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Non-linear visco-elasto-plastic rheology of a viscous vertex model

This paper establishes a general mean-field framework linking cell- and tissue-scale mechanics by deriving non-linear visco-elasto-plastic rheological relations for a viscous vertex model that incorporates explicit friction and active anisotropic stresses, validated through large-amplitude oscillatory shear predictions.

Original authors: Shalabh Kumar Anand, Matthias Merkel

Published 2026-02-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shalabh Kumar Anand, Matthias Merkel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a biological tissue, like the skin on your arm or the lining of your gut, not as a solid block, but as a bustling crowd of thousands of tiny, jiggly balloons packed tightly together. These "balloons" are cells. When your body grows, heals a wound, or a tumor spreads, these cells have to squeeze, stretch, and swap places with their neighbors to change the shape of the whole crowd.

This paper is about figuring out the rules of the road for how this crowd moves and flows. The authors call this "rheology" (the study of how things flow and deform), but you can think of it as the tissue's personality: Is it stiff like a rock? Stretchy like a rubber band? Or gooey like honey?

Here is the breakdown of their discovery, using simple analogies:

1. The Problem: The "Slippery Floor" vs. The "Crowded Room"

In the past, scientists modeled these tissues by imagining the cells were sliding on a rough floor (like a table). They assumed the friction came from the cells rubbing against the ground.

The New Insight:
The authors realized that many tissues (like a 3D ball of stem cells) aren't sitting on a floor at all. They are floating in a soup of other cells. In this case, the friction doesn't come from the ground; it comes from the cells rubbing against each other.

Analogy: Imagine trying to walk through a crowded dance floor. If you are on a smooth stage, you slide easily. But if you are in a mosh pit, the friction comes from bumping into other people. The authors built a model that accounts for this "mosh pit" friction, ensuring that if the whole crowd spins or moves in a straight line, the internal rules of the dance don't change. This is called Galilean invariance (a fancy way of saying the physics works the same whether you are standing still or moving).

2. The Two Ways Tissues Move: Stretching vs. Swapping

When you pull on a piece of tissue, it changes shape in two distinct ways. The authors separated these two behaviors like a chef separating egg whites from yolks:

The Stretch (Reversible): The cells get squished or elongated, like stretching a rubber band. If you let go, they snap back. This is elasticity.
The Swap (Irreversible): Sometimes, the cells get so crowded that they have to let go of one neighbor and grab a new one to fit. This is called a T1 transition. Once they swap, they don't go back. This is plasticity (permanent change).

The authors created a mathematical recipe that tracks both: How much is the tissue stretching right now? and How many cells are swapping places?

3. The "Active" Energy: Cells with a Mind of Their Own

Living tissues aren't just passive rubber bands; the cells are alive and active. They can push and pull. The authors tested two specific ways cells do this:

The "Anisotropic Tension": Imagine a cell that decides to tighten its belt only on the left and right sides, making it squeeze horizontally.
The "Crawling" Force: Imagine a cell grabbing its own top and bottom and pulling them together, like a caterpillar inching forward.

They found that these active forces act like a bias. They don't just make the tissue move faster; they shift the entire "personality" of the tissue, making it easier or harder to stretch in certain directions.

4. The Big Surprise: Viscosity Changes When Cells Swap

This is the most exciting finding. The authors discovered that the "thickness" (viscosity) of the tissue changes depending on what the cells are doing.

Scenario A: If the tissue is just stretching and snapping back (elastic), it has a certain thickness.
Scenario B: If the tissue is actively swapping cells (plastic), it suddenly becomes three times thicker/more resistant.

The Analogy: Think of a crowd of people trying to walk through a hallway.

If they just lean against each other and sway (elastic), it's easy to push them.
But if they start shoving past each other, grabbing new hand-holds, and rearranging the whole line (plastic/T1 transitions), the crowd suddenly feels much "stickier" and harder to push through. The act of rearranging creates extra resistance.

