Journal of Probability and Statistics

Volume 2015, Article ID 298647, 13 pages

http://dx.doi.org/10.1155/2015/298647

## Confidence Region Approach for Assessing Bioequivalence and Biosimilarity Accounting for Heterogeneity of Variability

School of Medicine, Duke University, Durham, NC 27705, USA

Received 27 August 2015; Accepted 23 November 2015

Academic Editor: Steve Su

Copyright © 2015 Jianghao Li and Shein-Chung Chow. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For approval of generic drugs, the FDA requires that evidence of bioequivalence in average bioequivalence in terms of drug absorption be provided through the conduct of a bioequivalence study. A test product is said to be average bioequivalent to a reference (innovative) product if the 90% confidence interval of the ratio of means (after log-transformation) is totally within (80%, 125%). This approach is considered a one-parameter approach, which does not account for possible heterogeneity of variability between drug products. In this paper, we study a two-parameter approach (i.e., confidence region approach) for assessing bioequivalence, which can also be applied to assessing biosimilarity of biosimilar products. The proposed confidence region approach is compared with the traditional one-parameter approach both theoretically and numerically (i.e., simulation study) for finite sample performance.

#### 1. Introduction

For approval of generic drug products, the United States (U.S.) Food and Drug Administration (FDA) requires that evidence of average bioequivalence (ABE) in terms of drug absorption in blood stream be provided. The evidence can only be obtained through the conduct of a bioequivalence study. Two drug products are claimed to be average bioequivalent if the 90% confidence interval for the ratio of means of the two drug products (i.e., /, where and are means of the test product and the reference product, resp.) based on log-transformed data is totally within the bioequivalence limits of (80%, 125%). We will refer to this approach as the one-parameter approach. One of the major criticisms of this approach is that it ignores possible heterogeneity of variability between the two drug products.

Alternatively, Chow and Shao [1] proposed considering a confidence region approach (which we will refer to as a two-parameter approach) for assessing average bioequivalence. In other words, we consider confidence region (ellipse) for (two-parameter approach) rather than confidence interval for (one-parameter approach). Hsu and Lu [2] studied the relationship between the one-parameter approach and the two-parameter approach. As indicated by Chow and Shao and Hsu and Lu, the one-parameter approach is a special case of the two-parameter approach. In this paper, the proposed confidence region method is compared with the one-parameter approach by comparing the slopes of the tangent lines that hit the confidence region (ellipse). In addition, the probability of consistency (i.e., the probability of claiming bioequivalence based on the proposed method given that and are claimed to be bioequivalent based on the traditional one-parameter approach) will be evaluated.

In the next section, the traditional one-parameter approach is briefly outlined. The proposed confidence region approach is described in Section 3. Section 4 compares the proposed method with the traditional one-parameter approach both theoretically and via simulation study for finite sample performance. Also included in this section is the evaluation of the probability of inconsistency. An example is presented to illustrate the proposed method in Section 5. Some concluding remarks are given in the last section.

#### 2. Confidence Interval Approach for Assessing Average Bioequivalence

In practice, the classic (shortest) confidence interval (CI) of is usually derived using the one-parameter method. According to the 2003 FDA guidance [3], bioequivalence should be assessed based on the log-transformed data of primary pharmacokinetic (PK) parameters such as AUC (area under the blood concentration-time curves) and (maximum concentration). This means the calculation of CI for can be obtained from the CI of . Assuming that the data follow a lognormal distribution, let and be the sample means for the test and reference products, respectively (after log-transformation); the classic confidence interval can then be obtained based on the following statistics:where and are the numbers of subjects in the test and reference group, respectively. Define as , where and are the responses of subject who receives test formulation and reference product, respectively. Under the assumption of normality and homogeneity, it can be verified that follows a central distribution with degrees of freedom. Thus, the 90% confidence interval for can be obtained,where is the 95th quantile of a central distribution with degrees of freedom. The exact confidence interval for can thus be obtained by taking exponential of and as follows:Let and be the respective lower and upper bioequivalence limits. Then average bioequivalence is claimed ifor

Meanwhile, we could also derive the criteria of confidence interval approach based on raw data. Assuming that , , and we could get

#### 3. Confidence Region Approach for Assessing Bioequivalence and Biosimilarity

Chow and Shao [1] proposed a confidence region approach in assessing average bioequivalence. They performed bioequivalence testing under a crossover design and derived a confidence region for based on the raw data. Their method is similar to statistics derived in the third case below. In what follows, we will further expand Chow and Shao’s notion more specifically working on the raw data and then discuss the relationship between using raw data and log-transformed data. We will mainly focus on the results in parallel design and crossover design whose period effects and sequence effects have been ruled out.

We will first look into the cases of raw data before eventually moving on to log-transformed data in Scenario 6. From Scenarios 1 to 5, we suppose that subjects receive the test product () and subjects receive the reference product (). Assume that the response of () follows a normal distribution, that is, , and the response of also follows a normal distribution, that is, . Denote by and the sample mean responses of and , respectively. We will consider the following possible scenarios.

*Scenario 1 ( and are independent with and known). * () and () are independent and identically distributed (i.i.d.); that is, , . Thus, we haveIt can be verified that follows a chi-square distribution with 2 degrees of freedom. Suppose is the th quantile of the distribution. Then is the confidence region for *μ*. Thus, we haveIn order to compare with the one-parameter approach, that is, 90% confidence interval of is totally within (80%, 125%) after log-transformation, consider the tangent line with the form of intercepting the ellipse. With some calculation, it can be verified thatThat gives the upper bound and lower bound of . Bioequivalence is concluded if falls within the given bioequivalence limits . Besides, the center of the ellipse should be within and . With some calculation (the details are shown in the Appendix), the criteria are the same as the following:Figure 1 shows the criteria for bioequivalence of the confidence region method. The red lines are and , which are the lower and upper bound of the tangent lines. Bioequivalence is concluded only when the confidence region (the ellipse) is completely within the boundaries. That is, the center of the ellipse is within the area bounded by the two red lines while the slope of the tangent lines is within . As is shown in Figure 1, the two red lines indicate and , respectively. The dotted lines indicate two tangent lines of the central ellipse that pass through the origin. Unlike the other two ellipses, which intersect with at least one of the two red lines, the central ellipse is right inside the area bordered by the red lines. Bioequivalence can only be claimed in this situation.

In order to calculate the possibility of inconsistency, we need to evaluate the distribution of relevant statistics. As and are constants, that is, and , respectively, we could evaluate two statistics and . It can be verified that both statistics follow normal distributions , . This is useful in the next section.

If we want to use mean difference () to test bioequivalence, we can use the test statistics described in Scenario 6.