From AUC and Dose to an Absolute Bioavailability Estimate
An oral dose does not necessarily deliver the same amount of drug to systemic circulation as an intravenous (IV) dose. Absolute bioavailability, denoted \(F\), compares systemic exposure after the test route with exposure after IV administration, which serves as the reference route. In practice, this comparison uses the area under the concentration–time curve (AUC) and the administered dose for each route.
The calculation is most straightforward when the oral and IV data refer to the same drug, use compatible AUC measures, and have AUC and dose units that can be compared. The examples below treat the supplied AUC values as appropriate summaries of exposure for the comparison; they do not address how AUC is estimated from concentration measurements.
This is the operational form of the dose-normalized AUC relationship established in the earlier material on deriving absolute bioavailability. In a numerical calculation, retain the dose terms until you confirm that the administered doses are equal. If the dose terms cancel, the AUC ratio is appropriate; otherwise, the ratio of AUCs alone is not the stated calculation.
Assumptions and a Unit Check
A calculated value is useful only insofar as the measurements and comparison support its interpretation. Before using the formula, check the following conditions.
- Comparable exposure measures: The two AUCs should represent comparable exposure for the intended comparison. For example, do not casually divide an AUC over one time window by an AUC over a different window.
- Same drug and compatible measurements: The oral and IV AUCs should quantify the same drug using compatible concentration measurements. If AUC units differ, convert them before taking a ratio.
- Dose proportionality over the relevant range: The formula interprets dose-normalized AUCs as comparable measures of systemic exposure per administered dose. This interpretation requires that the relevant pharmacokinetics support dose normalization; marked dose-dependent exposure can undermine it.
- Comparable conditions: Conditions that materially affect exposure, such as clearance or the timing and conduct of the comparison, should not differ in a way that distorts the route comparison. A numerical calculation alone cannot establish that this condition holds.
For a concentration measured in mg/L and time in hours, AUC has units mg·h/L. Dividing by a dose in mg gives h/L. Thus, the numerator and denominator of the \(F\) formula have matching units, and their ratio is dimensionless. The unit cancellation is a practical check: a result that still carries units usually signals an incompatible or incorrectly entered quantity.
A short justification clarifies why equal doses permit the simpler calculation. Starting from the dose-normalized ratio, divide by a fraction by multiplying by its reciprocal:
If \(\mathrm{Dose}_{\mathrm{IV}}=\mathrm{Dose}_{\mathrm{oral}}\), the dose ratio is 1, leaving the AUC ratio. If the doses are not equal, that cancellation does not occur. The formula therefore gives a direct way to check whether an AUC-only calculation is justified.
Worked Example: Equal Doses With AUC in Matching Units
In a hypothetical comparison, the oral and IV doses are each 100 mg. The oral AUC is 12.6 mg·h/L, and the IV AUC is 18.0 mg·h/L. Calculate \(F\).
The AUCs are for the same drug, have the same units, and are assumed to be comparable exposure measures. The doses are equal, so the dose terms cancel:
The estimated absolute bioavailability is \(F=0.700\), or 70.0%. Under the assumptions of the comparison, the oral route produced 70.0% of the IV reference exposure per administered dose. The fraction is unitless; 70.0% is its percentage expression.
Worked Calculations and Unit Conversions
A formula can be applied correctly and still produce a wrong result if the numerical inputs are not on a common scale. Convert units before calculating, and keep track of the numerator and denominator in the dose-normalized ratio.
Worked Example: Converting AUC Units Before the Ratio
Suppose, hypothetically, both routes use a dose of 80 mg. The oral AUC is reported as 8,000 ng·h/mL, while the IV AUC is 16 mg·h/L. Calculate \(F\).
