Differentiable Voronoi Diagrams for Efficient Simulation of Cell-Based Mechanical Systems

Core Concepts

A novel cell-centered approach based on differentiable Voronoi diagrams that substantially reduces the number of problem variables, eliminates the need for explicit contact handling, and ensures continuous geometry changes during topological transitions in cell-based mechanical systems.

Abstract

The paper presents a novel simulation approach for mechanical cellular systems based on differentiable Voronoi diagrams. The key highlights are:

Representation: Each cell is represented by a Voronoi site, and the topology and shape of the cells are defined implicitly. This substantially reduces the number of degrees of freedom compared to explicit cell models.

Continuity: The Voronoi representation ensures continuous cell geometry changes during topological transitions, eliminating the need for explicit handling of contact between cells.

Derivatives: The authors derive closed-form expressions for the first and second derivatives of cell geometry with respect to site positions, enabling the use of Newton-type optimization solvers for a wide range of per-cell energies.

Boundary Coupling: The differentiable Voronoi formulation is extended to enable coupling with arbitrary rigid and deformable boundaries, allowing the simulation of cellular systems interacting with external environments.

Applications: The method is demonstrated on a diverse set of examples, including tissue growth, foam coarsening, and inverse problems matching soap foam simulations to real-world images. Comparisons with explicit cell models show that the differentiable Voronoi approach can achieve qualitatively similar results with significantly faster computation times.

Stats

"Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods."
"Representing each cell with a Voronoi site, our method defines shape and topology of the interface network implicitly."
"Closed-form derivatives of network positions facilitate simulation with Newton-type methods for a wide range of per-cell energies."
"Comparative analysis with explicit cell models reveals that our method achieves qualitatively comparable results at significantly faster computation times."

Quotes

"Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods."
"Representing each cell with a Voronoi site, our method defines shape and topology of the interface network implicitly."
"Closed-form derivatives of network positions facilitate simulation with Newton-type methods for a wide range of per-cell energies."

Key Insights Distilled From

Differentiable Voronoi Diagrams for Simulation of Cell-Based Mechanical Systems

by Logan Numero... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18629.pdf

Deeper Inquiries

How could the differentiable Voronoi formulation be extended to handle more complex cell shapes, such as the non-convex scutoid cells observed in epithelial tissues?

The extension of the differentiable Voronoi formulation to handle more complex cell shapes, such as non-convex scutoid cells, would require modifications to the distance metric used in defining the Voronoi cells. One approach could involve introducing spatially varying distance metrics that can accommodate the irregular shapes of scutoid cells. By adapting the distance metric to capture the unique geometry of scutoid cells, the Voronoi diagram generation process can be tailored to produce more accurate representations of these complex cell shapes. Additionally, incorporating constraints or additional terms in the energy function that penalize deviations from the desired cell shapes can help guide the optimization process towards configurations that closely resemble non-convex cell geometries.

How could the differentiable Voronoi formulation be extended to handle more complex cell shapes, such as the non-convex scutoid cells observed in epithelial tissues?

To handle more complex cell shapes like non-convex scutoid cells, the differentiable Voronoi formulation can be extended by introducing adaptive distance metrics that can capture the intricate geometry of such cells. By incorporating spatially varying distance metrics that account for the non-convexity and irregularity of scutoid cells, the Voronoi diagram generation process can be customized to accurately represent these complex shapes. Additionally, integrating shape constraints or regularization terms into the energy function can help guide the optimization process towards configurations that closely resemble the desired non-convex cell shapes. This extension would enable the simulation of cellular systems with diverse and intricate geometries, enhancing the applicability of the differentiable Voronoi approach in modeling complex biological tissues.

How could the differentiable Voronoi formulation be extended to handle more complex cell shapes, such as the non-convex scutoid cells observed in epithelial tissues?

To handle more complex cell shapes like non-convex scutoid cells observed in epithelial tissues, the differentiable Voronoi formulation can be extended by incorporating adaptive distance metrics that can accurately represent the irregular geometry of such cells. By introducing spatially varying distance metrics tailored to the specific characteristics of non-convex shapes, the Voronoi diagram generation process can be optimized to capture the intricate details of scutoid cells. Additionally, integrating shape constraints or regularization terms into the energy function can guide the optimization process towards configurations that closely resemble the desired non-convex cell shapes. This extension would enhance the versatility of the differentiable Voronoi approach in simulating complex biological tissues with diverse and challenging geometries.