title: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching.
abstract: We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.
authors.affiliation: The University of Texas MD Anderson Cancer Center, School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA.
authors.forename: Vittorio
authors.initials: V
authors.lastname: Cristini
authors.forename: Xiangrong
authors.initials: X
authors.lastname: Li
authors.forename: John S
authors.initials: JS
authors.lastname: Lowengrub
authors.forename: Steven M
authors.initials: SM
authors.lastname: Wise
mesh_terms: D000465:Algorithms; D000818:Animals; D001834:Body Water / Q000502:physiology; D002448:Cell Adhesion / Q000502:physiology; D049109:Cell Proliferation; D006801:Humans; D016014:Linear Models; D055641:Mathematical Concepts; D008954:Models, Biological; D009361:Neoplasm Invasiveness / Q000473:pathology* / Q000503:physiopathology; D009369:Neoplasms / Q000473:pathology* / Q000503:physiopathology; D017711:Nonlinear Dynamics; D018874:Spheroids, Cellular / Q000473:pathology; D013816:Thermodynamics; D014407:Tumor Cells, Cultured
references.citation: Cancer Res. 2006 Feb 1;66(3):1597-604
references.pmid: 16452218
references.citation: Bull Math Biol. 1998 Sep;60(5):857-99
references.pmid: 9739618
references.citation: J Theor Biol. 2006 Nov 7;243(1):98-113
references.pmid: 16860344
references.citation: Biophys J. 2005 Dec;89(6):3884-94
references.pmid: 16199495
references.citation: J Math Biol. 2002 Aug;45(2):125-52
references.pmid: 12181602
references.citation: J Theor Biol. 2002 May 7;216(1):85-100
references.pmid: 12076130
references.citation: J Theor Biol. 2006 Aug 21;241(4):903-18
references.pmid: 16545398
references.citation: Clin Chim Acta. 2005 Jul 24;357(2):173-9
references.pmid: 15907826
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2007 May;75(5 Pt 1):051911
references.pmid: 17677102
references.citation: J Theor Biol. 2007 Apr 21;245(4):677-704
references.pmid: 17239903
references.citation: J Theor Biol. 2005 Apr 21;233(4):469-81
references.pmid: 15748909
references.citation: J Theor Biol. 1976 Jan;56(1):229-42
references.pmid: 1263527
references.citation: Clin Cancer Res. 2005 Oct 1;11(19 Pt 1):6772-9
references.pmid: 16203763
references.citation: Bull Math Biol. 2004 Sep;66(5):1039-91
references.pmid: 15294418
references.citation: J Theor Biol. 2008 Aug 7;253(3):524-43
references.pmid: 18485374
references.citation: J Theor Biol. 2002 Dec 7;219(3):343-70
references.pmid: 12419662
references.citation: Cell. 2006 Dec 1;127(5):905-15
references.pmid: 17129778
references.citation: Math Med Biol. 2003 Mar;20(1):47-89
references.pmid: 12974498
references.citation: Nat Rev Cancer. 2003 May;3(5):362-74
references.pmid: 12724734
references.citation: Neuroimage. 2007;37 Suppl 1:S59-70
references.pmid: 17475515
references.citation: J Theor Biol. 2003 Nov 21;225(2):257-74
references.pmid: 14575659
references.citation: Math Biosci. 1995 Dec;130(2):151-81
references.pmid: 8527869
references.citation: J Theor Biol. 2000 Apr 21;203(4):367-82
references.pmid: 10736214
references.citation: Math Med Biol. 2005 Jun;22(2):163-86
references.pmid: 15781426
references.citation: Phys Rev Lett. 2006 May 12;96(18):188103
references.pmid: 16712401
references.citation: J Math Biol. 2003 Mar;46(3):191-224
references.pmid: 12728333
references.citation: Microvasc Res. 2003 Nov;66(3):204-12
references.pmid: 14609526
references.citation: J Theor Biol. 2006 Apr 7;239(3):334-50
references.pmid: 16169016
references.citation: Magn Reson Med. 2005 Sep;54(3):616-24
references.pmid: 16088879
references.citation: Math Med Biol. 2003 Sep;20(3):277-308
references.pmid: 14667048
references.citation: In Silico Biol. 2002;2(3):393-406
references.pmid: 12542422
references.citation: Bull Math Biol. 2005 Mar;67(2):211-59
references.pmid: 15710180
references.citation: J Neurol Sci. 2003 Dec 15;216(1):1-10
references.pmid: 14607296
references.citation: Math Biosci. 1996 Jul 15;135(2):187-216
references.pmid: 8768220
references.citation: J Math Biol. 2006 Jul;53(1):86-134
references.pmid: 16791651
references.citation: J Math Biol. 2001 Sep;43(3):191-220
references.pmid: 11681526
references.citation: Phys Rev Lett. 2006 Feb 10;96(5):058104
references.pmid: 16486998
references.citation: Cancer Cell. 2003 Apr;3(4):347-61
references.pmid: 12726861
references.citation: J Theor Biol. 2007 Jun 21;246(4):583-603
references.pmid: 17374383
references.citation: Math Med Biol. 2006 Sep;23(3):197-229
references.pmid: 16648146
references.citation: IMA J Math Appl Med Biol. 2002 Mar;19(1):1-29
references.pmid: 12408222
references.citation: J Math Biol. 2009 Apr;58(4-5):625-56
references.pmid: 18853162
references.citation: Philos Trans A Math Phys Eng Sci. 2006 Jun 15;364(1843):1563-78
references.pmid: 16766361
references.citation: J Math Biol. 1999 Jul;39(1):59-89
references.pmid: 10444851
references.citation: Math Med Biol. 2003 Dec;20(4):341-66
references.pmid: 14969384
references.citation: J Math Biol. 2003 Nov;47(5):424-52
references.pmid: 14605857