5. The Prediction Machine

Finally, the authors built a "crystal ball" (a mean-field model). They took all these rules—the stretching, the swapping, the active pushing, and the changing thickness—and combined them into a single set of equations.

They tested this crystal ball by simulating a tissue being shaken back and forth violently (large-amplitude oscillatory shear).

The Result: The math predicted exactly how the tissue would behave, matching their computer simulations perfectly.

Why Does This Matter?

This paper provides a universal "instruction manual" for how biological tissues flow.

For Doctors: It helps explain how tumors invade other tissues or how embryos fold themselves into complex shapes during development.
For Engineers: It helps design better artificial tissues for regenerative medicine.
For Scientists: It shows that the "rules" of tissue flow depend heavily on how the cells are interacting (friction) and what they are doing (active forces). You can't just use one simple rule for all tissues; you have to look at the specific details of the cell crowd.

In a nutshell: The authors figured out that biological tissues are like a smart, sticky, active crowd. They stretch, they swap places, and they get "thicker" and harder to move exactly when they are busy rearranging themselves. They wrote the math to predict exactly how this crowd will flow under any condition.

1. Problem Statement

Understanding the mechanical behavior of biological tissues is essential for deciphering developmental processes, tissue engineering, and cancer metastasis. While tissue-scale mechanics arise from collective cellular behavior, a comprehensive theoretical framework linking cell-scale dynamics to tissue-scale rheology is lacking.

Gap in Literature: Previous vertex model studies have largely focused on linear rheology or elastic properties. Stress dissipation and the non-linear visco-plastic regime (involving cell rearrangements) are less explored.
Specific Limitations: Many existing models rely on substrate friction (friction against an external surface), which violates Galilean invariance. This is problematic for tissues like 3D aggregates or embryos that lack an external substrate and rely on internal dissipation. Furthermore, few models simultaneously account for explicit internal friction, non-linear elasticity, and active cellular forces in a mean-field framework.

2. Methodology

The authors constructed a 2D vertex model with explicit viscous friction and active forces, deriving mean-field rheological relations.

A. The Vertex Model Setup

Geometry: Tissues are modeled as a tiling of polygons (cells) defined by vertex positions.
Energy Functional: Uses a classical form including area elasticity ( $k_A$ ) and perimeter elasticity ( $p_0$ ).
Dynamics (Over-damped): Force balance on each vertex includes elastic, frictional, and active forces ( $F^{el} + F^{fr} + F^{act} = 0$ ).
Galilean Invariance: To ensure physical realism for tissues without substrates, the model implements internal friction that depends on velocity differences (relative motion) rather than absolute velocity. Friction is applied to:
1. Cell areas ( $\sigma_a$ ).
2. Cell perimeters ( $T_p$ ).
3. Cell-cell interface lengths ( $T_\ell$ ).
  This formulation satisfies both translational and rotational Galilean invariance.

B. Active Mechanisms

Two distinct active mechanisms were implemented to generate anisotropic stresses:

Anisotropic Interface Tension: Active tension on interfaces depends on the angle relative to the shear axis ( $T^{act} \propto \cos 2\phi$ ).
Active Crawling: Pulling forces acting on specific vertices within a cell (mimicking convergent extension).

C. Strain Decomposition Framework

The core theoretical approach decomposes the tissue shear rate ( $\tilde{v}$ ) into two components (based on Ref. [63]):
$\tilde{v} = \frac{DQ}{Dt} + R$

$\frac{DQ}{Dt}$ : Reversible contribution from cell shape changes (elastic/visco-elastic).
$R$ : Irreversible contribution from cell rearrangements (T1 transitions/plasticity).
This allows the construction of two separate constitutive relations:

Stress Relation: $\tilde{\sigma}(Q, \tilde{v})$ linking stress to shape and strain rate.
Rearrangement Relation: $R(Q, \tilde{v})$ describing yielding behavior.

