D. Ambrosi and L. Preziosi, ON THE CLOSURE OF MASS BALANCE MODELS FOR TUMOR GROWTH, Mathematical Models and Methods in Applied Sciences, vol.12, issue.05, pp.734-754, 2002.
DOI : 10.1142/S0218202502001878

D. Ambrosi and L. Preziosi, Cell adhesion mechanisms and stress relaxation in the mechanics of tumours, Biomechanics and Modeling in Mechanobiology, vol.2, issue.5, pp.397-413, 2009.
DOI : 10.1007/s10237-008-0145-y

D. Ambrosi, L. Preziosi, and C. Verdier, An elasto-visco-plastic model of cell aggregates, J. Theor. Biol, vol.262, pp.35-47, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00554642

A. R. Anderson and M. Chaplain, Continuous and Discrete Mathematical Models of Tumor-induced Angiogenesis, Bulletin of Mathematical Biology, vol.60, issue.5, pp.857-900, 1998.
DOI : 10.1006/bulm.1998.0042

M. Andreeff, W. David, . Goodrich, B. Arthur, and . Pardee, Cancer Medicine, chapter Cell Proliferation, Differentiation, and Apoptosis, 2000.

P. Angot, C. Bruneau, and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numerische Mathematik, vol.81, issue.4, pp.497-520, 1999.
DOI : 10.1007/s002110050401

P. Araujo and D. L. Mcelwain, A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues I: A General Formulation, SIAM Journal on Applied Mathematics, vol.65, issue.4, pp.1261-1284, 2005.
DOI : 10.1137/040607113

E. Arquis and J. P. Caltagirone, Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieux poreux: application la convection naturelle, C.R. Acad. Sci. Paris II, vol.299, pp.1-4, 1984.

I. Babuska, Die Methode der finiten Elemente f??r elliptische Gleichungen mit diskontinuierlichen Koeffizienten, Computing, vol.19, issue.3, pp.207-213, 1970.
DOI : 10.1007/BF02248021

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp et al., PETSc users manual, 2008.

S. Balay, W. D. Gropp, L. C. Mcinnes, and B. F. Smith, Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries, Modern Software Tools in Scientific Computing, pp.163-202, 1997.
DOI : 10.1007/978-1-4612-1986-6_8

E. L. Bearer, J. S. Lowengrub, Y. L. Chuang, H. B. Friboes, F. Jin et al., Multiparameter Computational Modeling of Tumor Invasion, Cancer Research, vol.69, issue.10, pp.4493-4501, 2009.
DOI : 10.1158/0008-5472.CAN-08-3834

URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2835777

P. Berthelsen, A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, Journal of Computational Physics, vol.197, issue.1, pp.364-386, 2004.
DOI : 10.1016/j.jcp.2003.12.003

F. Billy, B. Ribba, O. Saut, H. Morre-trouilhet, T. Colin et al., A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy, Journal of Theoretical Biology, vol.260, issue.4, pp.545-562, 2009.
DOI : 10.1016/j.jtbi.2009.06.026

URL : https://hal.archives-ouvertes.fr/inria-00440447

J. Bramble and J. King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Advances in Computational Mathematics, vol.58, issue.1, pp.109-138, 1996.
DOI : 10.1007/BF02127700

D. Bresch, T. Colin, E. Grenier, B. Ribba, and O. Saut, Computational Modeling of Solid Tumor Growth: The Avascular Stage, SIAM Journal on Scientific Computing, vol.32, issue.4, pp.2321-2344, 2009.
DOI : 10.1137/070708895

URL : https://hal.archives-ouvertes.fr/inria-00148610

F. Buret, N. Faure, L. Nicolas, R. Perussel, and C. Poignard, Numerical studies on the effect of electric pulses on an egg-shaped cell with a spherical nucleus, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00477495

J. T. Bushberg, J. A. Seibert, E. M. Jr, and J. M. Boone, The Essential Physics of Medical Imaging, Medical Physics, vol.30, issue.7, 2001.
DOI : 10.1118/1.1585033

F. Chantalat, C. H. Bruneau, C. Galusinski, and A. Iollo, Level-set, penalization and cartesian meshes: A paradigm for inverse problems and optimal design, Journal of Computational Physics, vol.228, issue.17, pp.6291-6315, 2009.
DOI : 10.1016/j.jcp.2009.05.017

URL : https://hal.archives-ouvertes.fr/hal-00385460

M. A. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development, Mathematical and Computer Modelling, vol.23, issue.6, pp.47-87, 1996.
DOI : 10.1016/0895-7177(96)00019-2

