# Ovarian follicle population dynamics along lifetime

Romain Yvinec

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Romain Yvinec. Ovarian follicle population dynamics along lifetime. JMBS 2025 - Journées Math Bio Santé, Nov 2025, Montpellier, France. hal-05444857

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Submitted on 6 Jan 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Ovarian follicle population dynamics along lifetime

Romain Yvinec

BIOS Team, Physiologie de la Reproduction et des Comportements,
INRAE CNRS, Université de Tours, (Tours, France)
MUSCA Team, INRIA-INRAE-CNRS (Saclay, France)
★ INRAE Rennes : Violette Thermes

★ INSERM - Paris Cité : Céline Guigon

★ INRIA Saclay : Frédérique Clément, Rosario Medina-Rodriguez, Guillaume Ballif, Chiara Villa

★ INRAE Tours : RY, Louis Fostier

<figure><img src="image_2.png" /></figure>

ANR OVOPAUSE

Dynamics and control of female germ cell populations : understanding aging through population dynamics models
<figure><img src="image_3.png" /></figure>

## Female gametogenesis

* Germ cell **proliferation** and follicle formation (peri-natal)

* Pool of Quiescent follicles **Ovarian reserve**. Slow activation

* Basal growth **Dynamic reserve** (starting at birth). Spanning over several ovarian cycles

* Terminal growth (After puberty) : **ovulation** within an ovarian cycle

$$ \text{Ovarian follicles} = \text{female germ cells} $$
<figure><img src="image_4.png" /></figure>

## Female gametogenesis

* **Longevity** of the ovarian reserve (iatrogenic disorder, Premature Ovarian failure, extending fertility lifespan)

* **Sensitivity** to environmental conditions (EDs)

* **Fertility** disorders (PCOS), Assisted reproductive technology, Contraception,

* Animal breeding reproduction management

Ovarian follicles = female germ cells
# How female germ cells population achieves homeostasis during reproductive lifespan ?

Ovarian follicles = female germ cells

<figure><img src="image_5.png" /><figcaption>Coxworth and Hawkes 2010</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Courtesy of Danielle Monniaux.</figcaption></figure>

<figure><img src="image_7.png" /></figure>
# How female germ cells population achieves homeostasis during reproductive lifespan?

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

* How to describe the **ovarian reserve** decay?

* How to explain a "stable" **dynamic reserve**?

* How to explain the emergence of cyclicity?
# Population dynamics in female gametogenesis

* Asynchronous growth

* Several timescales: Ten of years / Months / Weeks

* Interactions between subpopulations

<figure><img src="image_12.png" /></figure>
• Quiescent follicles ≫ Growing follicles and Slow activation / Fast growth

$$ \begin{array}{r@{\;}c@{\;}l@{\quad}l@{\quad}l} \underbrace{\frac{1}{\epsilon} X}_{\emptyset} & \xrightarrow{\varepsilon \lambda_0(\mathbf{Y})} & \multicolumn{2}{c}{\text{Slow}} \\[-1.5ex] \underbrace{\downarrow}_{\emptyset} & \xrightarrow{\varepsilon \mu_0(\mathbf{Y})} & \multicolumn{2}{c}{\text{Fast}} \\[-1.5ex] \emptyset & \xrightarrow{\phantom{\varepsilon \mu_0(\mathbf{Y})}} & \multicolumn{2}{c}{\text{ activation}} \end{array} $$
## Quiescent follicles ≫ Growing follicles and Slow activation / Fast growth

$$ \begin{array}{ccc} \frac{\partial}{\partial t} X & \rightarrow & \varepsilon \lambda_0(\mathbf{Y}) \\ \downarrow & & Y_1 \\ \emptyset & \rightarrow & \mu_1(\mathbf{Y}) \\ & & \downarrow \\ \emptyset & \rightarrow & \mu_d(\mathbf{Y}) \end{array} $$

In the limit $\varepsilon \to 0$, (under "reasonable assumptions") $X$ and $\mathbf{Y} = (Y_1, \dots, Y_d)$ converge to a differential-algebraic equation :

$$ \begin{cases} \frac{dx}{dt}(t) = -\lambda_0(\mathbf{y}(t))x(t) - \mu_0(\mathbf{y}(t))x(t), & x(0) = x^{\text{in}}, \\ 0 = G(x(t), \mathbf{y}(t)), & t > 0 \end{cases} $$

* Bonnet, Chahour, Clément, Postel, Y., Multiscale population dynamics in reproductive biology : singular perturbation reduction in deterministic and stochastic models, ESAIM : PROCEEDINGS AND SURVEYS, 2020.

* Ballif, Clément, Y., Averaging of a Stochastic Slow-Fast Model for Population Dynamics : Application to the Development of Ovarian Follicles, SIAP 2022
Slow-fast dynamics in practice ?

<figure><img src="image_13.png" /></figure>
Slow-fast dynamics in practice ?

