Developing Similarity Criteria of Dissolution Profiles through the Weibull Model and a Statistical Design Consideration Jessica Riley, Shells Artwork from Reflections Art in Health Stan Altan, Hans Coppenolle Wim Van der Elst Manufacturing and Applied Statistical Sciences (MAS) Simil
Outline • Part 1 Weibull Model (restricted to IR formulations) Advantages/Disadvantages of profile modeling • Proposed paradigm for establishing criteria during • product development Case study • • Part 2 Application of Statistical Designs to dissolution experiments Latin square design • Incomplete Block Design • • Summary • References 2
Part 1 Pros/Cons of Profile modeling Advantages Disadvantages Flexible hierarchical model to Some complexity, modeling represent known sources of may require special variability statistical tools for – Batch, Tablet, Analytical predictive calculations Allows estimation of non- Practicality of 3 observed time points parameters, when is a 4 th Permits assessment of time necessary – change or criterion, Q, for No optimal design USP/NF dissolution testing considerations carried out Parameters related to rate in practice and extent of dissolution in a Early part of the time first order process dependent profile Permits a concise comparison frequently not well in relation to parameters of characterized the model which have a natural interpretation 3
Weibull Model 𝜄3 𝑢 − + 𝜁 𝑢𝑘 i,j indexes tablet and 𝑍 𝑢𝑘 |𝑢, 𝜄 1 , 𝜄 2 , 𝜄 3 = 𝜄 1 ∗ 1 − 𝑓 𝜄2 time 𝜄 1 - dissolution extent parameter – 𝜄 2 - time to achieve 62.5 %, a rate parameter – 𝜄 3 - shape parameter – Can rewrite to shift the rate parameter to a desired 𝛿 *100% 𝜄3 𝑢 −𝜐 1 let 𝜐 = 𝑚𝑜 1−𝛿 , 0 < 𝛿 < 1 , then 𝑍 – 𝑢𝑘 |𝑢, 𝜄 1 , 𝜄 2 , 𝜄 3 , 𝜐 = 𝜄 1 ∗ 1 − 𝑓 𝜄2 + 𝜁 𝑢𝑘 For computational purposes, it is sometimes easier to fit the following reparameterized form: ∗ ) + 𝜁 𝑢𝑘 ∗(𝑚𝑝𝑢−𝜄2 ∗ = 𝜄 1 ∗ 1 − 𝑓 −𝑓 𝜄3 ∗ , 𝜄 3 𝑍 𝑢𝑘 |𝑢, 𝜄 1 , 𝜄 2 and if we include the 𝜐 parameter, this can be rewritten as ∗ ) + 𝜁 𝑢𝑘 ∗(𝑚𝑝𝑢−𝜄2 ∗ , 𝜐 = 𝜄 1 ∗ 1 − 𝑓 −𝑓 log 𝜐+𝜄3 ∗ , 𝜄 3 𝑍 𝑢𝑘 |𝑢, 𝜄 1 , 𝜄 2 4
Weibull Model, interpretation of parameters: Dissolution extent parameter Here, 𝜄 1 = 100 5
Weibull Model, interpretation of parameters: time to achieve 62.5 % Here, 𝜄 2 = 10 6
Weibull Model, interpretation of parameters: Impact of different shape parameters Here, 𝜄 3 = {0.6, 0.8, 1.0} 7
A few normative requirements from 3 authors Tsong et al (1996) - A well-defined similarity limit of the pre change product is established before comparing the dissolution data of the test and reference batches. The similarity limit is set either by the knowledge of the characteristics of the product or by the empirical experience on the batch- to-batch and the within-batch difference of the existing reference product. Global similarity Local similarity Eaton et al. (2003) - A specified function of population parameters (not involving data or experimental design) should be used to define dissolution profile similarity. Leblond et al (2016) - The test for similarity should make clear the inference space for the conclusion. For instance, does the conclusion apply to the populations of test and reference batches or only to those batches providing data for the comparison. 8
Tsong et al (1997) Multipoint Dissolution Specification and acceptance sampling based on profile modeling Proposed a release testing strategy based on a nonlinear modeling approach – Example using the Weibull model – Multivariate confidence region on location and shape parameters Compares individual tablet Weibull fit parameter estimates with the multivariate confidence region for decision rule 9
Shen et al (2011) A Bayesian Approach to Equivalence Testing Proposed a Bayesian aproach to equivalence testing of dissolution profiles through a Nonlinear mixed effects model (3-parameter Weibull) with respect to a similarity factor g 2 . – random batch component associated with the Upper Bound parameter . p 1 – Similarity factor g d , p 4 2 i p i 1 – Equivalence claimed between 2 processes if Pr( g ) 0 . 95 2 i.e. the equivalence criterion exceeds a predefined limit with a prespecified probability. Offers a statistically appropriate alternative to the f2 approach. 10
