Sensitivity analysis of a novel mathematical model identifies factors determining bone resorption rates

The development of pharmaceutical treatments for bone disease can be enhanced by computational models that predict their effects on resorption and rates of remodeling. Therefore, a simple mathematical model was formulated to simulate erosion depth and duration of resorption, using Michaelis-Menten (M-M) equations to describe changing rates of cellular activity during the two phases of bone resorption. The model was based on histomorphometric data and cellular interactions that occur in the bone microenvironment cited from the literature. Availability of bone substrate for osteoclastic activity during Phase I was assumed to be limited by the ratio of RANKL (ligand for receptor activator for nuclear factor kappaB) to osteoprotegerin (OPG) ('effective RANKL'). The required presence of marrow stromal cell produced macrophage-colony stimulating factor (M-CSF) for osteoclast action was represented as a factor equal to 1 for healthy bone. Growth factors released from the matrix during Phase I were assumed to cause two negative feedback effects: (1) the inhibitory effect of transforming growth factor-beta1 (TGFbeta1)-induced production of OPG by marrow osteoblast stromal cells, reducing effective RANKL; (2) the apoptosis of osteoclast nuclei assumed to occur at high concentrations of TGFbeta. This signaled the end of Phase I. During Phase II, cellular activity to remove the collagen fibrils left behind by osteoclasts was also simulated by Michaelis-Menten kinetic equations. Results of sensitivity analysis revealed variation in resorption depth and duration to fluctuate within 6% and 7% of the baseline value for changes in most input parameters. However, resorption depth was reduced and the duration of resorption lengthened by both a decrease in matrix TGFbeta and an increase the apoptotic threshold. Furthermore, the duration of resorption, but not erosion depth, was sensitive to changes in the maximum rate of cellular activity during removal of collagen fibrils. This mathematical model, which simulates the changing rates of cellular activity, has identified factors that reduce the duration and depth of resorption. It also suggests new targets for modeling therapeutic intervention to slow the rate of bone remodeling.