doi: 10.3934/dcdsb.2020284

Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence

1. 

Kanazawa University, Faculty of Mathematics & Physics, Kakuma, Kanazawa 920-1192, Japan

2. 

University of St. Andrews, School of Mathematics & Statistics, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland, UK

3. 

Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, Germany

4. 

Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany

5. 

Johannes Gutenberg-Universität Mainz, Institut für Mathematik, Staudingerweg 9, 55128 Mainz, Germany

Received  May 2020 Revised  August 2020 Published  September 2020

We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding tissue. Transitions between the two cell subpopulations are influenced by subcellular (receptor binding) dynamics, thus conferring the setting a multiscale character.

We prove global existence of weak solutions to a simplified model version and perform numerical simulations for the full setting under several phenotype switching and motility scenarios. We also compare (via simulations) this model with the corresponding haptotaxis-chemotaxis one featuring indirect chemorepellent production and provide a discussion about possible model extensions and mathematical challenges.

Citation: Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu, Jonas Lenz. Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020284
References:
[1]

N. Bellomo, Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach, Birkhäuser Boston, Inc., Boston, MA, 2008.  Google Scholar

[2]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[3]

X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Zeitschrift für Angewandte Mathematik und Physik, 67 (2016), Art. 11, 13 pp. doi: 10.1007/s00033-015-0601-3.  Google Scholar

[4]

S. B. Carter, Haptotaxis and the mechanism of cell motility, Nature, 213 (1967), 256-260.  doi: 10.1038/213256a0.  Google Scholar

[5]

F. ChalubY. Dolak-StrussP. MarkowichD. OelzC. Schmeiser and A. Soreff, Model hierarchies for cell aggregation by chemotaxis, Mathematical Models and Methods in Applied Sciences, 16 (2006), 1173-1197.  doi: 10.1142/S0218202506001509.  Google Scholar

[6]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734.  doi: 10.1142/S0218202505000947.  Google Scholar

[7]

M. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399.  Google Scholar

[8]

A. ChauvièreT. Hillen and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues, Netw. Heterog. Media, 2 (2007), 333-357.  doi: 10.3934/nhm.2007.2.333.  Google Scholar

[9]

L. Chen, K. Painter, C. Surulescu and A. Zhigun, Mathematical models for cell migration: A nonlocal perspective, Philosophical Transactions of the Royal Society B: Biological Sciences, 375 (2020), 20190379. Google Scholar

[10]

Z. Chen and Y. Tao, Large-data solutions in a three-dimensional chemotaxis-haptotaxis system with remodeling of non-diffusible attractant: The role of sub-linear production of diffusible signal, Acta Applicandae Mathematicae, 163 (2018), 129-143.  doi: 10.1007/s10440-018-0216-8.  Google Scholar

[11]

M. Conte and C. Surulescu, Mathematical modeling of vascularized glioma development under the go-or-grow dichotomy, arXiv: 2007.12204. Google Scholar

[12]

G. Corbin, C. Engwer, A. Klar, J. Nieto, J. Soler, C. Surulescu and M. Wenske, On a model for glioma invasion with anisotropy- and hypoxia-triggered motility enhancement, arXiv: 2006.12322. Google Scholar

[13]

G. CorbinA. HuntA. KlarF. Schneider and C. Surulescu, Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum, Mathematical Models and Methods in Applied Sciences, 28 (2018), 1771-1800.  doi: 10.1142/S0218202518400055.  Google Scholar

[14]

P. DomschkeD. TrucuA. Gerisch and M. A. J. Chaplain, Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns, Journal of Theoretical Biology, 361 (2014), 41-60.  doi: 10.1016/j.jtbi.2014.07.010.  Google Scholar

[15]

P. DomschkeD. TrucuA. Gerisch and M. A. J. Chaplain, Structured models of cell migration incorporating molecular binding processes, J. Math. Biol., 75 (2017), 1517-1561.  doi: 10.1007/s00285-017-1120-y.  Google Scholar

[16]

C. EngwerT. HillenM. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, Journal of Mathematical Biology, 71 (2014), 551-582.  doi: 10.1007/s00285-014-0822-7.  Google Scholar

[17]

C. EngwerA. Hunt and C. Surulescu, Effective equations for anisotropic glioma spread with proliferation: A multiscale approach and comparisons with previous settings, Mathematical Medicine and Biology, 33 (2015), 435-459.  doi: 10.1093/imammb/dqv030.  Google Scholar

[18]

C. EngwerM. Knappitsch and C. Surulescu, A multiscale model for glioma spread including cell-tissue interactions and proliferation, Mathematical Biosciences and Engineering, 13 (2016), 443-460.  doi: 10.3934/mbe.2015011.  Google Scholar

[19]

C. EngwerC. Stinner and C. Surulescu, On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation, Math. Models Methods Appl. Sci., 27 (2017), 1355-1390.  doi: 10.1142/S0218202517400188.  Google Scholar

[20]

S. M. Frisch and H. Francis, Disruption of epithelial cell-matrix interactions induces apoptosis, J. Cell Biol., 124 (1994), 619-626.  doi: 10.1083/jcb.124.4.619.  Google Scholar

[21]

A. Gerisch and M. A. J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion, J. Theoret. Biol., 250 (2008), 684-704.  doi: 10.1016/j.jtbi.2007.10.026.  Google Scholar

[22]

A. GieseR. BjerkvigM. Berens and M. Westphal, Cost of migration: Invasion of malignant gliomas and implications for treatment, Journal of Clinical Oncology, 21 (2003), 1624-1636.  doi: 10.1200/JCO.2003.05.063.  Google Scholar

[23]

A. GieseL. KluweB. LaubeH. MeissnerM. E. Berens and M. Westphal, Migration of human glioma cells on myelin, Neurosurgery, 38 (1996), 755-764.  doi: 10.1227/00006123-199604000-00026.  Google Scholar

[24]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674.  doi: 10.1016/j.cell.2011.02.013.  Google Scholar

[25]

D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society (EMS), Zürich, 2008.  Google Scholar

[26]

H. HatzikirouD. BasantaM. SimonK. Schaller and A. Deutsch, Go or grow: The key to the emergence of invasion in tumour progression?, Mathematical Medicine and Biology, 29 (2010), 49-65.  doi: 10.1093/imammb/dqq011.  Google Scholar

