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## Introduction to Population Analysis

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**Introduction to Population Analysis**Joga Gobburu Pharmacometrics Office of Clinical Pharmacology Food and Drug Administration Introduction to Population Analysis**Pharmacometrics Training**Introduction to Population Analysis**Agenda**• Introduction to population PK-PD • Application of population PK-PD in drug development and regulatory decision making • Pharmacometrics @ FDA • Introduction to population modeling • Linear and nonlinear regression • Introduction to mixed effects modeling • Mixed effects modeling applied to population PK • Different methods of analysis • Bayesian theory • Maximum likelihood • Sources of variability • Variance (Error) models Introduction to Population Analysis**Agenda**• Introduction to population PK-PD • Application of population PK-PD in drug development and regulatory decision making • Pharmacometrics @ FDA • Introduction to population modeling • Linear and nonlinear regression • Introduction to mixed effects modeling • Mixed effects modeling applied to population PK • Different methods of analysis • Bayesian theory • Maximum likelihood • Sources of variability • Variance (Error) models Introduction to Population Analysis**Definition: Modeling**Mathematical (conceptual) modeling is describing a physical phenomenon by logical principles characterized with quantitative relationships, e.g., formulas, whose parameters may be measured (or experimentally determined) http://www.hawcc.hawaii.edu/math/Courses/Math100/Chapter0/Glossary/Glossary.htm Introduction to Population Analysis**Uses of Models**Yates FE (1975) On the mathematical modeling of biological systems: a qualified “pro”, in Physiological Adaptation to the Environment (Vernberg FJ ed), Intext Educational Publishers, New York. • Conceptualize the system • Codify current facts • Test competing hypotheses • Identify controlling factors • Estimate inaccessible system variables • Predict system response under new conditions Introduction to Population Analysis**DV=Dependent variable**IDV=Independent variables P=Parameters IDV DV P DV? Model and its parts • Parametric or Mechanistic model parameters reflect biological processes • Non-parametric or empiric model parameters do NOT reflect biological processes • Deterministic models do not account for variability • Stochastic models account for variability Whether a quantity is DV, IDV or P depends on the context Introduction to Population Analysis**Model and its parts**Structural Model Structural Model (covariate) Stochastic Model (BSV) Stochastic Model (Residual Var) Introduction to Population Analysis**Variability versus Uncertainty**Confidence Interval Confidence Interval (Lower CI, Mean, Upper CI) (Lower CI, Variance, Upper CI) Point estimate Point estimate Confidence interval is a measure of the uncertainty on the point estimate. We obtain point estimates of both population means and variances. Introduction to Population Analysis**Between Subject Variability**Residual Variability (Individual-Pop Mean CL,V) - Pred-Obs Conc + 0 - + 0 Mixed-effects concept ith patient ij i (CL,i & V,i) Pop Avg Between-occasion variability = zero Introduction to Population Analysis**Mixed-effects concept**Fixed effects Fixed effects Fixed effects Random effects Random effects Random effects Introduction to Population Analysis**Types of data**• Continuous • A variable can take any value (physically possible). • E.g.: concentrations, time, dose, glucose levels • Discrete • A variable can take one of many pre-specified values • Binary, ordinal • Binary – Yes or No type response (e.g.: death, pain/no pain) • Ordinal – Graded response (e.g.: mild/severe pain, minor/major bleeding) • Frequency – how often does the event occur? • E.g.: seizures, vomiting • Time to event – when does the event occur? • E.g.: time to death, time to MI Introduction to Population Analysis**PKPD Data**• Experimental • Rich data are collected under controlled conditions, usually small • Best data for building structural models • Example: Dose-proportionality • Observational • Sparse data are collected under ‘real’ life conditions, usually large • Best data for building statistical models • Example: Pivotal or registration trials Introduction to Population Analysis**DV**IDV DV IDV Linear versus Nonlinear models • Whether a model is linear or nonlinear will need to be determined relative to the parameters NOT the variables. For example: • Which of the two is linear? • DV = a·IDV • DV = a·IDV + b·IDV2 • Linear models • Partial derivative of DV w.r.t parameters is independent of parameters • Estimate parameters using linear regression • Nonlinear models • Partial derivative of DV w.r.t parameters is NOT independent of parameters • Estimate parameters using non-linear regression Introduction to Population Analysis**DV**IDV Objective Function Estimation via optimization • Linear regression: Goal is to find a line that goes as close to the observations as possible. • Comment on the goodness-of-fit of red, blue and black lines shown on the right. • Linear models can be analytically solved for intercept and slope estimates. Ideal value of the SSR is zero Introduction to Population Analysis**Objective**Function Estimation via optimization • Nonlinear models do not have analytical solutions, so