Exploring various measures of the area under the curve for the assessment of dose-proportionality and estimation of bioavailability

Dion R Brocks, Elisabeth A Minthorn, Brian E Davies, Exploring various measures of the area under the curve for the assessment of dose-proportionality and estimation of bioavailability, Journal of Pharmacy and Pharmacology, Volume 76, Issue 3, March 2024, Pages 245–256, https://doi.org/10.1093/jpp/rgae004

Abstract

In pharmacokinetics, the area under the concentration versus time curve (AUC) extrapolated to infinity (AUC_0−∞) is the preferred metric but it is not always possible to have a reliable estimate of the terminal phase half-life. Here we sought to explore the accuracy of three different area measures to accurately identify dose proportionality and bioavailability.

One to three compartment model simulations with different doses for dose-proportionality or different rates and/or extents of bioavailability. Area measures evaluated were AUC_0−∞, to the last quantifiable concentration (AUC_tlast), and to a common time value (AUC_t’).

Under linear pharmacokinetics, AUC_t’ provided the most accurate measure of dose proportionality. Except for the one compartment model where AUC_0−∞ provided the best predictor of the true measure, there was no clear advantage to the use of either of the three measures of AUC.

With uncertainty about the terminal phase half-life, the use of AUC_t’ can be a very useful and even the preferred measure of exposure for use in assessing proportionality in exposure between doses. The choice of AUC measure in bioavailability is less clear and may depend on compartmental nature of the drug, and study parameters including assay sensitivity and sampling protocols.

Introduction

Ideally, in estimating pharmacokinetic parameters, it is optimal to use a complete area under the concentration versus time curve (AUC) of a drug from the time of dosing extrapolated to infinity (AUC_0−∞). This requires an accurate estimate of the terminal-phase rate constant (λ_z). Unfortunately, it is not always possible to determine λ_z from the available data for distinct reasons including, the presence of a multiphasic decline in concentrations, the limit of quantitation of the assay, and practical limitations in the duration of sampling possible after a dose. Fig. 1 depicts study data profiles where the terminal phases are incompletely characterized. In such cases, there is some risk that the usually overestimated value of λ_z will lead to significant downstream error in the estimation of other pharmacokinetic parameters. While this may hinder accurate calculation of the pharmacokinetic parameters, it is also of concern in comparing profiles between different doses or formulations of a drug. If the error is consistent between the doses used or the formulations under investigation, then accurate conclusions regarding dose-proportionality or bioavailability are possible. On the other hand, if the proportional error in estimating the measure of the AUC is too variable, then inaccurate conclusions would be the result.

Examples from various unspecified studies where differences in the last measured time point of an AUC determination can occur in different types of comparative pharmacokinetic studies, e.g.: (A and B) Absolute bioavailability assessments; (C and D) relative bioavailability studies of oral formulations; (E–G) Dose rising studies. Linear and logarithmic transformed profiles are shown.

A potential avenue in dealing with incomplete measures of AUC in these types of comparative AUC assessments is to compare partial areas. This may be in the form of AUC from the time of dosing to the last measured time point, a parameter that is commonly provided in the results of clinical studies, or to a set time after dosing. A number of publications [ 1–6] have explored the use of various AUC measures in bioequivalence studies. Each of them proposed the use of AUC up to a specific time point (AUC_0−t) as an alternative to the use of AUC_0−∞. While the use of truncated AUC in relative bioavailability estimates has received most of the attention, incomplete AUC values are also commonly encountered in dose proportionality assessment. In dose proportionality studies it may be realistic that at lower dose levels the AUC values are more likely to be truncated at earlier time points when compared to higher dose levels because of limitations in assay sensitivity.

While the available literature has suggested that reasonable estimates of bioavailability are possible using partial area measurements, it is not entirely clear as to what partial area is the best to use. For example, a percent of the true AUC_0−∞,λz (reliant on the half-life) needed to accurately estimate bioavailability has been explored [ 3]. However, in practice this requires an accurate measure of the true total AUC, which cannot be determined if λ_z is not precisely calculable. This is particularly likely in the face of a multiphasic exponential decline in concentrations.

