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Ken Kowalski, Ann Arbor Pharmacometrics Group (A2PG)

A General Framework for Model-Based Drug Development Using Probability Metrics for Quantitative Decision Making. Ken Kowalski, Ann Arbor Pharmacometrics Group (A2PG). Outline. Population Models Basic Notation and Key Concepts Basic Probabilistic Concepts

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Ken Kowalski, Ann Arbor Pharmacometrics Group (A2PG)

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  1. A General Framework for Model-Based Drug Development Using Probability Metrics for Quantitative Decision Making Ken Kowalski, Ann Arbor Pharmacometrics Group (A2PG)

  2. Outline • Population Models • Basic Notation and Key Concepts • Basic Probabilistic Concepts • General Framework for Model-Based Drug Development (MBDD) • Examples • Final Remarks/Discussion • Bibliography PaSiPhIC 2012

  3. Population ModelsBasic Notation • General Form of a Two-Level Hierarchical Mixed Effects Model: • Definitions: PaSiPhIC 2012

  4. Population ModelsKey Concepts • Typical Individual Prediction: • Easy to compute, same functional form as f • Population Mean Prediction: • Integral is often intractable when f is nonlinear • Typically requires Monte-Carlo integration (simulation) • The typical individual and population mean predictions are not the same when f is nonlinear • Cannot observe a ‘typical individual’ • Can observe a sample mean PaSiPhIC 2012

  5. Basic Probabilistic Concepts • Statistical intervals (i.e., confidence and prediction intervals) • Statistical power • Probability of achieving the target value (PTV) • Probability of success (POS) • Probability of correct decision (POCD) PaSiPhIC 2012

  6. What’s the difference between a confidence interval and prediction interval? • A confidenceinterval (CI) is used to make inference about the true (unknown) quantity (e.g., population mean) • Reflects uncertainty in the parameter estimates • Typically used to summarize the current state of knowledge regarding the quantity of interest based on all available data used in the estimation of the quantity • A predictioninterval (PI) is used to make inference for a future observation (or summary statistic of future observations) • Reflects both uncertainty in the parameter estimates as well as the sampling variation for the future observation PaSiPhIC 2012

  7. Relationship Between CIs and PIs Prediction Limits Recognizing Uncertainty in E( ) Prediction Limits if E( ) Located Here Distribution of sampling variation Confidence Limits for Note: Prediction intervals are always wider than confidence intervals. PaSiPhIC 2012

  8. Confidence interval for the mean based on a sample of N observations Sample mean (parameter estimate) Standard error of the mean (parameter uncertainty) PaSiPhIC 2012

  9. Prediction interval for a single future observation Sample variance of a future observation (sampling variation) Sample mean (parameter estimate) Sample variance of the mean (parameter uncertainty) Note: The sample mean based on N previous observations is the best estimate for a single future observation. PaSiPhIC 2012

  10. Prediction interval for the mean of M future observations Sample variance of the mean of M future observations (sampling variation) Sample mean (parameter estimate) Sample variance of the mean (parameter uncertainty) Note 1: The sample mean based on N previous observations is the best estimate for the mean of M future observations. Note 2: A prediction interval for M=∞ future observations is equivalent to a confidence interval (see Slide 8). This will also be referred to as ‘averagingout’ the sampling variation. PaSiPhIC 2012

  11. A Conceptual Extension of Confidence and Prediction Intervals to Population Modeling * Note for the simple mean model the standard error of the mean does not take into account uncertainty in the sampling variation (s) whereas in population models we typically take into account the uncertainty in Ω and . PaSiPhIC 2012

  12. Quantifying Parameter Uncertainty in Population Models – Nonparametric Bootstrap • Randomly sample with replacement subject data vectors to preserve within-subject correlations to construct bootstrap datasets • Re-estimate model parameters for each bootstrap dataset to obtain an empirical (posterior) distribution of the parameter estimates (, Ω, ) • May require stratified-resampling procedure (by design covariates) for a pooled-analysis with very heterogeneous study designs • E.g., limited data at a high dose in one study may be critical to estimation of Emax PaSiPhIC 2012

