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Erschienen in:
Two-Scale Moving Boundary Dynamics of Cancer Invasion: Heterotypic Cell Populations’ Evolution in Heterogeneous ECM Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

From Random Walks to Fully Anisotropic Diffusion Models for Cell and Animal Movement

verfasst von : Kevin J. Painter, Thomas Hillen

Erschienen in: Cell Movement

Verlag: Springer International Publishing

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Abstract

This chapter provides an introduction on how anisotropic diffusion models can be derived from position-jump and velocity-jump random walks. We show how the availability of measurement data can guide the choice of the appropriate model. We further present two new applications, respectively to cell movement on micro-fabricated surfaces and magnetic compass orientation by sea turtle hatchlings.

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Vorheriges Kapitel Bayesian Uncertainty Quantification for Particle-Based Simulation of Lipid Bilayer Membranes
Nächstes Kapitel Bacterial Chemotaxis: A Classic Example of Multiscale Modeling in Biology
Fußnoten
1
We note that this particular form assumes there is no net force on the particles, and thus no inertia on them.
 
2
It is worth noting that this is a key distinction from the kinetic theory of gas molecules, where \(V=\mathbb {R}^n\) permits (at least theoretically) individual molecules to acquire infinite momentum [8].
 
Literatur
1.
Alt, W.: Biased random walk model for chemotaxis and related diffusion approximation. J. Math. Biol. 9, 147–177 (1980)MathSciNetCrossRef
2.
Batschelet, E.: Circular Statistics in Biology. Academic Press, London (1981)MATH
3.
Bellomo, N., Schiavo, M.: Lecture Notes on the Mathematical Theory of Generalized Boltzmann Methods. World Scientific, Singapore (2000)CrossRef
4.
Berens, P.: Circstat: a MATLAB toolbox for circular statistics. J. Stat. Softw. 31, 1–21 (2009)CrossRef
5.
Berg, H.: Random Walks in Biology. Princeton University Press (1983)
6.
Bleck, R.: An oceanic general circulation model framed in hybrid isopycnic-cartesian coordinates. Ocean Mod. 4, 55–88 (2002)CrossRef
7.
Cagnacci, F., Boitani, L., Powell, R.A., Boyce, M.S.: Animal ecology meets GPS-based radiotelemetry: a perfect storm of opportunities and challenges. Phil. Trans. R. Soc. B 365, 21572162 (2010)
8.
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Diluted Gases. Springer, New York (1994)CrossRef
9.
Codling, E.A., Plank, M.J., Benhamou, S.: Random walk models in biology. J. Roy. Soc. Interface 5, 813–834 (2008)CrossRef
10.
Davenport, J. and Clough, W.: Swimming and diving in young loggerhead sea turtles (Caretta caretta L.). Copeia, 1986, 53–57 (1986)CrossRef
11.
Dawes, A., Iron, D.: Cortical geometry may influence placement of interface between par protein domains in early caenorhabditis elegans embryos. J. Theor. Bio. 333, 27–37 (2013)MathSciNetCrossRef
12.
Deutsch, A., Dormann, S.: Cellular Automaton Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis. Birkaeuser, Boston (2005)MATH
13.
Dickinson, R.B., Guido, S., Tranquillo, R.T.: Biased cell migration of fibroblasts exhibiting contact guidance in oriented collagen gels. Ann. Biomed. Eng. 22, 342–356 (1994)CrossRef
14.
Dunn, G.A., Heath, J.P.: A new hypothesis of contact guidance in tissue cells. Exp. Cell Res. 101, 1–14 (1976)CrossRef
15.