Here, the capability of various AUC estimates to accurately estimate bioavailability and dose proportionality was explored using simulations of drugs following compartmental pharmacokinetic models. In particular, the influence of incomplete characterization of the terminal phase on the assessment of the dose–AUC relationship has received little attention in the literature. Hence, our focus was not strictly on the criteria used to assess bioequivalence but also to delineate how close the measures would be to the actual values using available truncated AUC measures, and if possible, to make recommendations regarding the best AUC measure to use.

Methods

Concentration versus time profiles of simulated drugs were constructed that followed either a one, two, or three compartment model ( Supplementary Table S1 ; Appendix ). All drugs had similar clearance (CL) values, and in all cases the absolute bioavailability values of the simulated drugs were given as a value of one. The drug formulations were dosed either as intravenous bolus, infusion, or oral routes. Data were simulated using various doses and absorption rate constants (k_a).

Dose proportionality

Simulations were devised using three dose levels ranging by a factor of six from the highest to the lowest dose. To illustrate the impact of truncated AUC calculations on the dose proportionality assessments, arbitrary limits of quantitation were imposed on the concentration measures. The AUC values used in the assessments were as follows:

• 1) AUC from time of dose to the last measured time point (AUC_tlast).
• 2) AUC truncated to the time of the last measured time point of the lowest dose level that had the shortest t_last value (AUC_t’).
• 3) AUC extrapolated to infinity (AUC_tlast + C_last/λ_z = AUC_{0−∞, λz}) where λ_z was estimated from three spaced terminal phase concentration versus time values providing r 2 values in the excellent range of >0.98.

The simulated last measured concentration and time to that concentration were determined by setting assay sensitivity limits to a lower limit of quantitation. This had an influence not just on the ability to measure concentrations in the terminal phase, but also in some cases on the initial concentrations as the dose was being input into the blood by infusion or oral absorption ( Figs. 2 and 3).

degree of linearity (r2 >

0.98) for the two and three compartment drugs. Any points in the terminal phase of the one compartment drug would provide accurate estimates of the terminal phase rate constant. The last measured concentration was either 24 h or the last quantifiable concentration according to the lower limit of quantitation of the assay. The lower limits of assay quantitation were set to levels that would mimic insufficient time durations for discernment of the full compartmental nature of the drug using a non-compartmental approach." />

Relative AUC after three dose levels of drug given as short intravenous infusion doses. Symbols show the concentration versus time points used to estimate the terminal phase λ_z with a high degree of linearity (r 2 > 0.98) for the two and three compartment drugs. Any points in the terminal phase of the one compartment drug would provide accurate estimates of the terminal phase rate constant. The last measured concentration was either 24 h or the last quantifiable concentration according to the lower limit of quantitation of the assay. The lower limits of assay quantitation were set to levels that would mimic insufficient time durations for discernment of the full compartmental nature of the drug using a non-compartmental approach.

degree of linearity (r2 >

0.98). The last measured concentration was either 24 h or the last quantifiable concentration according to the lower limit of quantitation of the assay. The lower limits of assay quantitation were set to levels that would mimic insufficient time to assess the full compartmental nature of the drug using a non-compartmental approach." />

Results of an oral dose proportionality simulation of three drugs differing in their compartmental nature. Top three panels show the individual concentration versus time profiles (eight subjects per dose). The three lower panels show subjects with average pharmacokinetic parameters for each drug. The symbols show the concentration versus time points used to estimate the terminal phase λ_z with a high degree of linearity (r 2 > 0.98). The last measured concentration was either 24 h or the last quantifiable concentration according to the lower limit of quantitation of the assay. The lower limits of assay quantitation were set to levels that would mimic insufficient time to assess the full compartmental nature of the drug using a non-compartmental approach.

For the infusion simulations, a single-dose rising experiment involving a single-subject possessing average pharmacokinetic parameters for each model drug was examined. For oral dosing, given its greater complexity involving a first-order input phase, sets of eight patients for each of the one, two, and three compartment models, were simulated ( Fig. 3). To create different patients, randomness was introduced into each of the macro rate constants that made up the concentration versus time curve for each patient. The variability was restricted to between 80% and 125% of the average values. Variations were introduced into the k_a values to show the impact of the speed of absorption on the progression to the true F as time progressed.