  13. Quantifying Parameter Uncertainty in Population Models – Parametric Bootstrap • Draw random samples from multivariate normal distribution with • Mean vector = [ ] • Covariance matrix = Cov( ) • Obtained from Hessian or other procedure (e.g., COV step in NONMEM) • Based on Fisher’s theory (Efron, 1982) • Assumes asymptotic theory (large sample size) that maximum likelihood estimates converge to a MVN distribution • See Vonesh and Chinchilli (1997) • Based on Wald’s approximation that likelihood surface can be approximated by a quadratic model locally around the maximum likelihood estimates • Approximations are dependent on parameterization • Improved approximations may occur by estimating the natural logarithm of the parameter for parameters that must be positive PaSiPhIC 2012

  14. Non-parametric Versus Parametric Bootstrap Procedures • The non-parametric bootstrap procedure is widely used in pharmacometrics • Often used as a back-up procedure to quantify parameter uncertainty when difficulties arise in estimating the covariance matrix (eg., NONMEM COV step failure) • In this setting issues with a large number of convergence failures in the bootstrap runs may call into question the validity of the confidence intervals (i.e., Do they have the right coverage probabilities?) • This form of parametric bootstrap procedure is less computationally intensive than other bootstrap procedures that require re-estimation • Requires successful estimation of the covariance matrix (NONMEM COV step) but can lead to random draws outside the feasible range of the parameters unless appropriate transformations are applied PaSiPhIC 2012

  15. Non-parametric Versus Parametric Bootstrap Procedures (2) • Developing stable models that avoid extremely high pairwise correlations (>0.95) between parameter estimates and have low condition numbers (<1000) can help • Ensure successful covariance matrix estimation • Reduce convergence failures in non-parametric bootstrap runs • Choice of bootstrap procedure should focus on the adequacy of the parametric assumption • Random draws from MVN versus the more computationally intensive re-estimation approaches (e.g., non-parametric bootstrap) PaSiPhIC 2012

  16. Simulation Procedure to Construct Statistical Intervals for Population Model Predictions Obtain random draw of , Ω,  from bootstrap procedure for kth trial Simulate subject level data Yi | , Ω,  for M subjects Summarize predictions (e.g., mean) stratified by design (dose ,time, etc.) Use percentile method to obtain statistical interval from K predictions k<K Repeat for k=1,…,K trials Note 1: To construct confidence interval consider sufficiently large M (e.g., ≥2000 subjects) to average out sampling variation in Ω and . Note 2: For prediction intervals, M is chosen based on observed or planned sample size. k=K PaSiPhIC 2012

  17. To describe other probabilistic concepts we need to define some additional quantities • True (unknown) treatment effect or quantity () • Target value (TV) • A reference value for  • Data-analytic decision rule (e.g., Go/No-Go criteria) • Based on an observed treatment effect or quantity (T) PaSiPhIC 2012

  18. Treatment Effect () •  is the true (unknown) treatment effect • Models provide a prediction of  • Uncertainty in the parameter estimates of the model provides uncertainty in the prediction of  • P( ) denotes the distribution of predictions of  PaSiPhIC 2012

  19. Example of Model-Predicted Dose-Response Model for Fasted Plasma Glucose (FPG) • Semi-mechanistic model of inhibition of glucose production Mean Model Fit of FPG Versus Dose(integrates data across dose and time) Model-Predicted Placebo-Corrected FPG Versus Dose at Week 12 Week 0 Week2 Week 4 Week 6 Delta FPG (mg/dL) Placebo-Corrected Delta FPG (mg/dL) Week8 Week 12 Dose (mg) Dose (mg) Population Mean Prediction PaSiPhIC 2012 Observed Mean 5th Percentile (90% LCL) Typical Individual Prediction (PRED) 95th Percentile (90% UCL)

