12th, July 2007DEASE meeting - Vienna PDEs in Laser Waves and Biology Presentation of my research fields Marie Doumic Jauffret

12th, July 2007DEASE meeting - Vienna Outline I.Laser Wave Propagation  Modelling Laser Waves: what for ?  Approximation of the Physical Model  Theoretical Resolution  Numerical Simulations II.Transport Equations for Biology  Presentation of the BANG project team  Modelling Leukaemia: the « ARC ModLMC » network  Modelling the cell cycle: a macroscopic model and some results

I. Laser Wave Propagation Laser MEGAJOULE: the biggest in the world in 2009 Our goal: to model the Laser-Plasma interaction Work directed by François GOLSE and Rémi SENTIS

The physical problem Laser: Maxwell Equation + Plasma : mass and impulse conservation = Klein-Gordon Equation: ω 0 Laser impulse, ν absorption coefficient due to electron-ion collision N adimensioned electronic density

2 Main difficulties to model Laser- Plasma interaction -> very different orders of magnitude -> the ray propagates non perpendicularly to the boundary of the domain α k x y cf. M.D. Feit, J.A. Fleck, Beam non paraxiality, J. Opt.Soc.Am. B 5, p (1988). Only α < 15° and lack of mathematical justification

12th, July 2007DEASE meeting - Vienna 1st step: choose the correct small parameter ε

12th, July 2007DEASE meeting - Vienna 2nd step: approximation of K-G equation (Chapman-Enskog method) 1st order: Hamilton-Jacobi + transport equation

Second order: « paraxial approximation » « Advection-Schrödinger equation » 3rd Step: theoretical analysis (whole space) We prove that -> the problem is well-posed -> it is a correct approximation of the exact problem Cf. PhD Thesis of M. Doumic, available on HAL.

4th step: study in a bounded domain Preceding equation but -> time dependancy is neglected -> linear propagation along a fixed vector k, -> arbitrary angle α -> boundary conditions on (x=0) and (y=0) have to be found α k x y Oblique Schrödinger equation:

Half-space problem Fourier transform: for

12th, July 2007DEASE meeting - Vienna 5th step: numerical scheme with interaction with the plasma.

12th, July 2007DEASE meeting - Vienna Numerical scheme: Initializing: cf. preceding formula: FFT of g -> multiply by -> IFFT

12th, July 2007DEASE meeting - Vienna 1st stage: solving and then Simultaneously: we have: FFT of -> multiply by -> IFFT

12th, July 2007DEASE meeting - Vienna 2 nd stage: solving Standard upwind decentered scheme: With and

Second order scheme: Flux limiter of Van Leer: 2 rays crossing: we solve for p=1,2:

12th, July 2007DEASE meeting - Vienna Properties of the scheme stability: non-increasing scheme: Convergence towards Schrödinger eq.: If the scheme converges towards the solution of:

12th, July 2007DEASE meeting - Vienna 6th step: numerical tests Convergence of the scheme Fig. 1: reference , 0  1 , , angle 45 ° u in  exp(-(k.x/L) 2 ), L ,  x =  y =0.4. (CFL=1) We get L foc =60.0 and Max (|u| 2 )= °

12th, July 2007DEASE meeting - Vienna Convergence of the 1st order scheme Fig. 2: low precision  x =  y =0.8 (CFL=1) We get L foc =61.5 and Max (|u| 2 )=2.16

12th, July 2007DEASE meeting - Vienna Convergence of the 1st order scheme Fig. 3: high precision  x =  y =0.1 (CFL=1) We get L foc =59.4 and Max (|u| 2 )=2.14

12th, July 2007DEASE meeting - Vienna Convergence of the 2nd order scheme Fig. 3: low precision  x =0.16  y =0.4 (CFL=0.4) We get L foc =50.7 and Max (|u| 2 )=1.24

12th, July 2007DEASE meeting - Vienna Convergence of the 2nd order scheme Fig. 3: high precision  x =0.04  y =0.1 (CFL=0.4) We get L foc =60.5 and Max (|u| 2 )=2.06

12th, July 2007DEASE meeting - Vienna Variation of the incidence angle Fig. 3: Angle 5° We get L foc =60.6 and Max (|u| 2 )=2.2

12th, July 2007DEASE meeting - Vienna Variation of the incidence angle Fig. 3: Angle 60° We get L foc =59.7 and Max (|u| 2 )=2.10

Rays crossing incidence +/-45°, u 2 in = 0.8 exp(-(Y 2 /5) 2 ), u 1 in = exp(-(Y/40) 6 )(1+0.3cos(2pY/10))

12th, July 2007DEASE meeting - Vienna Rays crossing Interaction: Max (|u| 1 2 +|u| 2 2 )=12.3 No interaction: Max (|u| 1 2 +|u| 2 2 )=10.6

12th, July 2007DEASE meeting - Vienna 7th step: coupling with hydrodynamics (work of Frédéric DUBOC) Introduction of the scheme in the HERA code of CEA (here: angle = 15°)

