SIMULATION OF GROWTH AND DIVISION OF 3D CELLS BASED ON FINITE ELEMENT METHOD

Adequate proliferating model including each cell's growth and division in 3D is key for the simulation of tissue growth in early stages as embryo, avascular tumor and so on. This paper proposed a novel model to perform the simulation based on a nonlinear finite element method. With the mechanism of cellular growth controlled by total energy of volume and surface of cells, the surface of each cell is divided by triangular elements and the nodal displacements determine the variation of the cellular surface and volume uniquely when cells grow. The nonlinear finite element equations of nodal displacements were deduced by the minimum energy increment in each growth's step. Further, by using the volume-based algorithm to determine cellular division and the modified penalty function method, the nonlinear procedure of the cellular proliferation (growth-division-regrowth) was solved. The numerical results show that the method can simulate both tissue and tissue's every cell well in each growth-division step, and will do great help to model the growth of varied tissues in their early stages.

[1]  Jelena Pjesivac-Grbovic,et al.  A multiscale model for avascular tumor growth. , 2005, Biophysical journal.

[2]  A. Anderson,et al.  A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion , 2005 .

[3]  John R. King,et al.  Mathematical Modelling of the Effects of Mitotic Inhibitors on Avascular Tumour Growth , 1999 .

[4]  J. Reddy.  An introduction to the finite element method , 1989 .

[5]  Vittorio Cristini,et al.  Multiscale cancer modeling. , 2010, Annual review of biomedical engineering.

[6]  K. Hwang,et al.  Coarse grained modeling of biopolymers and proteins: Methods and applications , 2009 .

[7]  Luigi Preziosi,et al.  Multiscale Developments of the Cellular Potts Model , 2012, Multiscale Model. Simul..

[8]  Nicola Bellomo,et al.  Selected Topics In Cancer Modeling , 2008 .

[9]  Helen M. Byrne,et al.  A Multiple Scale Model for Tumor Growth , 2005, Multiscale Model. Simul..

[10]  Vittorio Cristini,et al.  Three-dimensional multispecies nonlinear tumor growth-II: Tumor invasion and angiogenesis. , 2010, Journal of theoretical biology.

[11]  J. Murray,et al.  The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas. , 1997, Journal of neuropathology and experimental neurology.

[12]  J. Freyer,et al.  Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply. , 1986, Cancer research.

[13]  J P Freyer,et al.  Proliferative and clonogenic heterogeneity of cells from EMT6/Ro multicellular spheroids induced by the glucose and oxygen supply. , 1986, Cancer research.

[14]  S. H. Lo,et al.  Finite element mesh generation over analytical curved surfaces , 1996 .

[15]  H. Othmer,et al.  A HYBRID MODEL FOR TUMOR SPHEROID GROWTH IN VITRO I: THEORETICAL DEVELOPMENT AND EARLY RESULTS , 2007 .

[16]  S. Schnell,et al.  A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies , 2006, Theoretical Biology and Medical Modelling.

[17]  Alexander R. A. Anderson,et al.  Computational Methods and Results for Structured Multiscale Models of Tumor Invasion , 2005, Multiscale Model. Simul..

[18]  Christophe Deroulers,et al.  Modeling tumor cell migration: From microscopic to macroscopic models. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Enrique Fernández-Cara,et al.  Fictitious domains and level sets for moving boundary problems. Applications to the numerical simulation of tumor growth , 2011, J. Comput. Phys..

[20]  G. Webb,et al.  A nonlinear structured population model of tumor growth with quiescence , 1990, Journal of mathematical biology.

[21]  Glazier,et al.  Simulation of the differential adhesion driven rearrangement of biological cells. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  Luigi Preziosi,et al.  Multiscale modeling and mathematical problems related to tumor evolution and medical therapy. , 2003 .

[23]  B. Grammaticos,et al.  A cellular automaton model for the migration of glioma cells , 2006, Physical biology.

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

[25]  J. Reddy,et al.  Micro-Constituent Based Viscoelastic Finite Element Analysis of Biological Cells , 2010 .

[26]  L. Preziosi,et al.  A Cellular Potts Model simulating cell migration on and in matrix environments. , 2012, Mathematical biosciences and engineering : MBE.

[27]  D. Drasdo,et al.  A single-cell-based model of tumor growth in vitro: monolayers and spheroids , 2005, Physical biology.

[28]  Markus J. Buehler,et al.  ROBUSTNESS-STRENGTH PERFORMANCE OF HIERARCHICAL ALPHA-HELICAL PROTEIN FILAMENTS , 2009 .

[29]  J. Shaw,et al.  A formula for scoring human embryo growth rates in in vitro fertilization: Its value in predicting pregnancy and in comparison with visual estimates of embryo quality , 1986, Journal of in Vitro Fertilization and Embryo Transfer.