Computational modeling of hormone- and cytokine-dependent proliferation of endometrial cells in 3D co-culture

This study develops and calibrates coupled ordinary and partial differential equation models to simulate and quantify the hormone- and cytokine-dependent proliferation dynamics of human endometrial epithelial and stromal cells in 3D co-culture, thereby elucidating their mutual interactions and the homogeneity of molecular exposure across physiological and pathological conditions.

The Gist

Imagine the human uterus as a bustling, high-rise apartment building that undergoes a massive renovation every single month. The inner lining (the endometrium) is the apartment complex itself. Every month, it prepares for a potential new tenant (a baby). If no one moves in, the building is partially demolished (menstruation) and then rebuilt from scratch.

This process is incredibly complex. It involves two main types of "construction workers":

1. Epithelial Cells: The architects and builders who form the rooms (glands).
2. Stromal Cells: The foundation crew and property managers who support the structure.

These workers don't just follow orders; they talk to each other, influenced by chemical "emails" (hormones like estrogen and progesterone) and "emergency alerts" (cytokines like IL-1β, which signal inflammation).

The problem is that we can't easily study this building in real life, and mice don't have monthly renovations like humans do. So, scientists built a miniature, 3D model of this building in a lab using a jelly-like substance (hydrogel) and real human cells. But even with a model, it's hard to see exactly how the workers are reacting to the emails and alerts, especially when there are so many variables.

Enter the "Digital Twins" (The Computer Models)

The authors of this paper created two computer programs to act as "digital twins" of this lab experiment. Think of them as a video game simulation that predicts exactly what will happen to the building under different conditions.

1. The "Population Counter" (The ODE Model)

The first model is like a sophisticated spreadsheet that tracks the population of the two types of workers over time.

• How it works: It uses math to ask: "If we add more estrogen, do the architects build faster? If we add an inflammation alert (IL-1β), do the foundation crew get scared and stop working?"
• The Twist: The researchers tested cells from three different human donors. Just like real people, these cells had different personalities. Some were super-fast builders; others were slow. Some were easily stressed by inflammation; others were tough.
• The Discovery: The model revealed that the two types of workers are constantly influencing each other. For example, in some cases, the foundation crew (stromal cells) actually helps the architects (epithelial cells) grow faster when they are in the same room. But if you add an inflammation alert, the foundation crew might shrink, which accidentally lets the architects grow even bigger because they aren't being held back anymore. It's a delicate dance of push-and-pull.

2. The "Smoke Detector" (The PDE Model)

The second model addresses a tricky physical problem: The "Pizza Delivery" Issue.
Imagine you order a pizza (a hormone or cytokine) to a large, dense apartment building. If the building is small, the delivery guy can get the pizza to every floor instantly. But if the building is huge and the residents are hungry (proliferating cells), the pizza might get eaten on the ground floor before it ever reaches the top floor. This creates a "gradient"—some people get a full meal, others get crumbs.

• How it works: This model simulates how the "pizza" (specifically the IL-1β molecule) diffuses through the jelly and gets "eaten" (absorbed) by the cells.
• The Discovery: The researchers found that for the small jelly blobs they were using in the lab (about the size of a grain of rice), the pizza delivery was perfect. Everyone got the same amount of food, regardless of how fast the cells were growing.
• The Warning: However, if they used a larger jelly blob, the delivery would fail. The cells on the outside would eat all the signals, leaving the cells in the center starving. This means that for future experiments, keeping the 3D cultures small is crucial to ensure everyone gets the same instructions.

Why Does This Matter?

This paper is like giving scientists a GPS and a weather forecast for studying uterine diseases.

1. Personalized Medicine: It shows that cells from different people react differently. A treatment that works for one person's "construction crew" might fail for another's. These models can help predict which patients will respond to which drugs.
2. Better Experiments: It tells researchers, "Hey, don't make your 3D cultures too big, or your chemical signals won't reach the center." This saves time and money on failed experiments.
3. Understanding Disease: By understanding how these cells talk to each other, we can better understand diseases like endometriosis (where the building grows outside the uterus) or adenomyosis (where it grows into the walls). The model suggests that the "conversation" between the two cell types is often broken in these diseases.

In a Nutshell:
The scientists built a computer simulation of a uterine "construction site" to figure out how different human cells react to hormones and inflammation. They found that the cells have unique personalities, they constantly influence each other, and that keeping the lab experiments small ensures everyone gets the same instructions. This helps us design better treatments for painful menstrual disorders.

Technical Summary

1. Problem Statement

The human endometrium is a complex, dynamic tissue that undergoes cyclic regeneration, shedding, and decidualization driven by interactions between multiple cell types (epithelial, stromal, vascular, immune) and hormonal signals (estradiol, progesterone).

• Limitations of Current Models: Animal models (e.g., mice) do not naturally menstruate, making them poor proxies for human endometrial disorders like endometriosis and adenomyosis.
• Experimental Challenges: While 3D co-culture systems using primary human cells (epithelial organoids and stromal cells) have advanced, they introduce complexity. It is difficult to isolate the specific influence of cell-cell crosstalk, donor variability, and molecular gradients (diffusion limitations) on experimental outcomes.
• The Gap: There is a lack of mechanistic computational models that can quantify how hormones and cytokines (specifically IL-1$\beta$) regulate the proliferation and death of distinct endometrial cell types in 3D environments, nor is there a clear understanding of whether cell proliferation creates spatial gradients of signaling molecules that could confound experimental results.

