Linearity of an analytical method

Providing evidence of the linearity of an analytical method is necessary for quantitative determinations. 

The linearity of an analytical method is its ability to elicit test results that are directly, or by means of well-defined mathematical transformation, proportional to the concentration of analytes in samples within a given range. Linearity is determined by a series of three to six injections of five or more standards whose concentrations span 80-120% of the expected concentration range. The response should be—directly or by means of a well-defined mathematical calculation—proportional to the concentrations of the analytes. A linear regression equation applied to the results should have an intercept not significantly differ-... 

Determination of Linearity and Range Determine the linearity of an analytical method by mathematically treating test results obtained from analysis of samples with analyte concentrations across the claimed range of the method. The treatment is normally a calculation of a regression line by the method of least squares of test results versus analyte concentrations. In some cases, to obtain proportionality between assays and sample concentrations, the test data may have to be subjected to a mathematical transformation before the regression analysis. The slope of the regression line and its variance (correlation coefficient) provide a mathematical measure of linearity the y-intercept is a measure of the potential assay bias. 

Other features of an analytical method that should be borne in mind are its linear range, which should be as large as possible to allow samples containing a wide range of analyte concentrations to be analysed without further manipulation, and its precision and accuracy. Method development and validation require all of these parameters to be studied and assessed quantitatively. 

According to USP 28 [1], the range of an analytical method can be defined as the interval between upper and lower levels (in the Pharmaceutical Industry usually a range from 80 to 120% of the target concentrations tested) of the analyte that have been demonstrated to be determined with a acceptable level of precision, accuracy, and linearity. Routine analyses should be conducted in this permitted range. For pharmacokinetic measurements, a wide range should be tested, where the maximum value exceeds the highest expected body fluid concentration, and the minimum value is the QL. 

The range of an analytical method is the interval between the upper and lower analytical concentration of a sample where the method has been shown to demonstrate acceptable accuracy, precision, and linearity. 

The purpose of an analytical method is the deliverance of a qualitative and/or quantitative result with an acceptable uncertainty level. Therefore, theoretically, validation boils down to measuring uncertainty . In practice, method validation is done by evaluating a series of method performance characteristics, such as precision, trueness, selectivity/specificity, linearity, operating range, recovery, LOD, limit of quantification (LOQ), sensitivity, ruggedness/robustness, and applicability. Calibration and traceability have been mentioned also as performance characteristics of a method [2, 4]. To these performance parameters, MU can be added, although MU is a key indicator for both fitness for purpose of a method and constant reliability of analytical results achieved in a laboratory (IQC). MU is a comprehensive parameter covering all sources of error and thus more than method validation alone. 

The working range of an analytical method is the interval between the upper and lower concentrations of the analyte in the sample for which it has been demonstrated that the method has acceptable precision, accuracy and linearity. This interval is normally derived from linearity studies and depends on the intended application of the method. However, validating over a range wider than actually needed provides confidence that the routine standard levels are well removed from nonlinear response concentrations, and allows quantitation of crude samples in support of process development. The range is normally expressed in the same units as the test results obtained by the analytical method. 

The linear range of an analytical method is the analyte concentration range over which response is proportional to concentration. A related quantity defined in Figure 4-12 is dynamic range—the concentration range over which there is a measurable response to analyte, even if the response is not linear. 

The selectivity a in linear HPLC does not express selectivity in the same sense as we have defined the selectivity of an analytical method above. It is still a reasonable measure of analytical selectivity, because the resolution / of two neighboring peaks (in this case the analyte and the interferent peak, respectively) is directly proportional... 

The selection of reference materials is therefore critical in validating the performance of an analytical method (see Chapter 1). CRMs should be used at least in the initial evaluation studies and in establishing the acceptability of calibrators used in routine service. The specific characteristics of calibrators should be documented, along with the number of different concentrations of calibrating solutions and the frequency of their use. These latter choices depend on the characteristics of tlie analytical method, particularly the stability, reproducibility, and linearity. 

The dynamic range of a mass spectrometer is defined as the range over which a linear response is observed for an analyte as a function of analyte concentration. It is a critical instrument performance parameter, particularly for quantitative applications, because it defines the concentration range over which analytes can be determined without sample dilution or preconcentration, which effects the accuracy and precision of an analytical method. Dynamic range is limited by physiochemical processes, such as sample preparation and ionization, and instrumental design, such as the type of mass analyzer used and the ion detection scheme. 

Sensitivity. The sensitivity of an analytical method is equal to the slope of the calibration line in a linear system. 

Accuracy and precision are the most important characteristics of an analytical method they give the best indication of random and systematic error associated with the analytical measurement. Systematic error refers to the deviation of an analytical result from the true value, and therefore affects the accuracy of a method. One the other hand, random errors influence the precision of a method (Kallner et al., 1999). Ideally, accuracy and precision should be assessed at multiple concentrations within the linear range of the assay (low, medium and high concentration). 

Procedurally, dilutional linearity should not be confused with the MRD of an assay. The MRD is a designed integral part of an analytical method and involves a predefined dilution of test samples, QC samples and, often, calibrators usually with a buffer-based matrix. In contrast, dilution linearity is used only to support analysis of study samples that exceed the assay s ULOQ and involves dilution(s) intended to result in an analyte concentration within the standard curve s validated range. Another notable difference is that, while MRD is usually performed in buffer, dilutional linearity is performed in matrix, often the same one used to prepare the standard curve. 

In this chapter, we have covered the way s to understand, estimate, and interpret various criteria required for assessing the validity of an analytical method. Whatever the complexity or simplicity of computations and models needed, the primary obj ective of an analytical method should never be forgotten Can each measurement be trusted or, equivalently, is the measurement error acceptable All the information needed to make a decision is contained in the measurement error profile. The key performance characteristics such as the linearity, accuracy, precision, limits of quantification, and sensitivity are readily obtained from this profile and can easily be understood and interpreted by an analyst. 

Range Ihe range of an analytical method refers to the interval between the upper and lower concentration for which it has been demonstrated that there is a suitable level of accuracy, precision, and linearity Repeatability A measme of the precision of the method over a short period of time using the same sample solution Resistance to mass transfer The time taken for the analyte to transfer from the mobile to the stationary phase... 

The previous section described how the conditions for postulating a linear regression function as a calibration function are checked, and how the parameters of an analytical method are determined. In routine analysis, the scientist now has the task of proving that the procedural characteristic data which he has obtained are not significantly different from the prescribed data. For this purpose the information values iy[) and the procedural standard deviation Sxo(R) are determined from 10 equally spaced concentration levels (xj). 

Validation of an analytical method establishes in laboratory studies that the performance characteristics of the method meet the requirements for the intended application, thus the method does what it is expected to do. The following items are listed by ICH precision, accuracy, limit of detection, limit of quantification, specificity, range, linearity, and ruggedness. Of these, accuracy is probably the most difficult to document or obtain, at least for solid formulations. This has to do with the fact that recovery experiments are difficult to design in such a way that they resemble the process conditions. The reactions there can create interactions that are not obtained in an experiment where the analyte has only been mixed or spiked to the sample. 

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