R Code to Estimate Probability of Passing USP Dissolution Test

Sunday, October 1, 2023

Prasanth Sambaraju- Independent Researcher

Abstract

Dissolution testing is required by the United States Pharmacopeia (USP) to ensure that the drug products meet the established standards of strength, quality and stability. A manufacturing process is considered to be in compliance only if it meets the respective acceptance criteria for tests like content uniformity and dissolution at different times during the process. The article proposes a custom-written code in R based on a reported method to estimate the probability of passing the multistage USP dissolution test.

Keywords Dissolution, USP, Acceptance criteria, Simulation,

Introduction

Dissolution testing for solid oral drug formulations was introduced in the 1960s and is used to predict the in vivo drug release. Dissolution testing provides useful information during the entire drug development process. It is used to select the most suitable formulation, test for batch-to-batch reproducibility and performance throughout its shelf life. USP <711> ¹ provides guidance for acceptance criteria for immediate-release dosage forms, which consists of three stages of testing. In the case of S₁testing six units are tested and are considered a pass if all six units are ≥ Q + 5%. The probability of passing can be directly computed P{Pass S₁ } = p^₆ , where p is the probability of one unit greater than Q + 5%. In case of S₂ and S₃ , no exact formulas are available due to the conditional nature of the test.²

Table 1. USP <711> dissolution test acceptance criteria for
immediate-release dosage forms

A Monte Carlo simulation method was employed to estimate the probability of passing the dissolution test. It was observed that the probability of passing is dependent on the mean and standard deviation (expressed as a percentage of labeled content) of the representative samples tested from the manufactured batch. Normal distribution was considered a suitable model since the amount of drug dissolved from each unit tested is dependent on a large number of variables.³ A parametric bootstrap method in combination with Monte Carlo simulation was proposed to obtain sampling distribution of estimated probabilities of passing the USP dissolution test and its confidence intervals.⁴

Method

Wang proposed a close-form approximation to estimate the probability of passing the USP dissolution test, under the assumption that the results of dissolution follow normal distribution with known mean and standard deviation.⁵

If y_i , i = 1, 2, . . . , 24 represent the sample dissolution results and ȳ_k represents the average of y₁ , y₂ , . . .,y_k . The events of passing three stages for the USP dissolution test are expressed as

T₁ , T₂ and T₃ represent the events that meet the criteria for S1, S2_, and S3 stages respectively. The event of passing the USP dissolution test is given by T₁ ⏝T₂ ⏝ T₃ .

When the sample dissolution results are normally distributed with a mean (µ) and standard deviation (σ). The probability of passing the USP dissolution test is given by:

For certain ranges of µ and σ P(T₁ ) and P(T₁ ⌒ T₂_⌒ T₃ ) are small when compared to other probabilities in Eq (2). So, the probability of passing the USP dissolution test can be approximated by:

The probability of passing the dissolution test is given by the following equation:

Where:

Φ = cumulative distribution function of standard normal distribution.

The probability of P(T₂ ⌒ T₃ .) can be approximated by P(D₂₂⌒ D₃₃). The event D₂₂ ⌒ D₃₃ is given by:

x₁ and x₂ are independent and identically random variables from a normal distribution with mean (µ) and standard deviation (σ/√12).

The probability of P(T2 ⌒ T3 ) is given by:

The estimated probability using this method was calculated by using Mathematica software.⁵ This paper aims to estimate the probability from this method using a custom-written code in R. R is an open-source programming language and has distinct advantages over other statistical software like being platform-independent. It is well suited for performing statistical analysis.

Results and Discussions

The accuracy of the above method was tested by comparing the results obtained from this code with reported values

a) Mean of dissolution was 75, Q was 75

b) Mean of dissolution was 80, and Q was 75.

Table 2. Comparison probabilities, reported vs using R code.

Figure 1. Probability of passing USP dissolution test with mean (75) and standard deviation (3 to 10 with increments of 0.5)

The results obtained using the R code were comparable to the reported results from the Wang method.⁵ The code can be used to estimate probability for different combinations of sample mean and standard deviations. The raw R code is included in Figure 3 for reference.

Figure 2. Probability of passing USP dissolution test with mean (80) and standard deviation (3 to 10 with increments of 0.5)

Table 3. Comparison probabilities, reported vs using R code

References

The United States Pharmacopeial Convention. USP Dissolution information page. Available at: https://www.usp.org/sites/default/fi les/usp/document/harmonization/gen-method/ stage_6_monograph_25_feb_2011.pdf. Accessed June 05, 2023.
Dumont ML, Berry MR, Nickerson B. Probability of passing dissolution acceptance criteria for an immediate release tablet. J Pharm Biomed Anal. 2007;44(1):79-84. doi:10.1016/j. jpba.2007.01.047.
Saccone CD, Tessore J, Olivera SA, Meneces NS. Statistical Properties of the Dissolution Test of USP. Dissolution Technol. 2004;11(3):25-28. doi:https://doi.org/10.14227/ dt110304p25.
Chiang C, Chen CF, Huang MY, Liu JP. An inferential procedure for the probability of passing the USP dissolution test. Pharm Stat. 2012;11(1):32-38. doi:10.1002/pst.492.
Wang H. Estimation of the probability of passing the USP dissolution test. J Biopharm Stat. 2007;17(3):407-413. doi:10.1080/10543400701199536.

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