Modeling Ovarian Folliculogenesis: Morphogenesis and Population Dynamics

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Modeling Ovarian Folliculogenesis: Morphogenesis and Population Dynamics

Celine Bonnet, Keltoum Chahour, Frederique Clement, Marie Postel, Frédérique Robin, Romain Yvinec

To cite this version:

Celine Bonnet, Keltoum Chahour, Frederique Clement, Marie Postel, Frédérique Robin, et al.. Mod-eling Ovarian Folliculogenesis: Morphogenesis and Population Dynamics. REPROSCIENCES 2019, Apr 2019, Toulouse, France. �hal-03115058�

M

ODELING OVARIAN FOLLICULOGENESIS

:

M

ORPHOGENESIS AND POPULATION DYNAMICS

C. B

ONNET 1

, K. C

HAHOUR 2

, F. C

LÉMENT 3

, M. P

OSTEL 4

, F. R

OBIN 3 AND

R. Y

VINEC 5

1

Ecole Polytechnique

2

Université Côte d’Azur

3

Inria Paris-Saclay

4

Sorbonne Université

5

INRA Tours

I

NTRODUCTION

We present stochastic population dynamic models applied to several situations to understand ovarian folliculogenesis

 Somatic cell population dynamics in a single follicle (either in the activating or basal growing phase)

 Whole follicle population dynamics during the reproductive lifespan

C

ELL DYNAMICS IN AN ACTIVATING FOLLICLE

Somatic cell transition and proliferation during follicle activation

Key questions:

What are the kinetics of follicle activation ?

Are transition and proliferation concomitant ?

Data:

Cell counts

(Lundy et al., J. Reprod. Fertil., 115 (1999);

Wilson et al. Biol Reprod 64 (2011))

0 5 10 15 Wild-Type 0 10 20 30 C 0 5 10 15 Mutant F

Model

:

Flattened (F) and

Cu-boidal (C) cell dynamics

Stochastic Nonlinear Model

Cell events

Rate

Spontaneous transition

(F, C) → (F − 1, C + 1)

αF

Auto-amplified transition

(F, C) → (F − 1, C + 1)

β

_{F +C}F C

Cuboidal Proliferation

(F, C) → (F, C + 1)

γC

Self-renewing asymmetric divisions

(F, C) → (F, C + 1)

δF

Data Fitting

0 5 10 15 20 25 30 35 C 0 5 10 15 20 F Wild-Type 0 5 10 15 20 25 30 35 C Mutant −8 −7 −6 −5 −4 −3 −2 −1 0 1

Results

: see [3]

Clear separation

bet-ween transition and

proliferation in

Wild-type.

Auto-amplification is

not mandatory,

espe-cially for the Mutant

dataset

W

HOLE FOLLICLE POPULATION DYNAMICS

Nonlinear interactions between follicular populations

Key questions:

How a quasi-stable maturity repartition is achieved ?

What is the role of nonlinear interactions ?

Data:

Follicle counts according to maturity class

Faddy and Gosden, Human Reproduction, 10 (1995); Thibault and Levasseur 2001MJ.Faddy and R.G.Gosden

number 100000" 10000 " 1000 " -100 100000" 10000 ' 1000 " K r 10000 • 1000 -100 " umber 1 0 2 ( a ) 29 ( b ) 29 ( c ) 29 stage -34 stage « 34 s tage 34 I II ~ - ^ I I I X f o l l i c l e s 39 f o i l 39 f o l -39 -" g 44 ic l e s -9 44 h c l e s -44 a g e m -a g e • K a g e -: 49 (years) H 49 (years) 49 (years)

Figure 1. A mathematical model fitted to follicle counts from 43 human ovaries aged from 19 to 50 years of age. Each panel shows a spline-smoothed regression ( ) to the raw data representing pairs of ovaries (X) and the fitted model (—). Panels (a), (b) and (c) are follicle stages I, II and III respectively.

were balanced, however, because no follicles were dying at this stage.

The final column in Table I shows the total numbers of follicles leaving stage III and therefore summarizes the outcome of earlier stages of follicular growth leading towards ovulation. The egress declined with age and this reflected the smaller

7.284 (1.253)

7.284 (1.253)

Figure 2. Schematic diagram showing the model of follicle dynamics in humans for adult ages up to and above 38 years of age (phases 1 and 2 respectively). Follicles leave stages I, II and III at the indicated growth and death rates (with SE) expressed as number per year per number of follicles present.

Table I. The i from 24 to 50 Age (years) 24-25 25-26 26-27 27-28 28-29 29-30 30-31 31-32 32-33 33-34 34-35 35-36 36-37 37-38 38-39 3 9 ^ 0 40-41 4 1 ^ 2 4 2 ^ 3 43-44 44-45 45-46 46-47 47-48 4 8 ^ 9 49-50

numbers of follicles moving from stage i years of age I-> 13617 12 099 10 750 9552 8487 7541 6700 5953 5290 4700 4176 3710 3297 2929 6099 4506 3329 2459 1817 1342 991 732 541 400 295 218

