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university-logo Numerical simulation of the selection process of ovarian follicles : From biology to algorithms Aymard Benjamin, Clément Frédérique, Coquel Frédéric and Postel Marie. Inria Sisyphe Junior Seminar December 13 th 2012 Inria Rocquencourt 1/24 [email protected] Selection process of ovarian follicles
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Numerical simulation of the selection process ofovarian follicles :
From biology to algorithms
Aymard Benjamin, Clément Frédérique,Coquel Frédéric and Postel Marie.
Inria Sisyphe
Junior SeminarDecember 13th 2012Inria Rocquencourt
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Introduction : Biological background
Macro scale : Pituitary gland, ovary.
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Introduction : Selection process
Micro scale : Chronology of the follicular development.
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Outline
Multiscale Biological ModelSelection processMicro : cell kineticsMacro : closed loop control
AlgorithmsMacro : Parallelization strategyMicro : Finite volume scheme
Numerical simulationsMovieMacro variablesParallelization performancesInterfaces performances
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Model : Follicle
Scheme of a follicle.
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Model : Population dynamics
Granulosa Cell phases.
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Model : Unknown of the problem
Cell densities of the follicles φ1(a, γ, t)
...
φNf (a, γ, t)
with a ageγ maturityt timeNf number of follicles
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Model : Example of a set of two follicles
Initial condition for one follicle.8/24 [email protected] Selection process of ovarian follicles
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Model : Closed loop control
Pituitary Gland
Ovary
FSHM1
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Model : Closed loop control
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Model : Hyperbolic system of conservation laws
Dynamic∂tφ1 + ∂ag(a, γ, u1(t))φ1 + ∂γh(a, γ, u1(t))φ1 = −Λ(a, γ,U(t))φ1
...
∂tφNf + ∂ag(a, γ, uNf (t))φNf + ∂γh(a, γ, uNf (t))φNf = −Λ(a, γ,U(t))φNf
Initialization φ1(a, γ, 0) = φ0
1(a, γ)
...
φNf (a, γ, 0) = φ0Nf
(a, γ)
Periodic boundary conditions.
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Quick reminder on transport equations
a) t = 0 b) t = 0.5∂tφ+ ∂xφ = 0
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Model : Moments
Zero order moment (local masses)m0(1, t) =
∫∫Ωφ1dadγ
...
m0(Nf , t) =∫∫
ΩφNf dadγ
First order moment (local maturities)m1(1, t) =
∫∫Ωγφ1dadγ
...
m1(Nf , t) =∫∫
ΩγφNf dadγ
Sum of the first order moments (global maturity)
M1(t) =
Nf∑f =1
m1(f , t)
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Model : Control
Global control U = S(M1) (global FSH level)Local control uf = b(mf )U (local FSH level)Particularity of the model : transport equation with controlled speeds
∂tφ+ ∂ag(u)φ+ ∂γh(u)φ = −Λ(U)φ
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Numerical method : general strategy
Analogy with biology.
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Numerical method : discretization
Age step ∆a and maturity step ∆γ.Time steps ∆t(n) such that tn = ∆t(1) + ...+ ∆t(n)
Mean value approximation in each grid cell
φnk,l ≈
1∆a∆γ
∫∫φ(a, γ, tn)dadγ
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Numerical method : computation grid
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Numerical method : finite volume scheme (Micro)
Finite volume scheme (we note ∆x = ∆a = ∆γ)φn+1
k,l = φnk,l − ∆t(n)
∆x Dnk,l
Dnk,l numericaldivergence
Stability condition
∆t(n) ≤ ∆x2max(|g |, |h|)
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Parallelization : communications
High granularity : Few communications at each time step.Computation of global maturity : reduction operation (sum).
Mn =
Nf∑i=1
mni
Synchronization of the time steps : reduction operation (min).
∆t(n) = min∆t(n)1 , ...,∆t(n)
Nf
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Numerical simulations : Movie
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Numerical simulations : Macro variables
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
m0(f
,t)
t
g2
0.500.550.600.650.700.750.800.850.900.95
Macro : Masses during the selection process.
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Numerical simulations : Parallelization performances
CPU time with respect to the number of follicles.
Aymard, Clément, Coquel, Postel, Cemracs Proceedings, 2011.Numerical simulation of the selection process of ovarian follicles.
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Numerical simulations : Interfaces performances
1e-05
0.0001
0.001
0.01
0.1
1
0.001 0.01
E1(
∆x)
∆x
gL = 0.5gL = 1gL = 2gL = 3
Test case : Density crossing a doubling interface. Comparison with exactsolution (without control). The convergence rate is 2.4.
Aymard, Clément, Coquel, Postel, Submitted, 2012.Numerics in cell dynamics : kinetic equations with discontinuouscoefficients
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End
Thank you for your attention.
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