A Negative feedback model for a mechanism based description of longitudinal observations: application for bone turnover biomarkers.
Massoud A. Boroujerdi (1), Stephan Schmidt (1), Oscar E. Della Pasqua (1,2) Meindert Danhof (1)
(1) Division of Pharmacology, Leiden-Amsterdam Center for Drug Research, Leiden, The Netherlands, ( 2) Clinical Pharmacology Modelling & Simulation, GlaxoSmithKline, Stockley Park, UK
Objectives: In modern medicine the initial diagnosis of an abnormal metabolic condition is based on blood borne measurements often involving multiple biomarkers. The biomarkers concentrations themselves would not be sufficient for providing a mechanism based description of mode of action. Estimates of clearance rates are the minimum requirement for a model based description of longitudinal observations.
Methods: A minimal negative feedback model is proposed for the description of longitudinal observations in clinical trials for treatment of osteoporosis in postmenopausal women. The negative feedback model is consisting of blood biomarker and a companion latent controller. By considering the ab ove basal biomarkers values it is shown that the kinetics can be described by a second order differential equation without the involvement of biomarkers production rates. The second order differential equation is also analogous to classical servomechanism model with two parameters ωn and ζ, the position being the above basal concentration. With the assumption that the rate constants defining the negative feedback model being equal ζ would be 0.707 with only ωn to be estimated. The parameters of the servomechanism model namely ωn was estimated for each biomarker providing values for their clearance rates [1].
Results: The parameter of the negative feedback model was estimated for both lumbar spine bone mineral density and bone-specific alkaline phosphatase for the mean observations in f our treatments groups over three years. The third year of treatment was a switch over among therapies (1) Placebo, (2) Aldendronate, (3) Conjugated Estrogen (4) Combination therapy.
Conclusions: The clearance rate of bone-specific alkaline phosphatase was higher than the bone mineral density in all treatment groups. The linkage of the models indicate that the changes in bone mineral density modulates the feedback flux rate constant of bone-specific alkaline phosphatase with the modulation coefficient being slower in the switch over year from Alendronate to other therapies [2].
References:
[1] Ackerman, E., Rosevear, J. W., McGuckin, W. F. A Mathematical Model of the Glucose-tolerance test. phys med biol 9:203-213; 1964.
[2] Greenspan, S. L., Emkey, R. D., Bone, H. G., Weiss, S. R., Bell, N. H., Downs, R. W., McKeever, C., Miller, S. S., Davidson, M., Bolognese, M. A., Mulloy, A. L., Heyden, N., Wu, M., Kaur, A., and Lombardi, A. Significant differential effects of alendronate, estrogen, or combination therapy on the rate of bone loss after discontinuation of treatment of postmenopausal osteoporosis. A randomized, double-blind, placebo-controlled trial. Ann Intern Med 137:875-83; 2002.
