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Journal of Long-Term Effects of Medical Implants

ISSN Print: 1050-6934
ISSN Online: 1940-4379

Journal of Long-Term Effects of Medical Implants

DOI: 10.1615/JLongTermEffMedImplants.2013006644
pages 253-262

Derivation and Application of a Mathematical Model for Long Bone Growth

Suneil Seetharam
Harvard University, School of Engineering and Applied Sciences, Cambridge, Massachusetts, 02138
Sujata K. Bhatia
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138


The objective of this work was to develop a mathematical model of long bone growth and to gain insights regarding growth disorders. A cell balance (mass balance of moving cells) assessment was performed on the three regions of the growth plate, to determine the variables (including number of proliferating cells, and division rate of proliferating cells) that influence tibia growth rate. Once this relationship was established, clinical data were used to understand how tibia growth rate and number of proliferating cells change with time. These equations were then inserted into the model to determine how cell division rate changes with time. The model was utilized to determine the influence of growth time, and to measure changes in vitamin C deficiency, Indian hedgehog (IHH) expression, and bone morphogenetic protein-2 (BMP-2) implants on tibia length. According to the model, a 10-month discrepancy in growth time between the two tibias is required to produce clinically significant leg asymmetry. In addition, vitamin C deficiency, IHH overexpression, and BMP-2 implants can all affect tibia length. These bioactive molecules have the greatest effect on tibia growth rate when these perturbations occur early in life for extended periods of time. The results are significant for modeling and predicting the effects of perturbations, including bioactive implants, on long bone growth.