References¶

pkpdutils implements the standard methods of pharmacokinetic data analysis. These are the textbooks, guidances and publications behind them; cite them when you report an analysis, and cite pkpdutils itself as described in Home. Every user guide page cites the entries it builds on.

Textbooks¶

Gabrielsson & Weiner. The reference for the non-compartmental parameters and their interpretation.

Gabrielsson J, Weiner D. Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and Applications. 5^th edition. Swedish Pharmaceutical Press; 2016.

Rowland & Tozer. Clinical pharmacokinetics, the physiological meaning of clearance, volume and half-life.

Rowland M, Tozer TN. Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications. 4^th edition. Lippincott Williams & Wilkins; 2011.

Gibaldi & Perrier. The derivations of the area and moment methods.

Gibaldi M, Perrier D. Pharmacokinetics. 2^nd edition. Marcel Dekker; 1982.

Shargel & Yu. A general biopharmaceutics and pharmacokinetics textbook, parallel to Rowland & Tozer and Gibaldi & Perrier.

Ducharme MP, Shargel L, Yu ABC. Shargel and Yu's Applied Biopharmaceutics & Pharmacokinetics. 8^th edition. McGraw Hill; 2022. ISBN 978-1-260-14299-0.

Bonate. General pharmacokinetic-pharmacodynamic modeling and simulation, complementary to Gabrielsson & Weiner.

Bonate PL. Pharmacokinetic-Pharmacodynamic Modeling and Simulation. 2^nd edition. Springer; 2011. doi:10.1007/978-1-4419-9485-1

Non-compartmental analysis¶

Phoenix WinNonlin. The rules for the selection of the terminal phase (best fit by adjusted R²) and the linear-up/log-down trapezoidal rule follow the Phoenix NCA implementation. Its plasma models 200 to 202 (extravascular, intravenous bolus, infusion), the urine models 210 to 212 with the parameters of the excretion rate curve of Urinary excretion, and the standard errors of the sparse designs of Sparse sampling are the naming conventions of those pages.

Certara. Phoenix WinNonlin User's Guide: Noncompartmental Analysis. Certara USA, Inc.

Non-compartmental analysis review. The review-article companion to Gabrielsson & Weiner, an NCA methodology overview.

Gabrielsson J, Weiner D. Non-compartmental analysis. Methods in Molecular Biology. 2012;929:377-389. doi:10.1007/978-1-62703-050-2_16

AUC integration methods. The classic comparisons of numerical integration algorithms behind the trapezoidal rules of AUCMethod, and the direct justification of the linear-up/log-down rule.

Yeh KC, Kwan KC. A comparison of numerical integrating algorithms by trapezoidal, Lagrange, and spline approximation. Journal of Pharmacokinetics and Biopharmaceutics. 1978;6(1):79-98. doi:10.1007/BF01066064

Chiou WL. Critical evaluation of the potential error in pharmacokinetic studies of using the linear trapezoidal rule method for the calculation of the area under the plasma level-time curve. Journal of Pharmacokinetics and Biopharmaceutics. 1978;6(6):539-546. doi:10.1007/BF01062108

Purves RD. Optimum numerical integration methods for estimation of area-under-the-curve (AUC) and area-under-the-moment-curve (AUMC). Journal of Pharmacokinetics and Biopharmaceutics. 1992;20(3):211-226. doi:10.1007/BF01062525

Sparse and destructive sampling. The variance of an AUC estimated from group data with one time point per subject, its degrees of freedom, and the extension of both to a batch design in which a subject contributes to several time points.

Bailer AJ. Testing for the equality of area under the curves when using destructive measurement techniques. Journal of Pharmacokinetics and Biopharmaceutics. 1988;16(3):303-309. doi:10.1007/BF01062139

Nedelman JR, Gibiansky E, Lau DTW. Applying Bailer's method for AUC confidence intervals to sparse sampling. Pharmaceutical Research. 1995;12(1):124-128. doi:10.1023/A:1016255124336

Nedelman JR, Jia X. An extension of Satterthwaite's approximation applied to pharmacokinetics. Journal of Biopharmaceutical Statistics. 1998;8(2):317-328. doi:10.1080/10543409808835241

Holder DJ. Comments on Nedelman and Jia's extension of Satterthwaite's approximation applied to pharmacokinetics. Journal of Biopharmaceutical Statistics. 2001;11(1-2):75-79. doi:10.1081/BIP-100104199

Delta method vs. bootstrap for AUC ratios. A pharmacokinetics-specific comparison of confidence interval methods for an exposure metric.