D. Simulation Protocol

Linear Regime: Small-amplitude oscillatory shear to measure storage ( $G'$ ) and loss ( $G''$ ) moduli.
Non-linear Regime: Constant pure shear rate simulations to probe large deformations, T1 transitions, and active effects.
Validation: Large-amplitude oscillatory shear (LAOS) simulations were used to test the predictive power of the derived mean-field equations.

3. Key Contributions & Results

A. Constitutive Relation for Shear Stress ( $\tilde{\sigma}$ )

The authors derived a non-linear stress function dependent on cell shape ( $Q$ ) and shear rate ( $\tilde{v}$ ):

Linear Regime: $\tilde{\sigma} = 2G_0 Q + 2\eta_0 \tilde{v}$ $\tilde{σ} = 2 G_{0} Q + 2 η_{0} \tilde{v}$ .
- $G_0 \approx 0.82$ (Shear modulus).
- $\eta_0 \approx 1.16$ (Viscosity in the absence of rearrangements).
Non-linear Elasticity: At large strains, the elastic stress becomes non-linear:
$\tilde{\sigma}_{el}(Q) = 2G_0 Q + 4G_2 Q^3 + 6G_4 Q^5$
The authors found that higher-order terms are necessary to describe stress at large cell shapes ( $Q > 0.4$ ).
Viscosity Dependence on Rearrangements: A critical finding is that the effective viscosity depends on whether the tissue is deforming elastically or plastically.
- When T1 transitions occur (plastic flow), the viscosity increases significantly ( $\eta_s \approx 3.12$ ) compared to the purely elastic regime.
- The authors propose that this "extra" viscosity arises from the relaxation of cell shapes immediately following T1 transitions.
Active Stresses: Active forces introduce a linear shift in the stress curve:
$\tilde{\sigma} = \tilde{\sigma}_{passive} - \rho T^{act}$
where $\rho$ is a coefficient dependent on the type of activity (interface tension vs. crawling).

B. Constitutive Relation for Rearrangements ( $R$ )

The irreversible fraction ( $R/\tilde{v}$ ) was found to depend exponentially on cell shape:
$\frac{R(Q, \tilde{v})}{\tilde{v}} = \frac{1}{2} \exp\left( b(\tilde{v}) [Q - Q_0(\tilde{v})] \right)$

Activity Effects: While the slope parameter $b$ is largely independent of activity, the threshold parameter $Q_0$ shifts significantly based on the type and magnitude of active forces.
Mechanism Distinction: The "crawling" mechanism shifts $Q_0$ more strongly than anisotropic tension, leading to different steady-state cell shapes under shear.

C. Predictive Validation

The derived mean-field equations successfully predicted the time evolution of cell shape ( $Q$ ) and shear stress ( $\tilde{\sigma}$ ) in Large-Amplitude Oscillatory Shear (LAOS) simulations. The model accurately captured the hysteresis and yielding behavior, confirming that the decomposition into reversible (shape) and irreversible (rearrangement) parts is robust.

4. Significance

General Framework: The paper provides a systematic method to construct non-linear mean-field constitutive relations for any cell-based tissue model, not just the vertex model.
Internal Dissipation: By enforcing Galilean invariance, the model correctly describes tissues without external substrates (e.g., 3D aggregates), correcting a common flaw in previous vertex models.
Viscosity Mechanism: The discovery that viscosity is state-dependent (higher during plastic flow due to T1 transitions) offers a new physical explanation for the apparent viscosity of biological tissues.
Activity Integration: The work quantifies how different active cellular mechanisms (tension vs. crawling) distinctly alter tissue rheology and steady-state shapes, providing a tool to interpret experimental data on morphogenesis.
Bridging Scales: It successfully links microscopic cell parameters (friction coefficients, active tension) to macroscopic tissue properties (moduli, yield stress, viscosity).

In summary, this work establishes a comprehensive, non-linear, visco-elasto-plastic rheological theory for biological tissues, explicitly accounting for internal dissipation, cell shape evolution, and active cellular forces.