[PPT] - Marie Postel Laboratoire Jacques-Louis Lions COLLOQUE EDP-NORMANDIE PowerPoint Presentation

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Marie Postel

Laboratoire Jacques-Louis Lions

Modélisation et simulation numérique du développement terminal des follicules ovariens En collaboration avec B. Aymard, F. Clément,

F. Coquel, D. Monniaux

IVe Colloque EDP-Normandie 24-25 oct. 2013 Caen

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Framework : large scale initiative REGATE / Equipe-Projet MYCENAE

WP1 : Controlled conservation laws for structured cell populations : application to ovulation control principal investigator : F. Clément (Inria)

Reproductive physiology : D. Monniaux (INRA)
Dynamical systems : P. Michel (Centrale Lyon)
Control theory : J.-M. Coron (UPMC),
P. Shang (UPMC, Inria)
Numerical simulations : B. Aymard (UPMC & Inria),
F. Coquel (CMAP-X) and M. Postel (LJLL-UPMC)
Earlier contributions to the development of the

PDE model : N. Echenim, C. Hombourger

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Outline

1 Multiscale biological model 2 Mathematical model 3 Numerical method

High order Finite Volumes Transmission conditions Adaptive mesh refinement Parallelization

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Représenta*on ¡schéma*que ¡de ¡l’ovaire ¡humain ¡ (Driancourt ¡et ¡al., ¡2001) ¡

COLLOQUE EDP-NORMANDIE - CAEN 2013

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Hormone ¡folliculo-‑s*mulante ¡ (FSH) ¡ Hormone ¡lutéinisante ¡ (LH) ¡

(d’après ¡Monniaux ¡et ¡al., ¡1999) ¡

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E2 ¡/ ¡INH ¡

Granulosa ¡cells ¡ Oocyte ¡ Antrum ¡ ANTRAL ¡FOLLICLES ¡ HYPOTHALAMUS ¡ PITUITARY ¡

FSH ¡/ ¡LH ¡ GnRH ¡ 1 ¡ 2 ¡ 3 ¡

PITUITARY ¡ HYPOTHALAMUS ¡

GnRH ¡

PREOVULATORY ¡ FOLLICLE ¡ Granulosa ¡cells ¡

OVULATION ¡ LH ¡ E2 ¡

Oocyte ¡ Antrum ¡

(Clément ¡& ¡Monniaux, ¡2012) ¡

Boucles ¡de ¡régula*on ¡endocrine ¡entre ¡le ¡système ¡hypothalamo-‑hypophysaire ¡ et ¡les ¡follicules ¡ovariens ¡en ¡développement ¡terminal ¡

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1 ¡ 2 ¡ 3 ¡

E2 ¡

E2 ¡ FSH ¡

Time ¡ Blood ¡concentra*ons ¡

E2 ¡ FSH ¡

E2 ¡

(Clément ¡& ¡Monniaux, ¡2012) ¡

Sélec*on ¡d’un ¡follicule ¡dominant ¡à ¡par*r ¡d’une ¡cohorte ¡de ¡follicules ¡à ¡antrum ¡ et ¡son ¡développement ¡jusqu’au ¡stade ¡de ¡follicule ¡préovulatoire ¡

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Biology : structured population dynamics

D

G1 G1 SM SM D maturity mitosis age

G1 SM x2 apoptosis

Granulosa cell phases

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Math model : microscopic scale unknowns

Cell densities of the follicles      φ1(a, γ, t) ... φNf (a, γ, t) with            a age γ maturity t time Nf number of follicles

s

G1 G1 SM SM D

g h a 1 2 γ γ 9/31

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Math model : system of weakly coupled transport equations

∂tφ1 + ∂ag1(a, γ, u1(t))φ1 + ∂γh1(a, γ, u1(t))φ1 = −Λ(a, γ, U(t))φ1 . . . ∂tφNf + ∂agNf (a, γ, uNf (t))φNf + ∂γhNf (a, γ, uNf (t))φNf = −Λ(a, γ, U(t))φNf

Initialization        φ1(a, γ, 0) = φ0

1(a, γ)

. . . φNf (a, γ, 0) = φ0

Nf (a, γ)

Periodic boundary conditions on outer boundaries Transmission conditions on internal boundaries between biological phases

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Math model : Example of initialization

Uniform repartition of cells in the first cell proliferation cycle

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Math model : multi scale closed loop control

microscopic scale : granulosa cell density φ(t, a, γ) mesoscopic scale : follicle maturity mf (t), f = 1, . . . , Nf macroscopic scale : ovary maturity M1(t)

Pituitary Gland Ovary FSH M1

mf = γφf (t, a, γ)dγda M1 =

Nf

f =1

mf Global maturity M1 → global FSH/LH resource Local maturity (mf )f =1,Nf → sensitivity to FSH, secretion of estradiol and inhibin

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Math model : Closed loop control

Particularity of the model : transport equation with controlled speeds ∂tφf + ∂ag(uf )φf + ∂γh(uf )φf = −Λ(U)φf

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1 2 3 4 5 6 7 8 U M1 S(M1)

Global control U = S(M1) (global FSH level)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 b(mf) mf b(mf)

Local control uf = b(mf )U (local FSH level)

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Math model : Controlled speeds

Aging function :

g(a, γ, u) = γ1u + γ2 in phase G1 1 in phase SM ∪ D

s

G1 G1 SM SM D

g h a 1 2 γ γ

Maturation function :

h(a, γ, u) =    −γ2 + (c1γ + c2)(1 − exp( −u

¯ u ))

in phases G1 ∪ D in phase SM

0.35
0.3
0.25
0.2
0.15
0.1
0.05

0.05 0.1 0.15 0.2 0.4 0.6 0.8 1 h

h()

