Derivation of a Hele-Shaw type system from a cell model with active motion

We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of ‘mushy region’ prepares the tumor invasion.

[1]  M. Chaplain.  Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development , 1996 .

[2]  HU BEI.  STABILITY AND INSTABILITY OF LIAPUNOV-SCHMIDT AND HOPF BIFURCATION FOR A FREE BOUNDARY PROBLEM ARISING IN A TUMOR MODEL , 2008 .

[3]  Thierry Colin,et al.  A Multilayer Grow-or-Go Model for GBM: Effects of Invasive Cells and Anti-Angiogenesis on Growth , 2014, Bulletin of mathematical biology.

[4]  Luigi Preziosi,et al.  Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications , 2009, Journal of mathematical biology.

[5]  Stefan Hoehme,et al.  Modeling the impact of granular embedding media, and pulling versus pushing cells on growing cell clones , 2012 .

[6]  P Ciarletta,et al.  The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis , 2011, Journal of The Royal Society Interface.

[7]  La limite de la solution de ut = Δp um lorsque m → ∞ , 1995 .

[8]  H. Greenspan.  Models for the Growth of a Solid Tumor by Diffusion , 1972 .

[9]  Shangbin Cui,et al.  Asymptotic Behaviour of Solutions of a Multidimensional Moving Boundary Problem Modeling Tumor Growth , 2008 .

[10]  L. Preziosi,et al.  Modelling Solid Tumor Growth Using the Theory of Mixtures , 2001, Mathematical medicine and biology : a journal of the IMA.

[11]  S. Cui.  FORMATION OF NECROTIC CORES IN THE GROWTH OF TUMORS: ANALYTIC RESULTS * * Project supported by the N , 2006 .

[12]  B. Perthame,et al.  The Hele–Shaw Asymptotics for Mechanical Models of Tumor Growth , 2013, Archive for Rational Mechanics and Analysis.

[13]  Ing. Rafael Laguardia.  Boundary layer formation in the transition from the Porous Media Equation to a Hele-Shaw flow , 2000 .

[14]  A. Brú,et al.  The universal dynamics of tumor growth. , 2003, Biophysical journal.

[15]  A. Crowley.  On the weak solution of moving boundary problems , 1979 .

[16]  H. Frieboes,et al.  Nonlinear modelling of cancer: bridging the gap between cells and tumours , 2010, Nonlinearity.

[17]  D. Aronson.  The porous medium equation , 1986 .

[18]  Nicola Bellomo,et al.  On the foundations of cancer modelling: Selected topics, speculations, and perspectives , 2008 .

[19]  Dirk Drasdo,et al.  Individual-based and continuum models of growing cell populations: a comparison , 2009, Journal of mathematical biology.

[20]  J. Vázquez.  The Porous Medium Equation: Mathematical Theory , 2006 .

[21]  Helen M. Byrne,et al.  The impact of cell crowding and active cell movement on vascular tumour growth , 2006, Networks Heterog. Media.

[22]  H M Byrne,et al.  Growth of necrotic tumors in the presence and absence of inhibitors. , 1996, Mathematical biosciences.

[23]  B. Perthame,et al.  Composite waves for a cell population system modeling tumor growth and invasion , 2013, Chinese Annals of Mathematics, Series B.

[24]  Frank Jülicher,et al.  Fluidization of tissues by cell division and apoptosis , 2010, Proceedings of the National Academy of Sciences.

[25]  S. Jonathan Chapman,et al.  Mathematical Models of Avascular Tumor Growth , 2007, SIAM Rev..