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12th, July 2007DEASE meeting - Vienna PDEs in Laser Waves and Biology Presentation of my research...

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

PDEs in Laser Waves and Biology

Presentation of my research fields

Marie Doumic [email protected]

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

1st step:1st step:

choose the correct small parameter ε

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

Half-space problem

Fourier transform: for

12th, July 2007 DEASE meeting - Vienna

5th step: numerical scheme

with interaction with the plasma.

12th, July 2007 DEASE meeting - Vienna

Numerical scheme:

Initializing: cf. preceding formula:

FFT of g -> multiply by -> IFFT

12th, July 2007 DEASE meeting - Vienna

1st stage: solving

and then

Simultaneously: we have:

FFT of -> multiply by -> IFFT

12th, July 2007 DEASE meeting - Vienna

2nd stage: solving

Standard upwind decentered scheme:

With and

12th, July 2007 DEASE meeting - Vienna

Second order scheme: Flux limiter of Van Leer:

2 rays crossing: we solve for p=1,2:

12th, July 2007 DEASE meeting - Vienna

Properties of the scheme

• stability: non-increasing scheme:

• Convergence towards Schrödinger eq.:If the scheme converges towards the

solution of:

12th, July 2007 DEASE meeting - Vienna

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

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

12th, July 2007 DEASE meeting - Vienna

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

1. Eigenvalue linearised problem

Simplified in:

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

12th, July 2007 DEASE 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

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)

12th, July 2007 DEASE meeting - Vienna

Danke für Ihre Aufmerksamkeit !


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