<figure><img src="image_14.png" /></figure>

Faddy et al., J.
Exp. Zool. 1976

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

$$
\begin{align*}
x_0(t) &= X_0^\text{in} \exp\left(-\left(\lambda_0 + \mu_0\right)t\right) \\
\pi_{x_0}(t)(y) &= \prod_{i=1}^d \frac{(x_0(t)\alpha_i)^{y_i}}{y_i!} e^{-x_0(t)\alpha_i} \\
\alpha_i &= \prod_{j=0}^{i-1} \frac{\lambda_j}{\lambda_{j+1} + \mu_{j+1}}
\end{align*}
$$

* Timescale separation is coherent with published data on follicle counts in mice at the lifespan time scale, *away from a transient period*
# Slow-fast dynamics in practice ?

<figure><img src="image_20.png" /></figure>

Faddy et al., J. Exp. Zool. 1976

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

$$ \begin{cases} x_0(t) \\ \pi_{x_0}(t)y \end{cases} = \begin{cases} X_0^{in} \exp(-(\lambda_0 + \mu_0)t) \\ \Pi_{y_0}(t) \alpha_i y_i \end{cases} $$

$$ \alpha_i = \prod_{j=0}^{i-1} \frac{\lambda_j}{\lambda_{j+1} + \mu_{j+1}} $$

* Parameter estimation -> identifiability issue ?

* How to estimate follicle interactions ?
* AMH Knock-Out in vivo on mice

* Quiescent and Growing follicle counts at 2 ages :
  - 25 days
  - 120 days

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

Durlinger and al, Endocrinology, 1999
Two distinct population in pre-pubertal period

• Ovarian Reserve build-up

<figure><img src="image_28.png" /></figure>

Zheng, Zhang and Liu, Hum. Mol. Reprod, 2014

<figure><img src="image_29.png" /><figcaption>Tamoxifen was given at E16.5 and ovaries were analyzed at various ages</figcaption></figure>

<figure><img src="image_30.png" /></figure>
<figure><img src="image_31.png" /></figure>

12 variables, 26 parameters. $\lambda_1(y, \bar{y}) = m_1 + \frac{f_1}{1+K(y_3+\bar{y}_3+y_4+\bar{y}_4)^n}$
<figure><img src="image_32.png" /></figure>

• Ballif, Clément, Y., *Nonlinear compartmental modeling to monitor ovarian follicle population dynamics on the whole lifespan*, JMB 2024
<figure><img src="image_33.png" /></figure>

* Frequentist approach (MLE + PLE) : 20 of 31 parameters are practically identifiable.
<figure><img src="image_34.png" /></figure>

* For compartments 2,3,4, follicle atresia is small but cannot be estimated with high precision

* The first-wave follicles is approx. 5 times faster than the adult wave

* Mean activation time is around 200 days, while growing time is around 50 days ($\epsilon \approx 0.1$).
<figure><img src="image_35.png" /></figure>

* For compartments 2,3,4, follicle atresia is small but cannot be estimated with high precision

* The first-wave follicles is approx. 5 times faster than the adult wave

* Mean activation time is around 200 days, while growing time is around 50 days ($\epsilon \approx 0.1$).

* Additional Data on germ cell dynamics (and atresia) would greatly improve practical identifiability
<figure><img src="image_36.png" /></figure>

* There is a two-fold change of activation rate between no regulation and maximal regulation, with a sharp transition ($n > 10$)

$$ \lambda_1(y, \bar{y}) = m_1 + \frac{f_1}{1 + K(y_3 + \bar{y}_3 + y_4 + \bar{y}_4)^n} $$
Ovarian reserve decline and growing follicle count

<figure><img src="image_37.png" /></figure>

Mean GF +/- 2*std

<figure><img src="image_38.png" /></figure>

$$ \begin{array}{ccc} & \varepsilon\lambda_0(Y) & \\ \frac{1}{\varepsilon}X & \rightarrow & Y \\ \downarrow\varepsilon\mu_0 & & \downarrow\mu_1 \\ \emptyset & & \emptyset \end{array} $$

$$ \lambda_0(y) = m + \frac{f}{1+(Ky)^n} \quad \text{with } n \gg 1 $$

<figure><img src="image_39.png" /><figcaption>Coxworth and Hawkes 2010</figcaption></figure>

Zhou et al., 2023

⇒ Ovarian reserve decay and stable growing follicle count
Ovarian reserve decline and growing follicle count

<figure><img src="image_40.png" /></figure>

Mean GF +/- 2*std

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

Coxworth and Hawkes 2010

Zhou et al., 2023

⇒ This model points the limitations of any ovarian reserve size prediction based on growing follicle number (or AMH levels).
Application : prediction of AMH administration

<figure><img src="image_43.png" /></figure>
<figure><img src="image_44.png" /></figure>

Related works : Menopause timing, tissue cryopreservation and autologous transplantation
Lawley et al BMB 2025
* ✓ Lifespan nonlinear ovarian follicle population dynamics model