[27]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.  Google Scholar

[28]

T. Hillen, A classification of spikes and plateaus, SIAM Rev., 49 (2007), 35-51.  doi: 10.1137/050632427.  Google Scholar

[29]

T. Hillen and K. J. A. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2008), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[30]

T. HillenK. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198.  doi: 10.1142/S0218202512500480.  Google Scholar

[31]

S. Hiremath and C. Surulescu, A stochastic multiscale model for acid mediated cancer invasion, Nonlinear Analysis: Real World Applications, 22 (2015), 176-205.  doi: 10.1016/j.nonrwa.2014.08.008.  Google Scholar

[32]

S. A. Hiremath and C. Surulescu, A stochastic model featuring acid-induced gaps during tumor progression, Nonlinearity, 29 (2016), 851-914.  doi: 10.1088/0951-7715/29/3/851.  Google Scholar

[33]

K. S. HoekO. M. EichhoffN. C. SchlegelU. DobbelingN. KobertL. SchaererS. Hemmi and R. Dummer, In vivo switching of human melanoma cells between proliferative and invasive states, Cancer Research, 68 (2008), 650-656.  doi: 10.1158/0008-5472.CAN-07-2491.  Google Scholar

[34]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[35]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[36]

X. HuL. WangC. Mu and L. Li, Boundedness in a three-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Comptes Rendus Mathematique, 355 (2017), 181-186.  doi: 10.1016/j.crma.2016.12.005.  Google Scholar

[37]

A. Hunt and C. Surulescu, A multiscale modeling approach to glioma invasion with therapy, Vietnam Journal of Mathematics, 45 (2016), 221-240.  doi: 10.1007/s10013-016-0223-x.  Google Scholar

[38]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[39]

Z. Jia and Z. Yang, Global boundedness to a chemotaxis-haptotaxis model with nonlinear diffusion, Applied Mathematics Letters, 103 (2020), 106192, 6 pp. doi: 10.1016/j.aml.2019.106192.  Google Scholar

[40]

C. Jin, Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bulletin of the London Mathematical Society, 50 (2018), 598-618.  doi: 10.1112/blms.12160.  Google Scholar

[41]

Y. Ke and J. Zheng, A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, Nonlinearity, 31 (2018), 4602-4620.  doi: 10.1088/1361-6544/aad307.  Google Scholar

[42]

J. Kelkel and C. Surulescu, On some models for cancer cell migration through tissue networks, Mathematical Biosciences and Engineering, 8 (2011), 575-589.  doi: 10.3934/mbe.2011.8.575.  Google Scholar

[43]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150017, 25 pp. doi: 10.1142/S0218202511500175.  Google Scholar

[44]

E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[45]

C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), 139-181.  doi: 10.1016/S0168-9274(02)00138-1.  Google Scholar

[46]

P. KleihuesF. SoylemezogluB. SchäubleB. Scheithauer and P. Burger, Histopathology, classification and grading of gliomas, Glia, 5 (1995), 211-221.  doi: 10.1002/glia.440150303.  Google Scholar

[47]

D. A. Knopoff, J. Nieto and L. Urrutia, Numerical simulation of a multiscale cell motility model based on the kinetic theory of active particles, Symmetry, 11 (2019), 1003. doi: 10.3390/sym11081003.  Google Scholar

[48]

H. KnútsdóttirE. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, Journal of Theoretical Biology, 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.  Google Scholar

[49]

N. KolbeJ. Kat'uchováN. SfakianakisN. Hellmann and M. Lukáčová-Medvid'ová, A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model, Applied Mathematics and Computation, 273 (2016), 353-376.  doi: 10.1016/j.amc.2015.08.023.  Google Scholar

[50]

N. KolbeM. Lukáčová-Medvid'ováN. Sfakianakis and B. Wiebe, Numerical simulation of a contractivity based multiscale cancer invasion model, Multiscale Models in Mechano and Tumor Biology, Lect. Notes Comput. Sci. Eng., Springer, Cham, 122 (2017), 73-91.  doi: 10.1007/978-3-319-73371-5_4.  Google Scholar

[51]

M. Krasnianski, K. Painter, C. Surulescu and A. Zhigun, Nonlocal and local models for taxis in cell migration: A rigorous limit procedure, arXiv: 1908.10287v2. Google Scholar

[52]

J. Lenz, Global Existence for a Tumor Invasion Model with Repellent Taxis and Therapy, Master thesis, TU Darmstadt, 2019 Google Scholar

[53]

J. LiY. Ke and Y. Wang, Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Analysis: Real World Applications, 39 (2018), 261-277.  doi: 10.1016/j.nonrwa.2017.07.002.  Google Scholar

[54]

J. Li and Y. Wang, Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, Journal of Mathematical Analysis and Applications, 467 (2018), 1066-1079.  doi: 10.1016/j.jmaa.2018.07.051.  Google Scholar

[55]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.  doi: 10.1088/0951-7715/29/5/1564.  Google Scholar

[56]

J. Liu and Y. Wang, A quasilinear chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, Mathematical Methods in the Applied Sciences, 40 (2017), 2107-2121.  doi: 10.1002/mma.4126.  Google Scholar

[57]

J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), Art. 21, 33 pp. doi: 10.1007/s00033-016-0620-8.  Google Scholar

[58]

J. LoganP. WhiteB. Bentz and J. Powell, Model analysis of spatial patterns in mountain pine beetle outbreaks, Theoretical Population Biology, 53 (1998), 236-255.   Google Scholar

[59]

T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2383-2436.  doi: 10.1142/S0218202514500249.  Google Scholar

[60]

M. Luca, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.   Google Scholar

[61]

P. K. Maini, Spatial and spatio-temporal patterns in a cell-haptotaxis model, Journal of Mathematical Biology, 27 (1989), 507-522.  doi: 10.1007/BF00288431.  Google Scholar

[62]

A. MalandrinoM. MakR. Kamm and E. Moeendarbary, Complex mechanics of the heterogeneous extracellular matrix in cancer, Extreme Mechanics Letters, 21 (2018), 25-34.   Google Scholar

[63]

D. Mallet, Mathematical modelling of the role of haptotaxis in tumour growth and invasion, PhD thesis, Queensland University of Technology. Google Scholar

