Two conjectures on 3D Voronoi structures: a toolkit with biomedical case studies.

Area of Science:

• Additive Manufacturing
• Materials Science
• Computational Geometry

Background:

• 3D Voronoi scaffolds mimic natural and synthetic structures, offering lightweight resilience.
• Current design methods for these scaffolds are often unpredictable, requiring extensive trial and error.
• Key structural features, like vertex count, influence topological similarity but are hard to control.

Purpose of the Study:

• To develop a predictive toolkit for 3D Voronoi scaffold design.
• To enable efficient and accurate prediction of design features based on desired structural parameters.
• To introduce new mathematical conjectures relating structural properties to generating points.

Main Methods:

• Analysis of over 12,000 3D Voronoi structures.
• Development of predictive equations for design features (e.g., number of generating points, G).
• Validation of equations across diverse parameter values and network sizes.

Main Results:

• A toolkit of validated equations allows accurate prediction of design features (within ±3G).
• Two new mathematical conjectures are proposed regarding Voronoi structure properties.
• A design code facilitates selection, adjustment, and 3D printing of Voronoi structures.

Conclusions:

• The developed toolkit significantly improves the efficiency and accuracy of 3D Voronoi scaffold design.
• The findings offer new insights into the mathematical relationships within Voronoi structures.
• Biomedical applications, including T-cell culturing and bone scaffolds, are demonstrated using the design code.