* ✓ Separation of time scale explains slow decay of the reserve and quasi stable growing follicle repartition

* ✓ Follicle count data and perturbation experiments may reveal feedback mechanisms

## Next steps :

* Mixed-effect modeling (large individual variability in reserve size)

* AMH also inhibits follicle growth -> Multi-stability is possible in that case
* ✓ Lifespan nonlinear ovarian follicle population dynamics model

* ✓ Separation of time scale explains slow decay of the reserve and quasi stable growing follicle repartition

* ✓ Follicle count data and perturbation experiments may reveal feedback mechanisms

## Next steps :

* Mixed-effect modeling (large individual variability in reserve size)

* AMH also inhibits follicle growth -> Multi-stability is possible in that case

## Limitations :

* DATA!
Automatic follicle count (Mice)

<figure><img src="image_45.png" /></figure>

Rosario Medina-Rodriguez, Céline Guigon, on-going work
Automatic follicle count (Mice)

Increasing Ovarian Follicle Annotations

**Pseudo-labeling** uses predictions from a trained model on unlabeled tissue images as “pseudo” annotations to increase under-represented classes, such as primordial follicles.

<figure><img src="image_46.png" /></figure>
Automatic follicle count (Medaka)

<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>
<figure><img src="image_49.png" /></figure>

$$ egin{cases} 
\partial_t \rho(t, x) + \partial_z (f(x) \rho(t, x)) = 0 \\ 
f(x_m) \rho(t, x_m) = r(t) \\ 
r'(t) = (\alpha(P(t)) - \lambda_0)r(t) \\ 
P(t) = \int_{x_m}^{x_M} \omega(x) \rho(t, x) dx 
\end{cases} $$

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>
AMH negatively regulates precursor proliferation

<figure><img src="image_52.png" /></figure>

* Growth speed : different order of magnitude

* AMH inhibits precursor proliferation and leads to a stabilization of the recruitment rate
Cyclic behavior (fast time scale)

<figure><img src="image_53.png" /></figure>
<figure><img src="image_54.png" /></figure>

$$ egin{cases} 
\partial_t \rho(t, x) + \partial_x (f(x)g(P(t))\rho(t, x)) + \mu\rho(t, x) = 0, & x \in (x_m, x_M), \quad t \ge 0 \\ 
\Lambda(P(t), x_m) \rho(t, x_m) = r(P(t)), & t \ge 0 \\ 
P(t) = \int_{x_m}^{x_M} \omega(x)\rho(t, x)dx, & t \ge 0 \\ 
\rho(t=0, x) = \rho_0(x), & x \in (x_m, x_M) 
\end{cases} $$

$$ r(c_r, P) = \frac{c_r}{1 + P^n} \quad \Lambda(P) = \frac{1 + c_\lambda P^n}{1 + P^n} $$
<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /><figcaption>Données expérimentales : un cycle de ponte</figcaption></figure>

<figure><img src="image_57.png" /><figcaption>Simu : solution périodique stable</figcaption></figure>
# Ovarian aging -> perturbed cycles ?

<figure><img src="image_58.png" /><figcaption>Données expérimentales : un cycle de ponte</figcaption></figure>

<figure><img src="image_59.png" /><figcaption>Données: petits ovocytes en fonction de l'âge</figcaption></figure>

<figure><img src="image_60.png" /><figcaption>Simu : Mise en place et disparition du cycle de ponte avec taux basal de recrutement âge dépendant</figcaption></figure>

<figure><img src="image_61.png" /><figcaption>Simu : solution périodique stable</figcaption></figure>
* Negative regulation on ovarian reserve, both on Mice (activation) and Medaka (proliferation), potentially mediated by AMH

* Follicle dynamics is characterized by several time scales

* Cyclic behavior emerges from double positive / negative feedback

## Next steps :

* More inverse problem (PINN, mixed-effects, etc)

* Characterization of follicular waves within cycles, follicle selection (in mammals)

* Coupling with hormonal dynamics (Chiara Villa) and intra-cellular signaling (Chloé Weckel)

* Follicle morphogenesis (Frédérique Robin and Mingyue Zhang)
* INRAE Rennes : Violette Thermes

* INSERM - Paris Cité : Céline Guigon

* INRIA Saclay : Frédérique Clément, Rosario Medina-Rodriguez, Guillaume Ballif, Chiara Villa

* INRAE Tours : RY, Louis Fostier

<figure><img src="image_62.png" /></figure>

ANR OVOPAUSE

Dynamics and control of female germ cell populations : understanding aging through population dynamics models
# Random initial size and Menopause Time

Ovarian reserve $\bar{x}$

reserve decay for random I.C

<figure><img src="image_63.png" /></figure>

Growing follicle for random I.C

<figure><img src="image_64.png" /></figure>

Growing follicle distr. for random I.C

<figure><img src="image_65.png" /></figure>
<figure><img src="image_66.png" /></figure>

<figure><img src="image_67.png" /></figure>