[64]

MATLAB, Version 9.7.0.1216025 (R2019b) Update 1, The MathWorks Inc., Natick, Massachusetts, 2019b. Google Scholar

[65]

Y. MatsukadoC. MacCarty and J. Kernohan, The growth of glioblastoma multiforme (astrocytomas, grades 3 and 4) in neurosurgical practice, Journal of Neurosurgery, 18 (1961), 636-644.   Google Scholar

[66]

G. MeralC. Stinner and C. Surulescu, A multiscale model for acid-mediated tumor invasion: Therapy approaches, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 135-142.   Google Scholar

[67]

G. MeralC. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213.  doi: 10.3934/dcdsb.2015.20.189.  Google Scholar

[68]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

[69]

C. Morales-Rodrigo and J. I. Tello, Global existence and asymptotic behavior of a tumor angiogenesis model with chemotaxis and haptotaxis, Mathematical Models and Methods in Applied Sciences, 24 (2014), 427-464.  doi: 10.1142/S0218202513500553.  Google Scholar

[70]

J. Nieto and L. Urrutia, A multiscale model of cell mobility: From a kinetic to a hydrodynamic description, Journal of Mathematical Analysis and Applications, 433 (2016), 1055-1071.  doi: 10.1016/j.jmaa.2015.08.042.  Google Scholar

[71]

M. Orme and M. Chaplain, A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, Mathematical Medicine and Biology, 13 (1996), 73-98.   Google Scholar

[72]

K. J. Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, Journal of Theoretical Biology, 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019.  Google Scholar

[73]

K. J. PainterP. K. Maini and H. G. Othmer, Development and applications of a model for cellular response to multiple chemotactic cues, Journal of Mathematical Biology, 41 (2000), 285-314.  doi: 10.1007/s002850000035.  Google Scholar

[74]

P. Y. H. Pang and Y. Wang, Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, Journal of Differential Equations, 263 (2017), 1269-1292.  doi: 10.1016/j.jde.2017.03.016.  Google Scholar

[75]

P. Y. H. Pang and Y. Wang, Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2211-2235.  doi: 10.1142/S0218202518400134.  Google Scholar

[76]

P. Y. H. Pang and Y. Wang, Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis-haptotaxis, Mathematical Models and Methods in Applied Sciences, 29 (2019), 1387-1412.  doi: 10.1142/S0218202519500246.  Google Scholar

[77]

L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systemswith relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4.  Google Scholar

[78]

J. R. Potts and M. A. Lewis, Spatial memory and taxis-driven pattern formation in model ecosystems, Bulletin of Mathematical Biology, 81 (2019), 2725-2747.  doi: 10.1007/s11538-019-00626-9.  Google Scholar

[79]

N. SfakianakisN. Kolbe and M. Lukáčová-Medvid'ová, A multiscale approach to the migration of cancer stem cells: Mathematical modelling and simulations, Bull. Math. Biol., 79 (2016), 209-235.  doi: 10.1007/s11538-016-0233-6.  Google Scholar

[80]

N. SfakianakisA. Madzvamuse and M. A. J. Chaplain, A hybrid multiscale model for cancer invasion of the extracellular matrix, Multiscale Model Sim., 18 (2020), 824-850.  doi: 10.1137/18M1189026.  Google Scholar

[81]

A. Silchenko and P. Tass, Mathematical modeling of chemotaxis and glial scarring around implanted electrodes, New Journal of Physics, 17 (2015), 023009. Google Scholar

[82]

C. StinnerC. Surulescu and G. Meral, A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321.  doi: 10.1093/imamat/hxu055.  Google Scholar

[83]

C. StinnerC. Surulescu and A. Uatay, Global existence for a go-or-grow multiscale model for tumor invasion with therapy, Math. Models Methods Appl. Sci., 26 (2016), 2163-2201.  doi: 10.1142/S021820251640011X.  Google Scholar

[84]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X.  Google Scholar

[85]

M. StubbsP. McSheehyJ. Griffiths and L. Bashford, Causes and consequences of tumour acidity and implications for treatment, Molecular Medicine Today, 6 (2000), 15-19.  doi: 10.1016/S1357-4310(99)01615-9.  Google Scholar

[86]

C. Surulescu and M. Winkler, 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), European Journal of Applied Mathematics, in print, arXiv: 1904.11210. Google Scholar

[87]

S. TakumiJ. VerdoneJ. HuangU. KahlertJ. HernandezG. TorgaJ. ZarifT. EpsteinR. GatenbyA. McCartneyJ. ElisseeffS. MooneyS. An and K. Pienta, Glycolysis is the primary bioenergetic pathway for cell motility and cytoskeletal remodeling in human prostate and breast cancer cells, Oncotarget, 6 (2015), 130-143.   Google Scholar

[88]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, Journal of Mathematical Analysis and Applications, 354 (2009), 60-69.  doi: 10.1016/j.jmaa.2008.12.039.  Google Scholar

[89]

Y. Tao and C. Cui, A density-dependent chemotaxis-haptotaxis system modeling cancer invasion, Journal of Mathematical Analysis and Applications, 367 (2010), 612-624.  doi: 10.1016/j.jmaa.2010.02.015.  Google Scholar

[90]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.  doi: 10.1088/0951-7715/21/10/002.  Google Scholar

[91]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[92]

Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM Journal on Mathematical Analysis, 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.  Google Scholar

[93]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047-2067.  doi: 10.3934/cpaa.2019092.  Google Scholar

[94]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[95]

M. Van der HeidenL. Cantley and C. Thompson, Understanding the Warburg effect: The metabolic requirements of cell proliferation, Science, 324 (2009), 1029-1033.   Google Scholar

[96]

H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 13 (1992), 631-644.  doi: 10.1137/0913035.  Google Scholar

[97]

B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, Journal of Computational Physics, 32 (1979), 101-136.   Google Scholar

[98]

D. Vig and C. Wolgemuth, Spatiotemporal evolution of erythema migrans, the hallmark rash of lyme disease, Biophysical Journal, 106 (2014), 763-768.   Google Scholar

[99]

Y. Wang, Boundedness in a multi-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Applied Mathematics Letters, 59 (2016), 122-126.  doi: 10.1016/j.aml.2016.03.019.  Google Scholar

