{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T03:04:06Z","timestamp":1789700646717,"version":"4.0.1"},"reference-count":25,"publisher":"Wiley","issue":"9","license":[{"start":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T00:00:00Z","timestamp":1788307200000},"content-version":"vor","delay-in-days":1,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["ascpt.onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["CPT Pharmacom &amp; Syst Pharma"],"published-print":{"date-parts":[[2026,9]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>Maximum likelihood estimation in population pharmacokinetic\/pharmacodynamic (PK\/PD) nonlinear mixed\u2010effects (NLME) models targets fixed\u2010effect, interindividual\u2010variability, and residual\u2010variability parameters through a marginal likelihood that integrates each subject's contribution over individual random effects. Because these subject\u2010level integrals are usually unavailable analytically, Laplace estimation approximates each contribution using a second\u2010order 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\u2010size sensitive. We developed automatic differentiation (AD)\u2010based methods for Laplace NLME estimation and compared three ways to account for EBE sensitivities during updates: FULL\u2010implicit uses the EBE mode equations, FULL\u2010unroll 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\u2010compartment PK example with absorption\/elimination ambiguity, FULL\u2010implicit, FULL\u2010unroll, and FD matched the lowest OFV within  OFV units, whereas STOP had maximum  relative to . Median wall times were 0.045, 0.108, 0.068, and 0.758\u2009s for FULL\u2010implicit, FULL\u2010unroll, STOP, and FD, respectively, making FULL\u2010implicit about 17\u2010fold faster than FD. In 10 runs on public warfarin PK\/PD data using the ODE representation, FULL\u2010implicit achieved a lower best OFV than FD (1624.02 vs. 1628.60) and was 36\u2010fold faster by median wall time (0.45 vs. 16.3\u2009min). In both scenarios, FULL\u2010implicit provided a faster AD\u2010based alternative to FD while reaching comparable or lower OFVs.<\/jats:p>","DOI":"10.1002\/psp4.70333","type":"journal-article","created":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T07:58:00Z","timestamp":1788335880000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adjoint and Unrolled Automatic Differentiation for Laplace\u2010Approximated Likelihoods in Population\n                    <scp>PK<\/scp>\n                    and\n                    <scp>PK<\/scp>\n                    \/\n                    <scp>PD<\/scp>\n                    Models"],"prefix":"10.1002","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-0032-228X","authenticated-orcid":false,"given":"Guido H.","family":"Jajamovich","sequence":"first","affiliation":[{"name":"Clinical Pharmacology Modeling and Simulation Amgen  South San Francisco California USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6683-4647","authenticated-orcid":false,"given":"William","family":"Holmes","sequence":"additional","affiliation":[{"name":"Clinical Pharmacology Modeling and Simulation Amgen  South San Francisco California USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-3126-3981","authenticated-orcid":false,"given":"Chih\u2010Wei","family":"Lin","sequence":"additional","affiliation":[{"name":"Clinical Pharmacology Modeling and Simulation Amgen  Thousand Oaks California USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4494-7098","authenticated-orcid":false,"given":"Khamir","family":"Mehta","sequence":"additional","affiliation":[{"name":"Clinical Pharmacology Modeling and Simulation Amgen  South San Francisco California USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,9,2]]},"reference":[{"key":"e_1_2_10_2_1","volume-title":"NONMEM Users Guides","author":"Beal S.","year":"1992"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2004.07.002"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1002\/psp4.12422"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10928\u2010015\u20109409\u20101"},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1208\/s12248\u2010009\u20109112\u20105"},{"key":"e_1_2_10_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718027"},{"key":"e_1_2_10_8_1","first-page":"1","article-title":"Automatic Differentiation in Machine Learning: A Survey","volume":"18","author":"Baydin A. G.","year":"2018","journal-title":"Journal of Machine Learning Research"},{"key":"e_1_2_10_9_1","first-page":"318","volume-title":"Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics (AISTATS)","author":"Domke J.","year":"2012"},{"key":"e_1_2_10_10_1","first-page":"2113","volume-title":"Proceedings of the 32nd International Conference on Machine Learning (ICML)","author":"Maclaurin D.","year":"2015"},{"key":"e_1_2_10_11_1","doi-asserted-by":"publisher","DOI":"10.1109\/HPEC49654.2021.9622796"},{"key":"e_1_2_10_12_1","first-page":"6571","article-title":"Neural Ordinary Differential Equations","volume":"31","author":"Chen R. T. Q.","year":"2018","journal-title":"Advances in Neural Information Processing Systems (NeurIPS 2018)"},{"key":"e_1_2_10_13_1","first-page":"9086","article-title":"Hamiltonian Monte Carlo Using an Adjoint\u2010Differentiated Laplace Approximation","volume":"33","author":"Margossian C. C.","year":"2020","journal-title":"Advances in Neural Information Processing Systems (NeurIPS 2020)"},{"key":"e_1_2_10_14_1","doi-asserted-by":"publisher","DOI":"10.52202\/068431\u20100378"},{"key":"e_1_2_10_15_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10928\u2010025\u201009997\u20100"},{"key":"e_1_2_10_16_1","unstructured":"K.Nieforth R.Chen M.Sale et\u00a0al. \u201cADPO: Automatic\u2010Differentiation\u2010Assisted Parametric Optimization Algorithm (PAGE 2025 Poster\/Abstract) \u201dPopulation Approach Group in Europe (PAGE) (2025)."},{"key":"e_1_2_10_17_1","doi-asserted-by":"crossref","unstructured":"C.Rackauckas Y.Ma A.Noack et\u00a0al. \u201cAccelerated Predictive Healthcare Analytics With Pumas a High\u2010Performance Pharmaceutical Modeling and Simulation Platform \u201dbioRxiv: The Preprint Server for Biology (2020) https:\/\/doi.org\/10.1101\/2020.11.28.402297.","DOI":"10.1101\/2020.11.28.402297"},{"key":"e_1_2_10_18_1","unstructured":"Pumas\u2010AI \u201cPumas Documentation \u201dPumas\u2010AI https:\/\/docs.pumas.ai\/stable\/."},{"key":"e_1_2_10_19_1","unstructured":"Z.Tan A.Chaturvedula J.Desrochers K. R.Bergmann Y. K.Lyauk andR. V.Overgaard \u201cComparison of Pumas and NONMEM in Parameter Estimation of a Complex Pharmacodynamic Model \u201d(2023)."},{"key":"e_1_2_10_20_1","doi-asserted-by":"publisher","DOI":"10.4155\/tde.11.19"},{"key":"e_1_2_10_21_1","doi-asserted-by":"publisher","DOI":"10.1172\/JCI104839"},{"key":"e_1_2_10_22_1","doi-asserted-by":"publisher","DOI":"10.1161\/01.CIR.38.1.169"},{"key":"e_1_2_10_23_1","unstructured":"N.Holford \u201cWarfarin PK\/PD Workshop Materials and Dataset (Public) \u201dUniversity of Auckland https:\/\/holford.fmhs.auckland.ac.nz\/research\/nlmixr\/warfarin\/warfarin_dat.csv."},{"key":"e_1_2_10_24_1","volume-title":"The Matrix Cookbook","author":"Petersen K. 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