  20. Target Value (TV) • Suppose we desire to develop a compound if the true unknown treatment effect () is greater than or equal to some target value (TV) • TV may be chosen based on: • Target marketing profile • Clinically important difference • Competitor’s performance • If we knew truth we would make a Go/No-Go decision to develop the compound based on: • Go:  ≥ TV • No-Go:  < TV PaSiPhIC 2012

  21. Data-Analytic Decision Rule • But we don’t know truth… • So we conduct trials and collect data to obtain an estimate of the treatment effect (T) • T can be a point estimate or confidence limit on the estimate or prediction of  (e.g., placebo-corrected change from baseline FPG) • We might make a data-analytic Go/No-Go decision to advance the development of the compound if: • Go: T ≥ TV • No-Go: T < TV PaSiPhIC 2012

  22. Statistical Power • Power is a conditional probability based on an assumed fixed value of the treatment effect () • Power = P(T ≥ TV | ) where P(T ≥ TV |  = TV) =  (false positive) • TV=0 for statistical tests of a treatment effect • Power is an operating characteristic of the design based on a likely value of  • No formal assessment that the compound can achieve the assumed value of  PaSiPhIC 2012

  23. Simulation Procedure to Calculate Power Based on a Population Model-Predicted  Use the same final estimates (, Ω, ) for each simulated trial Simulate subject level data Yi | , Ω,  for M subjects Analyze simulated data as per SAP to test Ho:  = TV Ha:   TV Power is calculated as the fraction of the K trials in which Ho is rejected k<K Repeat for k=1,…,K trials Note 1: Typically TV=0 when assessing whether the compound has an overall treatment effect. Note 2: When using simulations to evaluate power it is good practice to also simulate data under the null (e.g., no treatment effect or placebo model) to verify that the Type 1 error () is maintained. k=K PaSiPhIC 2012

  24. Probability of Achieving the Target Value (PTV) • Probability of achieving the target value is defined as the proportion of trials where the true  ≥ TV • PTV = P( ≥ TV) • Does not depend on design or sample size • Based only on prior information through the model(s) and its assumptions • PTV is a measure of confidence in the compound at a given stage of development • Can change as compound progresses through development • PTV can be calculated from the same set of simulations used to construct confidence intervals of the predicted treatment effect () PaSiPhIC 2012

  25. Simulation Procedure to Calculate PTV Based on Population Model Predictions Obtain random draw of , Ω,  from bootstrap procedure for kth trial Simulate subject-level data Yi | , Ω,  for arbitrarily large M Summarize simulated data to obtain population mean predictions of  Calculate PTV as proportion of K trials in which  ≥ TV k<K Repeat for k=1,…,K trials Note: To calculate PTV use anarbitrarily large M (e.g., ≥2000 subjects) to average out sampling variation in Ω and . PTV should only reflect the parameter uncertainty based on all available data used in the model estimation. k=K PaSiPhIC 2012

  26. Probability of Success (POS) • Probability of success is defined as the proportion of trials where a data-analytic Go decision is made • POS = P(Go) = P(T ≥ TV) • POS is an operating characteristic that evaluates both the performance of the compound and the design • In contrast to Power = P(T ≥ TV | ) which is an operating characteristic of the performance of the design for a likely value of  • POS is sometimes referred to as ‘average power’ where a Go decision is based on a statistical hypothesis test PaSiPhIC 2012

  27. Simulation Procedure to Calculate POS Based on a Population Model-Predicted  Obtain random draw of , Ω,  from bootstrap procedure for kth trial Simulate subject-level data Yi | , Ω,  for planned sample size (M) Summarize simulated data to obtain estimate of  (T) and perform hypothesis test Calculate POS as proportion of K trials in which T ≥ TV k<K Repeat for k=1,…,K trials Note: POS integrates the conditional probability of a significant result over the distribution of plausible values of  reflected through the uncertainty in the parameter estimates for , Ω, and . k=K PaSiPhIC 2012

