where OCL is the osteoclast population, pOCL is the preosteoclast population, π X 1 act is the activation from signaling factor X 1 , π X 2 act is the activation from signaling factor X 2 , and α pOCL and β OCL represent differentiation and apoptosis rate constants, respectively. In , differentiation and apoptosis terms are both activated and not inhibited. X 1 is RANKL, and X 2 is TGF-β. Different combinations of activating and repressing π terms are proposed in models from other publications.