Adjoint and Unrolled Automatic Differentiation for Laplace-Approximated Likelihoods in Population PK and PK/PD Models

Maximum likelihood estimation in population pharmacokinetic/pharmacodynamic (PK/PD) nonlinear mixed-effects (NLME) models targets fixed-effect, interindividual-variability, and residual-variability parameters through a marginal likelihood that integrates each subject's contribution over individual random effects. Because these subject-level integrals are usually unavailable analytically, Laplace estimation approximates each contribution using a second-order Taylor expansion around the empirical Bayes estimate (EBE; conditional posterior mode of the individual's random effects), making EBE estimation and curvature evaluation recurring tasks during objective function value (OFV) and derivative evaluation. Finite differences (FD) are commonly used for these derivative calculations but are costly and step-size sensitive. We developed automatic differentiation (AD)-based methods for Laplace NLME estimation and compared three ways to account for EBE sensitivities during updates: FULL-implicit uses the EBE mode equations, FULL-unroll differentiates through the Newton steps used to find EBEs, and STOP omits EBE sensitivity during outer differentiation. In 100 matched starts for a synthetic one-compartment PK example with absorption/elimination ambiguity, FULL-implicit, FULL-unroll, and FD matched the lowest OFV within ${10}^{-4}$ OFV units, whereas STOP had maximum $\text{ΔOFV}=1.9$ relative to ${\mathrm{OFV}}_{\min }=-1594.62$ . Median wall times were 0.045, 0.108, 0.068, and 0.758 s for FULL-implicit, FULL-unroll, STOP, and FD, respectively, making FULL-implicit about 17-fold faster than FD. In 10 runs on public warfarin PK/PD data using the ODE representation, FULL-implicit achieved a lower best OFV than FD (1624.02 vs. 1628.60) and was 36-fold faster by median wall time (0.45 vs. 16.3 min). In both scenarios, FULL-implicit provided a faster AD-based alternative to FD while reaching comparable or lower OFVs.