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

Marie Postel Laboratoire Jacques-Louis Lions COLLOQUE EDP-NORMANDIE - CAEN 2013 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 1/31

Framework : large scale initiative REGATE / Equipe-Projet MYCENAE COLLOQUE EDP-NORMANDIE - CAEN 2013 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 2/31

Outline COLLOQUE EDP-NORMANDIE - CAEN 2013 1 Multiscale biological model 2 Mathematical model 3 Numerical method High order Finite Volumes Transmission conditions Adaptive mesh refinement Parallelization 3/31

COLLOQUE EDP-NORMANDIE - CAEN 2013 Représenta*on ¡schéma*que ¡de ¡l’ovaire ¡humain ¡ (Driancourt ¡et ¡al., ¡2001) ¡

Hormone ¡folliculo-­‑s*mulante ¡ (FSH) ¡ Hormone ¡lutéinisante ¡ (d’après ¡Monniaux ¡et ¡al., ¡1999) ¡ (LH) ¡

Boucles ¡de ¡régula*on ¡endocrine ¡entre ¡le ¡système ¡hypothalamo-­‑hypophysaire ¡ et ¡les ¡follicules ¡ovariens ¡en ¡développement ¡terminal ¡ HYPOTHALAMUS ¡ HYPOTHALAMUS ¡ E2 ¡ GnRH ¡ GnRH ¡ PITUITARY ¡ PITUITARY ¡ E2 ¡/ ¡INH ¡ FSH ¡/ ¡LH ¡ LH ¡ Granulosa ¡cells ¡ Granulosa ¡cells ¡ Antrum ¡ Antrum ¡ Oocyte ¡ Oocyte ¡ PREOVULATORY ¡ 1 ¡ 2 ¡ 3 ¡ FOLLICLE ¡ ANTRAL ¡FOLLICLES ¡ OVULATION ¡ (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 ¡ 3 ¡ 2 ¡ E2 ¡ E2 ¡ 1 ¡ E2 ¡ E2 ¡ Blood ¡concentra*ons ¡ FSH ¡ E2 ¡ E2 ¡ FSH ¡ Time ¡ (Clément ¡& ¡Monniaux, ¡2012) ¡

Biology : structured population dynamics COLLOQUE EDP-NORMANDIE - CAEN 2013 maturity D apoptosis D G1 SM SM G1 G1 SM x2 mitosis age Granulosa cell phases 8/31

Math model : microscopic scale unknowns Cell densities of the follicles COLLOQUE EDP-NORMANDIE - CAEN 2013  φ 1 ( a , γ, t )   ...  φ N f ( a , γ, t ) γ  h D g with γ s SM SM G1 G1  a age a   2 1   maturity γ  t time     N f number of follicles  9/31

Math model : system of weakly coupled transport equations COLLOQUE EDP-NORMANDIE - CAEN 2013 ∂ t φ 1 + ∂ a g 1 ( a , γ, u 1 ( t )) φ 1 + ∂ γ h 1 ( a , γ, u 1 ( t )) φ 1 = − Λ( a , γ, U ( t )) φ 1 . . . ∂ t φ N f + ∂ a g N f ( a , γ, u N f ( t )) φ N f + ∂ γ h N f ( a , γ, u N f ( t )) φ N f = − Λ( a , γ, U ( t )) φ N f Initialization  φ 1 ( a , γ, 0 ) = φ 0 1 ( a , γ )   .  . .  φ N f ( a , γ, 0 ) = φ 0  N f ( a , γ )  Periodic boundary conditions on outer boundaries Transmission conditions on internal boundaries between biological phases 10/31

Math model : Example of initialization Uniform repartition of cells in the first cell proliferation cycle COLLOQUE EDP-NORMANDIE - CAEN 2013 11/31

Math model : multi scale closed loop control microscopic scale : granulosa cell density φ ( t , a , γ ) COLLOQUE EDP-NORMANDIE - CAEN 2013 mesoscopic scale : follicle maturity m f ( t ) , f = 1 , . . . , Nf macroscopic scale : ovary maturity M 1 ( t ) � � Pituitary Gland m f = γφ f ( t , a , γ ) d γ da M1 FSH N f � M 1 = m f Ovary f = 1 Global maturity M 1 → global FSH/LH resource Local maturity ( m f ) f = 1 , N f → sensitivity to FSH, secretion of estradiol and inhibin 12/31

Math model : Closed loop control Particularity of the model : transport equation with controlled COLLOQUE EDP-NORMANDIE - CAEN 2013 speeds ∂ t φ f + ∂ a g ( u f ) φ f + ∂ γ h ( u f ) φ f = − Λ( U ) φ f 1 1 S(M1) b(m f ) 0.9 0.95 0.8 0.9 0.7 0.85 0.6 0.8 b(m f ) 0.5 U 0.75 0.4 0.7 0.3 0.65 0.2 0.6 0.1 0.55 0 0.5 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 m f M1 Global control U = S ( M 1 ) Local control u f = b ( m f ) U (global FSH level) (local FSH level) 13/31