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Math model : Source term

The linear source term −Λφ models apoptosis (cell death). Active

nly close to the boundary between the proliferation and

differentiated domains Λ(a, γ, U) = K exp(−(γ − γs)2 ¯ γ )(1 − U) ≥ 0

s

G1 G1 SM SM D

g h a 1 2 γ γ

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1

()

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Math model : transmission conditions between biological phases

Continuous flux on interface G1 −

→ SM φf (t, a = 0.5+, γ) = (γ1uf + γ2)φf (t, a = 0.5−, γ)

Mitosis −

→ Doubling flux on the interface SM − → G1 (γ1uf + γ2)φf (t, a = 1+, γ) = 2φf (t, a = 1−, γ),

Waterproof −

→ Homoge- neous Dirichlet condition north

f interface SM ↑ D

φf (t, a, γ+

s ) = 0,

1 2 ≤ a ≤ 1

s

G1 G1 SM SM D

g h a 1 2 γ γ 16/31

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Math model : well posedness

[P. Shang, Cauchy problem for multiscale conservation laws : Application to structured cell populations JMAA 2013] Main difficulties : nonlocal velocity flux discontinuities at internal boundaries coupling between different follicles in the model

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Math model : block diagram ([Clément & Monniaux, 2013])

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Numerical method : general strategy

Design of a dedicated Finite Volume method Parallelization Transmission conditions Adaptive mesh in age and maturity Analogy between biology and computer resources

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Numerical method : finite volume discretization

Age step ∆a and maturity step ∆γ. Time steps ∆tn such that tn = ∆t(1) + ... + ∆tn Mean value approximation in each grid cell φn

k,l ≈

1 ∆a∆γ

φ(a, γ, tn)dadγ

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Numerical method : finite volume scheme (Micro)

Time explicit finite volume scheme (∆x = ∆a = ∆γ) φn+1

k,l

= φn

k,l − ∆tn

∆x Dn

k,l − ∆tnΛn k,lφn k,l

Dn

k,l

= F n

k+1,l+1/2 − F n k,l+1/2 + F n k+1/2,l+1 − F n k+1/2,l

k+1/2,l+1

Φ Φ Φ Φ Φ k−1,l k,l k+1,l k,l−1 k,l+1 F F Fk+1,l

k,l +1/2 +1/2 k+1/2,l

F

Numerical divergence D Non linear third order computation of numerical fluxes Runge-Kutta in time

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Model microscale outputs for 4 follicles

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Model outputs : macro and micro for 4 follicles

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Niveau FSH u1 u2 u3 u4 U Mt 2 4 6 8 10 12 14 16 Maturité m1 m2 m3 m4 5 5 10 15 20 25 30 35 Perte de masse 1 2 3 4 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 Masse temps 1 2 3 4 0.2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 Fraction de croissance GF temps GF1 GF2 GF3 GF4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2 4 6 8 10 Maturité cellulaire temps 1 2 3 4 s 23/31

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Numerical method : selection simulation

vulation control

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 2 4 6 8 10 4 8 12 16 20 24 U Mt temps Niveau FSH et maturité ovarienne u1 u2 u3 u4 U

Mat. ov.

Seuil 2 4 6 8 10 12 14 16 2 4 6 8 10 m temps Maturités folliculaires m1 m2 m3 m4 Seuil

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Numerical method specificities

1 high order enhancement [Aymard, Clément, Coquel, Postel,

ESAIM Proc 2012]

2 transmission conditions [Aymard, Clément, Coquel, Postel,

SISC 2013]

3 adaptive mesh refinement with multiresolution [Aymard,

Clément, Postel, submitted]

4 parallel computing coupled with multiresolution

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Comp. issue 1 : High order enhancement in space

details

Numerical flux at interface k between cells k − 1 and k [LeVeque 1997] Fk = F Low

k

+ ℓ(rk)(F High

k

− F Low

k

)

First order flux Second order flux F Low

k

(φn) = (g n

k−1)+φn k−1 + (g n k )−φn k

F High

k

(φn) = 1 2

g n

k−1φn k−1 + g n k φn k

r measures the local smoothness of the solution

ℓ(r) calibrates the amount of high order

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Comp. issue 2 : Numerical treatment of transmission

conditions

details

a) Initial time b) Intermediate time c) Final time a) 1st order b) 3rd order c) Exact solution

[Aymard, Clément, Coquel, P. (SISC 2013)]

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Comp. issue 3 : adaptive mesh refinement

Adaptive mesh driven by multiresolution analysis [Cohen, Kaber, Müller, Postel (2003) ] Coupling with discontinuous transmission conditions [Aymard, Clément, Postel (submitted) ]

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Comp. issue 4 : parallelization and adaptive mesh refinement

details

High granularity : Few communications at each time step. Computation of global maturity : reduction operation (sum). Mn = Nf

i=1 mn i

Synchronization of the time steps : reduction operation (min). ∆tn = min{∆tn

1, ..., ∆tn Nf }

Each follicle evolves on its own adaptive grid -> load balancing

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Current and future developments

Optimal control (J.-M. Coron, P. Shang)

details

Uncertainty propagation on reduced models (S. Passot M2 internship, FC, MP)

details

Model calibration (B. Aymard, FC, DM, MP) Evaluation of Parallel / Adaptive Multiresolution performances (B. Aymard, FC, F. Coquel, MP) Reduced model (B. Aymard, FC, F. Coquel, MP)

details

User interface for biologists (S. Steer)

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❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥ ✦

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