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PDEs in Laser Waves and Biology

PDEs in Laser Waves and Biology. Presentation of my research fields Marie Doumic Jauffret doumic@dma.ens.fr. Outline. Laser Wave Propagation Modelling Laser Waves: what for ? Approximation of the Physical Model Theoretical Resolution Numerical Simulations Transport Equations for Biology

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PDEs in Laser Waves and Biology

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  1. PDEs in Laser Waves and Biology Presentation of my research fields Marie Doumic Jauffret doumic@dma.ens.fr DEASE meeting - Vienna

  2. Outline • Laser Wave Propagation • Modelling Laser Waves: what for ? • Approximation of the Physical Model • Theoretical Resolution • Numerical Simulations • 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 DEASE meeting - Vienna

  3. 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

  4. The physical problem Laser: Maxwell Equation + Plasma : mass and impulse conservation = Klein-Gordon Equation: ω0 Laser impulse, νabsorption coefficient due to electron-ion collision N adimensioned electronic density

  5. 2 Main difficulties to model Laser-Plasma interaction -> very different orders of magnitude -> the ray propagates non perpendicularly to the boundary of the domain y k α x cf. M.D. Feit, J.A. Fleck, Beam non paraxiality, J. Opt.Soc.Am. B 5, p633-640 (1988). Only α < 15° and lack of mathematical justification

  6. 1st step: choose the correct small parameter ε DEASE meeting - Vienna

  7. 2nd step:approximation of K-G equation(Chapman-Enskog method) 1st order: Hamilton-Jacobi + transport equation DEASE meeting - Vienna

  8. 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.

  9. 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 y k α x Oblique Schrödinger equation:

  10. Half-space problem Fourier transform: for

  11. 5th step:numerical scheme with interaction with the plasma. DEASE meeting - Vienna

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

  13. 1st stage: solving and then Simultaneously: we have: FFT of -> multiply by -> IFFT DEASE meeting - Vienna

  14. 2nd stage: solving Standard upwind decentered scheme: With and DEASE meeting - Vienna

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

  16. Properties of the scheme • stability: non-increasing scheme: • Convergence towards Schrödinger eq.: If the scheme converges towards the solution of: DEASE meeting - Vienna

  17. 6th step: numerical testsConvergence of the scheme 45° Fig. 1: referencee = 0.05, n0 = n1 = 0.0005, a = 0.05, angle 45° uin = exp(-(k.x/L)2), L=2.5, dx=dy=0.4. (CFL=1) We get Lfoc=60.0 and Max (|u|2)=2.14 DEASE meeting - Vienna

  18. Convergence of the 1st order scheme Fig. 2: low precisiondx=dy=0.8 (CFL=1) We get Lfoc=61.5 and Max (|u|2)=2.16 DEASE meeting - Vienna

  19. Convergence of the 1st order scheme Fig. 3: high precisiondx=dy=0.1 (CFL=1) We get Lfoc=59.4 and Max (|u|2)=2.14 DEASE meeting - Vienna

  20. Convergence of the 2nd order scheme Fig. 3: low precisiondx=0.16 dy=0.4 (CFL=0.4) We get Lfoc=50.7 and Max (|u|2)=1.24 DEASE meeting - Vienna

  21. Convergence of the 2nd order scheme Fig. 3: high precisiondx=0.04 dy=0.1 (CFL=0.4) We get Lfoc=60.5 and Max (|u|2)=2.06 DEASE meeting - Vienna

  22. Variation of the incidence angle Fig. 3: Angle 5° We get Lfoc=60.6 and Max (|u|2)=2.2 DEASE meeting - Vienna

  23. Variation of the incidence angle Fig. 3: Angle 60° We get Lfoc=59.7 and Max (|u|2)=2.10 DEASE meeting - Vienna

  24. Rays crossing incidence +/-45°, u2in = 0.8 exp(-(Y2/5)2), u1in = exp(-(Y/40)6)(1+0.3cos(2pY/10))

  25. Rays crossing Interaction: Max (|u|12+|u|22)=12.3 No interaction: Max (|u|12+|u|22)=10.6 DEASE meeting - Vienna

  26. 7th step: coupling with hydrodynamics (work of Frédéric DUBOC) Introduction of the scheme in the HERA code of CEA (here: angle = 15°) DEASE meeting - Vienna

  27. … 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… DEASE meeting - Vienna

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

  29. 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 DEASE meeting - Vienna

  30. 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 DEASE meeting - Vienna

  31. Cyclin B M G2 G1 S Cyclin A A focus on : Modelling the cell division cycle S: DNA synthesis G1,G2:Gap1,2 M: mitosis Mitosis=M phase Cyclin D Mitotic human HeLa cell (from LBCMCP-Toulouse) • Physiological / therapeutic control • - on transitions between phases • (G1/S, G2/M, M/G1) • on death rates inside phases • (apoptosis or necrosis) • on the inclusion into the cell cycle • (G0 to G1 recruitment) Cyclin E DEASE meeting - Vienna

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

  33. Models for the cell cycle 2. The age variable: McKendrick-Von Foerster: Birth rate DEASE meeting - Vienna

  34. An age and molecular-content structured model for the cell cycle d2 d1 L Q F P G Proliferating cells Quiescent cells 3 variables: time t, age a, cyclin-content x

  35. An age and molecular-content structured model for the cell cycle Proliferating cells =1 quiescent cells DIVISION Death rate Death rate recruitment Demobilisation 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 2007.

  36. + Initial conditions at t=0: pin(a,x) and qin(a,x) + Birth condition for a=0: with Mother cell (cyclin content Daughter cell Uniform repartition: DEASE meeting - Vienna

  37. 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 DEASE meeting - Vienna

  38. 1. Eigenvalue linearised problem Simplified in:

  39. Linearised & simplified problem:Reformulation with the characteristics N=0 x XM Γ1<0 Γ1=0 Γ1>0 X0 a

  40. 1. Linearised & simplified problem Reformulation of the problem with the characteristics: Key assumption: Which can also be formulated as : -> there exists a unique λ0>0 and a unique solution N such thatfor all

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

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

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

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

  45. Robust polynomial growth Link between λ and λ0: If d2=0 and α2=0 in the formula we can obtain (H4) and unlimited subpolynomial growth in a robust way: DEASE meeting - Vienna

  46. 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) DEASE meeting - Vienna

  47. Danke für Ihre Aufmerksamkeit ! DEASE meeting - Vienna

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