Deriving Relative Bioavailability Between Formulations

Why Compare Two Extravascular Formulations Directly?

Suppose the same active drug is given in two oral formulations. The practical question may be whether one formulation produces greater systemic exposure than the other. An intravenous (IV) reference is needed to estimate absolute bioavailability, but it is not needed to compare the formulations with each other. The reason is that, under appropriate conditions, both formulations’ exposures contain the same systemic-clearance term, which cancels in their ratio.

The comparison must account for dose. If one formulation is given at a larger dose, its AUC may be larger simply because more drug was administered. Dose-normalized AUC puts exposures on a common administered-dose scale. The resulting ratio describes relative exposure, not either formulation’s absolute fraction reaching the systemic circulation.

Definition: Let $T$ denote a test extravascular formulation and $R$ a reference extravascular formulation. Their relative bioavailability is the ratio $F_T/F_R$, where $F_T$ and $F_R$ are the fractions of the respective administered doses reaching systemic circulation. Under linear pharmacokinetics and shared systemic clearance, it is estimated by the ratio of dose-normalized AUCs: $$ F_{\mathrm{rel}}=\frac{F_T}{F_R} =\frac{\mathrm{AUC}_T/D_T}{\mathrm{AUC}_R/D_R}. $$

Deriving the Dose-Normalized AUC Ratio

For formulation $j$, where $j$ is $T$ or $R$, let $D_j$ be the administered dose and $F_j$ the fraction of that dose reaching systemic circulation. Assume the drug has linear pharmacokinetics over the doses being compared, the systemic clearance $CL$ is the same after either formulation, and AUC is measured over a period that captures the exposure from the dose. For a complete AUC to infinity, the systemic amount entering the circulation is $F_jD_j$. With linear clearance, the total amount eliminated is $CL$ times the integrated concentration. Equating these amounts gives:

F_jD_j=CL\cdot \mathrm{AUC}_{j,0-\infty} \quad\Longrightarrow\quad \mathrm{AUC}_{j,0-\infty}=\frac{F_jD_j}{CL}.

Applying this relation to both formulations and dividing their dose-normalized AUCs cancels the common clearance:

\frac{\mathrm{AUC}_{T,0-\infty}/D_T} {\mathrm{AUC}_{R,0-\infty}/D_R} = \frac{(F_T/CL)}{(F_R/CL)} = \frac{F_T}{F_R}.

This is the key justification for a direct extravascular comparison. The IV reference is not required because the shared clearance cancels; no separate estimate of $CL$ is needed. In contrast, an absolute bioavailability calculation compares an extravascular formulation with IV administration to estimate a formulation’s availability relative to complete systemic input. A relative comparison estimates only $F_T/F_R$. Without additional information, it does not identify $F_T$ and $F_R$ separately.

Assumptions for the interpretation: The formulations contain the same active drug, the doses are expressed in comparable amounts of active drug, systemic clearance is shared, and exposure is dose-proportional over the doses used. The AUCs must represent comparable exposure intervals and be measured consistently. A crossover comparison in the same participants helps control differences in clearance between people; a comparison across separate groups requires the groups to be sufficiently comparable for clearance and other relevant factors.

Calculating the Ratio

When doses are equal, dose normalization does not change the numerical ratio: the relative bioavailability equals the test-to-reference AUC ratio. When doses differ, each AUC must first be divided by its own dose. A ratio above 1 indicates greater dose-normalized AUC for $T$ than for $R$; a ratio below 1 indicates less. A ratio of 1 indicates equal dose-normalized AUC in the data, subject to study variability and the assumptions above.

Worked Example: Equal Doses

Hypothetical data. The same participants receive 100 mg of each of two extravascular formulations on separate occasions. The measured AUC to infinity is 2,400 ng·h/mL for $T$ and 2,000 ng·h/mL for $R$. Assume the doses are in the linear range, the same analyte is measured, and systemic clearance is unchanged between occasions.

The dose-normalized AUCs are: $$ \frac{\mathrm{AUC}_T}{D_T} =\frac{2{,}400\ \mathrm{ng\cdot h/mL}}{100\ \mathrm{mg}} =24.0\ \mathrm{ng\cdot h/mL/mg}, $$ $$ \frac{\mathrm{AUC}_R}{D_R} =\frac{2{,}000\ \mathrm{ng\cdot h/mL}}{100\ \mathrm{mg}} =20.0\ \mathrm{ng\cdot h/mL/mg}. $$

Therefore, $$ F_{\mathrm{rel}}=\frac{24.0}{20.0}=1.20. $$

The test formulation produced 1.20 times the dose-normalized AUC of the reference formulation, or 20% greater dose-normalized exposure in this comparison. This does not mean that 120% of the test dose reached systemic circulation; the result is a ratio between two formulations, not an absolute fraction.

Worked Example: Unequal Doses

Hypothetical data. Formulation $T$ is given at 120 mg and produces an AUC of 3,000 ng·h/mL. Formulation $R$ is given at 80 mg and produces an AUC of 1,800 ng·h/mL. Assume linear dose proportionality across 80–120 mg, shared clearance, and comparable AUC measurement.

