Response function

The theoretical basis for the establishment of the appropriate response function between the measured signal and the analyte concentration is described in Chapter 2. The Consensus Reports specify that a calibration curve should contain five to eight calibration standards (or more, depending on the nature of the function) with single or duplicate standards, should cover the whole intended range of concentrations, and be continuous and reproducible. Additionally, the simplest response function should be selected, the fit should be tested statistically, and an appropriate algorithm or graph presented.

What this means in practical terms is that during validation and/or the actual study the response function selected should remain constant, and not be changed from run to run. The common practice of splitting a calibration curve into two ranges for high and low concentrations should also be abandoned. The consensus reports lean very heavily towards the use of the simplest response function, i.e. linear calibration curve. However, this emphasis on linearity may cause problems. A subjective judgment whether or not a set of points represents a linear model may be at variance with statistical tests [24], Thus a linear calibration curve may be forced on data that are slightly, but nevertheless clearly non-linear.

There could be several causes of non-linearity in chromatographic assays; receptor binding assays are non-linear by nature (see Chapter 2). Certain kinds of detectors provide non-linear responses, like the electron capture detector in gas chromatography, some older types of fluorescence detectors, or in fact any fluorescence or electrochemical detector if the calibration curve range covers concentrations of several orders of magnitude. To check the detector linearity one needs to inject increasing amounts of an unextracted analyte solutions and record responses. The analytical process may be also responsible for non-linearity, due to variable extraction recovery (see 10.3.6) or adsorption. To detect and document non-linearity one may use a number of techniques [25-27]:

• Visual assessment - subjective and requires an expertise in analytical methodology,

• Conventional analysis stemming from least squares regression - several approaches can be used like components of variance, lack-of-fit testing, quadratic regression, etc.

• Analysis of consecutive differences - simulates the visual assessment of linearity.

• Comparison of observed values against expected results (residuals, see 2.2.2.1)

Another simple test for linearity based on residuals is called "sign test" [24], The signs of residuals should be distributed at random between plus and minus, if no systematic error is involved. In a sequence of signs (e.g.—++++++—), a curvature of the regression line could be suspected and a lack of linear fit.

This author finds particularly useful as a diagnostic tool the sensitivity plot [26], or rather a variation of it. Peak height (area) ratio or absolute peak height (area) divided by the nominal concentration gives a value which is called a "response factor" or "unit ratio". Assuming a zero intercept this value represents the slope of calibration curve at this point, and should be constant and equal to the overall slope of the linear calibration curve. If a decreasing/increasing trend in the value is observed, the response function is not linear. (Additionally, if response factors are constant over the whole calibration curve with the exception of the lowest standards, an interference hidden under of the analyte should be suspected).

Consider the authentic data presented in Tables 10.1 -10.3. Briefly, drug GG211 was extracted by protein precipitation from whole blood, the supernatant injected directly onto a reverse phase chromatographic column, and the drug detected by fluorescence detector after passing through a photochemical reactor, where its fluorescence was enhanced by UV irradiation. Absolute peak height (no internal standard) was used for quantitation.

Visual inspection of the graph in Figure 10.1 did not reveal curvature. The problem with this data set is that the calibration curve covers a wide range of concentrations and it is difficult to see the points with values which are close to the origin.

Table 10.1

Summary of representative experimental data

Table 10.1

Summary of representative experimental data

Standard concentration (ng/mL)

Peak height (mV)

Response factor

0.15

154

1027

0.30

289

963

1.00

1076

1076

6.00

7609

1268

30.00

41116

1371

60.00

83055

1384

100.00

138144

1381

100.00

141888

1419

60.00

86257

1438

30.00

41160

1372

6.00

7739

1290

1.00

1126

1126

0.30

345

1150

0.15

157

1047

Standard (ng/mL)

Residuals (% Deviation from nominal)

(a)

(b)

Ln transformed quadratic

(d)

e

f

£

e

f

£

e

f

£

e

f

0.15

139.3

21.2

4.5

139.1

17.5

2.5

0.9

-0.8

-0.9

-21.8

-10.6

-0.1

0.30

51.7

-7.2

-13.7

51.6

-8.2

-12.2

-8.7

-8.5

-8.2

-27.1

-17.7

-9.1

1.00

1.5

-15.8

-14.6

1.5

-14.6

-9.8

-5.2

-2.6

-2.3

-19.6

-11.7

-4.8

6.00

-5.6

-8.0

-3.4

-5.6

-5.9

2.4

1.2

3.4

1.8

-7.0

-1.9

1.9

30.00

-1.7

-1.6

3.8

-1.7

-0.1

5.5

1.1

-1.0

-5.4

-1.2

0.7

1.4

60.00

-1.1

-0.8

4.8

-1.1

-0.3

1.4

-0.9

-5.7 ■

-11.4

-0.9

-0.4

-1.0

100.00

-1.5

-1.0

4.6

-1.5

-1.8

-4.3

-3.2

-10.0 ■

-16.5

-1.6

-2.2

-3.6

100.00

1.2

1.7

7.4

1.2

0.7

-2.0

-0.6

-7.8

-14.5

1.1

0.4

-1.2

60.00

2.7

3.1

8.8

2.7

3.5

4.9

2.8

-2.4

-8.3

2.9

3.3

2.6

30.00

-1.6

-1.5

3.9

-1.5

0.0

5.6

1.2

-0.9

-5.4

-1.1

0.8

1.5

6.00

-4.1

-6.5

-1.7

-4.1

-4.3

4.1

2.8

5.0

3.4

-5.4

-0.3

3.6

1.00

5.1

-12.2

-10.8

5.0

-10.9

-5.8

-1.0

1.8

2.0

-15.9

-7.7

-0.6

0.30

65.0

6.2

0.4

64.9

5.5

3.0

7.9

8.7

9.0

-13.1

-2.3

7.5

0.15

140.8

22.6

6.0

140.5

18.9

4.2

2.7

1.1

1.0

-20.3

-8.9

1.7

Sum of residuals

422.9

109.4

88.4

422.0

92.2

67.7

40.2

59.7

90.1

139.0

68.9

40.6

y = absolute peak height, x = nominal concentration a - linear regression y = a + bx; b - quadratic regression y = a + bx + cx^ c - In transformed quadratic ln(y) = a + blnx + clnx^ d - power y = bxa weighting factors: e - none; f - 1/x; g - l/x2

Concentration (ng/mL)

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