Fuxjager, M.J., Eastwood, B.S., Lohmann, K.J.: Orientation of hatchling loggerhead sea turtles to regional magnetic fields along a transoceanic migratory pathway. J. Exp. Biol. 214, 2504–2508 (2011)CrossRef
16.
Gritsenko, P., Ilina, O., Friedl, P.: Interstitial guidance of cancer invasion. J. Pathol. 226, 185–199 (2012)CrossRef
17.
Gritsenko, P., Leenders, W., Friedl, P.: Recapitulating in vivo-like plasticity of glioma cell invasion along blood vessels and in astrocyte-rich stroma. Histochem. Cell Biol. (2017). doi: 10.1007/s00418-017-1604-2CrossRef
18.
Hadeler, K., Hillen, T., Lutscher, F.: The Langevin or Klein-Kramers approach to biological modeling. Math. Models Meth. Appl. Sci. 14(10), 1561–1583 (2004)CrossRef
19.
Hanahan, D., Weinberg, R.: Hallmarks of cancer: The next generation. Cell 144, 646–674 (2011)CrossRef
20.
21.
Hillen, T., Othmer, H.: The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61, 751–775 (2000)MathSciNetCrossRef
22.
Hillen, T., Painter, K.J.: Transport models for movement in oriented habitats and anisotropic diffusion. In: Lewis, M., Maini, P., Petrovskii,S. (Eds.), Dispersal, Individual Movement and Spatial Ecology: A Mathematical Perspective. Springer,Heidelberg. p. 46 (2013)
23.
Hillen, T., Painter, K.J., Swan, A.C., Murtha, A.D.: Moments of von Mises and Fisher distributions and applications. Math. Biosci. & Eng 14, 673–694 (2017)MathSciNetCrossRef
24.
Hundsdorfer, W., Verwer, J.G.: Numerical solution of time-dependent advection-diffusion-reaction equations, vol. 33. Springer Science & Business Media (2003)
25.
Jeon, H., Hidai, H., Hwang, D.J., Healy, K.E., Grigoropoulos, C.P.: The effect of micronscale anisotropic cross patterns on fibroblast migration. Biomaterials 31, 4286–4295 (2010)CrossRef
26.
Keener, J., Sneyd, J.: Mathematical Physiology. Springer (1994)
27.
Lohmann, K.J., Cain, S.D., Dodge, S.A., Lohmann, C.M.F.: Regional magnetic fields as navigational markers for sea turtles. Science 294, 364–366 (2001)CrossRef
28.
Lohmann, K.J., Putman, N.F., Lohmann, C.M.F.: The magnetic map of hatchling loggerhead sea turtles. Curr. Opin. Neurobiol. 22, 336–342 (2012)CrossRef
29.
Luschi, P.: Long-distance animal migrations in the oceanic environment: orientation and navigation correlates. ISRN Zool. (2013)
30.
Lutscher, F., Pachepsky, E., Lewis, M.: The effect of dispersal patterns on stream populations. SIAM J. Appl. Math. 65, 1305–1327 (2005)MathSciNetCrossRef
31.
Mardia, K., Jupp, P.: Directional Statistics. Wiley and Sons (2000)
32.
McKenzie, H., Lewis, M., Merrill, E.: First passage time analysis of animal movement and insights into the functional response. Bull. Math. Biol. 71, 107–129 (2009)MathSciNetCrossRef
33.
McKenzie, H.W., Merrill, E.H., Spiteri, R.J., Lewis, M.A.: How linear features alter predator movement and the functional response. Interface focus 2, 205–216 (2012)CrossRef
34.
Moorcroft, P., Lewis, M.: Mechanistic Home Range Analysis. Princeton University Press, Princeton (2006)
35.
Murray, J.D.: Mathematical Biology. I: An Introduction, 3rd edn. Springer-Verlag, New York (2002)
36.
Murray, J.D.: Mathematical biology II: Spatial models and biochemical applications, Springer-Verlag, New York (2003)
37.
Okubo, A., Levin, S.: Diffusion and Ecological Problems: Modern Perspectives. Springer (2002)
38.
Othmer, H.G., Dunbar, S., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)MathSciNetCrossRef
39.
Othmer, H.G., Stevens, A: Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math., 57, 1044–1081 (1997).MathSciNetCrossRef
