Model-fitting and Prediction of Bone Mineral Density in Postmenopausal Women Using an Exponential-type Nonlinear Mixed-effect Model.

Abstract

An exponential-type nonlinear mixed-effect model was fitted to estimate and predict bone mineral density (BMD) at three sites -the lumbar spine, femoral neck, and radius -in 89 postmenopausal women. Years since menopause (YSM) and age at menopause (AAM) were used as explanatory variables. The model was assumed as (BMD)ij=α+ai+(β+bi)exp(γ· (YSM)ij) + δ ((AAM)i -sigma) + eij (i: subject; j: time point; α,β,γ and δ are the population means; ai and bi are the random effects among subjects; eij are random error). Because BMD decreases rapidly after menopause and changes slowly thereafter, an exponential model was assumed. The fitted models were as follows: lumbar spine: (BMD)ij=0.792+ai+(0.179+bi)exp(-0.185 x (YSM)ij)-0.00251((AAM)i -50.6), sigma = (0.106)2, sigma =(0.0442)2, sigmaab = 0.0000720, sigma = (0.0185)2 ,femoral neck: (BMD)ij =0.594+ai+(0.161+bi)exp(-0.108 x (YSM)ij)-0.00605((AAM)i -50.6), sigma=(0.108)2, sigma = (0.108)2, sigmaab=-0.00619, sigma = (0.0165)2 ,radius: (BMD)ij = 0.458 +ai+ 0.152+bi)exp(-0.0885 x (YSM)ij)-0.00248((AAM)i -50.6) , sigma = (0.108)2 , sigma = (0.106)2, sigmaab=-0.00987, sigma =(0.0115)2. The maximum likelihood model was done with a specific algorithm which took account of highly unbalanced data. These formulae were also used to derive a prediction formula for each individual using a Bayesian approach. This prediction formula with individual longitudinal data makes it possible to predict the future BMD and to estimate the risk of osteoporosis more accurately in individual subjects than the prediction model in cross sectional data.