[100]

Y. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Journal of Differential Equations, 260 (2016), 1975-1989.  doi: 10.1016/j.jde.2015.09.051.  Google Scholar

[101]

Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, Journal of Differential Equations, 260 (2016), 6960-6988.  doi: 10.1016/j.jde.2016.01.017.  Google Scholar

[102]

B. A. WebbM. ChimentiM. P. Jacobson and D. L. Barber, Dysregulated pH: A perfect storm for cancer progression, Nature Reviews Cancer, 11 (2011), 671-677.  doi: 10.1038/nrc3110.  Google Scholar

[103]

M. Winkler, Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, Journal de Mathématiques Pures et Appliquées, 112 (2018), 118-169.  doi: 10.1016/j.matpur.2017.11.002.  Google Scholar

[104]

M. Winkler and C. Surulescu, Global weak solutions to a strongly degenerate haptotaxis model, Comm. Math. Sci., 15 (2017), 1581-1616.  doi: 10.4310/CMS.2017.v15.n6.a5.  Google Scholar

[105]

T. Xiang and J. Zheng, A new result for 2D boundedness of solutions to a chemotaxis-haptotaxis model with/without sub-logistic source, Nonlinearity, 32 (2019), 4890-4911.  doi: 10.1088/1361-6544/ab41d5.  Google Scholar

[106]

G.-F. Xiong and R. Xu, Function of cancer cell-derived extracellular matrix in tumor progression, Journal of Cancer Metastasis and Treatment, 2 (2016), 357-364. doi: 10.20517/2394-4722.2016.08.  Google Scholar

[107]

P. ZhengC. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Discr. Cont. Dyn. Syst. A, 36 (2016), 1737-1757.  doi: 10.3934/dcds.2016.36.1737.  Google Scholar

[108]

P.-P. ZhengL.-A. SeverijnenM. van der WeidenR. Willemsen and J. Kros, Cell proliferation and migration are mutually exclusive cellular phenomena in vivo: Implications for cancer therapeutic strategies, Cell Cycle, 8 (2009), 950-951.  doi: 10.4161/cc.8.6.7851.  Google Scholar

[109]

A. ZhigunC. Surulescu and A. Hunt, A strongly degenerate diffusion-haptotaxis model of tumour invasion under the go-or-grow dichotomy hypothesis, Math Meth Appl Sci., 41 (2018), 2403-2428.  doi: 10.1002/mma.4749.  Google Scholar

[110]

A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), Art. 146, 29 pp. doi: 10.1007/s00033-016-0741-0.  Google Scholar

show all references

References:
[1]

N. Bellomo, Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach, Birkhäuser Boston, Inc., Boston, MA, 2008.  Google Scholar

[2]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[3]

X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Zeitschrift für Angewandte Mathematik und Physik, 67 (2016), Art. 11, 13 pp. doi: 10.1007/s00033-015-0601-3.  Google Scholar

[4]

S. B. Carter, Haptotaxis and the mechanism of cell motility, Nature, 213 (1967), 256-260.  doi: 10.1038/213256a0.  Google Scholar

[5]

F. ChalubY. Dolak-StrussP. MarkowichD. OelzC. Schmeiser and A. Soreff, Model hierarchies for cell aggregation by chemotaxis, Mathematical Models and Methods in Applied Sciences, 16 (2006), 1173-1197.  doi: 10.1142/S0218202506001509.  Google Scholar

[6]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734.  doi: 10.1142/S0218202505000947.  Google Scholar

[7]

M. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399.  Google Scholar

[8]

A. ChauvièreT. Hillen and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues, Netw. Heterog. Media, 2 (2007), 333-357.  doi: 10.3934/nhm.2007.2.333.  Google Scholar

[9]

L. Chen, K. Painter, C. Surulescu and A. Zhigun, Mathematical models for cell migration: A nonlocal perspective, Philosophical Transactions of the Royal Society B: Biological Sciences, 375 (2020), 20190379. Google Scholar

[10]

Z. Chen and Y. Tao, Large-data solutions in a three-dimensional chemotaxis-haptotaxis system with remodeling of non-diffusible attractant: The role of sub-linear production of diffusible signal, Acta Applicandae Mathematicae, 163 (2018), 129-143.  doi: 10.1007/s10440-018-0216-8.  Google Scholar

[11]

M. Conte and C. Surulescu, Mathematical modeling of vascularized glioma development under the go-or-grow dichotomy, arXiv: 2007.12204. Google Scholar

[12]

G. Corbin, C. Engwer, A. Klar, J. Nieto, J. Soler, C. Surulescu and M. Wenske, On a model for glioma invasion with anisotropy- and hypoxia-triggered motility enhancement, arXiv: 2006.12322. Google Scholar

[13]

G. CorbinA. HuntA. KlarF. Schneider and C. Surulescu, Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum, Mathematical Models and Methods in Applied Sciences, 28 (2018), 1771-1800.  doi: 10.1142/S0218202518400055.  Google Scholar

[14]

P. DomschkeD. TrucuA. Gerisch and M. A. J. Chaplain, Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns, Journal of Theoretical Biology, 361 (2014), 41-60.  doi: 10.1016/j.jtbi.2014.07.010.  Google Scholar

[15]

P. DomschkeD. TrucuA. Gerisch and M. A. J. Chaplain, Structured models of cell migration incorporating molecular binding processes, J. Math. Biol., 75 (2017), 1517-1561.  doi: 10.1007/s00285-017-1120-y.  Google Scholar

[16]

C. EngwerT. HillenM. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, Journal of Mathematical Biology, 71 (2014), 551-582.  doi: 10.1007/s00285-014-0822-7.  Google Scholar

[17]

C. EngwerA. Hunt and C. Surulescu, Effective equations for anisotropic glioma spread with proliferation: A multiscale approach and comparisons with previous settings, Mathematical Medicine and Biology, 33 (2015), 435-459.  doi: 10.1093/imammb/dqv030.  Google Scholar

[18]

C. EngwerM. Knappitsch and C. Surulescu, A multiscale model for glioma spread including cell-tissue interactions and proliferation, Mathematical Biosciences and Engineering, 13 (2016), 443-460.  doi: 10.3934/mbe.2015011.  Google Scholar