  28. Probability of Correct Decision (POCD) • A correct data-analytic Go decision is made when • T ≥ TV and  ≥ TV • A correct data-analytic No-Go decision is made when • T < TV and  < TV • Probability of a correct decision is calculated as the proportion of trials where correct decisions are made • POCD = P(T ≥ TV and  ≥ TV) + P(T < TV and  < TV) • POCD can only be evaluated through simulation where the underlying truth () is known based on the data-generation model used to simulate the data PaSiPhIC 2012

  29. Simulation Procedure to Calculate POCD Based on a Population Model-Predicted  Classify Go: ≥TV No Go: <TV Under Truth Obtain random draw of , Ω,  from bootstrap procedure for kth trial Classify Go: T≥TV No Go: T<TV Under Trial Data Compare Truth Versus Data-Analytic Decision Classify Go: ≥TV No Go: <TV Under Truth Simulate subject-level data Yi | , Ω,  for planned sample size (M) Summarize simulated data to obtain estimate of  (T) Calculate POS as proportion of K trials in which T ≥ TV Repeat for k=1,…,K trials k<K k=K Note: Classification of trial under truth is obtained from the PTV simulations. PaSiPhIC 2012

  30. General Framework for MBDD • Basic assumptions of MBDD • Six components of MBDD • Clinical trial simulations (CTS) as a tool to integrate MBDD activities • Table of trial performance metrics • Improving POCD • Setting performance targets • Comparing performance targets between early and late stage clinical drug development PaSiPhIC 2012

  31. Basic Assumptions of MBDD • Predicated on the assumptions: • That we can and should develop predictive models • That these models can be used in CTS to predict trial outcomes • Think of MBDD as a series of learn-predict-confirm cycles • Update models based on new data (learn) • Conduct CTS to predict trial outcomes (predict) • Conduct trial to obtain actual outcomes and evaluate predictions (confirm) Learn Predict Confirm PaSiPhIC 2012

  32. Six Components of MBDD Quantitative Decision Criteria PK/PD & Disease Models Trial Performance Metrics Evaluate probability of achieving target value (PTV), success (POS), correct decisions (POCD) Leverage understanding of pharmacology/disease – useful for extrapolation Implement SAP, evaluate alternative analysis methods – ANCOVA, MMRM, regression, NLME Evaluate designs and dose selection; incorporate trial execution models such as dropout models Understand competitive landscape from a dose-response perspective Explicitly and quantitatively defined criteria “draw line in the sand” MBDD Meta-Analytic Models (Meta-Data from Public Domain) Data-Analytic Models Design & Trial Execution Models PaSiPhIC 2012

  33. Clinical Trial Simulations (CTS) • Just as a clinical trial is the basic building block of a clinical drug development program, clinical trial simulations should be the cornerstone of an MBDD program • CTS allows us to assume (know) truth for a hypothetical trial • Based on simulation model we know  • Mimic all relevant design features of a proposed clinical trial • Sample size, treatments (doses), covariate distributions, drop out rates, etc. • Analyze simulated data based on the proposed statistical analysis plan (SAP) • Calculate T (test statistic for treatment effect) and apply data-analytic decision rule • CTS should be distinguished from other forms of stochastic simulations • E.g., CIs for dose predictions, PTV calculations, etc. • CTS can be used to integrate the components of MBDD and the various probabilistic concepts (including POS and POCD) PaSiPhIC 2012

  34. Table of Trial Performance Metrics Trial No Go Trial Go Total Correct No Go Incorrect Go P(True No Go) “True” No Go Incorrect No Go Correct Go P(True Go) “True” Go P(Trial No Go) P(Trial Go) 1.0 Total POCD POS PTV PaSiPhIC 2012

  35. Improving the probability of making correct decisions • Change the design •  n/group • Regression-based designs ( # of dose groups,  n/group) • Consider other design constraints (cross-over, titration, etc.) • Change the data-analytic model • Regression versus ANOVA • Longitudinal versus landmark analysis • Change the data-analytic decision rule • Alternative choices for T • Point estimate, confidence limit, etc. • All of the above can be evaluated in a CTS PaSiPhIC 2012