we need to solve them numerically. Obj Fn Maximum Likelihood Estimate 0 CL Introduction to Population Analysis**Maximum Likelihood Estimation**• Non-linear mixed effects model • Likelihood for individual i Introduction to Population Analysis**Technical goals of Population analyses**• Estimate population mean and variance • Population mean CL, V • Between subject variability of CL, V • Residual variability of concentrations • Explain between subject variability using patient covariates such as body size, age, organ function • Estimate individual CL and V to impute concentrations to perform PKPD analyses • Sometimes PK and PD measurements are not performed at the same time • PD change could be delayed from PK • Modeling PD using differential equations mandates a functional form (model) for PK Introduction to Population Analysis**Population mean versus Typical value**• Population mean is the naïve overall mean of a parameter • For example, the population mean CL is 10 L/h. • When there are influential covariates that explain meaningful variability in PK parameters, then Typical value is the mean of a group of similar subjects. • For example, the typical value of CL for a 70 kg subject is 10 L/h. Similarly, for a 35 kg it is 5 L/h. Introduction to Population Analysis**Methods of Population analyses**• Naïve averaged • Naïve pooled • Two-Stage • One-Stage Introduction to Population Analysis**Cp**Cp Time Naïve Averaged • Average concentration at each time point is calculated using all subjects’ observed concentrations. • Average calculation does not take into the number of observations at each time point are equal or not; also subjects’ characteristics (heavy/light) are not considered – hence called ‘naïve’. • Average time course of concentrations is then modelled to obtain naïve average PK parameters. Time Introduction to Population Analysis**Cp**Cp Time Time Naïve Pooled • Individual observations from each subject are ‘pooled’ to obtain average PK parameters. • Estimation does not take into the number of observations at each time point are equal or not; also subjects’ characteristics (heavy/light) are not considered – hence called ‘naïve’. Introduction to Population Analysis**Cp**Cp Time Cp Time Time Two-Stage • Individual observations from each subject are modelled separately to obtain average PK parameters for each subject. • Uncertainty in individual parameter estimates is ignored. • Each subject’s covariates and PK parameters are correlated to explain BSV. • Population mean (or typical value) and variance are calculated. Introduction to Population Analysis**Cp**Time Two-Stage Uncertainty in individual parameter estimates is ignored. Cp Time Which subject’s PK parameters are estimated with more certainty - Red or Blue? Say, CL = 10±5 L/h and 10±1 L/h. When calculating the mean only the point estimate is considered, the two-stage analysis does not account for the different uncertainty Introduction to Population Analysis**Cp**Cp Cp Time Time Time One-Stage • Data from all subjects are simultaneously modeled. Population mean and variance are estimated simultaneously, including covariate modeling. • Individual subject’s PK parameters are calculated subsequent to ‘one-stage’ estimation. There is no model ‘optimization’ in this step – hence called ‘post-hoc’ step. Introduction to Population Analysis**Methods of Population analyses**Introduction to Population Analysis**Bayes Theorem**Future = Past ·Present Posterior = Prior · Likelihood P( ) Probability Model Parameter y Data Introduction to Population Analysis**-**+ 0 Posterior Bayes Theorem – Uninformative Prior - - + + 0 0 Current Prior Introduction to Population Analysis**-**+ 0 Posterior Bayes Theorem – Informative Prior - - + + 0 0 Current Prior Introduction to Population Analysis**Bayes Theorem – One-Stage analysis**• ML estimation (such as that in NONMEM) uses an empirical approach in obtaining the individual PK estimates. It uses the maximum likelihood estimates (population parameters: mean and variance) as PRIOR and the individual observations as LIKELIHOOD (CURRENT) to calculate POSTERIOR. For this reason, these individual estimates are called – ‘post hoc’, ‘empiric bayesian’ estimates. • According to pure Bayesian estimation, POSTERIOR is a distribution. ML only estimates the MODE (central tendency) of that POSTERIOR distribution. Newer versions of NONMEM are able to estimate the POSTERIOR distribution (never used it myself). WinBUGS is a full fledged bayesian estimation program. Introduction to Population Analysis**Bayes Theorem – One-Stage analysis**Posterior = Prior · Likelihood Individual ‘post-hoc’ Parameters Population Parameters Individual Observations Indv estimates close to indv Pop Estimates Rich obs/subject Indv estimates close to pop Pop Estimates Few obs/subject Shrinkage Introduction to Population Analysis**Sources of ‘random’ variability**• Between subject variability (BSV) • Signifies deviance among subjects • For example, CL varies between two ‘clones’ • Between occasion variability (BOV) • Signifies deviance between occasions within a subject • For example, CL varies between day 1 and 14 for subject#1 • Residual (or within subject) variability (WSV) • Signifies deviance between predicted and observed in each subject. This is at the observation level. Usually not assumed to be different at the subject level also. • For example, predicted Cp at time=0 is 10 ug/L, obs Cp=12 ug/L. Introduction to Population Analysis**Sources of ‘random’ variability**• All variability is typically assumed to be centered at zero. This is so because if the deviation from mean is truly random, then when the experiment is performed enough number of times, observations will be some times above mean, sometimes below mean with equal probability. • Random variability is also ‘modeled’. Variability models also need to be carefully considered. Differences between individual and mean are generally described using normal or lognormal distribution models. Introduction to Population Analysis**CL**0 CL ln(CL) Normal GFR = 120 mL/min Is GFR=60 mL/min possible? Is GFR=240 mL/min possible? 