A mathematical model of curvature controlled tissue growth incorporating mechanical cell interactions

This study presents a novel discrete mathematical model and its derived continuum reaction-diffusion limit to simulate curvature-controlled biological tissue growth, demonstrating how mechanical cell interactions and spatial constraints drive interface smoothing consistent with experimental observations.

The Gist

Imagine you are watching a crowd of people trying to fill a room. If the room is a perfect circle, everyone moves in smoothly. But if the room is a square with sharp corners, something interesting happens: people naturally crowd into the corners, and the room slowly rounds itself out until it looks like a circle.

This paper is about building a mathematical "video game" to understand exactly how biological tissues (like skin, bone, or tumors) grow and reshape themselves, especially when they are trying to fill in a hole or a gap.

Here is the story of the research, broken down into simple concepts:

1. The Problem: Why Do Tissues Round Out?

Scientists have long known that when new tissue grows, it doesn't just expand in a straight line. It tends to smooth out. If you have a square-shaped hole in a scaffold (a structure used to grow tissue), the corners fill up faster than the flat sides. Eventually, the square becomes a circle.

Why? Is it because the cells "know" they are in a corner? Or is it just physics? The authors wanted to find out by creating a model that simulates the behavior of individual cells.

2. The Two Models: The "Ants" vs. The "River"

The researchers built two different ways to simulate this growth:

• The Discrete Model (The "Ants"): Imagine a chain of ants holding hands, forming a line around a hole. Each ant is a cell. They are connected by invisible springs. As they grow, they push new material outward.

  • If the line is straight, the ants move forward evenly.
  • If the line hits a sharp corner, the ants get squished together (crowded).
  • Because they are squished, the springs between them get compressed. The springs push back, forcing the ants to spread out sideways to relieve the pressure.
  • The Magic: Even though the ants don't have a "brain" telling them to round the corner, the simple physics of them pushing against each other naturally causes the sharp corner to smooth out.

• The Continuum Model (The "River"): Instead of tracking every single ant, imagine the line of ants as a flowing river of water. This is a smooth, continuous flow. The researchers used math to show that if you have enough ants (cells), their collective behavior looks exactly like this flowing river.

3. The Big Discovery: Curvature is a Side Effect

The most exciting part of the paper is a "aha!" moment.

In the "Ant" model, the researchers did not program the ants to care about corners or curves. They only programmed them to:

1. Push outward to grow.
2. Push against their neighbors if they get too close (mechanical stress).

Yet, when they ran the simulation, the tissue naturally smoothed out. When they translated this into the "River" (continuum) model, they found that curvature dependence emerged automatically.

The Analogy: Think of a crowd of people trying to exit a stadium. If the exit is a wide, flat wall, people leave evenly. If the exit is a sharp, narrow alley, people get jammed. The jamming forces people to move sideways to find space. The "curvature" of the exit didn't tell them to move; the crowding did. The math proves that tissue growth smoothing is just a result of cells bumping into each other and finding space.

4. Why This Matters

• For Doctors and Engineers: When building artificial bones or healing wounds, we want the tissue to grow smoothly. This model helps predict how fast a hole will close based on its shape.
• The "Bridging Time" Formula: The authors came up with a simple rule for how long it takes to fill a hole. It turns out the time depends on the ratio of the hole's Area (how much space needs filling) to its Perimeter (how many cells are on the edge pushing in).
  • Simple version: A bigger hole takes longer, but a hole with a longer edge (more cells pushing) fills up faster.
• Connecting the Scales: This is rare in science. Usually, you have to choose between a model that looks at individual cells (too messy for big tissues) or a model that looks at the whole tissue (too simple to see individual details). This paper bridges the gap, showing how the messy behavior of individual "springy" cells creates the smooth, predictable behavior of the whole tissue.

Summary

The paper tells us that tissue growth is like a game of "musical chairs" played by cells. They push outward to grow, but when they get crowded in corners, they push sideways to relieve pressure. This simple mechanical push-and-pull is enough to explain why biological tissues naturally smooth out sharp edges, without needing any complex biological instructions. It's a beautiful example of how complex shapes can emerge from simple, local rules.

Technical Summary

1. Problem Statement

Biological tissue growth is fundamentally influenced by the geometry of the supporting substrate and the mechanical interactions between cells. Experimental observations (e.g., in bone formation and porous scaffolds) show that tissues tend to grow faster in concave regions and slower in convex regions, leading to a "smoothing" of the tissue interface over time. This phenomenon is often described as curvature-controlled growth.

Existing mathematical models face limitations:

• Continuum models (e.g., morphoelastic theories or mean curvature flows) often rely on phenomenological assumptions that are difficult to link to individual cell properties or experimental measurements of cell trajectories.
• Discrete models often lack a rigorous connection to the macroscopic tissue-scale behavior or fail to explicitly derive how microscopic mechanical interactions lead to emergent curvature dependence.

The authors aim to bridge this gap by developing a discrete mathematical model that explicitly incorporates mechanical cell interactions (modeled as springs) and tissue generation, and then rigorously deriving its continuum limit to explain how curvature dependence emerges from cell-scale mechanics.

2. Methodology

A. Discrete Model Formulation

The authors represent the tissue interface as a closed chain of $N$ cells. Each cell is subdivided into $m$ sub-cellular components (springs) that interact mechanically.