A. Chauviere, T. Hillen, and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues, Networks Heterogen. Media, pp.333-357, 2007.
DOI : 10.3934/nhm.2007.2.333

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik, vol.79, issue.2, pp.175-202, 1998.
DOI : 10.1007/s002110050336

I. Chern and Y. Shu, A coupling interface method for elliptic interface problems, Journal of Computational Physics, vol.225, issue.2, pp.2138-2174, 2007.
DOI : 10.1016/j.jcp.2007.03.012

M. Cisternino and L. Weynans, Abstract, Communications in Computational Physics, vol.39, issue.05
DOI : 10.1016/j.jcp.2006.02.014

T. Colin, A. Iollo, D. Lombardi, and O. Saut, SYSTEM IDENTIFICATION IN TUMOR GROWTH MODELING USING SEMI-EMPIRICAL EIGENFUNCTIONS, Mathematical Models and Methods in Applied Sciences, vol.22, issue.06, 2011.
DOI : 10.1142/S0218202512500030

J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Camb, pp.50-67, 1947.

V. Cristini, X. Li, J. S. Lowengrub, and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching, Journal of Mathematical Biology, vol.67, issue.4-5, pp.723-763, 2009.
DOI : 10.1007/s00285-008-0215-x

C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, vol.77, issue.2, pp.439-471, 1988.
DOI : 10.1016/0021-9991(88)90177-5

C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, Journal of Computational Physics, vol.83, issue.1, pp.32-87, 1989.
DOI : 10.1016/0021-9991(89)90222-2

A. Deutsch and S. Dormann, Cellular Automaton Modeling of Biological Pattern Formation, Birkhäuser, 2005.

D. Drasdo and S. Hömme, Individual-based approaches to birth and death in avascu1ar tumors, Mathematical and Computer Modelling, vol.37, issue.11, pp.1163-1175, 2003.
DOI : 10.1016/S0895-7177(03)00128-6

URL : http://doi.org/10.1016/s0895-7177(03)00128-6

R. E. Ewing, Z. Li, T. Lin, and Y. Lin, The immersed finite volume element methods for the elliptic interface problems, Mathematics and Computers in Simulation, vol.50, issue.1-4, pp.63-76, 1999.
DOI : 10.1016/S0378-4754(99)00061-0

R. P. Fedkiw, Coupling an Eulerian Fluid Calculation to a Lagrangian Solid Calculation with the Ghost Fluid Method, Journal of Computational Physics, vol.175, issue.1, pp.200-224, 2002.
DOI : 10.1006/jcph.2001.6935

R. P. Fedkiw, T. Aslam, B. Merriman, and S. Osher, A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method), Journal of Computational Physics, vol.152, issue.2
DOI : 10.1006/jcph.1999.6236

J. Folkman, Cancer Medicine, chapter Tumor Angiogenesis, 2000.

M. Garbey and G. Zouridakis, Modeling tumor growth: from differential deformable models to growth prediction of tumors detected in PET images, Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE Cat. No.03CH37439), pp.2687-2690, 2003.
DOI : 10.1109/IEMBS.2003.1280470

P. Gerlee and A. R. Anderson, An evolutionary hybrid cellular automaton model of solid tumour growth, Journal of Theoretical Biology, vol.246, issue.4, pp.583-603, 2007.
DOI : 10.1016/j.jtbi.2007.01.027

F. Gibou, R. P. Fedkiw, L. T. Cheng, and M. Kang, A Second-Order-Accurate Symmetric Discretization of the Poisson Equation on Irregular Domains, Journal of Computational Physics, vol.176, issue.1, pp.205-227, 2002.
DOI : 10.1006/jcph.2001.6977

F. Gibou and R. P. Fedkiw, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, Journal of Computational Physics, vol.202, issue.2, pp.577-601, 2005.
DOI : 10.1016/j.jcp.2004.07.018

B. Gompertz, On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies, Philosophical Transactions of the Royal Society of London, vol.115, issue.0, pp.513-585
DOI : 10.1098/rstl.1825.0026

S. Gottlieb and C. Shu, Total variation diminishing Runge-Kutta schemes, Mathematics of Computation of the American Mathematical Society, vol.67, issue.221, pp.73-85, 1998.
DOI : 10.1090/S0025-5718-98-00913-2

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.105.4521

B. Gustafsson, A fourth order accurate discretization for the laplace and heat equations on arbitrary domains, with applications to the stefan problem, SIAM Journal of Numerical Analysis, vol.39, pp.396-406, 1975.

B. Gustafsson, The Convergence Rate for Difference Approximations to General Mixed Initial-Boundary Value Problems, SIAM Journal on Numerical Analysis, vol.18, issue.2, pp.179-190, 1981.
DOI : 10.1137/0718014

P. Hahnfeldt, D. Panigrahy, J. Folkman, and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, vol.59, pp.4770-4775, 1999.

