In the regulated environment of pharmaceutical and biotechnology laboratories, the role of Parallel Line Analysis (PLA) cannot be overstated. A crucial analytical method in Good Manufacturing Practice (GMP) and Good Laboratory Practice (GLP) settings, PLA is a method for evaluating the effect of biological assays. It not only facilitates comparisons between dose-response curves but also serves as a foundation for relative potency calculations. This article elaborates on the methodologies, statistical tests, and software tools required to carry out effective PLA.

Figure 1. Parallel line analysis of dose response data sets with a constrained global 4-parameter curve fit.

The Concept of Parallelism

The notion of parallelism is underpinned by mathematical functions. Specifically, two curves are considered parallel if they are dose shifted along the x-axis. The scaling factor, known as the relative potency, is essential for understanding how an unknown agent compares to a known reference. While linear regressions straightforwardly align with this methodology, complications arise when dealing with non-linear regressions like 4-parameter and 5-parameter logistic regressions.

Testing Methodologies for Parallelism

Testing for parallelism hinges on two fundamental approaches:

• Response Comparison Tests: These tests fit multiple curves into a statistical model to evaluate their parallel nature.
• Parameter Comparison Tests: This approach individually scrutinizes curve parameters to see if they align within a defined confidence interval.

Both methods are supported in SoftMax Pro and SoftMax Pro GxP software, which comes with built-in protocols using the F-test and chi-squared statistical tests.

Addressing Variability and Noise

Before diving into parallelism analysis, selecting an accurate curve fit model is imperative. Biological systems may be noisy, which can lead to incorrect conclusions if the data are not appropriately managed. The choice of a weighting factor can also make a significant difference in your analysis results.

Statistical Tests to Validate the Null Hypothesis

Upon selecting the response comparison method, various statistical techniques can be applied to validate the null hypothesis, which assumes that the curves are parallel. The F-test and chi-squared tests are commonly employed, each with its advantages and drawbacks, particularly in the context of noise. These tests have been integrated into SoftMax Pro Software, making it easier for users to calculate relevant probabilities.

PLA Implementation: SoftMax Pro GxP Software

SoftMax Pro GxP Software allows users to implement PLA for various curve fits, barring some exceptions like point-to-point and cubic spline. It offers the flexibility to adjust parameters, choose between constrained or unconstrained models, and even define custom weighting factors.

Figure 2. How to apply PLA in SoftMax Pro GxP and Standard Software and estimate relative potency. Select a graph section with multiple plots. Click Curve Fit in the Graph Tools section on the Home tab in the ribbon (A) or in the toolbar at the top of the graph section (B). (C) In the Curve Fit Settings dialog, select Global Fit (PLA). (D) Select any curve fit option except point-to-point, log-logit, or cubic spline from the drop down list. (E) Select a plot for the Reference Plot list. (F) If applicable, select the curve fit parameters and the Relative Potency Confidence Intervals. (G) If applicable, click the Weighting tab. See Application note “Selecting the best weighting factor in SoftMax Pro GxP and Standard Software”. You may also directly select the inverse of the variance weighting factor. (H) If applicable, click the Statistics tab. (I) When all curve fit options have been selected, click OK. The curves tested are fitted to the constrained model. The parameters describing the curves are identical for all curves except for the parameter describing the X-value as shown in Figure 3. For non-linear functions, the minimum and maximum responses (lower and upper asymptote, respectively) are also constrained to be the same for all curves.

Parameter Comparison Method

This approach often relies on equivalence testing, a robust method to compare individual parameters of each curve. One of the advanced statistical methods available for parameter comparison is Fieller’s theorem, which calculates the confidence interval for the ratio of two parameters.

Fieller’s Theorem in Detail

Incorporated into the preconfigured protocol “Parallelism Test” in the Protocol Library of SoftMax Pro Software, Fieller's theorem helps determine whether two curves can be considered parallel based on a predefined confidence level. This protocol automatically sets certain parameters, like the lower asymptote, to zero to circumvent mathematical limitations.

Figure 3. Response comparison method in SoftMax Pro Software to assess parallelism. The confidence level and the probability are set to 90 % and 0.1 respectively, but can be adjusted as needed. Once the lower (rBCILower and rDCILower) and upper (rBCIUpper and rDCIUpper) values of the confidence interval for the parameter ratio have been calculated, they are compared to a defined confidence interval (lval and uval). If the calculated confidence interval values are within that defined confidence interval, then the reference and the test curves can be considered parallel for that parameter.

Ensuring Precision and Regulatory Compliance

SoftMax Pro Software offers an all-in-one solution for those seeking comprehensive tools to assess parallelism in biological assays. It is fully compliant with regulatory standards like FDA 21 CFR Part 11 and EudraLex Annex 11, ensuring that your work meets the highest levels of data integrity and quality control.

Take Your Analysis to the Next Level

Ready to enhance your data analysis with the powerful capabilities of SoftMax Pro GxP Software? Experience the benefits of advanced curve fitting, flexible parameter adjustments, and built-in statistical tests for reliable PLA.

• Comprehensive Curve Fit Models: Choose from a wide range of options to suit your specific data needs.
• Robust Statistical Tools: Employ F-test, chi-squared tests, and Fieller's theorem for thorough validation.
• Regulatory Compliance: Ensure your analysis meets stringent regulatory standards for data integrity and quality.

Join Our Four-Part Tutorial Series!

Unlock the full potential of your SoftMax Pro Software with our four-part webinar tutorial series. Register now to dive deep into methodologies, gain insights from industry experts, and participate in hands-on demonstrations. One of the sessions will focus specifically on "How to perform a Parallel Line Analysis (PLA) in SoftMax Pro Software," providing you with practical, step-by-step guidance.

References

1. Gottschalk, P.D. and Dunn, J.R. (2005). Measuring parallelism, linearity, and relative potency in bioassay and immunoassay data. Journal of Biopharmaceutical Statistics, 15(3), 437-463.
2. Bates D. M. and Watts D. G. (1988). NonLinear Regression Analysis and its Applications. New York, Wiley.
3. Draper, N. R. and Smith H. (1998). Applied Regression Analysis. 3rd Ed. New York, Wiley.
4. Buonaccorsi, J. P. (2005). Fieller’s Theorem. In: Armitage, P., Colton, T., editors. Encyclopedia of Biostatistics. Vol. 3. New York, Wiley.
5. United States Department of Agriculture Center for Veterinary Biologics Standard Operating Policy/Procedure. (2015). Using Software to Estimate Relative Potency. USDA Publication No. CVBSOP0102.03. Ames, IA.

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