40.
Othmer, H.G., Hillen, T.: The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1122–1250 (2002)MathSciNetCrossRef
41.
Othmer, H.G., Xue, C.: The mathematical analysis of biological aggregation and dispersal: progress, problems and perspectives. In: Lewis, M., Maini, P., Petrovskii, S. (Eds.), Dispersal, Individual Movement and Spatial Ecology: A Mathematical Perspective. Springer,Heidelberg. 79–127 (2013)CrossRef
42.
Painter, K.J.: Modelling migration strategies in the extracellular matrix. J. Math. Biol. 58, 511–543 (2009)MathSciNetCrossRef
43.
Painter, K.J., Hillen, T.: Navigating the flow: Individual and continuum models for homing in flowing environments. Royal Soc. Interface 12, 20150,647 (2015)CrossRef
44.
Painter, K.J.: Multiscale models for movement in oriented environments and their application to hilltopping in butterflies. Theor. Ecol. 7, 53–75 (2014)CrossRef
45.
Painter, K.J., Hillen, T.: Mathematical modelling of glioma growth: the use of diffusion tensor imaging (DTI) data to predict the anisotropic pathways of cancer invasion. J. Theor. Biol. 323, 25–39 (2013)MathSciNetCrossRef
46.
47.
Perthame, B.: Transport Equations in Biology. Birkhäuser (2007)
48.
Preziosi, L. (ed.): Cancer Modelling and Simulation. Chapman Hall/CRC Press (2003)
49.
Provenzano, P.P., Eliceiri, K.W., Campbell, J.M., Inman, D.R., White, J.G., Keely, P.J.: Collagen reorganization at the tumor-stromal interface facilitates local invasion. BMC medicine 4, 38 (2006)CrossRef
50.
Putman, N.F., Endres, C.S., Lohmann, C.M.F., Lohmann, K.J.: Longitude perception and bicoordinate magnetic maps in sea turtles. Curr. Biol. 21, 463–466 (2011)CrossRef
51.
Putman, N.F., Verley, P., Shay, T.J., Lohmann, K.J.: Simulating transoceanic migrations of young loggerhead sea turtles: merging magnetic navigation behavior with an ocean circulation model. J. Exp. Biol. 215, 1863–1870 (2012)CrossRef
52.
Saxton, M.J., Jacobson, K.: Single-particle tracking: applications to membrane dynamics. Ann. Rev. Biophys. & Biomol. Struct. 26, 373–399 (1997)CrossRef
53.
Sobel, D.: Longitude: The true story of a lone genius who solved the greatest scientific problem of his time. Bloomsbury Publishing USA (1995)
54.
Stevens, A.: The derivation of chemotaxis-equations as limit dynamics of moderately interacting stochastic many particle systems. SIAM J. Appl. Math. 61(1), 183–212 (2000)MathSciNetCrossRef
55.
Stevens, A., Othmer, H.G.: Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081 (1997)MathSciNetCrossRef
56.
Swan, A., Hillen, T., Bowman, J.C., Murtha, A.D.: A patient-specific anisotropic diffusion model for brain tumour spread. Bull. Math. Biol. pp. 1–33 (2017)
57.
Turchin, P.: Quantitative Analysis of Movement. Sinauer Assoc., Sunderland (1998)
58.
Weickert, J.: Anisotropic diffusion in image processing. Teubner, Stuttgart (1998)MATH
59.
Wolf, K., Müller, R., Borgmann, S., Bröcker, E.B., Friedl, P.: Amoeboid shape change and contact guidance: T-lymphocyte crawling through fibrillar collagen is independent of matrix remodeling by MMPs and other proteases. Blood 102, 3262–3269 (2003)CrossRef
Metadaten
Titel
From Random Walks to Fully Anisotropic Diffusion Models for Cell and Animal Movement
verfasst von
Kevin J. Painter
Thomas Hillen
Copyright-Jahr
2018
Buch
Cell Movement

Print ISBN: 978-3-319-96841-4

Electronic ISBN: 978-3-319-96842-1

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-96842-1_5

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