Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Modeling Spatial Effects in Carcinogenesis: Stochastic and Deterministic Reaction-Diffusion Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

A Cell Population Model Structured by Cell Age Incorporating Cell–Cell Adhesion

verfasst von : Janet Dyson, Glenn F. Webb

Erschienen in: Mathematical Oncology 2013

Verlag: Springer New York

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

An analysis is given of a continuum model of a proliferating cell population, which incorporates cell movement in space and cell progression through the cell cycle. The model consists of a nonlinear partial differential equation for the cell density in the spatial position and the cell age coordinates. The equation contains a diffusion term corresponding to random cell movement, a nonlocal dispersion term corresponding to cell–cell adhesion, a cell age-dependent boundary condition corresponding to cell division, and a nonlinear logistic term corresponding to constrained population growth. Basic properties of the solutions are proved, including existence, uniqueness, positivity, and long-term behavior dependent on parametric input. The model is illustrated by simulations applicable to in vitro wound closure experiments, which are widely used for experimental testing of cancer therapies.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Avascular Tumor Growth Modelling: Physical Insights to Skin Cancer
Nächstes Kapitel A General Framework for Multiscale Modeling of Tumor–Immune System Interactions
Literatur
1.
J.C. Arciero, Q. Mi, M.F. Branco, D.J. Hackam, D. Swigon, Continuum model of collective cell migration in wound healing and colony expansion. Biophys. J. 100, 535–543 (2011)CrossRef
2.
N.J. Armstrong, K.J. Painter, J.A. Sherratt, A continuum approach to modelling cell–cell adhesion. J. Theor. Biol. 243, 98–113 (2006)MathSciNetCrossRef
3.
N.J. Armstrong, K.J. Painter, J.A. Sherratt, Adding adhesion to a chemical signalling model for somite formation. Bull. Math. Biol. 71, 1–24 (2009)MathSciNetMATHCrossRef
4.
H. Byrne, D. Draso, Individual based and continuum models of growing cell populations: a comparison. J. Math. Biol. 58, 657–687 (2009)MathSciNetCrossRef
5.
H. Byrne, M.A.J. Chaplain, D.L. Evans, I. Hopkinson, Mathematical modelling of angiogenesis in wound healing: comparison of theory and experiment. J. Theor. Med. 2, 175–197 (2000)MATHCrossRef
6.
X. Chen, A. Friedman: A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal. 32(4), 778–800 (2000)MathSciNetMATHCrossRef
7.
V. Christini, J. Lowengrub, Multi-Scale Modeling of Cancer (Cambridge University Press, Cambrigde, 2010)CrossRef
8.
G. Di Blasio, Mathematical analysis for an epidemic model with spatial and age structure. J. Evol. Equat. 10(4), 929–953 (2010)MATHCrossRef
9.
G. Di Blasio, A. Lorenzi, An identification problem in age-dependent population diffusion. Num. Funct. Anal. Optim. 34(1), 36–73 (2013)MATHCrossRef
10.
G.J. Doherty, H.T. McMahon, Mediation, modulation and consequences of membrane-cytoskeleton interactions. Ann. Rev. Biophys. 37, 65–95 (2008)CrossRef
11.
A. Ducrot, P. Magal, S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models. Arch. Ration. Mech. Anal. 195(1), 311–331 (2010)MathSciNetMATHCrossRef
12.
A. Ducrot, Travelling waves for a size and space structured model in population dynamics: point to sustained oscillating solution connections. J. Differ. Equat. 250(1), 410–449 (2011)MathSciNetMATHCrossRef
13.
14.
J. Dyson, E. Sánchez, R. Villella-Bressan, G.F. Webb, An age and spatially structured model of tumor invasion with haptotaxis. Discrete Contin. Dyn. Syst. -B 8(1), 45–60 (2007)MATHCrossRef
15.
J. Dyson, R. Villella-Bressan, G.F. Webb, A spatially structured model of tumor growth with cell age, cell size and mutation of cell phenotypes. Math. Model. Nat. Phenom. 2(3), 69–100 (2007)MathSciNetCrossRef
16.
J. Dyson, R. Villella-Bressan, G.F. Webb, An age and spatially structured model of tumor invasion with haptotaxis II. Math. Pop. Stud. 15, 73–95 (2008)MathSciNetMATHCrossRef
17.
J. Dyson, S. Gourley, R. Villella-Bressan, G.F. Webb, Existence and asymptotic properties of solutions of a nonlocal evolution equation modelling cell–cell adhesion. SIAM J. Math. Anal. 42(4), 1784–1804 (2010)MathSciNetMATHCrossRef
18.