[19]

C. EngwerC. Stinner and C. Surulescu, On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation, Math. Models Methods Appl. Sci., 27 (2017), 1355-1390.  doi: 10.1142/S0218202517400188.  Google Scholar

[20]

S. M. Frisch and H. Francis, Disruption of epithelial cell-matrix interactions induces apoptosis, J. Cell Biol., 124 (1994), 619-626.  doi: 10.1083/jcb.124.4.619.  Google Scholar

[21]

A. Gerisch and M. A. J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion, J. Theoret. Biol., 250 (2008), 684-704.  doi: 10.1016/j.jtbi.2007.10.026.  Google Scholar

[22]

A. GieseR. BjerkvigM. Berens and M. Westphal, Cost of migration: Invasion of malignant gliomas and implications for treatment, Journal of Clinical Oncology, 21 (2003), 1624-1636.  doi: 10.1200/JCO.2003.05.063.  Google Scholar

[23]

A. GieseL. KluweB. LaubeH. MeissnerM. E. Berens and M. Westphal, Migration of human glioma cells on myelin, Neurosurgery, 38 (1996), 755-764.  doi: 10.1227/00006123-199604000-00026.  Google Scholar

[24]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674.  doi: 10.1016/j.cell.2011.02.013.  Google Scholar

[25]

D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society (EMS), Zürich, 2008.  Google Scholar

[26]

H. HatzikirouD. BasantaM. SimonK. Schaller and A. Deutsch, Go or grow: The key to the emergence of invasion in tumour progression?, Mathematical Medicine and Biology, 29 (2010), 49-65.  doi: 10.1093/imammb/dqq011.  Google Scholar

[27]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.  Google Scholar

[28]

T. Hillen, A classification of spikes and plateaus, SIAM Rev., 49 (2007), 35-51.  doi: 10.1137/050632427.  Google Scholar

[29]

T. Hillen and K. J. A. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2008), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[30]

T. HillenK. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198.  doi: 10.1142/S0218202512500480.  Google Scholar

[31]

S. Hiremath and C. Surulescu, A stochastic multiscale model for acid mediated cancer invasion, Nonlinear Analysis: Real World Applications, 22 (2015), 176-205.  doi: 10.1016/j.nonrwa.2014.08.008.  Google Scholar

[32]

S. A. Hiremath and C. Surulescu, A stochastic model featuring acid-induced gaps during tumor progression, Nonlinearity, 29 (2016), 851-914.  doi: 10.1088/0951-7715/29/3/851.  Google Scholar

[33]

K. S. HoekO. M. EichhoffN. C. SchlegelU. DobbelingN. KobertL. SchaererS. Hemmi and R. Dummer, In vivo switching of human melanoma cells between proliferative and invasive states, Cancer Research, 68 (2008), 650-656.  doi: 10.1158/0008-5472.CAN-07-2491.  Google Scholar

[34]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[35]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[36]

X. HuL. WangC. Mu and L. Li, Boundedness in a three-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Comptes Rendus Mathematique, 355 (2017), 181-186.  doi: 10.1016/j.crma.2016.12.005.  Google Scholar

[37]

A. Hunt and C. Surulescu, A multiscale modeling approach to glioma invasion with therapy, Vietnam Journal of Mathematics, 45 (2016), 221-240.  doi: 10.1007/s10013-016-0223-x.  Google Scholar

[38]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[39]

Z. Jia and Z. Yang, Global boundedness to a chemotaxis-haptotaxis model with nonlinear diffusion, Applied Mathematics Letters, 103 (2020), 106192, 6 pp. doi: 10.1016/j.aml.2019.106192.  Google Scholar

[40]

C. Jin, Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bulletin of the London Mathematical Society, 50 (2018), 598-618.  doi: 10.1112/blms.12160.  Google Scholar

[41]

Y. Ke and J. Zheng, A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, Nonlinearity, 31 (2018), 4602-4620.  doi: 10.1088/1361-6544/aad307.  Google Scholar

[42]

J. Kelkel and C. Surulescu, On some models for cancer cell migration through tissue networks, Mathematical Biosciences and Engineering, 8 (2011), 575-589.  doi: 10.3934/mbe.2011.8.575.  Google Scholar

[43]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150017, 25 pp. doi: 10.1142/S0218202511500175.  Google Scholar

[44]

E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[45]

C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), 139-181.  doi: 10.1016/S0168-9274(02)00138-1.  Google Scholar

[46]

P. KleihuesF. SoylemezogluB. SchäubleB. Scheithauer and P. Burger, Histopathology, classification and grading of gliomas, Glia, 5 (1995), 211-221.  doi: 10.1002/glia.440150303.  Google Scholar

[47]

D. A. Knopoff, J. Nieto and L. Urrutia, Numerical simulation of a multiscale cell motility model based on the kinetic theory of active particles, Symmetry, 11 (2019), 1003. doi: 10.3390/sym11081003.  Google Scholar

[48]

H. KnútsdóttirE. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, Journal of Theoretical Biology, 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.  Google Scholar

[49]

N. KolbeJ. Kat'uchováN. SfakianakisN. Hellmann and M. Lukáčová-Medvid'ová, A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model, Applied Mathematics and Computation, 273 (2016), 353-376.  doi: 10.1016/j.amc.2015.08.023.  Google Scholar

[50]

N. KolbeM. Lukáčová-Medvid'ováN. Sfakianakis and B. Wiebe, Numerical simulation of a contractivity based multiscale cancer invasion model, Multiscale Models in Mechano and Tumor Biology, Lect. Notes Comput. Sci. Eng., Springer, Cham, 122 (2017), 73-91.  doi: 10.1007/978-3-319-73371-5_4.  Google Scholar

[51]

M. Krasnianski, K. Painter, C. Surulescu and A. Zhigun, Nonlocal and local models for taxis in cell migration: A rigorous limit procedure, arXiv: 1908.10287v2. Google Scholar

[52]

J. Lenz, Global Existence for a Tumor Invasion Model with Repellent Taxis and Therapy, Master thesis, TU Darmstadt, 2019 Google Scholar

[53]

J. LiY. Ke and Y. Wang, Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Analysis: Real World Applications, 39 (2018), 261-277.  doi: 10.1016/j.nonrwa.2017.07.002.  Google Scholar

