where OCL represents the number of osteoclasts, OBL represents the number of osteoblasts, g 11 represents autocrine (osteoclast to osteoclast) signaling action, g 21 represents paracrine (osteoblast to osteoclast) signaling action, and α OCL and β OCL represent proliferation and degradation rate constants, respectively. The proliferation of osteoclasts (j = 1) depends on autocrine from osteoclasts (i = 1) and paracrine from osteoblasts (i = 2) signaling effects. The degradation rate of osteoclasts is assumed to be proportional to the current population.