, 2019, Vol. 479, pp. 73–80. 
     
  35. Xiang T., Zheng J. A new result for 2D boundedness of solutions to a chemotaxis–haptotaxis model with/without sub-logistic source. Nonlinearity, 2019, Vol. 32, article 4890.
     
  36. Tao Y., Winkler M. Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy. J. Diff. Equ., 2020, Vol. 268, No. 9, article 4973.
     
  37. Perumpanani A. J., Sherratt J. A., Norbury J., Byrne H. A two parameter family of travelling waves with a singular barrier arising from the modelling of matrix mediated malignant invasion. Physica. Ser. D: Nonlinear Phenomena, 1999, Vol. 126, pp. 145–159.
     
  38. Marchant B. P., Norbury J., Sherratt J. A. Travelling wave solutions to a haptotaxis-dominated model of malignant invasion. Nonlinearity, 2001, Vol. 14, No. 6, pp. 1653–1671.
     
  39. Sherratt J. On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion. Math. Model. Nat. Phenom., 2010, Vol. 5, No. 5, pp. 64–79.
     
  40. Harley K., Van Heijster P., Marangell R., Pettet G. J., Wechselberger M. Existence of traveling wave solutions for a model of tumor invasion. J. Appl. Dynam. Syst., 2014, Vol. 13, No. 1, pp. 366–396.
     
  41. Olver P. J. Applications of Lie Groups to Differential Equations. Springer-Verl., 1986.
     
  42. Bateman H., Erdélyi A. Higher Transcendental Functions. V. 2. N. Y.; Toronto; London: McGraw-Hill Book Company, 1953.