  36. Setting Performance Targets • PTV will change over time as model is refined and new data emerge • Bring forward compounds/treatments with higher PTV as compound moves through development • PTV may be low in early development • Industry average Phase 3 failure rate is approximately 50% • It is difficult to improve on this average unless we can routinely quantify PTV • Strive to achieve PTV>50% before entering Phase 3 • Strive to achieve high POCD in decision-making throughout development PaSiPhIC 2012

  37. Comparing performance targets between early and late stage clinical drug development Trial No Go Trial No Go Trial Go Trial Go Total Total True No Go True No Go True Go True Go Total Total Late Development POCD should be high PTV should be high Advance good compounds / treatments to registration Early Development POCD should be high PTV may be low Kill poor compounds / treatments early PaSiPhIC 2012

  38. Examples • Rheumatoid Arthritis Example • Phase 3 development decision • Urinary Incontinence Example • Potency-scaling for back-up to by-pass Phase 2a POC trial and proceed to a Phase 2b dose-ranging trial • Acute Pain Differentiation Case Study • Decision to change development strategy to pursue acute pain differentiation hypothesis PaSiPhIC 2012

  39. Example – Rheumatoid Arthritis • Endpoints: • DAS28 remission (DAS28 < 2.6) • ACR20 response (20% improvement in ACR score) • Models developed based on Phase 2a study: • Continuous DAS28 longitudinal PK/PD model with Emax direct-effect drug model • ACR20 logistic regression PK/PD model with Emax drug model • Both direct and indirect-response models evaluated • Conducted clinical trial simulations for a 24-week Phase 2b placebo-controlled dose-ranging study (placebo, low, medium and high doses) • At Week 12 non-responders assigned to open label extension at medium dose level • Primary analysis at Week 24; Week 12 responses for non-responders carried forward to Week 24 • Evaluated No-Go/Hold/Go criteria for Phase 3 development PaSiPhIC 2012

  40. Example – Rheumatoid Arthritis (2) • No Go: Stop development • Hold: Wait for results of separate Phase 2b active comparator trial • Go: Proceed with Phase 3 development without waiting for results from comparator trial PaSiPhIC 2012

  41. Example – Rheumatoid Arthritis (3) • CTS results suggest a high probability that the team will have to wait for results from the Phase 2b active comparator trial before making a decision to proceed to Phase 3. Very low probability of taking low dose into Phase 3. PaSiPhIC 2012

  42. Example – Urinary Incontinence • Endpoint: • Daily micturition (MIC) counts • Models developed: • Longitudinal Poisson-Normal model developed for daily MIC counts for lead compound • Time-dependent Emax drug model using AUC0-24 as measure of exposure • Potency scaling for back-up based on: • In vitro potency estimates for lead and back-up (back-up more potent than lead) • Equipotency assumption between lead and back-up • Conducted CTS to evaluate Phase 2b study designs for back-up compound (placebo and four active dose levels) • Evaluated various dose scenarios of low (L), medium #1 (M1), medium #2 (M2) and high (H) doses levels • Implemented SAP (constrained MMRM analysis with step down trend tests) • Quantified POS for the L, M1, M2 and H doses for the various dose scenarios and potency assumptions PaSiPhIC 2012

  43. Example – Urinary Incontinence (2) Note: Low (L) dose was selected to be a sub-therapeutic response. Design was not powered to detect a significant treatment effect at this dose. PaSiPhIC 2012

  44. Example – Urinary Incontinence (3) • CTS results: • High POS (>95%) demonstrating statistical significance at the H dose for all 6 dose scenarios • Insensitive to potency assumptions • High POS (>88%) demonstrating statistical significance at the M2 dose for all 6 dose scenarios • Insensitive to potency assumptions • POS varied substantially for demonstrating statistical significance of the M1 dose • Depending on dose scenario and potency assumption • POS < 60% for demonstrating statistical significance at the L dose • Except for dose scenarios 4 – 6 for the in vitro potency assumption CTS results provided guidance to the team to select a range of doses that would have a high probability of demonstrating dose-response while being robust to the uncertainty in the relative potency between the back-up and lead compounds. Provided confidence to bypass POC and move directly to a Phase 2b trial for the back-up. PaSiPhIC 2012