J. Dyson, S. Gourley, G.F. Webb, A nonlocal evolution equation model of cell–cell adhesion in higher dimensional space. J. Biol. Dyn. 7(Suppl 1), 68–87 (2013). doi:10.1080/17513758.2012.755572MathSciNetCrossRef
19.
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics (American Mathematical Society, Providence, 2004)
20.
W.E. Fitzgibbon, M.E. Parrott, G.F. Webb, Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete Contin. Dyn. Syst. 1(1), 35–57 (1995)MathSciNetMATH
21.
W.E. Fitzgibbon, M.E. Parrott, G.F. Webb, A diffusive age-structured SEIRS epidemic model. Meth. Appl. Anal. 3(3), 358–369 (1996)MathSciNetMATH
22.
A. Friedman, Tutorials in Mathematical Biosciences, II: Cell Cycle, Proliferation, and Cancer. Springer Lecture Notes in Mathematics, vol. 1872 (Springer, New York, 2005)
23.
A. Friedman, Mathematical analysis and challenges arising from models of tumor growth. Math. Mod. Meth. Appl. Sci. 17, 1751–1772 (2007)MATHCrossRef
24.
A. Friedman, B. Hu, C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds. SIAM J. Math. Anal. 42, 2013–2040 (2010)MathSciNetMATHCrossRef
25.
E.A. Gaffney, P.K. Maini, J.A. Sherratt, P.D. Dale, Wound healing in the corneal epithelium: biological mechanisms and mathematical models. J. Theor. Med. 1, 13–23 (1997)MATHCrossRef
26.
E.A. Gaffney, P.K. Maini, J.A. Sherratt, S. Tutt, The mathematical modelling of cell kinetics in corneal epithelial wound healing. J. Theor. Biol. 197, 111–141 (1999)CrossRef
27.
A. Gandolfi, M. Iannelli, G. Marnoschi, An age-structured model of epidermis growth. J. Math. Biol. 62, 15–40 (2011)CrossRef
28.
A. Gandolfi, M. Iannelli, G. Marinoschi, Time evolution for a model of epidermis growth. J. Evol. Equat. 13, 509–533 (2013)MathSciNetMATHCrossRef
29.
A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: local and nonlocal models and the effect of adhesion. J. Theor. Biol. 250, 684–704 (2008)MathSciNetCrossRef
30.
31.
H.J.A.M. Heijmans, The Dynamical Behaviour of the Age-Size-Distribution of a Cell Population (Centre for Mathematics and Computer Science, Amsterdam; Springer, Berlin, 1984)
32.
D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840 (Springer, New York, 1981)
33.
M. Kubo, M. Langlais, Periodic solutions for a population dynamics problem with age-dependence and spatial structure. J. Math. Biol. 29(4), 363–378 (1991)MathSciNetMATHCrossRef
34.
M. Kubo, M. Langlais, Periodic solutions for nonlinear population dynamics models with age-dependence and spatial structure. J. Differ. Equat. 109(2), 274–294 (1994)MathSciNetMATHCrossRef
35.
P. Laurençot, Ch. Walker, Proteus mirabilis swarm-colony development with drift. J. Math. Pures Appl. 92(5), 476–498 (2009)MathSciNetMATHCrossRef
36.
J.S. Lowengrub, H.B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S.M. Wise, V. Cristin, Nonlinear models of cancer: bridging the gap between cells and tumours. Nonlinearity 23, R1–R91 (2010)MATHCrossRef
37.
38.
39.
J.A.J. Metz, O. Diekmann, The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, vol. 68 (Springer, New York, 1986)
40.
X. Mora, Semilinear parabolic problems define semiflows on C k spaces. Trans. Am. Math. Soc. 278, 21–55 (1983)MathSciNetMATH
41.
P.J. Murray, A. Walter, A.G. Fletcher, C.M. Edwards, M.J. Tindall, P.K. Maini, Comparing a discrete and continuum model of the intestinal crypt. Phys. Biol. 8(2), 026011 (2010)
42.
P.J. Murray, J-W. Kang, G.R. Mirams, S.-Y. Shin, H.M. Byrne, P.K. Maini, K.-H. Cho, Modelling spatially regulated β-catenin dynamics and invasion in intestinal crypts. Biophys. J. 99(3), 716–725 (2010)
43.
K.J. Painter, N.J. Armstrong, J.A. Sherratt, The impact of adhesion on cellular invasion processes in cancer and development. J. Theor. Biol. 264, 1057–1067 (2010)MathSciNetCrossRef
44.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, Berlin, 1983)MATHCrossRef
45.
G.J. Pettet, M.A.J. Chaplain, D.S.L. McElwain, J. Norbury, A model of wound healing-angiogenesis in soft tissue. Math. Biosci. 136, 35–63 (1996)MATHCrossRef
46.
G.J. Pettet, M.A.J. Chaplain, D.S.L. McElwain, H.M. Byrne: On the role of angiogenesis in wound healing. Proc. Roy. Soc. Lond. B 263, 1487–1493 (1996)CrossRef
47.
J.S. Ross, J.A. Fletcher, G.P. Linette, J. Stec, E. Clark, M. Ayers, W. Fraser Symmans, L. Pusztai, K.J. Bloom, The Her-2/neu gene and protein in breast cancer 2003: biomarker and target of therapy. Oncologist 8(4), 307–25 (2003)CrossRef