• Geometry: The interface is defined by vertices $\mathbf{r}_i(t)$.
• Growth Mechanism: Cells generate new tissue material at a constant rate $k_f$. This induces a normal velocity to the interface.
• Mechanical Relaxation: Cells interact via spring forces (restoring forces) and move in a viscous medium (overdamped dynamics). The motion is governed by the balance of spring forces and drag.
• Evolution Equation: The velocity of each spring boundary $\mathbf{r}_i$ is the sum of two components:
  1. $\mathbf{v}^f_i$: Velocity due to new tissue formation (normal displacement).
  2. $\mathbf{v}^m_i$: Velocity due to mechanical relaxation (tangential redistribution to relieve stress).
     $$\frac{d\mathbf{r}_i}{dt} = \mathbf{v}^f_i + \mathbf{v}^m_i$$
• Force Laws: Two restoring force laws are considered:
  1. Hookean: Linear force $f(\ell) = k(\ell - a)$.
  2. Nonlinear: Saturating force $f(\ell) = k a^2 (1/a - 1/\ell)$, which better models cell body mechanics (diverging at zero length, saturating at large extension).

B. Continuum Limit Derivation

The authors derive a continuum description by taking the limit as the number of springs per cell $m \to \infty$.

• Scaling: Parameters (stiffness $k$, drag $\eta$, resting length $a$, secretion rate $k_f$) are rescaled with $m$ to ensure finite, well-defined macroscopic quantities.
• Resulting PDE: The derivation leads to a reaction-diffusion partial differential equation (PDE) governing the evolution of cell density $q(s,t)$ along the arc length $s$ of the interface:
  $$\left( \frac{\partial q}{\partial t} \right)_n = \frac{\partial}{\partial s} \left( D(q) \frac{\partial q}{\partial s} \right) - q V(q) \kappa$$
  Where:
  • $(\partial q / \partial t)_n$ is the rate of change following normal trajectories.
  • $D(q)$ is the density-dependent diffusivity derived from the mechanical properties of the springs.
  • $V(q)$ is the normal velocity function.
  • $\kappa$ is the signed curvature.

Key Insight: The curvature term ($\kappa$) is not explicitly included in the discrete model. It emerges naturally in the continuum limit as a consequence of spatial constraints and cell crowding during growth.

C. Numerical Simulations

• Discrete: Solved using an ODE solver (DifferentialEquations.jl) with operator splitting to handle growth and mechanical relaxation.
• Continuum: Solved using finite difference and finite volume methods (Kurganov-Tadmor scheme) depending on the diffusivity regime.
• Validation: Simulations were performed on various geometries (square, hexagonal, circular pores, and trench-like structures) to compare discrete and continuum results.

3. Key Contributions

1. Emergence of Curvature Dependence: The paper demonstrates that curvature-dependent growth is an emergent property of mechanical cell interactions and crowding. Even though the discrete model has no explicit curvature term, the continuum limit naturally produces a curvature-dependent reaction term ($-qV(q)\kappa$).
2. Linking Micro to Macro Mechanics: The study provides a rigorous mathematical link between individual cell mechanics (spring stiffness, resting length, drag) and macroscopic tissue properties (diffusivity $D(q)$).
   • Hookean springs $\to$ Nonlinear diffusion ($D(q) \propto 1/q^2$).
   • Nonlinear springs $\to$ Linear diffusion ($D(q) = \text{constant}$).
3. Novel Bridging Time Relationship: The authors derive a new analytical relationship for the time ($T_b$) required for a tissue to fill a porous scaffold pore:
   $$T_b(L) = \frac{1}{k_f^* q_0} \frac{A(L)}{P(L)}$$
   Where $A(L)$ is the pore area and $P(L)$ is the perimeter. This generalizes previous findings, showing that bridging time depends on the ratio of area to perimeter, explaining why different pore shapes (even with the same area) close at different rates.
4. Discrete Model as a Discretization Scheme: The discrete model is shown to be a robust numerical discretization of the continuum reaction-diffusion PDE, capable of handling low diffusivity regimes where standard continuum solvers struggle.

4. Results

• Interface Smoothing: Both discrete and continuum models successfully reproduce the experimental observation of interface smoothing (rounding of corners) in square pores.
• Diffusivity Effects:
  • Low Diffusivity (Slow relaxation): Cells accumulate rapidly at concave corners, creating high-density regions and "cusps" that propagate as shock waves.
  • Intermediate/High Diffusivity: The interface smooths out effectively, matching experimental observations of rounded tissue growth.
• Model Agreement: The discrete model (with sufficient springs per cell, e.g., $m=10$) produces visually identical results to the continuum model across all diffusivity regimes.
• Cell Trajectories: The discrete model provides unique insights into individual cell trajectories and stress states (tension vs. compression) based on the resting length $a^*$, which continuum models cannot capture.
• Pore Closure: The derived formula $T_b \propto A/P$ accurately predicts experimental data for square pores, confirming that the linear scaling of bridging time with pore size is a geometric consequence of crowding effects.

5. Significance

• Mechanistic Understanding: The work moves beyond phenomenological descriptions of tissue growth, providing a mechanistic explanation for why curvature controls growth rates: it is a direct result of mechanical crowding and the inability of cells to expand in concave regions without generating stress.
• Tissue Engineering Applications: The derived relationship between pore geometry (Area/Perimeter) and bridging time offers a predictive tool for designing porous scaffolds. It suggests that pore shape is a critical design parameter, not just pore size.
• Methodological Advancement: By validating a discrete model against a continuum limit, the authors provide a framework that can be used to interpret experimental data at the single-cell level (trajectories, stress) while retaining the computational efficiency of continuum descriptions for large-scale predictions.
• Future Directions: The model is easily extensible to include cell proliferation, differentiation, and death, offering a pathway to more complex, biologically realistic simulations of regenerative medicine processes.