To assess dose proportionality, an F-test was used to identify those dose-normalized AUC values that had unequal variances. If they did not share the same variances, this would suggest a lack of proportionality in AUC between doses. To determine for the level of significance of dose proportionality in the simulated oral-dosed subjects, single-factor ANOVA was applied to the dose-normalized AUC values across the three oral dose levels. This procedure was repeated for each of the AUC_{0−∞, λz}, AUC_tlast, and AUC_t’ values. The level of significance of differences in the dose-normalized AUC was set at 0.05. Any significant differences noted from AUC were followed up with a post hoc analysis (the Holm–Sidak method). Microsoft Excel 2016 (Microsoft, Redmond WA) and Sigmaplot for Windows v. 13 (Systat Software, Palo Alto CA) were used for statistical analysis.

Bioavailability simulations with various AUC measures

For the simulations used to examine the influence of truncated AUC on estimates of the F of oral formulations, a single dose of study drug given to an individual following the set pharmacokinetic parameters was examined. The AUC was calculated up to various times after the dose and compared to the AUC at the same time after an IV dose had been administered.

The oral dose t_max for each subject was determined using eq. 1. For the multi-compartment models, t_max was solved iteratively using the Solver (GRG Nonlinear) procedure in Microsoft Excel (Redmond, WA, USA) to minimize the square of differences between the input and disposition phases (eqs. 2 and 3).

Progression to true F (set to a value of 1) was followed using the AUC_0−t ratios of the oral to the intravenously dosed drugs. The same approach was used for assessing the relative F of formulations differing in k_a, where the AUC_tlast of the less rapidly absorbed formulation was divided by that of the more rapidly absorbed formulation. The times to reach a specific F of 0.8, 0.9, 0.95, or 0.99 were also determined and presented in tabular format. Unlike λ_z which may have limitations in its estimation, an estimate of t_max is always evaluable from any oral dose in a meticulously designed study. Hence, to understand the relationships, the partial AUC values from time of dosing to time t after dosing were expressed as a multiple of t_max (i.e. t/t_max).

Nonlinear elimination

Data meant to mimic a toxicokinetic study were simulated based on the results of unpublished data from a preclinical species given a drug previously under development that exhibited nonlinearity in its kinetics. Based on compartmental analysis of the mean concentration versus time data, the drug was found to best conform to a three-compartment model after its iv dosing of 1, 5, and 25 mg/kg. Eight dogs per dose were them simulated by introducing population variances into the parameters. Clearance, volume, and terminal t½ values of the simulated model drugs were assigned population coefficients of variation of 20%, 10%, and 15%, respectively. These variances were then then randomly assigned to each dog and the pharmacokinetic estimates of AUC were determined using a lower limit of assay quantitation as per the original study (0.008 mg/l).

Results

Dose proportionality

With proportionality in AUC with different doses, there would be no differences in the variance of the means of the dose-normalized AUC values. However, where a lower limit of quantitation hindered accurate estimation of the λ _z , differences were apparent in the variance of the means of the doses. The use of AUC_t’ provided the lowest relative standard deviation between the mean dose-normalized AUC values for all three compartmental models used. The variances of the dose-normalized AUC values using t_last or to infinity, using the points in the terminal phase, tended to have more variance between the doses compared to the use of AUC_t’. It was noted that the infusion data from the one compartment drug showed no inter-dose variance differences between AUC_t’ or the AUC_{0−∞, λz}. The AUC_tlast, however, displayed a variance that was greater and different from the other two AUC measures. However, for the two and three compartment drug infusions, the use of the AUC_t’ provided less variability in the inter-dose dose normalized values compared to the use of AUC_tlast or AUC_{0−∞, λz} ( Tables 1 and 2).

Intravenous infusion (0.5 h) dose proportionality with truncated AUC (units of mg h/l) dictated by either the last measured time point or the lower limit of assay quantitation. The true AUC, AUC from time zero to the last measured concentration, the AUC to the last measured concentration of the lowest dose, and the AUC extrapolated to infinity based on three concentrations appearing in the terminal phase are shown.

a, b or c: Symbols show groups with similar variance from the results of the F-test. Groups with different symbols from others differ in the variance. The percent relative standard deviation (RSD%) of the AUC measure for that dose is given with the results of the F-test where the superscript letter shows which values share the same variance (common letter means those RSD are similar to one another).