(predicted from model) 13617 12 099 10 750 9552 8487 7541 6700 5953 5290 4700 4176 3710 3297 2929 2381 1759 1299 960 709 524 387 286 211 156 115 85 II-> 18 399 16 764 15 189 13 704 12 324 11 055 9896 8845 7896 7042 6276 5589 4975 4427 3914 3366 2822 2322 1884 1511 1200 947 742 578 448 346 to stage per 18 399 16 764 15 189 13 704 12 324 11055 9896 8845 7896 7042 6276 5589 4975 4427 3914 3366 2822 2322 1884 1511 1200 947 742 578 448 346 annum IIl-> 18 626 16 986 15 401 13 903 12 508 11 223 10 050 8984 8022 7155 6377 5680 5056 4499 3986 3442 2895 2388 1941 1559 1240 979 767 598 464 359

numbers leaving stage I. It may seem surprising that the numbers in the final column exceeded those in the first, but this is due to there being many follicles already in stages II and III at these ages, contributing to eventual egress from stage III.

Although the numbers of follicles at stage III are falling, they actually increase when expressed as a fraction of those in stage I (Figure 3). This is, however, only an apparent 772

at INRA Institut National de la Recherche Agronomique on February 7, 2016

http://humrep.oxfordjournals.org/

Downloaded from

Model:

Population-dependent

follicle maturity and atresia

· · ·

−−−−−→ F

λj−1 _j−1

−−−−−→ · · ·

λj





y

µ_j

∅

Stochastic Nonlinear Model

Follicle events

Rate

Follicle activation

F

₀

→ F

₁

ε

λ0 1+K Pd_j=1 a_jF_j

F

0

Follicle maturation

F

_j

→ F

_j+1

λ

_j

F

_j

Follicle atresia

F

j

→ ∅

µ

_j

F

_j

Qualitative agreement

Results:

see [1]

Time-scale separation

explains quasi-stable

maturity repartition.

Removal

of

acti-vation

inhibition

explains acceleration

of reserve exhaustion

C

ELL DYNAMICS IN A GROWING FOLLICLE

Somatic cell proliferation during basal follicle growth

Key questions:

What is the rate of growth ?

Is proliferation oocyte-dependent ?

Data:

Total cell numbers, Oocyte and Follicle

diameters at three time points

(Lundy et al., J. Reprod. Fertil.,

115 (1999); Smith et al. J. Reprod. Fertil., 100 (1994))

Model

:

Cell proliferation and

repartition in successive layers

Stochastic Linear Model

Cell events

Rate

Both daughter cells stay on mother’s layer

N

j

→ 2N

j

p

j_2,0

b

_j

N

j

One daughter cell into next mother’s layer

N

j

→ N

j

+ N

j+1

p

j_1,1

b

_j

N

j

Both daughter cells into next mother’s layer

N

j

→ 2N

j+1

p

j_0,2

b

_j

N

j

Data Fitting

Results:

see [2]

Data are compatible with

an

exponential

growth phase.

a

layer-dependent

growth rate is

de-creasing with oocyte

distance

M

ATHEMATICAL

T

OOLBOX

• (Structured) Population dynamics model: deterministic and stochastic models.

• Long-time and asymptotic analysis, time-scale separation.

• Transient analysis: analytical and numerical solutions, first passage time.

• Statistical methods: parameter estimation and identifiability analysis.

C

ONCLUSION AND

P

ERSPECTIVE

Mathematical modeling helps:

 to test different follicle growth scenario and make prediction

 to challenge biological knowledge on follicle dynamics.

Current study and future projects will focus on different scales:

?

_{Intra-cellular scale}

_{: FSHR signaling network and its role in}

fol-licle selection

(for a review of gonadotropin signaling models, see [5]).

?

_{Follicle scale}

_{: coupling cell dynamics with biomechanics}

mo-del to momo-del antrum formation

(extending and revisiting previous model [4]).

?

_{Ovarian scale}

_{: Complement follicle population dynamics with}

(i) ovarian reserve formation and (ii) ovarian cycle dynamics.

Comparative physiology approaches will also be taken into account.

R

EFERENCES

[1] C. Bonnet, K. Chahour, F. Clément, M. Postel, and R. Yvinec. Multiscale population dynamics in reproductive biology: Singular perturbation reduction in deterministic and stochastic models. Arxiv preprint: 1903.08555v1, Mar. 2019.

[2] F. Clément, F. Robin, and R. Yvinec. Analysis and Calibration of a Linear Model for Structured Cell Populations with Unidirectional Motion: Application to the Morphogenesis of Ovarian Follicles. SIAM Journal on Applied Mathematics, 79(1):207–229, Jan. 2019.

[3] F. Clément, F. Robin, and R. Yvinec. Stochastic nonlinear model for somatic cell population dynamics during ovarian follicle activation. Arxiv preprint: 1903.01316v1, Mar. 2019.

[4] F. Clément, P. Michel, D. Monniaux, and T. Stiehl. Coupled somatic cell kinetics and germ cell growth: multiscale model-based insight on ovarian follicular development. Multiscale Modeling & Simulation, 11(3):719–746, 2013.

[5] R. Yvinec, P. Crépieux, E. Reiter, A. Poupon, and F. Clément. Advances in computational modeling approaches of pituitary gonadotropin signaling. Expert Opinion on Drug Discovery, 13(9):799–813, Sept. 2018.

Acknowledgement: We thank Ken McNatty for providing experimental datasets and Danielle