Implementation of Parallelism Testing for Four-Parameter Logistic Model in Bioassays

Four-Parameter Logistic Model

The parallelism analysis template is developed for bioassays with dose-response data that can be described through a 4PL model: where y_ij is the measure at log dose or concentration x_j of a preparation, and ε_ij is the measurement error following N(0, σ²), with i = 1 and 2 corresponding to the test sample and standard reference preparations, respectively, j = 1, … , n, and σ is the assay variability. Under the parameterization of model 1, a_i and d_i are the upper and lower asymptotes, respectively, b_i is Hill slopes and c_i is inflection point where curvature changes direction. The parameter c_i, often referred to as EC₅₀, is the dose corresponding to a mean response midway between the lower and upper asymptotes. The method by Jonkman and Sidik (8) tests equivalence of the lower, upper, and Hill slope, based on an intersection union test between a test sample and reference standard. In the development of the parallelism analysis template, we reparameterize (a_i, b_i, d_i) as (a_i, f_i, s_i) with f_i = d_i − a_i and s_i = −(d_i − a_i)b_i/4, respectively. Such reparameterization has two advantages. First of all, it offsets potentially large variability in the ratio estimate of lower asymptotes between the test sample and reference standard, as the lower asymptotes tend to be close to zero. Secondly, s_i represents the slope of the dose-response curve at EC₅₀. It makes both intuitive and practical sense to test equivalence in slope between two response curves as opposed to b_i, which is a factor of slope. In practice, f_i, the difference between lower and upper asymptotes, is often referred to as the effective window. Let r₁ = a₁/a₂, r₂ = f₁/f₂, and r₃ = s₁/s₂ be the ratios of upper asymptote, effective window and slope at EC₅₀, respectively. The dose-response curves of the test sample and standard reference are deemed to be parallel if the following null hypothesis H₀ is rejected in favor of the alternative hypothesis H₁: versus where D_Lk and D_Uk are equivalence limits k = 1, 2, 3. If D_Lk = 1/D_Uk, the limits are geometrically symmetric about one. Some researchers suggest using D_Lk = 0.8 and D_Uk = 1.25 for bioequivalence test as the limits (8) and others suggest establishing provisional limits based on testing reference standard against itself. These limits can be modified as we learn more about both the assay and product (4). In our development of the analysis template, we adopt a non-parametric method proposed by Hauck et al. (4) to establish the equivalence limits. The discussion of this approach is deferred to a later section.

Intersection-Union Test

The null hypothesis in hypothesis 2 is a union of six sub-hypotheses concerning the three ratios of the 4PL model, and the alternative is an intersection of six hypotheses. In this paper, we propose testing the hypotheses in hypothesis 2, using an intersection-union test (IUT) similar to what is described by Jonkman and Sidik (8). The test consists of six one-sided tests of the three ratios in hypothesis 2, each rejecting the component hypotheses in hypothesis 2 at a significance level of α. Overall the test results in a significance level no more than α. This property of IUT is fully discussed in Berger (15), Casella and Berger (16), and Berger and Hsu (13). Similar to the linear case, this approach is operationally equivalent to the (1 − 2α)100% CI of r_i being fully contained within the interval (D_Lk, D_Uk), k = 1, 2, 3. Here, for simplicity, we do not consider correlation among r_k. The confidence intervals are constructed based on estimates of the model parameters and their associated errors. Let θ = (a₁, b₁, c₁, d₁, a₂, b₂, c₂, d₂). We let θ^ = (â₁, b^₁, c^₁, d^₁, â₂, b^₂, c^₂, d^₂) denote the estimate of θ from fitting model 1, and Σ^(θ) the estimated associated variance-covariance matrix of θ. Then r_k, as a function of θ, can be denoted as r_k = g_k(θ). Let g′_k(θ) = ∂g_k(θ)/∂θ be the partial derivative of r_k with respect to the model parameters θ. Then r^_k = g_k(θ^) is an estimate of r_k with the variance of r^_k being estimated by var^[r^k]=[g′k(θ^)]T∑(θ)[g′k(θ^)] . An approximate (1 − 2α)100% CI of r_k is given by where z_1−α is the upper (100α)% critical value of the standard normal distribution. The null hypothesis in hypothesis 2 is rejected if the interval in eq 4 is contained in (D_Lk, D_Uk) for k = 1, 2, 3. As an illustration, we demonstrate how to construct the (1 − 2α)100% confidence interval for r₁. The partial derivative of r₁ is Thus The (1 − 2α)100% CI is obtained as with var^[r^1] being given in eq 5. Fieller's method (17) can also be used to construct the interval for this ratio.