Intravenous infusion (0.5 h) dose proportionality with truncated AUC (units of mg h/l) dictated by either the last measured time point or the lower limit of assay quantitation. The true AUC, AUC from time zero to the last measured concentration, the AUC to the last measured concentration of the lowest dose, and the AUC extrapolated to infinity based on three concentrations appearing in the terminal phase are shown.

a, b or c: Symbols show groups with similar variance from the results of the F-test. Groups with different symbols from others differ in the variance. The percent relative standard deviation (RSD%) of the AUC measure for that dose is given with the results of the F-test where the superscript letter shows which values share the same variance (common letter means those RSD are similar to one another).

Oral dose proportionality with truncated AUC (mg.h/l) dictated by either the last measured time point (24 h maximum) or the lower limit of assay quantitation. The study design was based on each subject receiving each of the three doses. The true AUC (AUC_0−∞), AUC from time zero to the last measured concentration (AUC_tlast), AUC to the last measured concentration of the lowest dose (AUC_tʹ), and the AUC extrapolated to infinity based on three concentrations appearing in the terminal phase (AUC_{0−∞, λz}) are shown.

ANOVA: significance between doses in dose-normalized AUC within the same AUC category followed by post-hoc analysis. Superscript letters after mean ± SD indicate groups similar to those with the same letter. The F-test results show which AUC measures share the same or different between-dose variance measure (the same letter denotes variances that are not different from those with the same letter).

Oral dose proportionality with truncated AUC (mg.h/l) dictated by either the last measured time point (24 h maximum) or the lower limit of assay quantitation. The study design was based on each subject receiving each of the three doses. The true AUC (AUC_0−∞), AUC from time zero to the last measured concentration (AUC_tlast), AUC to the last measured concentration of the lowest dose (AUC_tʹ), and the AUC extrapolated to infinity based on three concentrations appearing in the terminal phase (AUC_{0−∞, λz}) are shown.

ANOVA: significance between doses in dose-normalized AUC within the same AUC category followed by post-hoc analysis. Superscript letters after mean ± SD indicate groups similar to those with the same letter. The F-test results show which AUC measures share the same or different between-dose variance measure (the same letter denotes variances that are not different from those with the same letter).

After an infusion is completed, the concentrations of drug in the blood immediately fall according to the disposition kinetics of distribution and elimination ( Fig. 2). In contrast, after oral dosing, the absorption phase may persist for some time after the t_max is reached ( Fig. 3). Hence, to examine the effects of oral dosing on truncated AUC and bioavailability estimates, a pool of subjects was simulated by randomization of the pharmacokinetic measures. This allowed for determination of between-dose comparisons for each AUC measure used ( Fig. 3, Table 2).

After rising oral dose administration to the simulated subjects, for the one compartment model, regardless of the AUC measure used, no differences were present between doses in the measures of dose-normalized AUC ( Table 2). Hence, one would conclude that all doses were proportional with respect to systemic exposure. For the two-compartment model, dose proportionality was apparent for the use of AUC_t’ and AUC_{0−∞, λz}, but not for the AUC_tlast. For the three-compartment model, only with the use of AUC_t’ was exposure found to be proportional to the administered oral dose ( Table 2).

Dose proportionality and nonlinear elimination: toxicokinetic simulation

For the simulated example with nonlinear elimination, there were no clear advantages in estimating dose proportionality amongst the various AUC measures used ( Fig. 4, Table 3).

Dose proportionality study involving a preclinical species where nonlinearity was present in the CL. The study design was three sets of eight animals given one of three doses of the drug. The AUC (mg·h/l) were truncated dictated by either the last measured time point (12 h maximum) or the lower limit of assay quantitation (profiles are shown in Fig. 4). The true AUC (AUC_0−∞), AUC from time zero to the last measured concentration (AUC_tlast), AUC to the last measured concentration of the lowest dose (AUC_tʹ), and the AUC extrapolated to infinity based on three concentrations appearing in the terminal phase (AUC_{0−∞, λz}) are shown.

# Significantly different from the 1 and 5 mg doses (ANOVA followed by post hoc test). The F-test results show which AUC measures share the same or different between-dose variance measure (same letter denotes variances that are similar to others with the same letter).

