A Mathematical Model to Predict Instantaneous Bone Formation Rate from Temporal Data of Cellular Biomarkers

Although several models have been proposed to predict spatial patterns of new bone formation across different regions of a bone, to our knowledge no model has predicted bone formation temporally from cellular biomarkers. In this article, we predict bone formation rate (BFR) temporally from Col1a1 gene expression data and compare our predictions with the average BFR reported in the literature. The proposed mathematical model identifies key parameters influencing BFR at the celluar level and quantifies how biomarkers encode mechanical loading information. This model serves as an excellent starting point to understand how BFR changes over time relative to a given regimen of exogenous loading. We report that the simplest mathematical model explaining this phenonmenon with reasonable accuracy is a second order linear critically-damped system with a delay time. Since BFR reported in the literature typically represents an average value over the interlabel period, we also propose a method to convert this measure into an instantaneous one, which is essential for constructing an "ideal" mathematical model. Finally, we present our results and discuss limitations of the model along with directions for future improvement.