A paradigm for similarity testing Concept : Relate similarity criterion to a defined range of API concentrations (Goal posts) – Bioequivalence criterion is 80-125% – Content Uniformity internal limits is 90-110% – Therapeutic window Experimental Design recommendations – Manufacture batches at the goal posts and target – Include factors that impact dissolution, eg Particle Size, Compaction Force (dry blend process), Excipients – Use block designs to allocate batches to vessels to orthogonalize dissolution run, HPLC run and Batch effects Fit Weibull model to batch profiles, relate the parameters to regions representing similarity margins Future similarity tests whether at the process level or batch level must show a 90% CI fits within the limits 11
Case Study to illustrate the similarity testing paradigm Design – Concentrations 90, 100, 110% reflecting the range of allowable differences between batches – 2 batches at each concentration – Early experiments found MgStearate, Particle Size and other process parameters had negligible effect on dissolution 12
Study carried out - Plot of data For each batch: Bath Operator N A 1 21 2 21 B 1 21 2 21 Total of 6 (vessels) * 7 (time points) = 42 vessels per run 2 runs per batch, i.e., N = 84 13
Exploratory plot • At 60 min, slightly below the nominal %API level • B ath and Operator have little effect • Similar variability for all dissolution time points across batches • Some slightly deviating observations, but no ‘outliers’ Fit Bayesian fixed- • effects three- parameter Weibull models to the data of each batch separately : 14
Weibull fits for 6 batches 15
Batch-specific estimated Weibull parameters (fixed-effects models, means of posteriors): Batch %API UB lambda k 1 110 108.60 7.58 0.82 2 100 98.53 7.09 0.83 3 90 88.30 6.69 0.84 4 110 107.64 7.56 0.82 5 90 88.93 6.69 0.84 6 100 97.92 6.86 0.84 UB parameter differs substantially by %API . Also within a given %API, there is some batch-to-batch variability. Extent lambda parameter differs quite substantially by %API . Within a given %API, there is very limited batch-to-batch variability. Rate k parameter very similar for different %API 16
Plot of residuals all 6 batches combined: Homoscedasticity assumption is plausible 17
Exploratory analysis main conclusions 1. Dissolution profiles are Modelling strategy well characterized by fit mixed-effects Weibull the Weibull model - fits with: are very close to the Fixed-effect structure: • empirical data %API-specific UB, %API- 2. Model parameters specific Lambda parameter, 1. UB parameter differs by common k parameter %API. Batch-Batch Random-effects • variability apparent structure: random batch 2. lambda parameter differs effect for UB nested within by %API, Batch-Batch %API. variability small Residuals : • 3. k parametery similar homoscedasticity assumed %API: common k- parameter 3. Homoscedasticity assumption reasonable 18
Nonlinear mixed effects hierarchical three- parameter Weibull model: Prior distributions Weakly Informative Priors driven by exploratory fixed model – 𝑉𝐶 %𝐵𝑄𝐽 ~ 𝑂 100, 5 , – 𝜇 ~ 𝑂 7, 5 , – k ~ 𝑂 0.83, 5 , – 𝛾 1 , 𝛾 2 ~ 𝑂 ±10, 5 , 𝛾 3 , 𝛾 4 ~ 𝑂 0, 5 – 𝜁 𝑗𝑘 , 𝛿 𝑗(𝑒) ~ ℎ𝑏𝑚𝑔 𝑢(3, scale = 15) 19
Fitted model Posterior Distributions Parameter Median 95% lower CI 95% upper CI 𝑉𝐶 %𝐵𝑄𝐽=100 98.327 97.150 99.746 Lambda (%API = 100) 6.980 6.861 7.105 K 0.829 0.816 0.842 𝛾 1 -9.602 -11.374 -7.951 𝛾 2 9.689 7.894 11.464 𝛾 3 -0.289 -0.464 -0.111 𝛾 4 0.590 0.419 0.767 Parameter 5% PC 50% PC 95% PC SD batch nested %API 0.227 0.578 1.809 SD residual 1.442 1.516 1.604 20
Fitted models, batch-specific predictions 110% API 100% API 90% API 21
Residual plot Homoscedasticity assumption is plausible 22
Scatterplot posterior samples 23
Defining the region of similarity Center the posterior samples (around the %API-specific center points) and determine an ‘overall’ 99% prediction ellipse (based on all 6 batches): 24
Defining the region of similarity Model the relation between the %API-specific center points ( λ , UB) using a fractional polynomial of order 2 25
Defining the region of similarity Assume the same prediction ellipse (that was determined earlier) across the entire prediction line. Refer to this Simllarity Region as R A 26
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