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Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

Part of: Physiological, cellular and medical topics Equations of mathematical physics and other areas of application Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  17 July 2020

CHRISTINA SURULESCU Open the ORCID record for CHRISTINA SURULESCU [Opens in a new window]  and
MICHAEL WINKLER
CHRISTINA SURULESCU
Affiliation:
Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, 67663 Kaiserslautern, Germany email: surulescu@mathematik.uni-kl.de
MICHAEL WINKLER
Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany email: michael.winkler@math.uni-paderborn.de
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Abstract

We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Keywords

chemotaxishaptotaxisindirect signal productionglobal existence

MSC classification

Primary: 35Q92: PDEs in connection with biology and other natural sciences
Secondary: 35B44: Blow-up 35K55: Nonlinear parabolic equations 92C17: Cell movement (chemotaxis, etc.) 35A01: Existence problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 4 , August 2021 , pp. 618 - 651
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

https://www.dealii.org/.Google Scholar
Ahn, I. & Yoon, C. (2020) Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis. J. Diff. Equ. 268, 4222–4255.CrossRefGoogle Scholar
Alikakos, N. D. (1979) Lp bounds of solutions of reaction-diffusion equations. Comm. Part. Diff. Equ. 4, 827–868.CrossRefGoogle Scholar
Bellomo, N., Bellouquid, A., Tao, Y. & Winkler, M. (2015) Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25, 1663–1763.CrossRefGoogle Scholar
Biler, P., Hebisch, W. & Nadzieja, T. (1994) The Debye system: existence and large time behavior of solutions. Nonlinear Anal. 23(9), 1189–1209.CrossRefGoogle Scholar
Cao, X. (2016) Boundedness in a three-dimensional chemotaxis-haptotaxis system. Z. Angew. Math. Phys. 67, 11.CrossRefGoogle Scholar
Chaplain, M. A. J. & Lolas, G. (2005) Mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15, 1685–1734.CrossRefGoogle Scholar
Chen, L., Painter, K. J., Surulescu, C. & Zhigun, A. Mathematical models for cell migration: a nonlocal perspective. Phil. Trans. Royal Soc. B, in print, arXiv:1911.05200.Google Scholar
Cirri, P. & Chiarugi, P. Cancer associated fibroblasts: the dark side of the coin. Am. J. Cancer Res. 1, 482–497 (2011).Google ScholarPubMed
Cirri, P. & Chiarugi, P. Cancer associated fibroblasts and tumour cells: a diabolic liaison driving cancer progression. Cancer Metast. Rev. 31, 195–208 (2012).CrossRefGoogle ScholarPubMed
Corrias, L., Perthame, B. & Zaag, H. Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004).CrossRefGoogle Scholar
Duda, D. G., Duyverman, A. M., Kohno, M., Snuderl, M., Steller, E. J., Fukumura, D. & Jain, R. K. (2010) Malignant cells facilitate lung metastasis by bringing their own soil. Proc. Natl. Acad. Sci. USA 107, 21677–21682.CrossRefGoogle ScholarPubMed
Engwer, C., Knappitsch, M. & Surulescu, C. A multiscale model for glioma spread including cell-tissue interactions and proliferation. Math. Biosci. Eng. 13, 443–460 (2016).CrossRefGoogle ScholarPubMed
Erdogan, B., Ao, M., White, L. M., Means, A. L., Brewer, B. M., Yang, L., Washington, M. K., Shi, C., Franco, O. E., Weaver, A. M., Hayward, S. W., Li, D. & Webb, D. J. (2017) Cancer-associated fibroblasts promote directional cancer cell migration by aligning fibronectin. J. Cell Biol. 216, 3799–3816.CrossRefGoogle ScholarPubMed
Fontelos, M. A., Friedman, A. & Hu, B. (2002) Mathematical analysis of a model for the initiation of angiogenesis. SIAM J. Math. Anal. 33, 1330–1355.CrossRefGoogle Scholar
Friedman, A. & Tello, J. I. (2002) Stability of solutions of chemotaxis equations in reinforced random walks. J. Math. Anal. Appl. 272, 138–163.CrossRefGoogle Scholar
