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

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.