12th, July 2007DEASE meeting - Vienna … and scheme adapted to curving rays and time-dependent interaction model Here angle from 15° to 23° … and last step: comparison with the experiments of Laser Megajoule…

12th, July 2007DEASE meeting - Vienna II. PDEs in Biology The « B » part of the BANG project team: -Joint INRIA and ENS team -Directed by Benoît Perthame -Some renowned people:

12th, July 2007DEASE meeting - Vienna The « ARC ModLMC » Research network coordinated by Mostafa Adimy (Pr. at Pau University) Joint group of –Medical Doctors: 3 teams in Lyon and Bordeaux of oncologists –Applied Mathematicians: 2 INRIA project teams (BANG and ANUBIS) and 1 team of Institut Camille Jordan of Lyon

12th, July 2007DEASE meeting - Vienna The « ARC ModLMC » Goals: –Develop and analyse new mathematical models for Chronic Myelogenous Leukaemia (CML/LMC in French) –Explain the oscillations experimentally observed during the chronic phase –Optimise the medical treatment by Imatinib: to control drug resistance and toxicity for healthy tissues

12th, July 2007DEASE meeting - Vienna Cyclin D Cyclin E Cyclin A Cyclin B S G1G1 G2G2 M A focus on : Modelling the cell division cycle Physiological / therapeutic control - on transitions between phases (G 1 /S, G 2 /M, M/G 1 ) (G 1 /S, G 2 /M, M/G 1 ) - on death rates inside phases (apoptosis or necrosis) (apoptosis or necrosis) -on the inclusion into the cell cycle (G 0 to G 1 recruitment) (G 0 to G 1 recruitment) S: DNA synthesis G 1,G 2 :Gap1,2 M: mitosis Mitosis=M phase Mitotic human HeLa cell (from LBCMCP-Toulouse)

Models for the cell cycle Malthus parameter: Exponential growth Logistic growth (Verhulst): 1. Historical models of population growth: -> various ways to complexify this equation: Cf. B. Perthame, Transport Equations in Biology, Birkhäuser 2006.

12th, July 2007DEASE meeting - Vienna Models for the cell cycle 2. The age variable: McKendrick-Von Foerster: Birth rate

An age and molecular-content structured model for the cell cycle P Q Proliferating cellsQuiescent cells L G d1d1 d2d2 F 3 variables: time t, age a, cyclin-content x

An age and molecular-content structured model for the cell cycle Cf. F. Bekkal-Brikci, J. Clairambault, B. Perthame, Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Math. And Comp. Modelling, available on line, july quiescent cells Proliferating cells =1 Demobilisation DIVISIONDeath rate recruitmentDeath rate

12th, July 2007DEASE meeting - Vienna with Daughter cell Mother cell (cyclin content Uniform repartition: + Initial conditions at t=0: p in (a,x) and q in (a,x) + Birth condition for a=0:

12th, July 2007DEASE meeting - Vienna Goal: study the asymptotic behaviour of the model : the Malthus parameter 1. study of the eigenvalue linearised problem (and its adjoint) 2. Generalised Relative Entropy method Cf. Michel P., Mischler S., Perthame B., General relative entropy inequality: an illustration on growth models, J. Math. Pur. Appl. (2005). 3. Back to the non-linear problem 4. Numerical validation

1. Eigenvalue linearised problem Simplified in:

a x Γ 1 =0 Γ 1 >0 Γ 1 <0 XMXM X0X0 1.Linearised & simplified problem: Reformulation with the characteristics N=0

Reformulation of the problem with the characteristics: Key assumption: Which can also be formulated as : -> there exists a unique λ 0 >0 and a unique solution N such that for all 1. Linearised & simplified problem

12th, July 2007DEASE meeting - Vienna Theorem: Under the same assumptions than for existence and unicity in the eigenvalue problem, we have 2. Asymptotic convergence for the linearised problem

Back to the original non-linear problem Eigenvalue problem: Since G=G(N(t)) we have P=e λ[G(N(t))].t Study of the linearised problem in different values of G(N)

Healthy tissues: (H1) forwe have non-extinction (H2) for we have convergence towards a steady state The non-linear problem P=e λ[G(N(t))].t

Tumour growth: (H3) for we have unlimited exponential growth (H4) for we have subpolynomial growth (not robust) The non-linear problem P=e λ[G(N(t))].t

12th, July 2007DEASE meeting - Vienna Robust polynomial growth Link between λ and λ 0 : If d 2 =0 and α 2 =0 in the formula we can obtain (H4) and unlimited subpolynomial growth in a robust way:

12th, July 2007DEASE meeting - Vienna What is coming next…. -compare the model with data: inverse problems -Adapt the model to leukaemia (by distinction between mature cells and stem cells: at least 4 compartments)

12th, July 2007DEASE meeting - Vienna Danke für Ihre Aufmerksamkeit !