Free Boundary Problems Associated with Multiscale Tumor Models

A. Friedman

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 3, page 134-155
  • ISSN: 0973-5348

Abstract

top
The present paper introduces a tumor model with two time scales, the time t during which the tumor grows and the cycle time of individual cells. The model also includes the effects of gene mutations on the population density of the tumor cells. The model is formulated as a free boundary problem for a coupled system of elliptic, parabolic and hyperbolic equations within the tumor region, with nonlinear and nonlocal terms. Existence and uniqueness theorems are proved, and properties of the free boundary are established.

How to cite

top

Friedman, A.. "Free Boundary Problems Associated with Multiscale Tumor Models." Mathematical Modelling of Natural Phenomena 4.3 (2009): 134-155. <http://eudml.org/doc/222390>.

@article{Friedman2009,
abstract = { The present paper introduces a tumor model with two time scales, the time t during which the tumor grows and the cycle time of individual cells. The model also includes the effects of gene mutations on the population density of the tumor cells. The model is formulated as a free boundary problem for a coupled system of elliptic, parabolic and hyperbolic equations within the tumor region, with nonlinear and nonlocal terms. Existence and uniqueness theorems are proved, and properties of the free boundary are established. },
author = {Friedman, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {free boundary problems; system of PDEs; tumor; cell cycle; free boundary problem; tumor growth},
language = {eng},
month = {6},
number = {3},
pages = {134-155},
publisher = {EDP Sciences},
title = {Free Boundary Problems Associated with Multiscale Tumor Models},
url = {http://eudml.org/doc/222390},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Friedman, A.
TI - Free Boundary Problems Associated with Multiscale Tumor Models
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/6//
PB - EDP Sciences
VL - 4
IS - 3
SP - 134
EP - 155
AB - The present paper introduces a tumor model with two time scales, the time t during which the tumor grows and the cycle time of individual cells. The model also includes the effects of gene mutations on the population density of the tumor cells. The model is formulated as a free boundary problem for a coupled system of elliptic, parabolic and hyperbolic equations within the tumor region, with nonlinear and nonlocal terms. Existence and uniqueness theorems are proved, and properties of the free boundary are established.
LA - eng
KW - free boundary problems; system of PDEs; tumor; cell cycle; free boundary problem; tumor growth
UR - http://eudml.org/doc/222390
ER -