Fujie, K., Ito, A. & Yokota, T. (2014) Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type. Adv. Math. Sci. Appl. 24, 67–84.Google Scholar
Giese, A., Loo, M., Tran, N., Haskett, D., Coons, S. & Berens, M. (1996) Dichotomy of astrocytoma migration and proliferation. Int. J. Cancer. 67, 275–282.3.0.CO;2-9>CrossRefGoogle ScholarPubMed
Giga, Y. & Sohr, H. (1991) Abstract Lp estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94.CrossRefGoogle Scholar
Hanahan, D. & Weinberg, R. A. (2011) Hallmarks of cancer: the Next Generation. Cell 144, 646–674.CrossRefGoogle ScholarPubMed
Herrero, M. A. & Velázquez, J. J. L. (1997) A blow-up mechanism for a chemotaxis model. Ann. Scu. Norm. Sup. Pisa Cl. Sci. 24, 633–683.Google Scholar
Hillen, T., Painter, K. J. & Winkler, M. (2018) Global solvability and explicit bounds for a non-local adhesion model. Eur. J. Appl. Math. 29, 645–684.CrossRefGoogle Scholar
Horstmann, D. (2003) From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I. Jahresberichte DMV 105, 103–165.Google Scholar
Horstmann, D. & Winkler, M. (2005) Boundedness vs. blow-up in a chemotaxis system. J. Diff. Equ. 215, 52–107.CrossRefGoogle Scholar
Hu, B. & Tao, Y. (2016) To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production. Math. Models Meth. Appl. Sci. 26, 2111.CrossRefGoogle Scholar
Hunt, A. & Surulescu, C. (2017) A multiscale modeling approach to Glioma invasion with therapy. Vietn. J. Math. 45, 221–240.CrossRefGoogle Scholar
Ishii, G., Ochiai, A. & Neri, S. (2016) Phenotypic and functional heterogeneity of cancer-associated fibroblast within the tumor microenvironment. Adv. Drug Deliv. Rev. 99 B, 186–196.CrossRefGoogle ScholarPubMed
Jerby, L., Wolf, L., Denkert, C., Stein, G.Y., Hilvo, M., Oresic, M., Geiger, T. & Ruppin, E. (2012) Metabolic associations of reduced proliferation and oxidative stress in advanced breast cancer. Cancer Res. 72, 5712–5720.CrossRefGoogle ScholarPubMed
Jolly, L. A., Novitskiy, S., Owens, P., Massoll, N., Cheng, N., Fang, W., Moses, H. L. & Franco, A. T. (2016) Fibroblast-mediated collagen remodeling within the tumor microenvironment facilitates progression of thyroid cancers driven by BrafV600E and Pten loss. Cancer Res. 76, 1804–1813.CrossRefGoogle ScholarPubMed
Kalluri, R. & Zeisberg, M. (2006) Fibroblasts in cancer. Nat. Rev. Cancer 6, 392–401.CrossRefGoogle ScholarPubMed
Ke, Y. & Zheng, J. (2017) A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant. Nonlinearity 31, 4602–4620.CrossRefGoogle Scholar
Krasnianski, M., Painter, K.J., Surulescu, C. & Zhigun, A. Nonlocal and local models for taxis in cell migration: a rigorous limit procedure, arXiv:1908.10287.Google Scholar
Laurençot, Ph. (2019) Global bounded and unbounded solutions to a chemotaxis system with indirect signal production. Discr. Cont. Dyn. Syst. B 24, 6419–6444.Google Scholar
Lecomte, J., Masset, A., Edwards, D.R. & Noël, A. (2011) Tumor fibroblast-associated metalloproteases. In: Mueller, M. M. & Fusenig, N. E. (editors), Tumor-Associated Fibroblasts and their Matrix, Springer.Google Scholar
Madsen, D. H., Jürgensen, H. J., Siersbaek, M.S., Kuczek, M. D., Grey Cloud, L., Liu, S., Behrendt, N., Grøntved, L., Weigert, R. & Bugge, T. H. (2017) Tumor-associated macrophages derived from circulating inflammatory monocytes degrade collagen through cellular uptake. Cell Reports 21, 3662–3671.CrossRefGoogle ScholarPubMed
Matusiewicz, M. (2011) Extracellular matrix remodeling. In: Schwab, M. (editor), Encyclopedia of Cancer, Springer.Google Scholar
Mitra, A. K., Zillhardt, M., Hua, Y., Tiwari, P., Murmann, A. E., Peter, M. E. & Lengyel, E. (2012) MicroRNAs reprogram normal fibroblasts into cancer-associated fibroblasts in ovarian cancer. Cancer Discov. 2, 1100–1108.CrossRefGoogle ScholarPubMed
Neri, S., Miyashita, T., Hashimoto, H., Suda, Y., Ishibashi, M., Kii, H., Watanabe, H., Kuwata, T., Tsuboi, M., Goto, K., Menju, T., Sonobe, M., Date, H., Ochiai, A. & Ishii, G. (2017) Fibroblast-led cancer cell invasion is activated by epithelial-mesenchymal transition through platelet-derived growth factor BB secretion of lung adenocarcinoma. Cancer Lett. 395, 20–30.CrossRefGoogle ScholarPubMed