Jaki T, Wolfsegger MJ, Ploner M. Confidence intervals for ratios of AUCs in the case of serial sampling: a comparison of seven methods. Pharmaceutical Statistics. 2009;8(1):12-24. doi:10.1002/pst.321

NonCompart. A comparable open-source, CDISC SDTM-oriented non-compartmental analysis implementation.

Bae KS. NonCompart: Noncompartmental Analysis for Pharmacokinetic Data. CRAN package. cran.r-project.org/package=NonCompart

NonCompart validation report. The published per-subject Phoenix WinNonlin results for the theophylline and the indomethacin dataset, the reference values of Validation.

Han S. Validation of Noncompartmental Analysis Performed by NonCompart R package. 2018. asancpt.github.io/NonCompart-tests

Theophylline dataset. The twelve subject oral single dose study distributed as datasets::Theoph of R, one of the two validation datasets.

Boeckmann AJ, Sheiner LB, Beal SL. NONMEM Users Guide: Part V. NONMEM Project Group, University of California, San Francisco; 1994.

Indomethacin dataset. The six subject intravenous bolus study distributed as datasets::Indometh of R, the second validation dataset.

Kwan KC, Breault GO, Umbenhauer ER, McMahon FG, Duggan DE. Kinetics of indomethacin absorption, elimination, and enterohepatic circulation in man. Journal of Pharmacokinetics and Biopharmaceutics. 1976;4(3):255-280. doi:10.1007/BF01063617

Regulatory guidance¶

FDA drug interaction guidance. The thresholds of the classification of inhibitors, inducers and sensitive substrates.

U.S. Food and Drug Administration. Clinical Drug Interaction Studies - Cytochrome P450 Enzyme- and Transporter-Mediated Drug Interactions. Guidance for Industry. 2020.

EMA drug interaction guideline.

European Medicines Agency. Guideline on the investigation of drug interactions. CPMP/EWP/560/95/Rev. 1. 2012.

FDA bioequivalence guidance. The 80-125 % acceptance range of the 90 % confidence interval of the geometric mean ratio.

U.S. Food and Drug Administration. Statistical Approaches to Establishing Bioequivalence. Guidance for Industry. 2026.

FDA bioavailability guidance. General bioavailability studies and AUC-based exposure endpoints.

U.S. Food and Drug Administration. Bioavailability Studies Submitted in NDAs or INDs - General Considerations. Guidance for Industry. 2022. fda.gov

FDA bioequivalence guidance for ANDAs. The Cmax/AUC bioequivalence acceptance criteria for generic drugs.

U.S. Food and Drug Administration. Bioequivalence Studies With Pharmacokinetic Endpoints for Drugs Submitted Under an ANDA. Guidance for Industry. 2026. fda.gov

EMA bioequivalence guideline. European bioequivalence acceptance criteria and design requirements.

European Medicines Agency. Guideline on the Investigation of Bioequivalence. CPMP/EWP/QWP/1401/98 Rev. 1. 2010. ema.europa.eu

FDA reference-scaled average bioequivalence. The product-specific guidance which defines the scaled criterion of a highly variable drug: the switching condition \(s_{wR} = 0.294\), the regulatory constant \(\sigma_{w0} = 0.25\), the upper 95 % bound of \((\mu_T - \mu_R)^2 - \theta\sigma_{wR}^2\) by Howe's approximation and the point estimate within 80.00-125.00 %.

U.S. Food and Drug Administration. Draft Guidance on Progesterone. 2011 (recommended Feb 2011, revised). accessdata.fda.gov

FDA narrow therapeutic index bioequivalence. The product-specific guidance which defines the narrow therapeutic index approach: the fully replicate four period design, \(\sigma_{w0} = 0.10\), \(\Delta = 1/0.9\), the unscaled interval within 80.00-125.00 % and the upper 90 % bound of \(s_{wT}/s_{wR}\) at most 2.500.

