Bone remodelling is carried out by `bone multicellular units (BMUs) in which active osteoclasts and active osteoblasts are spatially and temporally coupled. The refilling of new bone by osteoblasts towards the back of the BMU occurs at a rate that depends both on the number of osteoblasts and on their secretory activity. In cortical bone, a linear phenomenological relationship between matrix apposition rate (MAR) and BMU cavity radius is found experimentally. How this relationship emerges from the combination of complex, nonlinear regulations of osteoblast number and secretory activity is unknown. Here, we extend our previous mathematical model of cell development within a single BMU to investigate how osteoblast number and osteoblast secretory activity vary along the BMUs closing cone. MARs predicted by the model are compared with data from tetracycline double labelling experiments. We find that the linear phenomenological relationship observed in these experiments between MAR and BMU cavity radius holds for most of the refilling phase simulated by our model, but not near the start and end of refilling. This suggests that at a particular bone site undergoing remodelling, bone formation starts and ends rapidly. Our model also suggests that part of the observed cross-sectional variability in tetracycline data may be due to different bone sites being refilled by BMUs at different stages of their lifetime. The different stages of a BMUs lifetime depend on whether the cell populations within the BMU are still developing or have reached a quasi-steady state while travelling through bone. We find that due to their longer lifespan, active osteoblasts reach a quasi-steady distribution more slowly than active osteoclasts. We suggest that this fact may locally enlarge the Haversian canal diameter (due to a local lack of osteoblasts compared to osteoclasts) near the BMUs point of origin.
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