Dose proportionality study involving a preclinical species where nonlinearity was present in the CL. The study design was three sets of eight animals given one of three doses of the drug. The AUC (mg·h/l) were truncated dictated by either the last measured time point (12 h maximum) or the lower limit of assay quantitation (profiles are shown in Fig. 4). The true AUC (AUC_0−∞), AUC from time zero to the last measured concentration (AUC_tlast), AUC to the last measured concentration of the lowest dose (AUC_tʹ), and the AUC extrapolated to infinity based on three concentrations appearing in the terminal phase (AUC_{0−∞, λz}) are shown.

# Significantly different from the 1 and 5 mg doses (ANOVA followed by post hoc test). The F-test results show which AUC measures share the same or different between-dose variance measure (same letter denotes variances that are similar to others with the same letter).

Simulated concentration versus time profiles of a drug displaying three compartment model and nonlinear CL. Intravenous doses were given ranging from 1 to 25 mg/kg to eight simulated animals (different animals for each dose). The lower limit of quantitation was 0.008 mg/l. For the mean profiles, the solid lines indicate the measurable profiles, whereas the dashed lines show the profile not measurable by the assay (LLQ being 0.008 mg/l).

Effect of various AUC measures for estimation of absolute bioavailability

Some general observations were apparent. In each case regardless of the number of compartments, the longer the measurement of AUC for both the iv and the oral formulation, the more accurate the estimation of F. Convergence to the true value (F = 1) is shown in Fig. 5. The time needed to obtain estimates of F that were within 0.8, 0.9, 0.95, and 0.99 of the true value of F was characterized (see Supplementary Table S2 ). To achieve an accuracy of at least 80% of the true measure, the time needed varied from 4.4 to 16.2 h for the formulations with the lowest k_a value. In contrast, for 99% accuracy, the times ranged from 23 to 147 h. Another identifiable pattern was that the more rapid the speed of absorption, the less the time needed to accurately estimate F ( Fig. 5). By normalizing the time required to achieve these defined limits of accurate estimation of F to the t_max values, it was seen that truncating the AUC to a time between 3 and 6 multiples of the t_max could provide estimates of F that were between 90% and 95% of the true values. The three-compartment model required ratios of time to t_max that were closer to 6-fold, whereas one and two compartment drugs needed times that were closer to 3-fold that of the t_max.

Relationship between the estimated absolute oral bioavailability of three drugs differing in disposition kinetics and absorption rate constant versus the time of AUC truncation normalized to the t_max of oral dose administration (labeled as fold of t_max). For each drug, the same dose of oral and intravenous formulation was given, and for each oral formulation, the true F is 1.

Assessment of relative oral bioavailability

The approach used to assess the factors affecting absolute bioavailability (F) was also applied to the issue of relative bioavailability (F_rel) of two formulations where the absorption rate between the two formulations differed by a factor of between 3 and 5. Representative concentration versus time profiles are shown in Fig. 6. Some of the same observations that were noticed in the estimation of F held true for F_rel, including a progression to a more accurate estimation of F_rel with longer ability to measure concentrations, and more accurate estimation and shorter truncation durations with more rapidly absorbed formulations ( Table 4, Fig. 7).

The effect of lower limits of quantitation on the estimates of absolute and relative F (ratio of tablet A to tablet B AUC) for drugs following various compartmental models and their respective intravenous or oral doses. The data is based on the simulations shown in Fig. 6. Values in parentheses show the error in the calculations of F. At the last measured concentration for the iv dose, the tʹ/oral t_max ratio was 2.9. At the last measured concentration for the more rapidly absorbed oral formulation, the tʹ/oral t_max ratio was 2.6.

AUC_tʹ: AUC truncated to the last quantifiable concentration of the formulation with the fastest rate of absorption, AUC_tlast: AUC truncated to the time of the last quantifiable concentration for both formulations, AUC_0−∞,λz: AUC extrapolated to infinity based on the estimated terminal phase λz.

The effect of lower limits of quantitation on the estimates of absolute and relative F (ratio of tablet A to tablet B AUC) for drugs following various compartmental models and their respective intravenous or oral doses. The data is based on the simulations shown in Fig. 6. Values in parentheses show the error in the calculations of F. At the last measured concentration for the iv dose, the tʹ/oral t_max ratio was 2.9. At the last measured concentration for the more rapidly absorbed oral formulation, the tʹ/oral t_max ratio was 2.6.

