Bioequivalence and Bioavailability Forum

Dear DLabes!

❝ i would like to hear your opinion about the calculation of terminal half life, especially of choosing the linear part.

❝

❝ Is it done by you via 'informed' look at the concentration time curve (half logarithmic of course) or do you use any automatic method?

Personally I’m sticking to ‘eyeball-PK’ in this respect. The standard textbook specialized in regression emphasizes the importance of visual inspection of the fit.^[1] Other rather old, but still valid references are given below.^[2,3]

❝ As i know, WINNONLIN has built in a method using the adjusted R².

Yes, quoting WinNonlin’s (v5.2, 2007) online-help:

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During the analysis, WinNonlin repeats regressions using the last three points with non-zero concentrations, then the last four points, last five, etc. Points prior to C_max or prior to the end of infusion are not used unless the user spe­cifically requests that time range. Points with a value of zero for the depen­dent variable are excluded. For each regression, an adjusted \(\small{R^2}\) is computed:
\(\text{Adjusted}\,R^2=\frac{(1-R^2)\,\times\,(n-1)}{(n-2)}\)
where \(\small{n}\) is the number of data points in the regression and \(\small{R^2}\) is the square of the correlation coefficient (aka the coefficient of determination).
WinNonlin estimates \(\small{\lambda_\text{z}}\) using the regression with the largest adjusted \(\small{R^2}\) and:

• If the adjusted \(\small{R^2}\) does not improve, but is within 0.0001 of the largest adjusted \(\small{R^2}\) value, the regression with the larger number of points is used.
  
• \(\small{\lambda_\text{z}}\) must be positive, and calculated from at least three data points.

[…] Using this methodology, WinNonlin will almost always compute an estimate for \(\small{\lambda_\text{z}}\). It is the user’s responsibility to evaluate the appropriateness of the esti­mated value.

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My emphasis…

❝ How are your experiences with that?

❝ Does it lead to reasonable choices?

In my experience this method shows a tendency to include too many points – regularly includes even C_max/t_max…
\(\small{R^2}\) is a terribly bad parameter in assessing the ‘quality of fit’; we had a rather lengthy discussion at David Bourne's PKPD-List in 2002. To quote myself: […] we can see the dependency of \(\small{R^2}\) from \(\small{n}\), e.g., \(\small{R^2=0.996}\) for \(\small{n=4}\) reflects the same ‘quality of fit’ than does \(\small{R^2=0.766}\) for \(\small{n=9}\)!
The adjusted \(\small{R^2}\) doesn't help – since \(\small{R^2_\text{adj}=0.994}\) (\(\small{n=4}\)), and \(\small{R^2_\text{adj}=0.733}\) (\(\small{n=9}\)).
Another discussion in the context of calibration almost became a flame war for two weeks and ended just yesterday.

Interestingly enough the automated method is not mentioned by a single word in Section ‘2.8.4 Strategies for estimation of lambda_z’.^[4] Dan Weiner is Pharsight’s Chief Technology Officer…

❝ Are there any other methods (automatic or half automatic) used?

IMHO no.

❝ I think in view of the standardization of the pharmacokinetic evaluation within the framework of bioequivalence studies an automated method would be desirable.

❝ This would remove the subjectivity in estimating the AUC values, especially the extrapolated part, if possible.

Yes, but on the other hand if the procedure is laid down in an SOP/the protocol no problems are to be expected (at least I hadn’t a single request from regulators in the last 27 years). You may also find this thread interesting.
My personal procedure:

• Inclusion of at least three points
  
• Fits are accepted if \(\small{p_R\leq0.05}\) (where \(\small{\widehat{t}}\) is one-sided tested against \(\small{t_{0.05,n-2}}\): \(\small{\widehat{t}=\left | R \right |\times\sqrt{\frac{n-2}{1-R^2}}}\)

Although Pharsight gives this nice sentence about responsibility of the user, all automated procedures may lead to ‘push-the-button-PK’ – or ‘Mickey Mouse pharmacokinetics’… ^© Nick Holford
IMHO they are waiving themself out from responsibility (…we have told you, that…) and are misleading users to apply the automated procedure. In the NCA wizard > Lambda z Ranges > Lambda z Calculation Method > ⦿ Best Fit is checked by default. I’m afraid to observe a tendency that unwary users simply love clicking themselves just through all windows as fast as possible…

An example (real data from a study with very little variability; only data following t_max:
+------+--------+
| time |  conc. |
+------+--------+
|  2.5 | 5.075  |
|  3   | 4.89   |
|  3.5 | 5.025  |
|  4   | 4.93   |
|  6   | 4.12   |
|  8   | 2.975  |
| 10   | 2.055  |
| 12   | 1.405  |
| 24   | 0.1895 |
+------+--------+

Comparison of fits:
+----------+-------------+-------------+------------+-------------+
| n        |      3      |      4      |      5     |      6      |
+----------+-------------+-------------+------------+-------------+
| lambda-z |  0.169106   |  0.170921   |  0.171285  |  0.167192   |
| SE       |  0.002800   |  0.002754   |  0.002018  |  0.004257   |
| CV%      |  1.655627   |  1.611108   |  1.177875  |  2.54598    |
| T1/2     |  4.098903   |  4.055373   |  4.046743  |  4.145812   |
| R^2      |  0.999726   |  0.999481   |  0.999584  |  0.997414   |
| R^2 adj  |  0.999452   |  0.999222   |  0.999445  |  0.996767   |
| t^       | 60.4000909  | 62.0691045  | 84.8986909 | 39.2775774  |
| p(R)     |  0.00526954 |  0.00012973 |  1.801E-06 |  1.2551E-06 |
+----------+-------------+-------------+------------+-------------+

WinNonlin chooses 5 data-points, but why?
R²_adj for 3 data points (0.999452) > R²_adj for 5 data points (0.999445). Obviously the rule of ‘less than 0.0001 difference → use the larger n’ was applied. But since we are interested in estimation of the terminal half life and not in getting a high R², IMHO we should apply Occam’s razor!
On the other hand, I would have chosen 5 data points as well. Hans Proost’s suggestion of using the minimum SE of lambda_z would also lead to n=5.

Final remarks:

• Inclusion of C_max/t_max should be avoided,
  
• selection of data points should be done blinded for treatment, and
  
• consistently across subjects - therefore some kind of EDA before fitting is strongly recommended.

References:

1. NR Draper and H Smith
   Applied Regression Analysis
   John Wiley, New York, 3^rd ed 1998
   
2. FJ Anscombe and JW Tukey
   The Examination and Analysis of Residuals
   Technometrics 5/2, 141-160 (1963)
   
3. Boxenbaum HG, Riegelman S and RM Elashoff
   Statistical Estimations in Pharmacokinetics
   J Pharmacokin Biopharm 2/2, 123-148 (1974)
   
4. J Gabrielsson and D Weiner
   Pharmacokinetic an Pharmacodynamic Data Analysis: Concepts and Applications
   Swedish Pharmaceutical Press, Stockholm pp 167-169 (4^th edition 2006)