Log-AUCs

Fig. 9. Correlation of VolSurf descriptors with systemic exposure after oral administration in rabbits for 49 structurally diverse MMP-8 inhibitors. Left. Predicted versus experimental systemic exposure -log(AUC(orai)) from the 4 component PLS model. Right PLS loadings showing the importance of VolSurf descriptors to the prediction of the systemic exposure in rabbits.

Intercept) Residual StdDev 0.3650006 0.04121098 Fixed effects log(auc) log(dose)... 

FIGURE 47.9 Diagnostic plots used in the model evaluation of the linear regression model that related log(AUC) to log(dose) incorporating subject type (1 for diseased subjects (patient), and 0 for healthy subject) as covariate. Top row The left and right panels are residuals showing that the adequacy of the model fit. Bottom row The left panel plot reinforces the fact that the model adequately describes the data. The right-hand plot shows the adequacy of the error model. 

An unbiased method to evaluate the quality of ranks for true positives is to compute the so-called ROC curves. The curves are obtained by plotting the number of true positives against the number of false positives in the score-ordered list. The model screening performance is most often evaluated numerically as the Area Under Curve, or AUC (43). However, the lack of focus on the low-scoring part of the ranked list, in our humble opinion, makes ROC AUC an inferior optimization function to the rank-square-root or log-AUC function. ROC AUC may still be a good function to report and compare screening performance. 

It is sometimes said that by log-transforming AUCs we are actually considering the ratios of medians rather than the ratio of means. The argument goes like this. If the log-AUCs are a sample from a Normally distributed population, then the mean and median of this distribution are identical. The AUCs themselves, however, will follow a log-Normal distribution and for this distribution the mean is higher than the median. 

However, if ijl is the mean (and hence also the median) of the log-AUCs and is the variance, then e is the median of the population of AUCs but not its mean, which equals Hence, an inference about the means of the log-AUCs corresponds to... 

If this is so, it implies that the expected value of the ratio of means will depend on the subjects recruited. This is, however, a quite undesirable property and shows that such a ratio is in fact meaningless. The only sensible thing to do is to measure on the additive scale. There are good reasons for supposing that this is more likely to be log-AUC and not AUC. To sum up, we are not interested in making inferences about ratios of means and medians but in making inferences about means (and medians) of ratios. 

Demonstrate that the semi-log plot makes the curve more linear during its rise and fall from baseline. The recirculation hump is still present but is discounted by measuring the area under the curve (AUC) enclosed by a tangent from the initial down stroke. This is the AUC that is used in the calculations. 

The semi-log transformation again makes the rise and fall of the graph linear. Note that this time there is no recirculation hump. As the fall on the initial plot was exponential, so the curve is transformed to a linear fall by plotting it as a semi-log. The AUC is still used in the calculations of cardiac output. 

The AUC from Figure 3.3 can be calculated as shown in Table 3.1. However, in a number of instances, the decrease in concentration is not linear but rather exponential. A more accurate method for calculating AUC is to use a log trapezoidal rule. Two consecutive observations on the exponential curve C(i ) and C(i +i) at times tj and f(i+i) are related to each other by... 

The reason why ratios of geometric means are used in this context is as discussed in Section 11.4 the distributions of AUC and tend to be positively skewed and the log transformation is applied to recover normality. 

Using log-transformed data, bioequivalence is established by showing that the 90% confidence interval of the ratio of geometric mean responses (usually AUC and Cmax) of the two formulations is contained within the limits of 0.8 to 1.25 [22]. Equivalently, it could be said that bioequivalence is established if the hypothesis that the ratio of geometric means is less than or equal to 0.8 is rejected with... 

Further support for the thesis that the observed drug-membrane interaction directly or indirectly affects the receptor and does not represent pharmacokinetic influences can be derived from preliminary data of a small set of five derivatives for which some pharmacokinetic parameters were determined in rats [41]. The pharmacokinetic parameters - area under the curve (AUC), elimination rate constant (kd ), half-life (to 5), the time of maximal concentration (tmax), and maximal concentration (cmax) - did not correlate significantly with either log 1/ED50(MES), log Al/T2, or log fC0i t. Instead, even for this small set of compounds, log 1 /ED50(MES) correlated again significantly with both parameters log Al/T2 and log K ocL (r = 0.998 and 0.973 respectively). 

All bioanalytical data, derived PK data, and safety data were listed and descriptive statistics calculated. Individual and median data were plotted. The log-transformed PK parameters AUC and Cmax were analyzed for dose proportionality. The PK parameters AUC(0-24) and Cmax were also descriptively analyzed for accumulation ratio. 

An exploratory analysis was performed using a four-factor ANOVA model, with treatment, period, and sequence as fixed factors and subject within sequence as random factor. The results from the ANOVA were used to calculate the back-transformed 90 % confidence intervals (Cl) for the differences between the fed and fasted condition in the log-transformed exposure measurements (Cmax, AUCo-t and AUCo-cc). For Cmax the difference between fasting and fed conditions was found to be statistically significant while this was not the case for the AUC parameters. 

Based on the planned Analysis of variance on log-transformed data, 90% confidence intervals for AUC ratios ethinylestradiol + Drug XYZ and ethinylestradiol alone, 20 subjects had to complete the study as planned. 

The relationship between age and pharmacokinetics were assessed by an analysis of variance (ANOVA) on AUCs, MRT and Cmax with adjustments for treatment, period, sequence and subject within sequence effects by age class using the natural log transformed values to compare treatments within age class. Point estimates and 95 % confidence intervals were calculated for me treatment ratios per age class. 

The software system WinNonlin (18) uses a combination of the trapezoidal and log-trapezoidal formulas to estimate AUC and AUMC, and the formulas resulting from them. As a result, no statistical information is available. 

Once the experimental part of a comparative BA study is completed and respective pharmacokinetic parameters are derived and compared, the products are declared bioequivalent when they meet the set and expected specifications for the parameters. The requirements and parameters and their specifications may vary from country to country. However, the most common standard followed is that of the U.S. Food and Drug Administration (FDA). In this case, a 90% confidence interval of the ratios of the log-transformed values of parameters (Cmax and AUC) should fall within the range 80-125. 

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