[54]

J. Li and Y. Wang, Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, Journal of Mathematical Analysis and Applications, 467 (2018), 1066-1079.  doi: 10.1016/j.jmaa.2018.07.051.  Google Scholar

[55]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.  doi: 10.1088/0951-7715/29/5/1564.  Google Scholar

[56]

J. Liu and Y. Wang, A quasilinear chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, Mathematical Methods in the Applied Sciences, 40 (2017), 2107-2121.  doi: 10.1002/mma.4126.  Google Scholar

[57]

J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), Art. 21, 33 pp. doi: 10.1007/s00033-016-0620-8.  Google Scholar

[58]

J. LoganP. WhiteB. Bentz and J. Powell, Model analysis of spatial patterns in mountain pine beetle outbreaks, Theoretical Population Biology, 53 (1998), 236-255.   Google Scholar

[59]

T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2383-2436.  doi: 10.1142/S0218202514500249.  Google Scholar

[60]

M. Luca, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.   Google Scholar

[61]

P. K. Maini, Spatial and spatio-temporal patterns in a cell-haptotaxis model, Journal of Mathematical Biology, 27 (1989), 507-522.  doi: 10.1007/BF00288431.  Google Scholar

[62]

A. MalandrinoM. MakR. Kamm and E. Moeendarbary, Complex mechanics of the heterogeneous extracellular matrix in cancer, Extreme Mechanics Letters, 21 (2018), 25-34.   Google Scholar

[63]

D. Mallet, Mathematical modelling of the role of haptotaxis in tumour growth and invasion, PhD thesis, Queensland University of Technology. Google Scholar

[64]

MATLAB, Version 9.7.0.1216025 (R2019b) Update 1, The MathWorks Inc., Natick, Massachusetts, 2019b. Google Scholar

[65]

Y. MatsukadoC. MacCarty and J. Kernohan, The growth of glioblastoma multiforme (astrocytomas, grades 3 and 4) in neurosurgical practice, Journal of Neurosurgery, 18 (1961), 636-644.   Google Scholar

[66]

G. MeralC. Stinner and C. Surulescu, A multiscale model for acid-mediated tumor invasion: Therapy approaches, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 135-142.   Google Scholar

[67]

G. MeralC. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213.  doi: 10.3934/dcdsb.2015.20.189.  Google Scholar

[68]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

[69]

C. Morales-Rodrigo and J. I. Tello, Global existence and asymptotic behavior of a tumor angiogenesis model with chemotaxis and haptotaxis, Mathematical Models and Methods in Applied Sciences, 24 (2014), 427-464.  doi: 10.1142/S0218202513500553.  Google Scholar

[70]

J. Nieto and L. Urrutia, A multiscale model of cell mobility: From a kinetic to a hydrodynamic description, Journal of Mathematical Analysis and Applications, 433 (2016), 1055-1071.  doi: 10.1016/j.jmaa.2015.08.042.  Google Scholar

[71]

M. Orme and M. Chaplain, A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, Mathematical Medicine and Biology, 13 (1996), 73-98.   Google Scholar

[72]

K. J. Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, Journal of Theoretical Biology, 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019.  Google Scholar

[73]

K. J. PainterP. K. Maini and H. G. Othmer, Development and applications of a model for cellular response to multiple chemotactic cues, Journal of Mathematical Biology, 41 (2000), 285-314.  doi: 10.1007/s002850000035.  Google Scholar

[74]

P. Y. H. Pang and Y. Wang, Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, Journal of Differential Equations, 263 (2017), 1269-1292.  doi: 10.1016/j.jde.2017.03.016.  Google Scholar

[75]

P. Y. H. Pang and Y. Wang, Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2211-2235.  doi: 10.1142/S0218202518400134.  Google Scholar

[76]

P. Y. H. Pang and Y. Wang, Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis-haptotaxis, Mathematical Models and Methods in Applied Sciences, 29 (2019), 1387-1412.  doi: 10.1142/S0218202519500246.  Google Scholar

[77]

L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systemswith relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4.  Google Scholar

[78]

J. R. Potts and M. A. Lewis, Spatial memory and taxis-driven pattern formation in model ecosystems, Bulletin of Mathematical Biology, 81 (2019), 2725-2747.  doi: 10.1007/s11538-019-00626-9.  Google Scholar

[79]

N. SfakianakisN. Kolbe and M. Lukáčová-Medvid'ová, A multiscale approach to the migration of cancer stem cells: Mathematical modelling and simulations, Bull. Math. Biol., 79 (2016), 209-235.  doi: 10.1007/s11538-016-0233-6.  Google Scholar

[80]

N. SfakianakisA. Madzvamuse and M. A. J. Chaplain, A hybrid multiscale model for cancer invasion of the extracellular matrix, Multiscale Model Sim., 18 (2020), 824-850.  doi: 10.1137/18M1189026.  Google Scholar

[81]

A. Silchenko and P. Tass, Mathematical modeling of chemotaxis and glial scarring around implanted electrodes, New Journal of Physics, 17 (2015), 023009. Google Scholar

[82]

C. StinnerC. Surulescu and G. Meral, A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321.  doi: 10.1093/imamat/hxu055.  Google Scholar

[83]

C. StinnerC. Surulescu and A. Uatay, Global existence for a go-or-grow multiscale model for tumor invasion with therapy, Math. Models Methods Appl. Sci., 26 (2016), 2163-2201.  doi: 10.1142/S021820251640011X.  Google Scholar

[84]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X.  Google Scholar

[85]

M. StubbsP. McSheehyJ. Griffiths and L. Bashford, Causes and consequences of tumour acidity and implications for treatment, Molecular Medicine Today, 6 (2000), 15-19.  doi: 10.1016/S1357-4310(99)01615-9.  Google Scholar

[86]

C. Surulescu and M. Winkler, 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), European Journal of Applied Mathematics, in print, arXiv: 1904.11210. Google Scholar

[87]

S. TakumiJ. VerdoneJ. HuangU. KahlertJ. HernandezG. TorgaJ. ZarifT. EpsteinR. GatenbyA. McCartneyJ. ElisseeffS. MooneyS. An and K. Pienta, Glycolysis is the primary bioenergetic pathway for cell motility and cytoskeletal remodeling in human prostate and breast cancer cells, Oncotarget, 6 (2015), 130-143.   Google Scholar