  45. Case Study – Acute Pain DifferentiationBackground • SC-75416 is a selective COX-2 inhibitor • Capsule dental pain study conducted • Poor pain response relative to active control (50 mg rofecoxib) • Lower than expected SC-75416 exposure with capsule relative to oral solution evaluated in Phase 1 PK studies • PK/PD models developed to assess whether greater efficacy would have been obtained if exposures were more like that observed for the oral solution • Pain relief scores (PR) modeled as an ordered-categorical logistic normal model • Dropouts due to rescue therapy modeled as a discrete survival endpoint dependent on the patient’s last observed PR • Assumes a missing at random (MAR) dropout mechanism PaSiPhIC 2012

  46. Case Study – Acute Pain DifferentiationBackground (2) • PK/PD modeling predicted greater efficacy with oral solution relative to capsules • A 6-fold higher SC-75416 dose (360 mg) than previously studied predicted to have clinically relevant improvement in pain relief relative to active control (400 mg ibuprofen) • Model extrapolates from capsules to oral solution and leverages in-house data from other COX-2s and NSAIDs • Project team considers change in development strategy to pursue a high-dose efficacy differentiation hypothesis • Original strategy was to determine an acute pain dose that was equivalent to an active control and then scale down the dose for chronic pain (osteoarthritis) • Based on well established relationships that chronic pain doses for NSAIDs tend to be about half of the acute pain dose PaSiPhIC 2012

  47. Case Study – Acute Pain DifferentiationProposed POC Dental Pain Trial • Proposed conducting a proof of concept oral solution dental pain study • Demonstrate improvement in pain relief for 360 mg SC relative to 400 mg ibuprofen • Primary endpoint is TOTPAR6 (SC vs. ibuprofen) • TOTPAR6 = 3 (TV) is considered clinically relevant • Perform ANOVA on observed LOCF-imputed TOTPAR6 response and calculate LS mean differences • T = LS mean (SC) – LS mean (ibuprofen) • LCL95 = 2-sided lower 95% confidence limit on T • Compound and data-analytic decision rule: • Truth: Go if ≥3, No-Go if <3 • Data: Go if T≥3 and LCL95>0, No-Go if T<3 or LCL95≤0 PaSiPhIC 2012

  48. Simulate PR Model Parameters (PR,2) ~ MVN Simulate PR Scores M=2,000 patients per treatment Perform LOCF Imputation and Calculate TOTPAR6 Simulate Dropout Times M=2,000 patients per treatment Simulate Dropout Model Parameters DO ~ MVN Calculate Population Mean TOTPAR6 & TOTPAR6 Across M=2,000 pts k<K Determine True Decision Go: 3 No Go: <3 k=K Repeat for k = 1,…,K=10,000 trials Summarize Distribution of TOTPAR6 () Case Study – Acute Pain DifferentiationSimulation Procedure to Calculate PTV PaSiPhIC 2012

  49. PTV = P(  3) = 67.2% Mean Prediction = 3.2 Case Study – Acute Pain DifferentiationPosterior Distribution of TOTPAR6 PTV = 67.2% sufficiently high to warrant recommendation to conduct oral solution dental pain study to test efficacy differentiation hypothesis. PaSiPhIC 2012

  50. Simulate PR Scores for k-th Trial n pts / treatment Calculate Mean TOTPAR6 (T), SEM & 95% LCL Perform LOCF Imputation & Calculate TOTPAR6 Simulate Dropout Times for k-th Trial n pts / treatment Apply Decision Rule Go: LCL>0 and T3 No Go: LCL0 or T<3 Compare Truth vs. Data-Analytic Decision Repeat for k=1,…,K=10,000 trials Calculate Metrics POS POCD Case Study – Acute Pain DifferentiationCTS Procedure to Evaluate POC Trial Designs k<K k=K PaSiPhIC 2012

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