Pang, P. Y. H. & Wang, Y. (2017). Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant. J. Diff. Equ. 263, 1269–1292.CrossRefGoogle Scholar
Pang, P. Y. H. & Wang, Y. (2018) Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling. Math. Mod. Meth. Appl. Sci. 28, 2211–2235.CrossRefGoogle Scholar
Qiu, S., Mu, C. & Wang, L. (2018) Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production. Comput. Math. Appl. 75, 3213–3223.CrossRefGoogle Scholar
Quail, D. F. & Joyce, J. A. (2013) Microenvironmental regulation of tumor progression and metastasis. Nat. Med. 19, 1423–143.CrossRefGoogle ScholarPubMed
Rivière, B. (2008) Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, SIAM, Philadelphia.CrossRefGoogle Scholar
Spano, D. & Zollo, M. (2012) Tumor microenvironment: a main actor in the metastasis process. Clin. Exp. Metastasis 29, 381–395.CrossRefGoogle ScholarPubMed
Stinner, C., Surulescu, C. & Uatay, A. (2016) Global existence for a go-or-grow multiscale model for tumor invasion with therapy. Math. Models Meth. Appl. Sci. 26, 2163–2201.CrossRefGoogle Scholar
Stinner, C., Surulescu, C. & Winkler, M. (2014) Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. 46, 1969–2007.CrossRefGoogle Scholar
Stuelten, C. H., DaCosta Byfield, S., Arany, P. R., Karpova, T. S., Stetler-Stevenson, W. G. & Roberts, A. B. (2005) Breast cancer cells induce stromal fibroblasts to express MMP-9 via secretion of TNF-alpha and TGF-beta. J. Cell Sci. 118, 2143–2153.CrossRefGoogle ScholarPubMed
Tao, Y. (2009) Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source. J. Math. Anal. Appl. 354, 60–69.CrossRefGoogle Scholar
Tao, Y. Boundedness in a two-dimensional chemotaxis-haptotaxis system. arXiv:1407.7382v1Google Scholar
Tao, L., Huang, G., Song, H., Chen, Y. & Chen, L. (2017) Cancer associated fibroblasts: an essential role in the tumor microenvironment (Review). Oncol. Lett. 14, 2611–2620.CrossRefGoogle Scholar
Tao, Y. & Wang, M. (2009) A combined chemotaxis-haptotaxis system: the role of logistic source. SIAM J. Math. Anal. 41, 1533–1558.CrossRefGoogle Scholar
Tao, Y. & Winkler, M. Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity. J. Diff. Equ. 252(1), 692–715 (2012).CrossRefGoogle Scholar
Tao, Y. & Winkler, M. (2014) Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant. J. Diff. Equ. 257, 784–815.CrossRefGoogle Scholar
Tao, Y. & Winkler, M. (2015) Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion. SIAM J. Math. Anal. 47, 4229–4250.CrossRefGoogle Scholar
Tao, Y. & Winkler, M. (2017) Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production. J. European Math. Soc. 19, 3641–3678.CrossRefGoogle Scholar
Tello, J. I. & Wrzosek, D. (2016) Predator-prey model with diffusion and indirect prey-taxis. Math. Meth. Appl. Sci. 26, 2129–2162.CrossRefGoogle Scholar
Wang, Y. & Ke, Y. (2016) Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions. J. Diff. Equ. 260, 6960–6988.CrossRefGoogle Scholar
Wang, J. & Wang, M. (2019) The dynamics of a predator-prey model with diffusion and indirect prey-taxis. J. Dyn. Diff. Equ. https://doi.org/10.1007/s10884-019-09778-7.CrossRefGoogle Scholar
Widmer, D., Hoek, K., Cheng, P., Eichhoff, O., Biedermann, T., Raaijmakers, M. I. G., Hemmi, S., Dummer, R. & Levesque, M. P. (2013) Hypoxia contributes to melanoma heterogeneity by triggering hif1α-dependent phenotype switching. J. Invest. Dermat. 133, 2436–2443.CrossRefGoogle ScholarPubMed
Winkler, M. (2010) Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905.CrossRefGoogle Scholar
Zhigun, A., Surulescu, C. & Hunt, A. (2018) A strongly degenerate diffusion-haptotaxis model of tumour invasion under the go-or-grow dichotomy hypothesis. Math. Meth. Appl. Sci. 41, 2403–2428.Google Scholar