Using the Lasso to simultaneously identify the covariate and variance-covariance structures of nonlinear mixed-effects models
Department of Anesthesiology, Leiden University Medical Center, The Netherlands
Objectives: Remifentanil is a synthetic opioid with ultrafast elimination. Its pharmacokinetics after a one-minute infusion in children (age range 5 days - 17 yr) have been described by Kinder Ross et al. using model-independent analysis . The first objective of the present study was to obtain a compartmental model of remifentanil pharmacokinetics with the possible inclusion of covariates age and weight.
The Lasso has been suggested as a method to find the covariate coefficients [2,3]. Interindividual variabilities of the structural pharmacokinetic parameters with possibly a smaller number of random effects can be described by a (not necessarily square) strength matrix transforming the random effects . The covariate and strength matrix coefficients are likely to be interdependent. Therefore, the second objective of this study was to explore the ability of the Lasso to simultaneously find optimal covariate and variance-covariance coefficients.
Methods: The remifentanil concentration data of study  were kindly provided by GlaxoSmithKline. Because of the fast elimination, concentrations were below the lower limit of quantitation within 30 minutes; 196 samples were available in 36 children. Two-compartment models were fit to the data using NONMEM (FOCE, SIGDIGITS=4); parameters (V1, K10, K12, and K21) were given by a population value times an exponential function of coefficients times the covariates and random effects. Age (plus 280 days) and weight were log-transformed and normalized to have mean zero and variance one (denoted by NLA and NLW respectively); the variances of random effects were also set to one.
A received model was identified by first finding a standard matrix Omega of the random effects and subsequently finding covariate coefficients significantly different from zero via backward elimination. Akaike's information-theoretic criterion (AIC) was used for model discrimination.
Penalized likelihood adds a penalty to the minus two log-likelihood, of the form lambda times the sum of absolute values of coefficients. In the model incorporating strength matrix A, there were a total of 24 coefficients: 8 for covariates and 16 for A; the coefficients of A are not identifiable without the penalty term. At a certain lambda, the coefficients different from zero (threshold 0.0001) were identified; subsequently, maximum likelihood estimates of those coefficients and the structural parameters were obtained and AIC computed. Using a grid search (lambda was varied between 0.1 and 100 in 300 increments) the value of lambda was located that identified the best model in terms of AIC.
Simulations were performed to assess the influence of the covariates and random effects (posterior predictive check) on the pharmacokinetic profile.
Results: Age and weight were (obviously) highly correlated - normalized weight (NLW) was well predicted by normalized age (NLA); therefore NLW and NLW-NLA were incorporated as covariates.
The received model consisted of a diagonal matrix Omega with non-zero elements for V1 and K10; four covariate coefficients were different from zero. The AIC of this model was 395.2. The model obtained via the Lasso consisted of six coefficients of the strength matrix using two random effects (from which a full (co)variance matrix of interindividual variabilities of the structural parameters can be constructed), and six covariate coefficients. The AIC of this model was 390.2.
The optimal model was found at 10 of the 300 evaluated values of lambda. 161 NONMEM runs were successful; 139 ended with rounding errors. AIC as a function of lambda displayed multiple local minima, and the number of parameters as a function of lambda was not monotonically decreasing.
Prediction intervals obtained from the optimal model were smaller than those from the received model, except during a few minutes after infusion, and after 30 min (extrapolating after the study period). Simulated concentrations from the optimal model, after administration of a dose relative to weight, were monotonically increasing with age; this was not so for simulations from the received model: it (most probably erroneously) displayed lowest concentrations at about 1 yr of age.
Conclusions: The Lasso was capable of simultaneously estimating covariate and random effects coefficients. While an exhaustive search of all combinations of 0-fixed and free coefficients is not feasible, both the methods of forward/backward selection and the Lasso are not guaranteed to find the optimal model parameter values. The fact that AIC as a (non-continuous) function of lambda displayed multiple local minima requires the model to be evaluated at many values of lambda. With the present data the Lasso, with respect to the standard approach, provided a better description of interindividual variability and the influence of covariates.
 Kinder Ross, A, et al., Pharmacokinetics of remifentanil in anesthetized pediatric patients undergoing elective surgery or diagnostic procedures, Anesth Analg; 93:1393, 2001.
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 Ribbing, J, et al., The LASSO: A novel method for predictive covariate model building in nonlinear mixed effects models, PAGE meetings 2006 (Abstract 936) and 2005 (Abstract 713).
 Olofsen E, et al., Population pharmacokinetics/pharmacodynamics of anesthetics, AAPS Journal 7, Article 39, 2005.