U.S. Food and Drug Administration. Draft Guidance on Warfarin Sodium. 2012 (recommended Dec 2012). accessdata.fda.gov

ICH M13A. The current harmonized (FDA/EMA/PMDA) bioequivalence design and analysis standard for immediate-release solid oral dosage forms.

International Council for Harmonisation. ICH Harmonised Guideline: Bioequivalence for Immediate-Release Solid Oral Dosage Forms M13A. 2024. database.ich.org

ICH S3A. The toxicokinetic guideline behind the sparse and microsampling designs of Sparse sampling.

International Council for Harmonisation. S3A Guideline: Note for Guidance on Toxicokinetics: The Assessment of Systemic Exposure in Toxicity Studies - Questions and Answers, Focus on Microsampling. 2017. database.ich.org

FDA population pharmacokinetics guidance. Context for group and batch analyses and the handoff from a non-compartmental to a population pharmacokinetic analysis.

U.S. Food and Drug Administration. Population Pharmacokinetics. Guidance for Industry. 2022. fda.gov

FDA exposure-response guidance. Regulatory context for exposure-response and pharmacodynamic modeling.

U.S. Food and Drug Administration. Exposure-Response Relationships - Study Design, Data Analysis, and Regulatory Applications. Guidance for Industry. 2003. fda.gov

ICH E9(R1). The conceptual framing of what a comparison or an effect estimate represents.

International Council for Harmonisation. ICH E9(R1) Addendum on Estimands and Sensitivity Analysis in Clinical Trials to the Guideline on Statistical Principles for Clinical Trials E9(R1). 2019. database.ich.org

FDA drug interaction table. The FDA's living reference tables of clinical index substrates, inhibitors and inducers.

U.S. Food and Drug Administration. Drug Development and Drug Interactions: Table of Substrates, Inhibitors and Inducers. fda.gov

Statistics¶

Two one-sided tests.

Schuirmann DJ. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics. 1987;15(6):657-680. doi:10.1007/BF01068419

Owen's Q function. The bivariate non-central t probability the exact power of the two one-sided tests is computed from.

Owen DB. A special case of a bivariate non-central t-distribution. Biometrika. 1965;52(3-4):437-446. doi:10.2307/2333696

Hodges-Lehmann estimator. The median of the Walsh averages and the distribution free confidence interval of the \(t_\mathrm{max}\) comparison.

Hodges JL, Lehmann EL. Estimates of location based on rank tests. The Annals of Mathematical Statistics. 1963;34(2):598-611. doi:10.1214/aoms/1177704172

Dose proportionality.

Smith BP, Vandenhende FR, DeSante KA, Farid NA, Welch PA, Callaghan JT, Forgue ST. Confidence interval criteria for assessment of dose proportionality. Pharmaceutical Research. 2000;17(10):1278-1283. doi:10.1023/A:1026451721686

Effect sizes. The small sample correction \(J\) of Hedges' g and its variance.

Hedges LV. Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics. 1981;6(2):107-128. doi:10.3102/10769986006002107

Random effects meta-analysis. The between-study variance \(\tau^2\) of the random effects model.

DerSimonian R, Laird N. Meta-analysis in clinical trials. Controlled Clinical Trials. 1986;7(3):177-188. doi:10.1016/0197-2456(86)90046-2

Crossover designs. The period-difference analysis of the 2x2 crossover, the tests of the period and the carryover effect and the within-subject CV.

Chow SC, Liu JP. Design and Analysis of Bioavailability and Bioequivalence Studies. 3^rd edition. Chapman & Hall/CRC; 2009.

Heterogeneity. \(I^2\) and \(H^2\) of a meta-analysis.

Higgins JPT, Thompson SG. Quantifying heterogeneity in a meta-analysis. Statistics in Medicine. 2002;21(11):1539-1558. doi:10.1002/sim.1186

Meta-analysis. The standard meta-analysis textbook, complementary to DerSimonian & Laird and Higgins & Thompson above.