A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III, Journal of Computational Physics, vol.71, issue.2, p.231, 1987.
DOI : 10.1016/0021-9991(87)90031-3

J. Huang and J. Zou, A mortar element method for elliptic problems with discontinuous coefficients. A mortar element method for elliptic problems with discontinuous coefficients, pp.549-576, 2002.

J. Huh and J. A. Sethian, Exact subgrid interface correction schemes for elliptic interface problems, Proceedings of the National Academy of Sciences of the United States of America, p.9874, 2008.
DOI : 10.1073/pnas.0707997105

H. Johansen and P. Colella, A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains, Journal of Computational Physics, vol.147, issue.1, pp.60-85, 1998.
DOI : 10.1006/jcph.1998.5965

J. S. Kim, M. A. Stolarska, and H. G. Othmer, I: THEORETICAL DEVELOPMENT AND EARLY RESULTS, Mathematical Models and Methods in Applied Sciences, vol.17, issue.supp01, pp.1773-1798, 2007.
DOI : 10.1142/S0218202507002479

A. K. Laird, Dynamics of Tumor Growth, British Journal of Cancer, vol.18, issue.3, pp.490-502, 1964.
DOI : 10.1038/bjc.1964.55

R. J. Leveque and L. Z. Li, The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources, SIAM Journal on Numerical Analysis, vol.31, issue.4, pp.1019-1044, 1994.
DOI : 10.1137/0731054

Z. L. Li, A Fast Iterative Algorithm for Elliptic Interface Problems, SIAM Journal on Numerical Analysis, vol.35, issue.1, pp.230-254, 1998.
DOI : 10.1137/S0036142995291329

Z. L. Li and K. Ito, Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients, SIAM Journal on Scientific Computing, vol.23, issue.1, pp.339-361, 2001.
DOI : 10.1137/S1064827500370160

A. Lance, . Liotta, C. Elise, and . Kohn, Cancer Medicine, chapter Invasion and Metastases, 2000.

X. Liu, R. P. Fedkiw, and M. Kang, A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains, Journal of Computational Physics, vol.160, issue.1, pp.151-178, 2000.
DOI : 10.1006/jcph.2000.6444

D. Lombardi, Probì emes Inverses pour les modèles de croissance tumorale, 2011.

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. Chuang, X. Li et al., Nonlinear modelling of cancer: bridging the gap between cells and tumours, Nonlinearity, vol.23, issue.1, pp.1-91, 2010.
DOI : 10.1088/0951-7715/23/1/R01

P. Macklin and J. S. Lowengrub, An improved geometry-aware curvature discretization for level set methods: Application to tumor growth, Journal of Computational Physics, vol.215, issue.2, pp.392-401, 2006.
DOI : 10.1016/j.jcp.2005.11.016

B. Maury, A fat boundary method for the poisson problem in a domain with holes, Journal of Scientific Computing, vol.16, issue.3, pp.319-339, 2001.
DOI : 10.1023/A:1012821728631

A. Mayo, The Fast Solution of Poisson???s and the Biharmonic Equations on Irregular Regions, SIAM Journal on Numerical Analysis, vol.21, issue.2, pp.285-299, 1984.
DOI : 10.1137/0721021

A. Mayo, The rapid evaluation of volume integrals of potential theory on general regions, Journal of Computational Physics, vol.100, issue.2, pp.236-245, 1992.
DOI : 10.1016/0021-9991(92)90231-M

A. Mayo and A. Greenbaum, Fast Parallel Iterative Solution of Poisson???s and the Biharmonic Equations on Irregular Regions, SIAM Journal on Scientific and Statistical Computing, vol.13, issue.1, pp.101-118, 1992.
DOI : 10.1137/0913006

P. Mccorquodale, P. Collela, and H. Johansen, A Cartesian Grid Embedded Boundary Method for the Heat Equation on Irregular Domains, Journal of Computational Physics, vol.173, issue.2, pp.620-635, 2001.
DOI : 10.1006/jcph.2001.6900

A. Bellouquid, N. Bellomo, and E. De-angelis, The modelling of the immune competition by generalized kinetic (boltzmann) models: Review and research perspectives, Mathematical and Computer Modelling, vol.37, pp.65-86, 2003.

M. Oevermann, C. Scharfenberg, and R. Klein, A sharp interface finite volume method for elliptic equations on Cartesian grids, Journal of Computational Physics, vol.228, issue.14, pp.5184-5206, 2009.
DOI : 10.1016/j.jcp.2009.04.018

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, 2003.

