A negative feedback model for a mechanism based description of longitudinal observations. Application for bone turnover biomarkers

In modern medicine the diagnosis and prognosis of an abnormal metabolic condition is based on blood borne measurements involving one or more biomarker. This paper reports the development of a minimal negative feedback model for the description of longitudinal biomarkers concentrations for treatment of osteoporosis in postmenopausal women. Literature data were obtained from double-blind, placebo-controlled clinical trial over three years. There were four treatment groups: 1) Placebo, 2) Alendronate, 3) Conjugated Estrogen, and/or 4) Combination therapy. The negative feedback model consists of a biomarker and a companion controller. By considering the above basal biomarker values it is shown that the dynamics can be described by a second order differential equation without the involvement of biomarker production rate. The second order differential equation is also analogous to classical negative feedback servomechanism model with two parameters ω(n) and ξ. It was assumed that the rate constants defining the negative feedback model were equal which would set ξ to 0.707 with only ω(n) to be estimated. ω(n) was estimated for both lumbar spine bone mineral density (BMD) and bone-specific alkaline phosphatase (BAP) in four treatments groups. The t(½) of BMD and BAP were estimated at 26.8 (0.30) and 9.4 (0.30) days respectively. The negative feedback model of BMD supports the mechanism whereby Conjugated Estrogen and Alendronate decrease the clearance rate constant of BMD analogous to increased apoptosis of osteoclasts. The linked negative feedback models facilitate a mechanism based prediction of BMD using the concentrations of the bone turnover marker BAP.