Frequentist and Bayesian tolerance intervals for setting specification limits for left-censored gamma distributed drug quality attributes.

Tolerance intervals from quality attribute measurements are used to establish specification limits for drug products. Some attribute measurements may be below the reporting limits, that is, left-censored data. When data has a long, right-skew tail, a gamma distribution may be applicable. This paper compares maximum likelihood estimation (MLE) and Bayesian methods to estimate shape and scale parameters of censored gamma distributions and to calculate tolerance intervals under varying sample sizes and extents of censoring. The noninformative reference prior and the maximal data information prior (MDIP) are used to compare the impact of prior choice. Metrics used are bias and root mean square error for the parameter estimation and average length and confidence coefficient for the tolerance interval evaluation. It will be shown that Bayesian method using a reference prior overall performs better than MLE for the scenarios evaluated. When sample size is small, the Bayesian method using MDIP yields conservatively too wide tolerance intervals that are unsuitable basis for specification setting. The metrics for all methods worsened with increasing extent of censoring but improved with increasing sample size, as expected. This study demonstrates that although MLE is relatively simple and available in user-friendly statistical software, it falls short in accurately and precisely producing tolerance limits that maintain the stated confidence depending on the scenario. The Bayesian method using noninformative prior, even though computationally intensive and requires considerable statistical programming, produces tolerance limits which are practically useful for specification setting. Real-world examples are provided to illustrate the findings from the simulation study.

Simple Approach to Calculate Random Effects Model Tolerance Intervals to Set Release and Shelf-Life Specification Limits of Pharmaceutical Products.

Tolerance intervals are used to statistically derive the acceptance limits to which drugs must conform upon manufacture (release) and throughout shelf-life. The single measurement per lot in release data and repeated measurements per lot longitudinally for stability data have to be considered in the calculation. Methods for the one-way random effects model by Hoffman and Kringle (HK) for two-sided intervals and Hoffman (H) for one-sided limits are extended to a random-intercept, fixed-slope model in this paper. The performance of HK and H was evaluated via simulation by varying the following factors: (a) magnitude of stability trend over time, (b) sample size, (c) percentage of lot-to-lot contribution to total variation, (d) targeted proportion, and (e) data inclusion. The performance metrics are average width (for two-sided) or average limit (for one-sided) and attained confidence level. HK and H maintained nominal confidence levels as originally developed, but H is too conservative (i.e., achieved confidence level exceeds the nominal level) in some situations. The HK method adapted for an attribute that changes over time performed comparably to the more computationally intensive generalized pivotal quantity and Bayesian posterior predictive methods. Mathematical formulas and example calculations as implemented using R statistical software functions are provided to assist practitioners in implementing the methods. The calculations for the proposed approach can also be easily performed in a spreadsheet given basic regression output from a statistical software package. Microsoft Excel spreadsheets are available from the authors upon request.LAY ABSTRACT: Tolerance intervals (a measure of what can be expected from the manufacturing process) calculated from attribute measurements of drug product lots are one of the factors considered when establishing acceptance limits to ensure drug product quality. The methods often used to calculate tolerance intervals when there are multiple measurements per lot and the attribute changes over time are either lacking in statistical rigor or statistically rigorous but computationally intensive to implement. The latter type requires simulations that have to be programmed using specialized statistical software, because closed-form mathematical formulas are not available. As a consequence, some quality practitioners and applied statisticians involved in setting acceptance limits may be hindered in using such computationally intensive methods. This paper aims to address this need by proposing an approach that is statistically rigorous yet simple enough to implement using spreadsheets. The approach builds upon previously published works developed for attributes that do not change over time and adapts the cited works for attributes that change over time. The proposed approach is demonstrated to have good statistical properties and compares favorably against the more computationally intensive alternative methods. The paper provides closed-form mathematical formulas, example data, and illustrative calculations as implemented in programmed R functions to facilitate implementation by practitioners. Alternatively, the calculations can be performed without requiring complex programming/simulation using Microsoft Excel spreadsheets that can be requested from the authors.

Variation transmission model for setting acceptance criteria in a multi-staged pharmaceutical manufacturing process.

Pharmaceutical manufacturing processes consist of a series of stages (e.g., reaction, workup, isolation) to generate the active pharmaceutical ingredient (API). Outputs at intermediate stages (in-process control) and API need to be controlled within acceptance criteria to assure final drug product quality. In this paper, two methods based on tolerance interval to derive such acceptance criteria will be evaluated. The first method is serial worst case (SWC), an industry risk minimization strategy, wherein input materials and process parameters of a stage are fixed at their worst-case settings to calculate the maximum level expected from the stage. This maximum output then becomes input to the next stage wherein process parameters are again fixed at worst-case setting. The procedure is serially repeated throughout the process until the final stage. The calculated limits using SWC can be artificially high and may not reflect the actual process performance. The second method is the variation transmission (VT) using autoregressive model, wherein variation transmitted up to a stage is estimated by accounting for the recursive structure of the errors at each stage. Computer simulations at varying extent of variation transmission and process stage variability are performed. For the scenarios tested, VT method is demonstrated to better maintain the simulated confidence level and more precisely estimate the true proportion parameter than SWC. Real data examples are also presented that corroborate the findings from the simulation. Overall, VT is recommended for setting acceptance criteria in a multi-staged pharmaceutical manufacturing process.

Ex vivo fludarabine exposure inhibits graft-versus-host activity of allogeneic T cells while preserving graft-versus-leukemia effects.

Allogeneic donor T cells in bone marrow transplantation (BMT) can contribute to beneficial graft-versus-leukemia (GVL) effects but can also cause detrimental graft-versus-host disease (GVHD). A successful method for the ex vivo treatment of donor T cells to limit their GVHD potential while retaining GVL activity would have broad clinical applications for patients undergoing allogeneic hematopoietic cell transplantation for malignant diseases. We hypothesized that donor lymphocyte infusions treated with fludarabine, an immunosuppressive nucleoside analog, would have reduced GVHD potential in a fully major histocompatibility complex-mismatched C57BL/6 --> B10.BR mouse BMT model. Recipients of fludarabine-treated donor lymphocyte infusions (F-DLI) had significantly reduced GVHD mortality, reduced histopathologic evidence of GVHD, and lower inflammatory serum cytokine levels than recipients of untreated DLI. Combined comparisons of GVHD incidence and donor-derived hematopoietic chimerism indicated that F-DLI had a therapeutic index superior to that of untreated DLI. Furthermore, adoptive immunotherapy of lymphoblastic lymphoma using F-DLI in the C57BL/6 --> B10.BR model demonstrated a broad therapeutic index with markedly reduced GVHD activity and preservation of GVL activity compared with untreated allogeneic T cells. Fludarabine exposure markedly reduced the CD4+CD44(low)-naive donor T-cell population within 48 hours of transplantation and altered the relative representation of cytokine-producing CD4+ T cells, consistent with T-helper type 2 polarization. However, proliferation of fludarabine-treated T cells in allogeneic recipient spleens was equivalent to that of untreated T cells. The results suggest that fludarabine reduces the GVHD potential of donor lymphocytes through effects on a CD4+CD44(low) T-cell population, with less effect on alloreactive T cells and CD4+CD44(high) memory T cells that are able to mediate GVL effects. Thus, F-DLI represents a novel method of immune modulation that may be useful to enhance immune reconstitution among allograft recipients with reduced risk of GVHD while retaining beneficial GVL effects.