[88]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, Journal of Mathematical Analysis and Applications, 354 (2009), 60-69.  doi: 10.1016/j.jmaa.2008.12.039.  Google Scholar

[89]

Y. Tao and C. Cui, A density-dependent chemotaxis-haptotaxis system modeling cancer invasion, Journal of Mathematical Analysis and Applications, 367 (2010), 612-624.  doi: 10.1016/j.jmaa.2010.02.015.  Google Scholar

[90]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.  doi: 10.1088/0951-7715/21/10/002.  Google Scholar

[91]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[92]

Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM Journal on Mathematical Analysis, 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.  Google Scholar

[93]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047-2067.  doi: 10.3934/cpaa.2019092.  Google Scholar

[94]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[95]

M. Van der HeidenL. Cantley and C. Thompson, Understanding the Warburg effect: The metabolic requirements of cell proliferation, Science, 324 (2009), 1029-1033.   Google Scholar

[96]

H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 13 (1992), 631-644.  doi: 10.1137/0913035.  Google Scholar

[97]

B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, Journal of Computational Physics, 32 (1979), 101-136.   Google Scholar

[98]

D. Vig and C. Wolgemuth, Spatiotemporal evolution of erythema migrans, the hallmark rash of lyme disease, Biophysical Journal, 106 (2014), 763-768.   Google Scholar

[99]

Y. Wang, Boundedness in a multi-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Applied Mathematics Letters, 59 (2016), 122-126.  doi: 10.1016/j.aml.2016.03.019.  Google Scholar

[100]

Y. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Journal of Differential Equations, 260 (2016), 1975-1989.  doi: 10.1016/j.jde.2015.09.051.  Google Scholar

[101]

Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, Journal of Differential Equations, 260 (2016), 6960-6988.  doi: 10.1016/j.jde.2016.01.017.  Google Scholar

[102]

B. A. WebbM. ChimentiM. P. Jacobson and D. L. Barber, Dysregulated pH: A perfect storm for cancer progression, Nature Reviews Cancer, 11 (2011), 671-677.  doi: 10.1038/nrc3110.  Google Scholar

[103]

M. Winkler, Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, Journal de Mathématiques Pures et Appliquées, 112 (2018), 118-169.  doi: 10.1016/j.matpur.2017.11.002.  Google Scholar

[104]

M. Winkler and C. Surulescu, Global weak solutions to a strongly degenerate haptotaxis model, Comm. Math. Sci., 15 (2017), 1581-1616.  doi: 10.4310/CMS.2017.v15.n6.a5.  Google Scholar

[105]

T. Xiang and J. Zheng, A new result for 2D boundedness of solutions to a chemotaxis-haptotaxis model with/without sub-logistic source, Nonlinearity, 32 (2019), 4890-4911.  doi: 10.1088/1361-6544/ab41d5.  Google Scholar

[106]

G.-F. Xiong and R. Xu, Function of cancer cell-derived extracellular matrix in tumor progression, Journal of Cancer Metastasis and Treatment, 2 (2016), 357-364. doi: 10.20517/2394-4722.2016.08.  Google Scholar

[107]

P. ZhengC. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Discr. Cont. Dyn. Syst. A, 36 (2016), 1737-1757.  doi: 10.3934/dcds.2016.36.1737.  Google Scholar

[108]

P.-P. ZhengL.-A. SeverijnenM. van der WeidenR. Willemsen and J. Kros, Cell proliferation and migration are mutually exclusive cellular phenomena in vivo: Implications for cancer therapeutic strategies, Cell Cycle, 8 (2009), 950-951.  doi: 10.4161/cc.8.6.7851.  Google Scholar

[109]

A. ZhigunC. Surulescu and A. Hunt, A strongly degenerate diffusion-haptotaxis model of tumour invasion under the go-or-grow dichotomy hypothesis, Math Meth Appl Sci., 41 (2018), 2403-2428.  doi: 10.1002/mma.4749.  Google Scholar

[110]

A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), Art. 146, 29 pp. doi: 10.1007/s00033-016-0741-0.  Google Scholar