Borenstein M, Hedges LV, Higgins JPT, Rothstein HR. Introduction to Meta-Analysis. 2^nd edition. Wiley; 2021. doi:10.1002/9781119558378

Multiple comparisons.

Holm S. A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics. 1979;6(2):65-70.

Benjamini Y, Hochberg Y. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society B. 1995;57(1):289-300. doi:10.1111/j.2517-6161.1995.tb02031.x

Agreement of two measurements. The Bland-Altman plot of plot_bland_altman.

Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. The Lancet. 1986;327(8476):307-310. doi:10.1016/S0140-6736(86)90837-8

Meta-analysis reference data. The BCG vaccine trials used as the regression reference of the meta-analysis (tests/data/reference/meta_bcg.json), analysed with the R package metafor.

Colditz GA, Brewer TF, Berkey CS, et al. Efficacy of BCG vaccine in the prevention of tuberculosis: meta-analysis of the published literature. JAMA. 1994;271(9):698-702.

Viechtbauer W. Conducting meta-analyses in R with the metafor package. Journal of Statistical Software. 2010;36(3):1-48. doi:10.18637/jss.v036.i03

Bootstrap. The parametric bootstrap and the delta method of the uncertainty of the NCA parameters (ch. 5 and 6) and the residual bootstrap of a fit (ch. 9).

Efron B, Tibshirani RJ. An Introduction to the Bootstrap. Chapman & Hall/CRC; 1993.

Nonlinear regression. The covariance of the parameters from the Jacobian, the t based confidence intervals and the delta method for derived parameters.

Seber GAF, Wild CJ. Nonlinear Regression. Wiley; 1989.

Model selection by AICc and Akaike weights. The ranking of the candidate models of a fit and the correction for the small sample size.

Burnham KP, Anderson DR. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. 2^nd edition. Springer; 2002.

Drug-drug interaction study design. The industry perspective on study design and classification that FDA's drug interaction guidance later formalized.

Bjornsson TD, Callaghan JT, Einolf HJ, et al. The conduct of in vitro and in vivo drug-drug interaction studies: a Pharmaceutical Research and Manufacturers of America (PhRMA) perspective. Journal of Clinical Pharmacology. 2003;43(5):443-469. doi:10.1177/0091270003252519

Draper and Smith. The confidence band of a simple linear regression, drawn around the terminal regression of the NCA figure.

Draper NR, Smith H. Applied Regression Analysis. 3^rd edition. Wiley; 1998. doi:10.1002/9781118625590

Data formats¶

The exchange formats of pkpdutils.io: the event records of NONMEM and Monolix, the two tables of PKNCA and the CDISC ADaM dataset of a non-compartmental analysis.

NONMEM event records. The one row per event data format, EVID, MDV, AMT, RATE, and the repeated doses of ADDL, II and SS.

Bauer RJ. NONMEM Tutorial Part I: Description of Commands and Options, with Simple Examples of Population Analysis. CPT: Pharmacometrics & Systems Pharmacology. 2019;8(8):525-537. doi:10.1002/psp4.12404

Monolix data format. The column names of the same format in the MonolixSuite (AMOUNT, OBSERVATION, INFUSION DURATION, ADDITIONAL DOSES, INTERDOSE INTERVAL, STEADY STATE).

Lixoft. MonolixSuite documentation: data format. monolix.lixoft.com/data-format

PKNCA. The two table layout of the concentrations and the doses of the R package for automatic non-compartmental analysis.

Denney W, Duvvuri S, Buckeridge C. Simple, automatic noncompartmental analysis: the PKNCA R package. Journal of Pharmacokinetics and Pharmacodynamics. 2015;42:S65. cran.r-project.org/package=PKNCA

CDISC ADaM ADNCA. The analysis dataset of the input data of a non-compartmental analysis (USUBJID, PARAMCD, AVAL, AFRLT, ARRLT, DOSEA, DTYPE).