1 0 BSV, BOV Variability models Residuals are normally distributed with a mean of zero Residuals are log-normally distributed with a mean of one Introduction to Population Analysis**Measured**True Measured What would be the SD at each true value for both scenarios? True Residual variability models • Spread of ‘measured’ values is constant across • true value range • Spread of ‘measured values is higher at higher • true values Introduction to Population Analysis**SD**True CV True Residual variability models • Variability (SD) is same at low and high true values • Called “additive” model • Variability (SD) increases with true values • Called “proportional” or “constant CV” model SD True Introduction to Population Analysis**Residual variability models**• Variability (SD) is constant at low true values, but • increases with true values at higher values • Called “combined additive-prop” model SD True Introduction to Population Analysis**Agenda**• Introduction to population PK-PD • Application of population PK-PD in drug development and regulatory decision making • Pharmacometrics @ FDA • Introduction to population modeling • Linear and nonlinear regression • Introduction to mixed effects modeling • Mixed effects modeling applied to population PK • Different methods of analysis • Bayesian theory • Maximum likelihood • Sources of variability • Variance (Error) models Introduction to Population Analysis**Pharmacometrics**• Includes • Population PK • Exposure-Response (or PKPD) for effectiveness, safety • Clinical trial simulations • Disease-drug-trial modeling Pharmacometrics is the science that deals with quantifying pharmacology and disease to influence drug development and regulatory decisions Introduction to Population Analysis**Regulatory Initiatives Dictating Pharmacometrics**• Guidances for Industry • Population PK • Exposure-Response • Dose-Response • Evidence for Effectiveness • Pediatrics Clinical Pharmacology • EOP2A Meetings (draft) • Critical Path Initiative • OCP Strategic Plan • Internal CDER Deliverables Introduction to Population Analysis**NDA Reviews**Protocols Dose-Finding trials Registration trials QT Reviews Central QT team EOP2A Meetings Disease Models Knowledge Management Evidence of Effectiveness Labeling Quantify benefit/risk Dose optimization Dose adjustments Trial design Pharmacometrics Scope Tasks Decisions Influenced 1. Bhattaram et al. AAPS Journal. 2005 2. Bhattaram et al. CPT. Feb 2007 3. Garnett et al. JCP. Jan 2008 4. Wang et al. JCP. 2008 (in press) Introduction to Population Analysis**FDA PharmacometricsDemand Increasing, Focus Expanding**Demand QT Resources Focus Introduction to Population Analysis**Gobburu, Sekar, Int.J.Clin.Pharm., 2002**Integration of Knowledge Dose Ranging Studies Bridging Studies Effectiveness Safety DDI, Age Gender, Disease Smoking, Food Model Effectiveness Safety Introduction to Population Analysis**Argatroban**• Synthetic Direct Thrombin Inhibitor • Approved in Adults • prophylaxis or treatment of thrombosis in patients with heparin‑induced thrombocytopenia (HIT) • Anticoagulant in PCI patiets with HIT or at risk for HIT • Dosing • Initial dose in HIT: 2 mcg/kg/min • Titrated to 1.5 – 3 times baseline aPTT (aPTT not to exceed 100s) at steady-state (1 – 3 hrs) Introduction to Population Analysis**PKPD in Adults**• Mainly distributed in ECF • Predominantly hepatically (CYP3A4/5) metabolized • Elimination half-life is 39 – 15 min • Direct relationship between argatroban plasma concentration and anticoagulant effects. • Steady-state reached in 1-3 hrs Introduction to Population Analysis**Pediatric PKPD Data**Introduction to Population Analysis**PKPD Data**• 15 of the 16 patients received 6-10 doses of argatroban over 14 days. • Serial concentration and aPTT measurements were available in each patient. In total, about 166concentration and 329 aPTT measurements were available over a concentration range of 100 to 10,000 ng/mL. • Argatroban plasma concentration and aPTT data from 5 healthy adult studies (N=52) were used for model development. • Infusion doses range from 1µg/kg/min – 40µg/kg/min Introduction to Population Analysis**Body weight reduces the between-patient variability from 70%**to 41% Introduction to Population Analysis**Patients with elevated bilirubin exhibit 75% lower CL than**normalsVariability reduces further to 30% upon adjusting for hepatic status, after body weight Elevated bilirubin was manifested by cardiac complications Introduction to Population Analysis**Effect on aPTT is concentration dependentConcentration-aPTT**relationship is similar between adults (healthy) and pediatrics (patients) Introduction to Population Analysis