Figure 1.  Simulation results of Experiment 5 — constant phenotypic switch rates with the ECM-with-stripes initial conditions (41). In this and the rest of the simulation results, the densities of MCCs and PCCs are represented with isolines colored according to the displayed colorbars. The density of the ECM is visualized by a variable-intensity color that follows the corresponding colorbar. The MCCs, in their taxis-biased random motion, follow the gradients of the ECM and accordingly their density increases over the stripes of the ECM. The ECM is depleted by both cell subpopulations, which also limit its reconstruction. The PCCs obey a logistic-type growth and fill the free space left by the ECM and MCCs; they moreover undergo phenotypic transitions back-and-forth to MCCs according to the PMT and MPT rates $ \lambda $ and $ \gamma $
Figure 2.  Simulation results of Experiment 1 — constant phenotypic switch rates with the randomly-structured ECM initial conditions (42). Through their haptotaxis-biased random migration, the MCCs identify the higher ECM density regions and accordingly invade the surrounding environment. The ECM and PCCs exhibit similar behavior as in the ECM-with-stripes case shown in Figures 1; the ECM is depleted by the action of both MCCs and PCCs while the PCCs fill the space left by the ECM and MCCs
Figure 3.  The PMT rate $ \lambda $ and its relation to the MPT rate $ \gamma $ with respect to the amount of occupied cell-cell receptors ($ \zeta $) and cell-tissue receptors ($ y $) in case $ \gamma_0 = 1 $
Figure 4.  Simulation results of Experiment 2 — dynamic phenotypic switch rates using the ECM-with-stripes initial conditions (41). When comparing with the corresponding simulation in Experiment 1 (constant phenotypic transition rates) shown in Figure 1, the MCCs can infer higher densities at sites with larger ECM gradients, and lower ones where cell-tissue interfaces are less sharp (e.g., at the center of fiber strands crossing), thus allowing for less cells to move beyond the main fiber tracts
Figure 5.  Simulation results of Experiment 2 — dynamic phenotypic switch rates with the randomly-structured ECM initial conditions (42). Comparing with the simulation of Experiment 1 (constant phenotypic transition), shown in Figure 2, the MCCs' invasion is here slightly more cohesive, allowing for fewer, but larger local maxima. The density of the PCCs is thereby slightly lower than in Figure 2
Figure 6.  Simulation results of Experiment 3 – acidity driven migration with the ECM-with-stripes initial conditions (41). In addition to the MCCs, PCCs, and ECM, we also visualize here the pH levels. When comparing with Experiment 2 (dynamic phenotypic switch without acidity), Figure 4, the effect of the acidity can be seen in the more extensive spread of MCCs, due to chemorepellence by a self-diffusing signal, along with reduced proliferation due to hypoxia, and enhanced ECM degradation throughout the domain
Figure 7.  Simulation results of Experiment 3 – acidity driven migration with the random-structured ECM initial conditions (42). Remarks analogous to those made in Figures 6 apply here as well, when correspondingly comparing with Experiment 2 (dynamic phenotypic switch without acidity), Figure 5
Figure 8.  Simulation results of Experiment 4 — degenerate diffusion, with the ECM-with-stripes initial conditions (41). Compared to the non-degenerate diffusion in Experiment 2, shown in Figure 4, we note that there is a similar extent of tumor spread, however with MCCs forming very localized, relatively large aggregates (see also e.g., the closeup in Figure 9), while the PCC density remains almost the same
Figure 9.  Closeup of the densities at $ t = 10 $ in Experiment 4 — degenerate diffusion, with the ECM-with-stripes initial conditions (41). Compare to Figure 8. MCCs form localized aggregates while the PCC density remains similar as in Experiment 2 shown in Figure 4
Figure 10.  Simulation results of Experiment 4 — degenerate diffusion with the randomly structured ECM (42). When comparing with the non-degenerate diffusion in Experiment 2, shown in Figure 5, we note that the degenerate case leads to higher, more localized MCC densities, mainly near the invasion front
Figure 11.  Simulation results of Experiment 5 — ECM remodeling by cancer cells with the ECM-with-stripes initial conditions (41). Compared to Experiment 2 (dynamic phenotypic transition with self-remodeling of the matrix) shown in Figure 4 we see only a slight impact of the cell reconstruction of tissue; the results are almost identical, maybe with a slightly higher concentration of the MCCs towards the invasion front and higher ECM degradation in the inner part of the tumor
Figure 12.  Simulation results of Experiment 5 — ECM remodeling by cancer cells on a randomly-structured ECM (42). When compared with Experiment 2 (dynamic phenotypic transition with self-remodeling of the matrix), shown in Figure 5, it is clear that the cell reconstruction of the ECM leads to a more fragmented invasion of the MCCs invasion and to higher concentrations along the propagating fronts. We moreover see that the PCCs exhibit a non-smooth boundary/periphery in their support, and that the reconstruction of the ECM is localized where the MCCs are located
Figure 13.  Simulation results of Experiment 6 — anoikis effect on an ECM with initial condition (41). Compared to Experiment 2, shown in Figure 4, the results are almost identical; no particular anoikis effect is visible
Figure 14.  Simulation results of Experiment 6 — anoikis effect on the randomly structured ECM initial conditions (42). Compared to Experiment 2, shown in Figure 5, the effect of anoikis becomes visible: the tumor pattern is more heterogeneous (mainly due to the evolution of MCCs) with correspondingly lower PCC density in regions with stronger degraded ECM
Figure 15.  Construction of the randomly-structured ECM with a sequence of grid refinements steps. The first stage of this process is the construction of a random $ 8\times8 $ grid (top left panel) with values normally distributed in $ [0,1] $. This grid is progressively refined to the final (for this case) resolution of $ 256\times256 $ (bottom right panel). At every refinement step the number of computational cells is doubled along each dimension and the new values are obtained by a) averaging the values of the neighboring cells of the coarser grid, and b) adding some random and normally distributed noise. Periodic interpolations are employed at the "boundary" the discretization domain. It can be clearly seen that the coarse structure of the ECM that was randomly chosen in the $ 8\times8 $ matrix is still visible in the refined $ 256\times 256 $ grid
Table 1.  Butcher tableau for the explicit (upper) and the implicit (lower) parts of the third order IMEX scheme ARK3(2)4L[2]SA we use in (B.5), see also [45]
$ 0 $
$ \frac{1767732205903}{2027836641118} $ $ \frac{1767732205903}{2027836641118} $
$ \frac{3}{5} $ $ \frac{5535828885825}{10492691773637} $ $ \frac{788022342437}{10882634858940} $
$ 1 $ $ \frac{6485989280629}{16251701735622} $ $ -\frac{4246266847089}{9704473918619} $ $ \frac{10755448449292}{10357097424841} $
$ \frac{1471266399579}{7840856788654} $ $ -\frac{4482444167858}{7529755066697} $ $ \frac{11266239266428}{11593286722821} $ $ \frac{1767732205903}{4055673282236} $
0 0
$\frac{1767732205903}{2027836641118}$ $\frac{1767732205903}{4055673282236}$ $\frac{1767732205903}{4055673282236}$
$\frac{3}{5}$ $\frac{2746238789719}{10658868560708}$ $-\frac{640167445237}{6845629431997}$ $\frac{1767732205903}{4055673282236}$
1 $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$
$\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$
$ 0 $
$ \frac{1767732205903}{2027836641118} $ $ \frac{1767732205903}{2027836641118} $
$ \frac{3}{5} $ $ \frac{5535828885825}{10492691773637} $ $ \frac{788022342437}{10882634858940} $
$ 1 $ $ \frac{6485989280629}{16251701735622} $ $ -\frac{4246266847089}{9704473918619} $ $ \frac{10755448449292}{10357097424841} $
$ \frac{1471266399579}{7840856788654} $ $ -\frac{4482444167858}{7529755066697} $ $ \frac{11266239266428}{11593286722821} $ $ \frac{1767732205903}{4055673282236} $
0 0
$\frac{1767732205903}{2027836641118}$ $\frac{1767732205903}{4055673282236}$ $\frac{1767732205903}{4055673282236}$
$\frac{3}{5}$ $\frac{2746238789719}{10658868560708}$ $-\frac{640167445237}{6845629431997}$ $\frac{1767732205903}{4055673282236}$
1 $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$
$\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$
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