Modifications of the Method for Calculating Absolute Drug Bioavailability.

PURPOSE
Absolute bioavailability (F) is calculated as the ratio of the area under the plasma drug concentration-time curve (AUC) between extravascular administration and intravenous injection. However, as distribution of a drug after intravenous administration does not reach an equilibrium in the body during the distribution phase, the plasma drug concentration at this phase does not reflect the total amount of drug in the body. The goal of this paper was to analyze the insufficiencies of the method for calculating on absolute bioavailability and to propose a modification to improve the calculation.


METHODS
Literature reporting absolute bioavailability published during 1983-2014 was searched for ten drug candidates. Plasma drug concentrations representing the amount of drug in the body were then calculated at each time point during the distribution phase according to the plasma drug concentration-time relationship during the elimination phase.


RESULTS
The AUC values based on the distribution equilibrium drug concentrations following intravenous injection were 75%±11% of the actually measured drug concentrations in the literature. The absolute bioavailability values in the literature were 76%±12% of the actual bioavailability based on the AUCs from distribution-equilibrium drug concentrations.


CONCLUSIONS
The present method underestimates the absolute drug bioavailability and should be modified to represent the data more accurately. This article is open to POST-PUBLICATION REVIEW. Registered readers (see "For Readers") may comment by clicking on ABSTRACT on the issue's contents page.


INTRODUCTION
As an important indicator in pharmacokinetics, absolute bioavailability is defined as the fraction of an administered drug that reaches the circulation system (F), which is the ratio between the amount of drug presented in the systemic circulation and the total administered dose. To calculate F, the area under the curve of the plasma drug concentration (AUC) after intravenous (iv) and extravascular (ev) administration is obtained [1][2][3][4]. The AUC following iv administration, however, reflects the amount of a drug in the blood circulation before distribution into the rest of the body. On the other hand, following ev administration, the drug will distribute while it is being absorbed so that, for the majority of drugs, the distribution process is masked by the absorption phase. Hence, as a marker of the amount of the drug in the body, the plasma drug concentration during the distribution phase is an over-estimation. This investigation aimed to analyze these insufficiencies and to suggest possible modifications for determining absolute drug bioavailability.
The concentrations of a drug in different organs or tissues are different. Nonetheless, when the drug distribution reaches pseudo-equilibrium, i.e., equal concentration in the permeable tissues, the amount of a drug in an organ or a tissue is equal the drug concentration multiply by the volume of the organ or the tissue. The apparent volume of distribution, Vd, is defined as the theoretical volume of body fluid that is required to dissolve the drug at the same concentration as that in plasma. In linear pharmacokinetics, Vd of a drug is constant regardless of the administration routes. Hence, the total amount of a drug (A) in the body equals Vd multiple by its blood drug concentration (C) at equilibrium:

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The fraction of an administered drug that reaches the circulatory system (F) is the ratio between the amount of drug being bioavailable and the total amount of drug administered. When a drug is injected via a vein, F equals 1. Therefore, Where, Cev and Civ are the plasma drug concentrations of ev and iv administration of the same dose, respectively. While during the pre-pseudo-equilibrium time, the amount in the reachable sites of the body is different between iv and ev administration, the concentration values are assumed to be markers of the drug in the body. This, however, is not true until the post-pseudoequilibrium has reached. Taking the two-compartment model for example, after iv administration, the drug reaches the circulatory system instantaneously. The plasma drug concentration-time curve shows a distribution phase and an elimination phase. First, the drug emerges in the central compartment, such as the vascular system, and those organs or tissues that take the drug in as fast as it is distributed in plasma. The distribution into the organs and/or tissues (deep tissue compartment) that are slow in letting the drug permeate occurs is significantly slower pace (Fig. 1). Graphically, this process is presented by two distinct phases, distribution (α) and elimination (β) (Fig. 2).  In contrast to iv route, with ev administration, a drug molecule gradually enters the central compartment to be distributed to other organs and/or tissues. In such cases, the pseudo-equilibrium point may not be seen if the absorption is slower than distribution process.
Absolute drug bioavailability is typically calculated using actually measured values of plasma drug concentrations, hence, to calculate AUC, the contribution of the effect of route of administration on the distribution kinetics is one often ignored.
From equation 3 below, which describes a log-linear process where log C is the drug concentration, t is the time, β is the elimination rate constant, log B is they-intercept of the log linear elimination line and B is the maximum plasma drug concentration when the distribution has reached pseudo-equilibrium.
AUC can be calculated from: Or B/β reflects the AUC of the dose after attainment of pseudo-equilibrium. For a drug with a two or multi-compatment model, this value is obviously smaller than AUC calculated using the actually measured drug concentrations that includes those during distribution phase. For example, we used the data of Muck et al. [5] to calculate the true F that was adjusted for the distribution process. Healthy young male volunteers received either a single 100 μg iv dose or 200 μg oral tablets of cerivastatin to determine the absolute bioavailability.
The plasma cerivastatin concentrations after iv administration were determined using the Microsoft Paint software according to the concentration-time curve ( Table 1). The data for the plasma cerivastatin concentrations and times were then input into the DAS 3.1.5 software (Shanghai, China). The results showed that the plasma concentration-time curve fits to the two-compartment model. The AUC∞, B, and β were 7.89 μg·h/L, 1.804 and 0.349, respectively. This AUC∞ is 1.02 times that reported in the literature [5], suggesting that the recovered data are very closed to the actual data. The values of B and β are entered into equation 5, and the AUC based on equilibrium is calculated as the follows: The result is 5.17 μg·h/L. In the literature, the oral AUC is 9.34 μg·h/L. Adjusting for the dose, the iv and oral AUCs are input into equation 4 to determine the absolute bioavailability F, which is 0.903. In addition, the AUC based on equilibrium can be calculated using another method. According to equation 3, after log C and t of every points in the elimination phase are linearly regressed, the β/2.303 (-0.152) and log B (0.256) can be obtained. Therefore, the equation representing the plasma cerivastatin concentration-time relationship at the elimination phase is as follows: By inputting t values from every time point into the linear equation 7, the corresponding drug concentrations is calculated (Table 1). We did so using DAS 3.1.5 to calculate the AUC following iv administration. The result was 5.22 μg·h/L. In the literature, the oral AUC was 9.34 μg·h/L for twice as much a dose. Hence, an F of 89.5% was obtained.