2. Methodology

The authors developed a two-part computational framework calibrated against experimental data from a 3D hydrogel co-culture system (based on Gnecco et al., 2023).

A. ODE-Based Cell Proliferation Model

• Structure: A system of Ordinary Differential Equations (ODEs) modeling the density of Endometrial Epithelial Organoid Cells (EEOCs) and Endometrial Stromal Cells (ESCs).
• Mechanism:
  • Proliferation: Modeled using a Hill-like function to account for density-dependent inhibition (contact inhibition) and cross-regulation between cell types.
  • Death: Modeled as a first-order reaction.
  • Perturbants: The model includes "on/off" multiplier functions ($FX$) to simulate the effects of specific conditions:
    • Hormones: Estradiol (E2), Medroxyprogesterone Acetate (MPA/progestin), and RU-486 (progesterone receptor inhibitor).
    • Cytokines: Interleukin-1$\beta$ (IL-1$\beta$).
    • Crosstalk: Parameters representing how the presence of one cell type affects the proliferation/death of the other.
• Calibration: The model was optimized using a non-linear least squares solver (lsqnonlin in MATLAB) against experimental data from three human donors (ages 27–42, varying pathologies). The data included time-course measurements of organoid radius and endpoint stromal cell density (Live/Dead staining).
• Uncertainty Quantification: The optimization was run 100 times with different initial conditions. Solutions were "gated" based on criteria such as error thresholds (<65% deviation from mean), lack of unbounded growth, and biological plausibility of stromal growth rates.

B. PDE-Based Reaction-Diffusion Model

• Purpose: To determine if cell proliferation creates spatial gradients of IL-1$\beta$ within the hydrogel, leading to heterogeneous cell exposure.
• Equation: Solved Fick's second law of diffusion with a reaction term (consumption) in spherical coordinates.
• Inputs: Used the cell density predictions from the ODE model as time-varying inputs for receptor-mediated uptake.
• Parameters: Included diffusion coefficients, receptor binding affinity ($K_D$), receptor density per cell, and internalization rates.
• Scenarios: Simulated IL-1$\beta$ distribution in small (6 $\mu$L post-swelling) and large (50 $\mu$L post-swelling) hydrogels under varying proliferation rates.

3. Key Contributions

1. Mechanistic ODE Model: Created the first mechanistic model capable of simulating the coupled dynamics of primary human endometrial epithelial and stromal cells in 3D co-culture, explicitly accounting for donor-to-donor variability and hormone/cytokine perturbations.
2. Quantification of Crosstalk: Successfully quantified the bidirectional influence of epithelial and stromal cells on each other's proliferation, revealing that these interactions are highly donor-specific.
3. Gradient Analysis: Developed a coupled PDE model to assess the validity of the "well-mixed" assumption in 3D cultures. It demonstrated that for small hydrogels (<50 $\mu$L), molecular gradients are negligible even with high proliferation, but become significant in larger volumes.
4. Experimental Design Guidance: Provided quantitative criteria for experimental design, suggesting that smaller hydrogel volumes are preferable for multi-day studies to ensure uniform cytokine exposure.

4. Key Results

• Cell Turnover Differences: Epithelial cells exhibited higher base proliferation rates ($g_{base}$) and higher base death rates ($d_{base}$) compared to stromal cells across all donors, resulting in higher net turnover for epithelia.
• Donor Variability:
  • Donor 260: Stromal cells slightly inhibited epithelial proliferation; epithelial cells strongly promoted stromal proliferation.
  • Donor 271: Epithelial cells strongly inhibited stromal proliferation.
  • Donor 272: Epithelial cells strongly promoted stromal proliferation.
• Hormone/Cytokine Effects:
  • Progestin (MPA): Effects were highly variable and context-dependent. In some donors, progestin inhibited epithelial growth in monoculture but increased it in co-culture due to the inhibition of stromal cells (which normally suppress epithelial growth).
  • IL-1$\beta$: Consistently increased epithelial proliferation (likely via stromal-mediated mechanisms) and decreased stromal density across all donors.
• Diffusion and Gradients:
  • In small hydrogels (6 $\mu$L), IL-1$\beta$ saturated the gel within ~5 hours, regardless of cell proliferation rates. The concentration remained homogeneous (>0.9 ng/mL), ensuring uniform exposure.
  • In larger hydrogels (50 $\mu$L), significant gradients formed, and cell proliferation rates began to impact the distribution of IL-1$\beta$, potentially leading to heterogeneous biological responses.
  • The diffusion coefficient ($D$) was identified as the most sensitive parameter affecting gradient formation.

5. Significance

• Personalized Medicine: The model demonstrates that computational frameworks can capture inter-individual variability in endometrial tissue responses, a crucial step toward personalized treatment strategies for endometriosis and infertility.
• Experimental Validation: By proving that small 3D cultures maintain homogeneous cytokine distribution, the study validates the use of small-volume hydrogels for studying signaling pathways without the confounding factor of diffusion gradients.
• Therapeutic Development: The models provide a tool to predict how immune-modulating therapies (e.g., IL-1 inhibitors) might behave in 3D tissue environments, aiding in the design of preclinical assays for endometriosis treatments.
• Methodological Advancement: The study bridges the gap between in vitro 3D culture and in silico modeling, offering a blueprint for integrating mechanistic biology with systems biology to study complex, multicellular human tissues.