48.
S. Ruan, J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in Spatial Ecology, S. Cantrell, C. Cosner, S. Ruan (Chapman & Hall CRC Press, Boca Raton, 2009), pp. 293–316
49.
J.A. Sherratt, J.C. Dallon, Theoretical models of wound healing: past successes and future challenges. Comp. Rend. Biol. 325(5), 557–564 (2002)CrossRef
50.
J.A. Sherratt, J.D. Murray, Mathematical analysis of a basic model for epidermal wound healing. Proc. Biol. Sci. 241, 29–36 (1990)CrossRef
51.
J.A. Sherratt, J.D. Murray, Models of epidermal wound healing. J. Math. Biol. 31, 703–716 (1993)MATHCrossRef
52.
J.A. Sherratt, S.A. Gourley, N.J. Armstrong, K.J. Painter, Boundedness of solutions of a nonlocal reaction-diffusion model for adhesion in cell aggregation and cancer invasion. Euro. J. Appl. Math. 20, 123–144 (2009)MathSciNetMATHCrossRef
53.
J.W.-H. So, J. Wu, X. Zou, A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains. Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457(2012),1841–1853 (2001)MathSciNetMATHCrossRef
54.
S.L. Tucker, S.O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM J. Appl. Math. 48(3), 549–591 (1988)MathSciNetMATHCrossRef
55.
S. Turner, J.A. Sherratt, Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model. J. Theor. Biol. 216, 85–100 (2002)MathSciNetCrossRef
56.
Ch. Walker, A haptotaxis model with age and spatial structure and nonlinear age-boundary conditions. PAMM 7(1), 1040601–1040602 (2007)CrossRef
57.
Ch. Walker, Global well-posedness of a haptotaxis model with spatial and age structure. Differ. Integral Equat. 20(9), 1053–1074 (2007)MATH
58.
Ch. Walker, Global existence for an age and spatially structured haptotaxis model with nonlinear age-boundary conditions. Eur. J. Appl. Math. 19, 113–147 (2008)MATHCrossRef
59.
Ch. Walker, Positive equilibrium solutions for age-and spatially-structured population models. SIAM J. Math. Anal. 41(4), 1366–1387 (2009)MathSciNetMATHCrossRef
60.
Ch. Walker, Global bifurcation of positive equilibria in nonlinear population models. J. Differ. Equat. 248(7), 1756–1776 (2010)MATHCrossRef
61.
Ch. Walker, Bifurcation of positive equilibria in nonlinear structured population models with varying mortality rates. Ann. Math. Pura Appl. 190(1), 1–19 (2011)MATHCrossRef
62.
Ch. Walker, On positive solutions of some systems of reaction-diffusion equations with nonlocal initial conditions. J. Reine Angew. Math. 660, 149–179 (2011)MathSciNetMATH
63.
Ch. Walker, On nonlocal parabolic steady-state equations of cooperative or competing systems. Nonlinear Anal. Real World Appl. 12(6), 3552–3571 (2011)MathSciNetMATHCrossRef
64.
65.
Ch. Walker, Global continua of positive solutions for some quasilinear parabolic equation with a nonlocal initial condition. J. Dyn. Differ. Equat. 25, 159–172 (2013). doi:10.1007/s10884-013-9292-7MATHCrossRef
66.
Ch. Walker, Some remarks on the asymptotic behavior of the semigroup associated with age-structured diffusive populations. Monatsh. Math. 170, 481–501 (2013). doi:10.1007/s00605-012-0428-3MathSciNetMATHCrossRef
67.
S.E. Wang, I. Shin, F.Y. Wu, D.B. Friedman, C.L. Arteaga, HER2/Neu (ErbB2) signaling to Rac1-Pak1 is temporally and spatially modulated by transforming growth factor β. Cancer Res. 66(19), 9591–9600 (2006)
68.
G.F. Webb, An age-dependent epidemic model with spatial diffusion. Arch. Ration. Mech. Anal. 75(1), 91–102 (1980)MATHCrossRef
69.
70.
G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and textbooks in Pure and Applied Mathematics (Marcel Dekker, New York, 1985)
71.
G.F. Webb, Population models structured by age, size and spatial position, in Structured Population Models in Biology and Epidemiology, ed. by P. Magal, S. Ruan. Lecture Notes in Mathematics, vol. 1936, Mathematical Biosciences Series (Springer, Berlin, 2008), pp.1–49
72.
C. Xue, A. Friedman, C.K. Sen, A mathematical model of ischemic cutaneous wounds. Proc. Natl. Acad. Sci. USA 106, 16782–16787 (2009)CrossRef
Metadaten
Titel
A Cell Population Model Structured by Cell Age Incorporating Cell–Cell Adhesion
verfasst von
Janet Dyson
Glenn F. Webb
Copyright-Jahr
2014
Verlag
Springer New York
Buch
Mathematical Oncology 2013

Print ISBN: 978-1-4939-0457-0

Electronic ISBN: 978-1-4939-0458-7

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-1-4939-0458-7_4

Premium Partner

    Bildnachweise