# A simple mathematical model for evaluation of non-fragmentation property of injectable calcium-phosphate cement

## TL;DR

This paper develops a mathematical model to understand how injectable bone cement avoids breaking apart during setting.

## Contribution

A novel phenomenological model using the Allen-Cahn equation is proposed to explain the non-fragmentation property of injectable calcium-phosphate cement.

## Key findings

- The model successfully represents the setting behavior of the cement paste through numerical simulations.
- The integration of mathematical modeling and experiments provides insight into the non-fragmentation mechanism.
- Phenomenological modeling offers a new approach to studying cement behavior without relying solely on physical laws.

## Abstract

In this study, the mechanism of fragmentation including void formation in an injectable calcium-phosphate cement of paste-like artificial bone grafting material, is clarified from the point of view of mathematical modeling. Instead of modeling from physical laws or chemical reactions, phenomenological modeling is used, and three experiments are performed to build the model. We then successfully use the results in terms of the equations and necessary assumptions to construct a simple mathematical model. Our proposed model is based on the mathematically well-known Allen-Cahn type equation. The state of the paste is used as an unknown function. After scaling according to the appropriate nondimensionalization, numerical simulations are performed. The numerical results represent the setting behavior of the paste, which is difficult to observe experimentally. The integration of mathematical and experimental results provides deep insight into the mechanism of non-fragmentation property of an injectable calcium-phosphate cement.