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, p.79, 1988.
DOI : 10.1016/0021-9991(88)90002-2

H. G. Othmer and A. Stevens, Aggregation, blowup and collapse: the abc's of taxis in reinforced random walks, SIAM J. Appl. Math, vol.57, issue.4, pp.1044-1081, 1997.

K. A. Rejniak, An immersed boundary framework for modelling the growth of individual cells: An application to the early tumour development, Journal of Theoretical Biology, vol.247, issue.1, pp.186-204, 2007.
DOI : 10.1016/j.jtbi.2007.02.019

B. Ribba, T. Colin, and S. Schnell, A multiscale mathematical model of cancer, and its use in analizing irradiation therapies, Theoretical Biology and Medical Modelling, vol.3, issue.1, p.7, 2006.
DOI : 10.1186/1742-4682-3-7

J. B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier et al., A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents, Journal of Theoretical Biology, vol.243, issue.4, pp.523-541, 2006.
DOI : 10.1016/j.jtbi.2006.07.013

URL : https://hal.archives-ouvertes.fr/hal-00428053

T. Roose, S. J. Chapman, and P. K. Maini, Mathematical Models of Avascular Tumor Growth, SIAM Review, vol.49, issue.2, pp.179-208, 2007.
DOI : 10.1137/S0036144504446291

T. Roose, P. A. Netti, L. L. Munn, Y. Boucher, and R. Jain, Solid stress generated by spheroid growth estimated using a linear poroelasticity model???, Microvascular Research, vol.66, issue.3, pp.204-212, 2003.
DOI : 10.1016/S0026-2862(03)00057-8

Y. Saad, Sparskit a basic tool-kit for sparse matrix computations

A. Sarthou, S. Vincent, P. Angot, and J. P. Caltagirone, The algebraic immersed interface and boundary method for elliptic equations with discontinuous coefficients, 2009.

J. A. Sethian, A fast marching level set method for monotonically advancing fronts., Proc. Nat. Acad. Sci, pp.1591-1595, 1996.
DOI : 10.1073/pnas.93.4.1591

J. A. Sethian, Level Set Methods and Fast Marching Methods, 1999.

J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science, 1999.

J. A. Sethian, Evolution, Implementation, and Application of Level Set and Fast Marching Methods for Advancing Fronts, Journal of Computational Physics, vol.169, issue.2, pp.503-555, 2001.
DOI : 10.1006/jcph.2000.6657

M. Svard and J. Nordstrom, On the order of accuracy for difference approximations of initial-boundary value problems, Journal of Computational Physics, vol.218, issue.1, pp.333-352, 2006.
DOI : 10.1016/j.jcp.2006.02.014

K. R. Swanson, C. Bridge, J. D. Murray, and E. C. Jr, Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion, Journal of the Neurological Sciences, vol.216, issue.1, pp.1-10, 2003.
DOI : 10.1016/j.jns.2003.06.001

Z. Wang, L. Zhang, J. Sagotsky, and T. S. Deisboeck, Simulating non-small cell lung cancer with a multiscale agent-based model, Theoretical Biology and Medical Modelling, vol.4, issue.1, p.50, 2007.
DOI : 10.1186/1742-4682-4-50

A. Wiegmann and K. Bube, The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions, SIAM Journal on Numerical Analysis, vol.37, issue.3, pp.827-862, 2000.
DOI : 10.1137/S0036142997328664

X. Liu, S. Osher, and C. T. , Weighted Essentially Non-oscillatory Schemes, Journal of Computational Physics, vol.115, issue.1, pp.202-212, 1996.
DOI : 10.1006/jcph.1994.1187

T. S. Yoo, M. J. Ackerman, W. E. Lorensen, W. Schroeder, V. Chalana et al., Engineering and algorithm design for an image processing api: A technical report on itk -the insight toolkit, Proc. of Medicine Meets Virtual, pp.586-592, 2002.

S. Yu and G. W. Wei, Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities, Journal of Computational Physics, vol.227, issue.1, pp.602-632, 2007.
DOI : 10.1016/j.jcp.2007.08.003

X. Zhong, A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity, Journal of Computational Physics, vol.225, issue.1, pp.1066-1099, 2007.
DOI : 10.1016/j.jcp.2007.01.017

Y. C. Zhou and G. W. Wei, On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, Journal of Computational Physics, vol.219, issue.1, pp.228-246, 2006.
DOI : 10.1016/j.jcp.2006.03.027

Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, Journal of Computational Physics, vol.213, issue.1, pp.1-30, 2006.
DOI : 10.1016/j.jcp.2005.07.022