Determination of Equivalence Limits

Unlike the significant test that seeks to support no difference claim by failure to find statistical difference, the equivalence method is oriented to demonstration that the two sets of parameters are equivalent for a specified difference that is of no practical consequence. To perform parallelism testing using the equivalence test, we need to select a priori the equivalence limits. Ideally the equivalence limits should be based on the impact of non-parallelism on the quality of product. However, as pointed out by Hauck et al. (4), this is not a trivial task because not all important product quality issues may be reflected in a parallelism measure, and not all differences in parallelism are necessarily indicative of an important quality issue. Because sufficient data for setting the equivalence bounds may not be available, Hauck et al. (4) suggest using capability-based approaches based on repeated testing of reference standard against itself to set provisional limits, and revise them as more information about the assay and product is gained. However, as noted by Liao et al. (18), the capability-based limits only control false rejection rate of parallelism or producer's risk, and renders little control over false acceptance of non-parallelism or consumer's risk. As a remedy, they develop a method that balances both types of risk. Two non-parametric tolerance interval methods are discussed for setting the equivalence limits (4). The first approach determines the equivalence limits based on ranked data. In general, let X₁ … X_n be a random sample of ratios with size n and X⁽¹⁾ … X⁽ⁿ⁾ be the ranked values. Let where r and m are two integers less than n so chosen as to produce desirable level of coverage of the population, and x_2(r+m)(1 − α) is the (1 − α) percentile of a chi-squared distribution of 2(r + m) degrees of freedom. With probability at least (1 − α), the tolerance limits X^(r) and X^(n+1−m) covers 100q% of the population (19). When n = 48, r = m = 2 correspond to the case where the second smallest and largest values are chosen as the tolerance limits, respectively. Setting α = 10%, we obtain q = 87%. The second method takes into account the precision with which each ratio is generated. It involves constructing a one-sided tolerance limit for the upper confidence limits of the ratios. The second largest value of the confidence limits is chosen as the upper equivalence limit. The reciprocal of this value is used as the lower equivalence limit. These two limits provide the same coverage of the content as the first method with higher level of confidence. In the template that we develop, we used the second method to construct equivalence limits, and these equivalence limits serve as input parameters to the template.

The methods described above provide protection against the producer's risk, in the sense that when the response curves of the sample and reference are parallel, they will have a high chance of passing parallelism testing. However, they offer little control over the probability of failing non-parallel samples. A method that accounts for both consumer's and producer's risks in determining the equivalence limits has been proposed by Yang and Zhang (20). However, detailed discussions of the method are beyond the scope of this paper, which is centered on automation of the parallelism testing method recommended in revised USP Chapters 〈1032〉 and 〈1034〉.

Parallelism Analysis Template

SMP v5 is a commercially available software package that is designed to control molecular device plate readers that measure the signals generated in the wells of assay plates. The software provides readings of kinetic and endpoint responses of test sample and reference standard from microplate assays, fits 4PL models to the response curves, and can be programmed to estimate relative potency in terms of EC₅₀ ratio between the reference standard and test sample. It also affords a utility that allows the end-user to program built-in algorithms to perform customized analyses such as parallelism testing. The model parameters are estimated in SMP v5, using maximum likelihood estimators. Both variance and covariance of the model parameter estimates can be derived from its output. Using the scripting function of the software, we are able to extract these estimates, and construct an approximate 90% confidence interval for each of the three ratios, using the method described previously. The IUT is performed, and parallelism or non-parallelism is claimed, checking the CIs against the input equivalence limits. The entire parallelism test has been automated. Forty-eight paired samples from reference materials were used to determine the equivalence limits as discussed previously. Forty-eight 90% upper confidence limits were calculated, and the second largest values were used as the upper equivalence limits, and their reciprocals were used as the lower equivalence limits.

As an illustration, we briefly discuss how to implement equivalence parallelism testing in our template. The equivalence limits were determined using historical data and stored in the template as input parameters. Conditional clauses are programmed in the template that check if the 90% CI for each ratio falls within the corresponding equivalence bound. If these are all true, then the two curves are parallel and the relative potency is calculated and output. Otherwise, the two curves are not parallel and the relative potency is not calculated.

After an assay is performed per the method standard operating procedure, the software captures the luminescence readout and fits the 4PL model. The template does the parallelism test. All the raw data of concentration and response, model fitting parameters, equivalence test result for each ratio, and the final relative potency are output to a .pdf document. Therefore, equivalence parallelism testing is incorporated in the assay analysis and is fully automated. The end-user only needs to run the assay and obtain the analysis results, including model fitting, parallelism testing, and relative potency, from the software package output.

Parallelism Testing Assessment with Known Inactive Product Variant

In our case study, a spiking study was performed using an inactive product variant (PV) that is capable of altering the product dose-response kinetics. Product (PD) samples with the expected relative potencies of 100%, 75%, 50%, and 25% were used for this spiking study. The product variant known to compete with the product and inhibit its bioactivity in the bioassay was spiked, volume per volume, to give rise to four simulated samples of the mixtures of 75% PD + 25% PV, 50% PD + 50% PV, 25% PD + 75% PV, and 0% PD + 100% PV per sample. Each of the four spiked sample was then tested in triplicate along with a reference standard, resulting in n = 12 assays. Also tested in triplicate were four PD samples of expected potencies at 100%, 75%, 50%, and 25%, producing n = 12 assays. The impact of PV on bioactivities and parallelism parameters was examined in comparison with the corresponding PD samples in the absence of PV (Figure 3). Unspiked samples were able to show similar full dose-responses kinetics, and each sample curve behaved as a dilution/concentration of the assay reference shown on Plot A. In contrast, PV-spiked samples failed to achieve the kinetic potential of the fully active product. The upper asymptotes failed to reach maximum bioactivity, causing failure to meet the parallelism criteria for some test samples shown in Plot B. For each of the 24 runs of the experiment, the parallelism between test sample and standard reference was assessed, using the tool. The results are summarized in Table I.

• Download figure
• Open in new tab
• Download powerpoint

Figure 3

Impact of PD and PV on bioactivity is demonstrated in Plots A and B, respectively.

As seen from Table I, when there is no product variant in the 100%, 75%, 50% and 25% simulated potency samples, the parallelism criteria were met in 11 out of 12 tested samples, representing the expected result 91.7% of the time. The 25% simulated potency sample failed the parallelism criteria in one test, possibly due to its being outside of the qualified assay range. By contrast, non-parallelism was demonstrated in 10 out of 12 cases in samples spiked with 25%, 50%, 75%, and 100% of PV, which amounts to the expected result 83.3% of the time. A detailed summary is given in Table II. Thus, the parallelism method used in this case study is shown to be highly accurate in detecting PV.

Conflict of Interest Declaration

The authors declare that they have no competing interests.