AUC_tʹ: AUC truncated to the last quantifiable concentration of the formulation with the fastest rate of absorption, AUC_tlast: AUC truncated to the time of the last quantifiable concentration for both formulations, AUC_0−∞,λz: AUC extrapolated to infinity based on the estimated terminal phase λz.

Simulations to examine the influence of a lower limit of assay sensitivity, causing differences in the duration of concentration measurements, on absolute and relative bioavailability estimates. Sampling was available for up to 24 h after the respective oral or intravenous bolus doses. The k_a of tablet A was 5, 3, and 4-fold higher than tablet B for the one, two, and three compartment simulations. Formulations differed by a factor of 5 in the F_rel evaluations. Solid lines depict the duration of measurement, dashed lines show the complete profile, and symbols show the three concentrations that were used to estimate the terminal phase half-lives. The same dose levels were used for each formulation although the bioavailability differed as shown above each panel. For the relative bioavailability examples, the relative bioavailability was calculated as the ratio of AUC of the less rapidly absorbed formulation (tablet A) to the more rapidly absorbed formulation.

Relationship between the AUC truncation time (normalized to the t_max of either tablet A, tablet B, or the mean of t_max of the two formulations) and the estimates of relative bioavailability of the formulations truncated to the same time values. Each drug differed in its compartmental model and the k_a to λ_z ratio. The true bioavailability was the same between the two formulations.

In viewing the times required to measure F_rel with accuracy ranging from 0.8 to 0.99 (see Supplementary Table S3 ), the times normalized to t_max could be assessed by three methods ( Fig. 7). In the first two methods, the times of AUC truncation were normalized to the t_max of the more rapidly absorbed and the less rapidly absorbed formulations, respectively. The third method was based on normalization of the time of AUC truncation to the average of the two t_max. Following the AUC for at least 5–8 t_max multiples of the formulation with a slower rate of absorption tended to provide accuracy to within 10% of the true value ( Table 2).

In looking at the effects of various AUC measures on estimates of F, AUC_t’ consistently underestimated the F and relative F_A/B. It was a consistent magnitude of error between levels of bioavailability. For a validated level of LLQ, it is expected that as the bioavailability increases, the estimates of F using AUC_tlast or AUC_{0−∞, λz} would become more precise.

For F_rel, if the absorption rates are the same (same t_max) the expectation would be that AUC_t’ would perform very well, as was seen in the dose proportionality assessments. When the absorption rates become different, there will be errors in using that particular measure of AUC. Similar to F, use of AUC_t’ will consistently underestimate the F_rel when comparing the ratios of the formulations with slower to faster rates of absorption. In the one compartment model comparisons, AUC_tlast and AUC_{0−∞, λz} showed a better performance in estimating the F_rel, with the latter showing a somewhat better ability to measure the F_rel.

For the simulated drugs defined by a multicompartmental model, the imposition of the LLQ caused some variances in its effect on the calculations of F_rel using AUC_tlast or AUC_{0−∞, λz}. For the drug following the two-compartment model, when the true value was 1, both AUC measures provided very accurate estimates of F_rel. On either side of a true value of 1, the estimates appeared to be likewise under- or overestimated, respectively. This pattern was also evident for the three-compartment drug for AUC_tlast. But for the use of AUC_{0−∞, λz} there was a progressively increasing overestimation of the true value of F_rel.

For the one compartment model, the formulation with the slower rate of absorption showed an overestimation of the terminal phase t½. This was caused by the t½ being affected by an incomplete absorption peak. For the multicompartmental drugs, the t_max occurred soon after the doses were administered and, therefore, there was no effect of the absorption peak on the estimation of the terminal t½. However, because of the LLQ, the true terminal phases were difficult to estimate with the available concentrations and, hence, an underestimation of the terminal t½ values was the result.