First convert the oral AUC. Since 1 ng/mL equals 1 \(\mu\)g/L, 8,000 ng·h/mL equals 8,000 \(\mu\)g·h/L, or 8.00 mg·h/L. Now both AUC values use mg·h/L. Equal doses permit cancellation:
The estimate is 0.500, or 50.0%. The conversion is essential: using the numerical values 8,000 and 16 directly would mix scales and give a meaningless ratio. The assumption that the AUCs are comparable and the dose response permits dose normalization remains necessary; converting units does not establish those assumptions.
Worked Example: Retaining the Dose Terms
In a hypothetical comparison, an oral dose of 120 mg produces an AUC of 15.0 mg·h/L. An IV dose of 60 mg produces an AUC of 10.0 mg·h/L. Assume the AUC measures are comparable, units match, and dose normalization is appropriate. Calculate \(F\).
The doses differ, so the AUCs cannot be divided on their own. Calculate the dose-normalized AUC for each route, then take their ratio:
The estimated absolute bioavailability is 0.750, or 75.0%. As an arithmetic check, the expanded form gives \((15.0/10.0)(60/120)=1.50\times0.500=0.750\). Dividing the AUCs alone would yield 1.50, which is not the dose-adjusted estimate.
Interpreting the Result and Finding Errors
An estimate such as \(F=0.700\) is interpreted as 70.0% absolute bioavailability under the assumptions used for the comparison. It is a route comparison of dose-normalized systemic exposure, not a direct count of drug molecules crossing the gut wall. It also is not automatically a statement about the fraction physically absorbed; that distinction requires additional concepts.
The value of \(F\) is a fraction, so a value of 0.70 is reported as 70%, not 0.70%. In a basic calculation, a value between 0 and 1 is readily expressed as a percentage between 0% and 100%. If the arithmetic gives a value outside that range, check the doses, units, AUC definitions, and calculations before interpreting it. A discrepancy can also reflect limitations in the comparison or assumptions; the arithmetic by itself does not identify the cause.
A useful diagnostic is to calculate the dose-normalized AUCs separately and inspect their units before taking the ratio. Another is to expand the formula and confirm the direction of the dose correction: the IV dose is in the numerator of the dose ratio, and the oral dose is in its denominator. This follows directly from dividing oral dose-normalized AUC by IV dose-normalized AUC.
The main shortcut is to compare AUCs directly when doses are equal and the AUC units match. Its advantage is simplicity; its limitation is that it is invalid when doses differ. The general dose-normalized formula handles unequal dose values arithmetically, but its interpretation still depends on the validity of dose normalization and comparable exposure measurements. Neither approach can repair poor comparability, inappropriate AUC windows, or a failure of the underlying assumptions.
In a numerical report, state the inputs, the calculation, and the result with its interpretation. For example: “Using comparable AUC measures and equal 100 mg doses, \(F=12.6/18.0=0.700\), corresponding to an estimated absolute bioavailability of 70.0%.” This makes clear whether doses were equal and how the result was obtained.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- With equal oral and IV doses, the AUCs are 9.0 and 15.0 mg·h/L, respectively. What is \(F\), and how is it expressed as a percentage?
- Oral AUC is 4.0 mg·h/L at a dose of 40 mg; IV AUC is 12.0 mg·h/L at a dose of 60 mg. Calculate \(F\).
- Why should an AUC in ng·h/mL be converted before it is compared with an AUC in mg·h/L?
- A calculation gives \(F=0.65\). State an appropriate interpretation and one assumption it depends on.
- Someone divides oral AUC by IV AUC when the oral dose is twice the IV dose. What is the problem with that reasoning?
Solutions. (1) \(F=9.0/15.0=0.600\), or 60.0%. (2) \(F=(4.0/40)/(12.0/60)=0.100/0.200=0.500\), or 50.0%; both dose-normalized AUCs are in h/L. (3) The numerical scales differ; conversion is needed so the AUC ratio is meaningful and unitless. (4) It is an estimated absolute bioavailability of 65.0% under the comparison assumptions, including appropriate dose normalization. (5) Unequal doses do not cancel. Use the ratio of dose-normalized AUCs, not the raw AUC ratio.