References

top
  1. B.P. Ayati, G.F. Webb, A.R.A Anderson. Computational methods and results for structured mutliscale methods of tumor invasion. Multiscale Model. Simul., 5 (2006), 1–20.  Zbl1182.92039
  2. H.M. Byrne. The importance of intercellular adhesion in the development of carcinomas. I. MA J. Math. Appl. Med. Biol., 14 (1997), 305–323.  Zbl0891.92017
  3. H.M. Byrne, M.A.J. Chaplain. Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci., 130 (1995), 130–151.  Zbl0836.92011
  4. H.M. Byrne, M.A.J. Chaplain. Modelling the role of cell-cell adhesion in the growth and development of carcinomas. Math. Comput. Modeling, 24 (1996), 1–17.  Zbl0883.92014
  5. X. Chen, S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior. Trans. Amer. Math. Soc., 357 (2005), 4771–4804.  Zbl1082.35166
  6. X. Chen, A. Friedman. A free boundary problem for elliptic-hyperbolic system: An application to tumor growth. SIAM J. Math. Anal., 35 (2003), 974–986.  Zbl1054.35144
  7. S. Cui, A. Friedman. A free boundary problem for a singular system of differential equations: An application to a model of tumor growth. Trans. Amer. Math. Soc., 355 (2003), 3537–3590.  Zbl1036.34018
  8. S. Cui, A. Friedman. A hyperbolic free boundary problem modeling tumor growth. Interfaces Free Bound., 5 (2003) , 159–182.  Zbl1040.35143
  9. S.J.H. Franks, H.M. Byrne, J.P. King, J.C.E. Underwood, C.E. Lewis. Modelling the early growth of ductal carcinoma in situ of the breast. J. Math. Biology, 47 (2003), 424–452.  Zbl1050.92030
  10. S.J.H. Franks, H.M. Byrne, J.P. King, J.C.E. Underwood, C.E. Lewis. Modelling the growth of comedo ductal carcinoma in situ. Mathematical Medicine & Biology, 20 (2003), 277–308.  Zbl1039.92021
  11. S.J.H. Franks, H.M. Byrne, J.C.E. Underwood, C.E. Lewis. Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast. J. Theoretical Biology, 232 (2005), 523–543.  
  12. S.J.H. Franks, J.P. King. Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties. Mathematical Medicine & Biology, 20 (2003), 47–89.  Zbl1044.92032
  13. A. Friedman. Cancer models and their mathematical analysis. In: Tutorials in Mathematical Biosciences III. Lecture Notes in Mathematics, 1872, 223-246. Springer, Berlin, 2006.  
  14. A. Friedman. A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth. Interfaces and Free Boundaries, 8 (2006), 247–261.  Zbl1106.35145
  15. A. Friedman. Mathematical analysis and challenges arising from models of tumor growth.Math. Models & Methods in Applied Sciences, 17 (2007), 1751–1772.  Zbl1135.92013
  16. A. Friedman. A multiscale tumor model. Interfaces and Free Boundaries, 10 (2008), 245–262.  Zbl1157.35501
  17. A. Friedman, B. Hu. The role of oxygen in tissue maintenance: Mathematical modeling and qualitative analysis. Math. Mod. Meth. Appl. Sci., 18 (2008), 1–33.  
  18. A. Friedman, B. Hu, C-Y. Kao. Cell cycle control at the first restriction point and its effect on tissue growth. Submitted for publication.  Zbl1202.92024
  19. A. Friedman, F. Reitich. Quasi-static motion of a capillary drop, II: The three-dimensional case. J. Diff. Eqs., 186 (2002), 509–557.  Zbl1146.76593
  20. Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, J.P. Freyer. A multiscale model for avascular tumor growth. Biophysical Journal, 89 (2005), 3884–3894.  
  21. N. Komaraova. Stochastic modeling of loss- and gain-of-function mutation in cancer. Bull. Math. Biology, 17 (2007), 1647–1673.  Zbl1135.92017
  22. H.A. Levine, S.L. Pamuk, B.D. Sleeman, M. Nilsen-Hamilton. Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull. Math. Biology, 63 (2001), 801–863.  Zbl1323.92029
  23. G. Lolas. Mathematical modelling of proteolysis and cancer cell invasion of tissue. In: Tutorials in Mathematical Biosciences III, Lecture Notes in Mathematics, 1872, 77–130. Springer, Berlin, 2006.  
  24. N. Mantzaris, S. Webb, H.G. Othmer. Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol., 49 (2004), 87–111.  Zbl1109.92020
  25. M.A. Nowak, K. Sigmund. Evolutionary dynamics of biological games. Science, 303 (2004), 793–799.  
  26. G.J. Pettet, C.P. Please, M.J. Tindall, D.L.S. McElwain. The migration of cells in multicell tumor spheroids. Bull. Math. Biol., 63 (2001), 231–257.  Zbl1323.92037
  27. R. Ribba, T. Colin, S. Schnell. A multiscale model of cancer, and its use in analyzing irradiation therapies. Theoretical Biology and Medical Modeling, 3 (2006), No. 7, 1–19.  
  28. V.A. Solonnikov. On quasistationary approximation in the problem of a capillary drop. In: J. Escher & G. Simonett (Eds.), Progress in Nonlinear Differential Equations and Their Applications, 35, 643-671. Birkhäuser Verlag, Basel, 1999.  Zbl0919.35103
  29. M.M. Vainberg, V.A. Trenogin. Theory of branching solutions of non-linear equations. Nordhoff International Publishing, Leyden, 1974.  Zbl0274.47033

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.