Discussion

The AUC_0−∞ and AUC_0−tlast have been commonly reported in clinical study results [ 7–11], and are commonly used for estimation of bioequivalence and dose-proportionality. The use of AUC_t’ is much less reported and examined as a metric of relative exposure. In particular, how various measures of AUC accurately predict dose proportionality has garnered little attention compared with bioequivalence. In comparing the various measures of AUC available, it was apparent that the use of the AUC_t’ was the superior measure in the assessment of dose proportionality in the presence of linear kinetics. In this case, the tʹ was defined as the longest period of time that the lowest dose could be measured for blood fluid concentrations. The variances associated with the use of AUC to the last measured time point, or extrapolated to infinity, seemed to be higher in the more complex three compartment model compared to the one and two compartment models. The lower limits of quantitation were set for the multi-compartmental models to mimic the situation where the true terminal phase would not be discernible. In cases where this would not be the case, the use of AUC extrapolated to infinity would be expected to provide the same level of precision in estimation of dose proportionality as AUC_t’.

With linear kinetics, conceptually it would be expected that the use of AUC_t’ would provide a reliable measure of dose proportionality because the same kinetic pattern of concentrations versus time would exist between doses. The use of AUC extrapolated to infinity could become problematic in the determination of the extrapolated area, for several reasons. These include when the true terminal phase had not been reached before the end of sampling, when concentrations had fallen below the limit of quantitation, or when the blood fluid concentrations available for determination of the terminal phase t½ included samples that were too close to the peak of the absorption phase for an oral formulation.

In the presence of nonlinear kinetics presented as a toxicokinetic simulation ( Table 3, Fig. 4), each of the AUC measures was equally adept at discerning the presence of the nonlinearity of a drug that is no longer in development to examine the use of AUC_t’ and AUC_{0−∞, λz}. Both AUC measures could identify the lack of dose proportionality equally. In this particular case, the terminal phase even after the lowest intravenous dose appeared to be relatively accurately estimated. Part of this was attributable to the very short distribution phase half-lives combined with use of an assay that, with the doses used, was able to measure concentrations that appeared in the elimination phase of the drug. When the terminal phase is not as accurately measured (e.g. lower doses given or use of a less sensitive assay method), caution must be exercised in interpreting the results of a dose-rising study. Failure to do so could lead to an erroneous conclusion about the presence of nonlinearity between doses. The use of AUC_t’ is not dependent on this assumption and, hence, could be advantageous.

When increasing oral doses are administered, a clue of the existence of nonlinearity is available from examining the t_max values. Should the t_max shift to the right, this would be an indication that there is nonlinear elimination with higher doses. If it is not shifted, then it would be unlikely that there is nonlinearity related to increased dose. In either case, the use of AUC_t’ may be advisable in assessing dose proportionality because it is not dependent on being able to estimate the terminal rate constant.

Other reports have appeared regarding the use of partial areas for bioequivalence determinations. Loverling et al. [ 1] considered data from several bioequivalence studies and reported that that partial areas could be used to estimate F_rel at time points preceding the last measured quantifiable concentration. In general, it was found that the ratio of two formulations appeared to merge to a common value as time progressed, using AUC_0−t. This was also seen in the simulations reported here ( Fig. 7). The report of Moreno et al. [ 12] found that the use of partial areas provided similar outcomes in estimating bioequivalence of formulations of drugs with longer half-lives, using a 6-fold range of truncated AUC. Others, using a similar approach in comparing confidence intervals (CIs) with the use of various AUC measures, found that in general, partial AUC for bioequivalence provided generally the same results as using AUC_0−∞,λz [ 13, 14]. Interestingly, for two drugs where bioequivalence varied by AUC measure used [ 13], there was no consistent trend regarding the duration of the measure and outcome. For digoxin, the lowest duration of AUC indicated bioequivalence, whereas all others were bioinequivalent [ 13]. On the other hand, for letrozole, all truncated measures gave virtually the same 90% CIs (all bioequivalent). It is of interest that letrozole is associated with a multiple compartment model and quite variable concentrations in its extended terminal phase [ 15, 16].

Rosenbaum et al. [ 2] suggested that when both the rate and extent of absorption of a test product were not equivalent to a reference product, AUC to a time before infinity could accurately reflect the F_rel for the test product in a pharmacokinetic-model-dependent fashion. This required that the proportions of the AUC_0−t minimally comprised 45% or 60% of the AUC_{0−∞, λz} for a one- and two-compartment model drug, respectively. Chen et al. [ 3] found using simulated one and two compartment drugs that, as the tʹ normalized to the terminal phase t½ increased, more accurate estimates of bioavailability were achievable. Martinez and Jackson [ 4] suggested that AUC of two formulations could be truncated only under certain conditions when they differed significantly in their absorption rates, including ensuring that both formulations have the time of the last measured concentration well within the terminal phase of the concentration versus time profiles. They also explored the use of AUC_{0−∞, λz}, AUC_t’ and AUC_tlast for calculation of F_rel of test and reference formulations of three drugs that differed in terminal phase t½. Similar to the work presented in this publication, the authors found that each AUC measure had some utility in the measurement of F_rel for each drug (baclofen, danazol, and oxazepam; no profiles were shown).