Math model : Controlled speeds Aging function : γ h COLLOQUE EDP-NORMANDIE - CAEN 2013 D � γ 1 u + γ 2 g in phase G 1 g ( a , γ, u ) = γ 1 in phase SM ∪ D s SM G1 SM G1 a 2 1 0.15 Maturation function : h( � ) 0.1 0 0.05 − γ 2 + ( c 1 γ + c 2 )( 1 − exp ( − u 0  u )) -0.05 ¯  -0.1 h h ( a , γ, u ) = in phases G 1 ∪ D -0.15 -0.2 0 in phase SM -0.25  -0.3 -0.35 0 0.2 0.4 0.6 0.8 1 � 14/31

Math model : Source term The linear source term − Λ φ models apoptosis (cell death). Active only close to the boundary between the proliferation and COLLOQUE EDP-NORMANDIE - CAEN 2013 differentiated domains Λ( a , γ, U ) = K exp ( − ( γ − γ s ) 2 )( 1 − U ) ≥ 0 γ ¯ 3 � ( � ) 2.5 γ h 2 D g 1.5 � γ 1 s SM SM G1 G1 0.5 a 0 1 2 0 0.2 0.4 0.6 0.8 1 � 15/31

Math model : transmission conditions between biological phases • Continuous flux on interface G 1 − → SM COLLOQUE EDP-NORMANDIE - CAEN 2013 φ f ( t , a = 0 . 5 + , γ ) = ( γ 1 u f + γ 2 ) φ f ( t , a = 0 . 5 − , γ ) • Mitosis − → Doubling flux on the interface SM − → G 1 ( γ 1 u f + γ 2 ) φ f ( t , a = 1 + , γ ) = 2 φ f ( t , a = 1 − , γ ) , • Waterproof − → Homoge- γ h neous Dirichlet condition north D of interface SM ↑ D g γ s 1 SM SM G1 G1 φ f ( t , a , γ + s ) = 0 , 2 ≤ a ≤ 1 a 2 1 16/31

Math model : well posedness COLLOQUE EDP-NORMANDIE - CAEN 2013 [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 Details 17/31

Math model : block diagram ([Clément & Monniaux, 2013]) COLLOQUE EDP-NORMANDIE - CAEN 2013 18/31

Numerical method : general strategy Design of a dedicated Finite Volume method Parallelization COLLOQUE EDP-NORMANDIE - CAEN 2013 Transmission conditions Adaptive mesh in age and maturity Analogy between biology and computer resources 19/31

Numerical method : finite volume discretization COLLOQUE EDP-NORMANDIE - CAEN 2013 Age step ∆ a and maturity step ∆ γ . Time steps ∆ t n such that t n = ∆ t ( 1 ) + ... + ∆ t n Mean value approximation in each grid cell 1 �� φ n φ ( a , γ, t n ) dad γ k , l ≈ ∆ a ∆ γ 20/31

Numerical method : finite volume scheme (Micro) Time explicit finite volume scheme ( ∆ x = ∆ a = ∆ γ ) k , l − ∆ t n COLLOQUE EDP-NORMANDIE - CAEN 2013 φ n + 1 φ n ∆ x D n k , l − ∆ t n Λ n k , l φ n = k , l k , l D n F n k + 1 , l + 1 / 2 − F n k , l + 1 / 2 + F n k + 1 / 2 , l + 1 − F n = k , l k + 1 / 2 , l Φ k,l+1 F k+1/2,l+1 F F k+1,l Non linear third order k,l +1/2 +1/2 Φ Φ Φ k−1,l k,l k+1,l computation of numerical F k+1/2,l fluxes Runge-Kutta in time Φ k,l−1 21/31 Numerical divergence D

Model microscale outputs for 4 follicles COLLOQUE EDP-NORMANDIE - CAEN 2013 22/31

Model outputs : macro and micro for 4 follicles 0.16 16 35 m 1 �� 1 COLLOQUE EDP-NORMANDIE - CAEN 2013 m 2 �� 2 0.14 14 30 m 3 �� 3 m 4 �� 4 0.12 12 25 Perte de masse �� u 1 Niveau FSH 0.1 10 20 u 2 Maturité u 3 0.08 8 15 u 4 U 0.06 6 10 M t 0.04 4 5 0.02 2 0 0 0 � 5 20 1 0.8 � 1 GF 1 � 1 18 � 2 GF 2 � 2 Fraction de croissance GF 0.8 0.7 � 3 GF 3 � 3 16 � 4 GF 4 Maturité cellulaire � � 4 14 � s 0.6 0.6 Masse � 12 10 0.4 0.5 8 0.2 0.4 6 4 0 0.3 2 0 � 0.2 0.2 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 temps temps temps 23/31

Numerical method : selection simulation ovulation control Niveau FSH et maturité ovarienne Maturités folliculaires 0.16 24 16 m 1 m 2 COLLOQUE EDP-NORMANDIE - CAEN 2013 m 3 0.14 14 m 4 20 Seuil 0.12 12 16 0.1 10 u 1 u 2 Mt U u 3 0.08 u 4 12 8 m U Mat. ov. Seuil 0.06 6 8 0.04 4 4 0.02 2 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 temps temps 24/31

Numerical method specificities COLLOQUE EDP-NORMANDIE - CAEN 2013 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 25/31