Post on 19-Dec-2015
12th, July 2007 DEASE meeting - Vienna
PDEs in Laser Waves and Biology
Presentation of my research fields
Marie Doumic Jauffretdoumic@dma.ens.fr
12th, July 2007 DEASE 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
12th, July 2007 DEASE meeting - Vienna
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
12th, July 2007 DEASE meeting - Vienna
The physical problem
Laser: Maxwell Equation+
Plasma : mass and impulse conservation=
Klein-Gordon Equation:
ω0 Laser impulse,
ν absorption coefficient due to electron-ion collision
N adimensioned electronic density
12th, July 2007 DEASE meeting - Vienna
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, p633-640 (1988).
Only α < 15° and lack of mathematical justification
12th, July 2007 DEASE meeting - Vienna
2nd step: approximation of K-G equation(Chapman-Enskog method)
1st order:
Hamilton-Jacobi + transport equation
12th, July 2007 DEASE meeting - Vienna
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 problemCf. PhD Thesis of M. Doumic, available on HAL.
12th, July 2007 DEASE meeting - Vienna
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:
12th, July 2007 DEASE meeting - Vienna
Numerical scheme:
Initializing: cf. preceding formula:
FFT of g -> multiply by -> IFFT
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1st stage: solving
and then
Simultaneously: we have:
FFT of -> multiply by -> IFFT
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2nd stage: solving
Standard upwind decentered scheme:
With and
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Second order scheme: Flux limiter of Van Leer:
2 rays crossing: we solve for p=1,2:
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Properties of the scheme
• stability: non-increasing scheme:
• Convergence towards Schrödinger eq.:If the scheme converges towards the
solution of:
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6th step: numerical testsConvergence of the scheme
Fig. 1: reference ,
0 1 , , angle
45°
uinexp(-(k.x/L)2), L, x=y=0.4.(CFL=1)
We get Lfoc=60.0 and Max (|u|2)=2.14
45°
12th, July 2007 DEASE meeting - Vienna
Convergence of the 1st order scheme
Fig. 2: low
precision x=y=0.8
(CFL=1)
We get Lfoc=61.5 and Max (|u|2)=2.16
12th, July 2007 DEASE meeting - Vienna
Convergence of the 1st order scheme
Fig. 3: high
precision x=y=0.1
(CFL=1)
We get Lfoc=59.4 and Max (|u|2)=2.14
12th, July 2007 DEASE meeting - Vienna
Convergence of the 2nd order scheme
Fig. 3: low
precision x=0.16 y=0.4
(CFL=0.4)
We get Lfoc=50.7 and Max (|u|2)=1.24
12th, July 2007 DEASE meeting - Vienna
Convergence of the 2nd order scheme
Fig. 3: high
precision x=0.04 y=0.1
(CFL=0.4)
We get Lfoc=60.5 and Max (|u|2)=2.06
12th, July 2007 DEASE meeting - Vienna
Variation of the incidence angle
Fig. 3: Angle 5°
We get Lfoc=60.6 and Max (|u|2)=2.2
12th, July 2007 DEASE meeting - Vienna
Variation of the incidence angle
Fig. 3:Angle 60°
We get Lfoc=59.7 and Max (|u|2)=2.10
12th, July 2007 DEASE meeting - Vienna
Rays crossing
incidence +/-45°, u2in = 0.8 exp(-(Y2/5)2),
u1in = exp(-(Y/40)6)(1+0.3cos(2pY/10))
12th, July 2007 DEASE meeting - Vienna
Rays crossing
Interaction: Max (|u|1
2+|u|22)=12.3
No interaction: Max (|u|1
2+|u|22)=10.6
12th, July 2007 DEASE 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 2007 DEASE 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 2007 DEASE 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 2007 DEASE 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 2007 DEASE 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 2007 DEASE meeting - Vienna
Cyclin DCyclin D
Cyclin ECyclin ECyclin ACyclin A
Cyclin BCyclin B
SG1
G2
M
A focus on : Modelling the cell division cycleA focus on : Modelling the cell division cycle
Physiological / therapeutic controlPhysiological / therapeutic control- on transitions between phases- on transitions between phases (G(G11/S, G/S, G22/M, M/G/M, M/G11))- on death rates inside phaseson death rates inside phases (apoptosis or necrosis) (apoptosis or necrosis) -on the inclusion into the cell cycleon the inclusion into the cell cycle (G(G00 to G to G1 1 recruitment)recruitment)
S: DNA synthesis S: DNA synthesis GG11,G,G22:Gap1,2 M: mitosis:Gap1,2 M: mitosis
Mitosis=M phaseMitosis=M phase
Mitotic human HeLa cell (from LBCMCP-Toulouse)Mitotic human HeLa cell (from LBCMCP-Toulouse)
12th, July 2007 DEASE meeting - Vienna
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 2007 DEASE meeting - Vienna
Models for the cell cycle2. The age variable: McKendrick-Von Foerster:
Birth rate
12th, July 2007 DEASE meeting - Vienna
An age and molecular-content
structured model for the cell cycle
P Q
Proliferating cells Quiescent cells
L
G
d1d2
F
3 variables: time t, age a, cyclin-content x
12th, July 2007 DEASE meeting - Vienna
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 2007.
quiescent cells
Proliferating cells =1
Demobilisation
DIVISION Death rate
recruitmentDeath rate
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with Daughter cell Mother cell (cyclin content
Uniform repartition:
+ Initial conditions at t=0: pin(a,x) and qin(a,x)
+ Birth condition for a=0:
12th, July 2007 DEASE 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
12th, July 2007 DEASE meeting - Vienna
a
x
Γ1=0
Γ1>0
Γ1<0XM
X0
1. Linearised & simplified problem:Reformulation with the characteristics
N=0
12th, July 2007 DEASE meeting - Vienna
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 that for all
1. Linearised & simplified problem
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Theorem: Under the same assumptions than for existence and unicity in the eigenvalue problem, we have
2. Asymptotic convergence for the linearised problem
12th, July 2007 DEASE meeting - Vienna
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)
12th, July 2007 DEASE meeting - Vienna
Healthy tissues:
(H1) for we have
non-extinction
(H2) for we have
convergence towards a steady state
The non-linear problem
P=eλ[G(N(t))].t
12th, July 2007 DEASE meeting - Vienna
Tumour growth: (H3) for we have
unlimited exponential growth
(H4) for we have
subpolynomial growth (not robust)
The non-linear problemP=eλ[G(N(t))].t
12th, July 2007 DEASE meeting - Vienna
Robust polynomial growthLink 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:
12th, July 2007 DEASE 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)