Bioequivalence and Bioavailability Forum

Merhaba Haydonat,

I agree with what ElMaestro has posted in the meantime. Some more stuff in the following:

The GL states in Section 4.1.4.:

Ideally, before carrying out the validation of the analytical method it should be known what concentration range is expected. This range should be covered by the calibration curve range, defined by the LLOQ being the lowest calibration standard and the upper limit of quanti­fi­ca­tion (ULOQ), being the highest calibration standard. The range should be established to allow adequate description of the pharmacokinetics of the analyte of interest.

Let’s concentrate on the last sentence which IMHO is the most important. The main principle of validation is to “demonstrate that the method is suitable for the intended use”. Even for the same calibration range the PK may be different. Example: Two studies of the antiepileptic valproic acid; (1) after a high single dose and (2) after lower doses in steady state. C_max is expected to be similar.

1. Estimation of elimination is important. Therefore, we may want to set the calibrators in such a way that the mean is on the lower end. This is what most analysts do in order to reduce the variability at the LLOQ anyway. We would set the medium QC maybe in the lower third of the range.
   
2. We have a high accumulation and a small %PTF. It is not uncommon that the C_max/C_min-ratio is only ~1.5. We have also low between subject variability (example). Of course we have to show that we have no pre-dose concentrations in period 1 and have to follow the time-course of trough-values in the saturation phases, but if we want to describe the PK as accurate as possible (GL!) it is a good idea to adapt the interval of calibrators (e.g., somewhere between evenly spaced and a geometric progression) and move the medium QC up.

I don’t know how many people writing the GL had a bioanalytical background, but “around 50% of the calibration range (medium QC)” as one-size-fits-all contradicts the leading principle of validation.

Let’s start with an excursion into validation in other fields of analytical chemistry. There are methods (mainly in environmental analysis, but also food and clinical chemistry) where you have to be accredited to perform them. In these cases we have a set of standards to follow (ISO, NIST in the US, or DIN in Germany). Many follow this approach:

• Define the calibration range.
  
• Start with a set of calibrators evenly spaced (!) throughout the range.
  
• Perform the calibration and assess linearity: Run two models – linear and quadratic. Run a statistical test (e.g., Mandel 1930) to decide which one fits the data better. For an example see this post.
  
• Regression requires additivity of effects and constant variance (aka homoscedasticity). Check the model residuals for a “funnel shape” (i.e., residuals increase with the concentration). Such a behavior is common in bioanalytics, especially if a wide range is covered – the variance increases, but the CV is roughly constant. Some standards require a test for homogeneity of variances based on ten replicates at LLOQ and ULOQ. If the test fails (or visual inspection suggests heteroscedasticity) the unweighted regression is not applicable. Go to the next step (weighting).
  
• Theoretically the best weighting scheme would be 1/σ² of calibrators. Weighting is an art in itself. I have seen methods, where a weighting function (depend on the concentration) was established by running ten (!) replicates over the entire range. The actual calibration was done with singlets with the weighting function from the validation. The most commonly applied ones are 1/σ² (requires at least duplicates at all levels; problematic if one of the calibrators is outside the acceptance criteria), 1/x, 1x², 1/y, and 1/y². The chosen weighting schema should be justified based on back-calculated concentrations. See also the excellent paper by Almeida et al. (2002)^*

Now for the more exotic stuff – where to place the calibrators within the range? Note: This is based on regression theory (e.g., Draper & Smith, Fox).

• Linear function: 50% of calibrators at the LLOQ and 50% at the ULOQ. Why? We know from the validation already that the function is linear. We get the most accurate and precise estimates of slope and intercept if we have as many data points at the extremes. Data points in between are not informative.
  
• Quadratic function: Evenly spaced throughout the range (since we have to additionally estimate the curvature).

Have you seen any of the above? I didn’t. Here enters tradition. Analysts (OK, the better ones of them) know that error in bioanalysis is multiplicative. We have serial dilutions of stock solutions and a wide calibration range. Before weighting became common practice analysts handled the problem with inaccuracy in the lower range (a single inaccurate measurement at the ULOQ may substantially alter the slope) by spreading calibrators in a geometric progression. Doing so, x|y (the “hub” of the function) is shifted downwards and inaccurate measurements in the upper range influence lower values to a lesser degree.

I’ll keep it simple: Ten calibrators, evenly spaced between 2 and 200, in a geometric progression (2.00, 3.34, 5.57, 9.28, 15.5, 25.8, 43.1, 71.9, 120, 200), or only five replicates at LLOQ and ULOQ each; linear decreasing error (from 10% CV at LLOQ to 7.5% at ULOQ = variance increasing from 0.04 to 225), lognormal, theoretical function y=0.05x. Typical example:



The dashed cross shows the “hub” (the narrowest CI). For unweighted regression this is at x|y. We see that geometric spacing pulls the range of the narrow CI towards lower values. The same is true for weighting. But which one is the “best”? I performed simulations (10⁵ per scenario) and examined the area of the confidence band (smaller ~ better):



method, weighting        min     Q1   median  mean    Q3   max
═══════════════════════════════════════════════════════════════
evenly spaced, w=1      2.398  13.34  16.81  17.30  20.69 51.73
50% LLOQ/ULOQ, w=1      0.8493 12.14  16.05  16.51  20.38 48.92
geometr. progr., w=1    0.6494  6.59   9.296 10.11  12.77 42.69
───────────────────────────────────────────────────────────────
evenly spaced, w=1/y    2.070   9.998 12.15  12.36  14.51 29.21
geometr. progr., w=1/y  1.329   8.649 11.07  11.52  13.91 34.72
50% LLOQ/ULOQ, w=1/y    0.7742  7.642 10.04  10.32  12.70 30.00

The variance is extremely increasing and only weighted models are justified. In my simulations 50% LLOQ/ULOQ performed “best” (smallest CI-area) – which I expected according to theory. It was a close shave with the geometric progression – the one most people use… But this is just a crude set showing the impact of locations of calibrators and weighting.

Where to set the medium QC? IMHO it depends on the purpose of the method (GL: “[…] adequate description of the pharmacokinetics”). I would suggest to set the medium QC at the median of concentrations expected in the study – see the valproic acid example at the very beginning and what ElMaestro said. I would reckon “around 50% of the calibration curve range” to be a rare exception. CROs will hate me for such a suggestion because it might require partial revalidation.

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• Almeida AM, Castel-Branco MM, Falcão AC. Linear regression for calibration lines revisited: weighting schemes for bioanalytical methods. J Chrom B. 2002;774:215–22.