METHODS
A search of the literature from 1983 to 2014 was undertaken using PubMed (https://www.ncbi.nlm.nih.gov/pubmed) using the keywords "absolute bioavailability". We downloaded 171 papers, and only two of which included data regarding the plasma drug concentrations after iv administration. Thirty-two papers showed legible figures of plasma drug concentration-time curves. Microsoft Paint software, which has a digital scale, was used to convert the plasma drug concentrations from the concentration-time curves to numerical values for each time point. The recovered numerical values of plasma drug concentrations were then input into the DAS 3.1.5 software, and the information about the compartment model and the pharmacokinetic parameters, including AUC, B, and β, were calculated. In this case, we used only the ratio of the AUC calculated from the recovered plasma drug concentrations and the actual measurements was larger than 0.95 and smaller than 1.05, indicating that the recovered plasma drug concentrations were close to the actual measured ones for further study. In this study, we only analyzed the deviation of F based on the two-compartment model. As a result, 10 data sets, 2 of which were original data from published literature [6] and 8 of which were transformed from the plasma concentration-time graphs [5,[7][8][9][10][11][12], met the preconditions and were further analyzed.
Information on the types of administration, doses, AUCs drugs after ev administration as well as absolute bioavailability were recorded for 10 selected studies (Table 2). They all were described as having the two-compartment model, and used the actually measured drug concentrations. Regarding experimental subjects, one study used broiler chickens [11], one used Sprague-Dawley rats [12], and the rest used human subjects. All of the studies used oral administration as the ev administration method, except for one that used inhalation [7]. The information from the 10 studies are depicted in Table 2. The distribution-equilibrium plasma drug concentrations were calculated using equation 3, and then applied to calculate the AUC of the drugs after iv administration. Equation 5 can be also used to calculate AUCs based on the distribution equilibrium (AUCb). Table 3 demonstrate that the AUCb values were lower than those based on actual measurements (AUCa). The AUCb/AUCa ratio was 0.75±0.11. The absolute bioavailability based on the distribution balance (Fb) was calculated using equation 2, and the results showed that Fb was higher than the absolute bioavailability based on the actually measured concentrations (Fa). The values of Fa/Fb for the 10 data sets ranged from 0.462 to 0.880. The mean and standard deviation was 76%±12%, which suggests that the F value of drugs is underestimated using the present method.

Data presented in
A drug administered via an iv route has an absolute bioavailability of 1.0, whereas the absolute bioavailability of drugs administered by other routes is less than one unless it is fully bioavailable. AUC reflects the amount of a drug in the blood, and AUCev/AUCiv presents the absolute bioavailability following ev administration. The assumption herein, is that in both iv and ev doses have reached equilibrium in the body. This assumption is reached for both routes of administration when the drug follows the conventional one compartment model right after administration, which happens.
After iv administration of multi-compartment model drugs, on the other hand, the circulating drug concentration measured reflect the amount in the body only when pseudo-equilibrium has reached. Hence, the AUC following iv doses of drugs with multi-compartment is an over-estimation of that that reflects the amount that reached the permeable pars of the entire body. These results in underestimation of absolute bioavailability measured based on AUCev/AUCiv. To further calculate corresponding AUCs, plasma drug concentrations in the distribution equilibrium are calculated based on the formula for the plasma drug concentration-time relationship in the elimination phase. In addition, B/β is a simpler method for calculating the AUC. Although some differences in the results of the two methods exist, these differences are very small and negligible. Our results showed that the AUC values based on drug concentrations at the distribution equilibrium is 0.75 times those of the AUCs in the corresponding published papers. The average Fa calculated from the AUCiv values based on the actually measured concentrations is 76% of the Fb from the AUCiv values based on the distribution equilibrium, indicating that the F was underestimated by the present method, and corrections should be performed.
Drug distribution model can impact the application of the method. For a drug with a one-compartment model, the drug distribution reaches equilibrium immediately. After intravascular administration, there is an ignorable or no distribution phase. Therefore, the result of this extrapolated method is the same or similar one of the current method. So, the method is suitable for drugs with a two-compartment and multi-compartment, but not for one-compartment model. However, the number of drugs with one-compartment model is less.

CONCLUSION
The distribution of a drug after intravenous administration does not reach an equilibrium in the body during the distribution phase, and the AUC values based on the actual measurements of drug concentrations does not reflect the total amount of drug in the body. The absolute bioavailability values based on the AUCs from actual measurements of drug concentrations are lower than the actual bioavailability based on the AUCs from distribution-equilibrium drug concentrations. The present method underestimates the absolute drug bioavailability and should be modified.