Bois et al. [ 5] used computer simulations to explore which AUC measurement should be used to assess bioequivalence. The authors concluded that AUC_tlast performed well in most cases, even though it underestimated the actual area. They also suggested that there was no advantage in using AUC to a common time point for all individuals. Model-based estimates of AUC which attempted to determine the unknown portion of the curves performed very poorly when two-compartment kinetics were treated as one-compartment data, particularly in the case of the vanishing exponential. Extrapolation of AUC_tlast using simple linear regression of log-transformed data also led to considerable error. The authors recommended AUC_tlast of the lower limit of quantitation to estimate the extent of bioequivalence to be used. Midha et al. [ 6] recommended the use of a ratio of the C_max to a partial AUC for estimations of bioequivalence of test and reference formulations. One author also suggested that the use of truncated AUC may have utility in bioequivalence studies [ 17].

In bioequivalence, the results highlight some of the complexities involved in choosing the AUC measures that best describe the bioavailability of different formulations. To explore the progression of the AUC_tʹ to the true F, tʹ was normalized to the observed t_max (time when elimination and absorption rates are in balance) for the formulations ( Figs. 5 and 7). The measures of bioavailability became more accurate with longer tʹ. This was in line with other authors exploring the use of partial AUC [ 1–3]. Generally, for those simulations focussing on absolute F, regardless of the number of compartments, the most rapidly absorbed oral formulation needed a shorter amount of time to obtain accurate estimates of F. To achieve measures of AUC that allowed estimation within 10% of the true F, for the formulation with the slowest rate of absorption that was between three (one compartment) and six multiples of t_max was needed. In examining the comparative performance of AUC_tʹ, AUC_tlast, and AUC_{0−∞, λz} depending on the model used, the latter two indices were able to better estimate the F of the slowly absorbed drug, especially when the true F approached a value of 1. Noticeably larger errors were seen when the bioavailability began to fall to as low as 0.5 ( Table 4).

In examining F_rel ( Fig. 7), the more slowly absorbed drug needed a longer tʹ to estimate F_rel with more accuracy, but also the least multiples of t_max (to reach 80%–99% of the true values). The determination of F_rel using AUC_tʹ was more complex than the calculation of F ( Table 4). Similar to F, the use of AUC_tʹ led to consistently underestimated values of F_rel. The use of AUC_tlast was associated with various measures of F_rel ranging from underestimate to overestimate. The use of AUC_{0ʹ−∞, λz} led to progressively increasing levels of overestimation of F_rel as the true value increased from 0.75 to 1.5.

Conclusions

For dose proportionality assessments, the use of AUC to a common time point (tʹ) in the presence of linear kinetics led to very accurate conclusions regarding dose proportionality. It was superior to the use of AUC to the last measured time point, and to the AUC extrapolated to infinity for drugs that followed a multi-compartment model. Regarding absolute bioavailability assessments, for the one compartment model, the use of AUC_{0−∞, λz} was the optimal measure for estimating F. For the estimation of the F_rel, each of the measures, including the AUC extrapolated to infinity, had a burden of error in assessing the F_rel of two formulations. When bioavailability comparisons are being made between formulations, careful consideration of the concentration versus time profile (one or multicompartment model, spacing of terminal phase concentrations from the t_max) is recommended in selecting the AUC metric for use in that situation.

Author contributions

D.B. was involved in the data generation, compiling of data, and writing the manuscript. E.M. was involved in the concept and data generation and editing of the manuscript. B.D. developed the concept and was involved in writing the manuscript.

Conflict of interest

B.D. and D.B. have no conflicts of interest regarding this work. E.M. is a salaried employee of Glaxo Smithkline Pharmaceuticals and receives stock options.