CDISC. ADaM Implementation Guide for Non-compartmental Analysis Input Data (ADNCA). 2021. cdisc.org/standards/foundational/adam/adamig-non-compartmental-analysis-input-data-v1-0

CDISC. Analysis Data Model (ADaM) Implementation Guide. cdisc.org/standards/foundational/adam

Software¶

PowerTOST. The reference implementation of the power and the sample size of the two one-sided tests, whose design constants, degrees of freedom and published examples pkpdutils.stats.power reproduces.

Labes D, Schütz H, Lang B. PowerTOST: Power and Sample Size for (Bio)Equivalence Studies. CRAN package. cran.r-project.org/package=PowerTOST

pint, xarray, scipy. The libraries the package is built on.

Hoyer S, Hamman J. xarray: N-D labeled arrays and datasets in Python. Journal of Open Research Software. 2017;5(1):10. doi:10.5334/jors.148

Virtanen P, Gommers R, Oliphant TE, et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods. 2020;17:261-272. doi:10.1038/s41592-019-0686-2

Further reading¶

• PKanalix documentation - Lixoft/Simulations Plus's documentation for PKanalix, a GUI application for non-compartmental and compartmental PK analysis covering NCA rules, custom parameters, bioequivalence and regulatory reporting, for analysts who want a graphical cross-check to code-based NCA.
• PKNCA package documentation site - the official documentation and vignette site for the PKNCA R package, with worked examples, AUC/half-life method articles, sparse sampling and bioequivalence vignettes, for analysts comparing pkpdutils's NCA conventions against PKNCA's.
• PKNCA - theophylline vignette - the worked pk.nca example on datasets::Theoph whose printed results are the PKNCA reference values of Validation.
• PKNCA - FDA-oriented introduction vignette - frames PKNCA's design goals (regulatory readiness, reproducibility, CDISC-aligned data structures) for a regulatory audience, for analysts preparing regulatory NCA submissions.
• PKNCA training session vignette - a step-by-step walkthrough of an NCA workflow in R, for analysts new to R-based NCA.
• Pumas - handling missing and BLQ data - the BLQ conventions of the Pumas NCA (:first, :middle, :last with :keep, :drop and numeric imputation), the positional axis BLQRules implements, for analysts porting an analysis between the two.
• NonCompart on CRAN - an alternative open-source, CDISC SDTM-oriented NCA implementation in R with automatic/manual slope selection and multiple trapezoidal methods, for readers comparing implementations of the same NCA rules pkpdutils implements.
• CDISC "Introduction to PK Analysis" course - CDISC's on-demand course introducing PK analysis concepts alongside CDISC data standards, for clinical data and statistical programmers who need the CDISC ADaM/ADNCA context that io.py's read_adnca targets.
• CDISC ADaMIG for Non-compartmental Analysis Input Data v1.0 - the implementation guide page itself, useful for programmers building ADNCA-compliant datasets, in addition to being cited above as the formal CDISC ADaM ADNCA reference.
• FDA "Drug Development and Drug Interactions" table - the FDA's living reference tables of clinical index substrates, inhibitors and inducers with strong/moderate/weak classification, the practical companion to stats/ddi.py's DDIThresholds.
• Holford NHG - Advanced Pharmacometrics teaching page - Nick Holford's pharmacometrics course materials at the University of Auckland, covering PK/PD modeling concepts beyond NCA, for readers wanting the population-modeling perspective on the same parameters pkpdutils computes non-compartmentally.
• Mould & Upton, "Basic concepts in population modeling, simulation, and model-based drug development." CPT:PSP. 2012;1(9):e6 - part 1 of a three-part introductory tutorial series for pharmacometrics newcomers.
• Mould & Upton, "...part 2: introduction to pharmacokinetic modeling methods." CPT:PSP. 2013;2(4):e38 - part 2, focused on PK modeling methods, a natural next step after this library's NCA and fitting docs.
• Upton & Mould, "...part 3: introduction to pharmacodynamic modeling methods." CPT:PSP. 2014;3:e88 - part 3, focused on PD modeling methods, a companion to Pharmacodynamics.
• PKNCA GitHub repository - the source repository, for readers who want to compare pkpdutils's NCA implementation choices against PKNCA's source directly.