Normalize each exposure using its own administered dose: $$ \frac{\mathrm{AUC}_T}{D_T} =\frac{3{,}000}{120} =25.0\ \mathrm{ng\cdot h/mL/mg}, \qquad \frac{\mathrm{AUC}_R}{D_R} =\frac{1{,}800}{80} =22.5\ \mathrm{ng\cdot h/mL/mg}. $$

The relative-bioavailability estimate is: $$ F_{\mathrm{rel}}=\frac{25.0}{22.5}=1.111. $$

Thus the test formulation’s dose-normalized AUC is about 1.11 times the reference’s, or about 11.1% greater. The raw AUC ratio is $3{,}000/1{,}800=1.667$, but that number does not isolate formulation differences because the test dose was also larger. The linearity assumption is essential: if AUC does not scale proportionally with dose, simple dose normalization may not make the comparison valid.

Why No IV Reference Is Needed

An IV reference would provide information about systemic exposure when the entire dose enters the circulation. For a relative comparison, however, both extravascular formulations are measured against each other. Their dose-normalized AUCs are $F_T/CL$ and $F_R/CL$. Since the systemic clearance is common, the ratio removes it. This remains true even when the clearance is unknown.

Worked Example: A Relative Estimate Without Knowing Clearance

Hypothetical data. In a study with no IV dosing, dose-normalized AUC is 40 ng·h/mL/mg for formulation $T$ and 32 ng·h/mL/mg for formulation $R$. Assume the two formulations share clearance and the other conditions for the ratio are met.

The relative estimate is: $$ F_{\mathrm{rel}}=\frac{40}{32}=1.25. $$

The test formulation therefore has 1.25 times the dose-normalized AUC of the reference in this comparison. The same result follows from the model: $40=F_T/CL$ and $32=F_R/CL$, so $40/32=(F_T/CL)/(F_R/CL)=F_T/F_R$. Because $CL$ and an IV exposure were not supplied, neither $F_T$ nor $F_R$ can be calculated separately from these data. The relative ratio is nevertheless calculable because the shared clearance cancels.

Conditions, Diagnostics, and Limitations

The algebra is simple, but it supports a formulation interpretation only when its assumptions are credible. A crossover study is often useful because each participant receives both formulations, so a participant’s clearance is shared across their two observations. Adequate separation between periods and appropriate study conduct are needed so that one period does not influence the next. In a parallel-group study, differences in participant characteristics or clearance can contribute to the exposure ratio, so the ratio may not reflect formulation alone.

Dose proportionality also matters. If pharmacokinetics are nonlinear, changing the dose can change AUC per unit dose. In that case, dose-normalized AUCs at unequal doses need not differ solely because of formulation. Similarly, if systemic clearance changes between treatment periods, the clearance terms do not cancel as assumed. Dose normalization cannot correct either problem.

AUC estimation should be comparable between formulations. For AUC to infinity, inspect the concentration-time profiles and the fraction of the estimated AUC that is extrapolated beyond the last observation. If one formulation has substantially less complete sampling or a much larger extrapolated fraction, its estimate may be less reliable. Consistent bioanalytical methods, adequate sampling, and correct dose records are also necessary. These checks address whether the inputs to the ratio are trustworthy; they do not, by themselves, prove that the model assumptions hold.

The AUC ratio describes the relative extent of systemic exposure under the stated assumptions. It does not, by itself, describe the rate of absorption, establish that two formulations are equivalent, or identify the absolute availability of either formulation. Those are different questions requiring other measures or inferential procedures.

Common pitfall: Dividing the test AUC by the reference AUC when doses differ treats the doses as if they were equal. The correction is to calculate $\mathrm{AUC}_T/D_T$ and $\mathrm{AUC}_R/D_R$ separately, then divide those normalized values. Another error is to call $F_T/F_R=1.20$ an absolute bioavailability of 120%; it means only that the test-to-reference ratio is 1.20.

Check Your Understanding

Use the model and definitions above. Solutions are included so that each calculation can be checked.

Two formulations are given at the same dose. Their AUCs are 1,500 and 1,250 ng·h/mL for $T$ and $R$, respectively. What is the relative-bioavailability estimate, and how should it be interpreted?
Formulation $T$ has an AUC of 2,400 ng·h/mL after 120 mg; formulation $R$ has an AUC of 1,500 ng·h/mL after 75 mg. Calculate the dose-normalized ratio.
Why can an IV reference be omitted when estimating $F_T/F_R$, and what cannot be learned from this ratio alone?
A ratio is calculated from two extravascular AUCs at unequal doses. The drug shows nonlinear pharmacokinetics over that dose range. Is simple dose normalization enough to isolate a formulation effect? Explain.

Solutions. (1) Equal doses cancel, so $F_{\mathrm{rel}}=1{,}500/1{,}250=1.20$. The test formulation has 1.20 times the reference’s dose-normalized AUC in this comparison; this is not an absolute bioavailability of 120%. (2) The normalized values are $2{,}400/120=20.0$ and $1{,}500/75=20.0$ ng·h/mL/mg. Their ratio is $20.0/20.0=1.00$, indicating equal dose-normalized AUCs in these data. (3) Each dose-normalized AUC is $F/CL$; with shared clearance, $(F_T/CL)/(F_R/CL)=F_T/F_R$. The ratio alone does not identify either absolute $F$. (4) No. With nonlinear pharmacokinetics, AUC per unit dose may change with dose, so dose normalization does not necessarily remove the dose effect. The resulting ratio cannot be attributed solely to formulation without a model or design that accounts for the nonlinearity.