Constraining Bioavailability Across Parallel Pathways

Why Dose Allocation Is Not Bioavailability

In a parallel absorption model, an administered dose may be divided between two routes of input, such as a prompt pathway and a delayed pathway. The fraction assigned to a pathway describes where the dose is allocated in the model; it does not by itself say how much of that dose reaches systemic circulation. A portion assigned to either pathway may be unavailable because of incomplete absorption or presystemic loss.

Keeping these quantities separate matters when interpreting parameters and predicting exposure. A pathway can receive most of the dose but contribute less systemic drug than a smaller pathway if its systemic availability is lower. The model must also respect the physical constraint that total systemic availability cannot exceed the administered dose.

Definition: Let $D$ be the administered dose and $q$ the fraction allocated to pathway 1, so $1-q$ is allocated to pathway 2. Let $a_1$ and $a_2$ be the fractions of the dose assigned to each pathway that reach systemic circulation through that pathway. The total absolute bioavailability is $$ F_{\mathrm{tot}}=q a_1+(1-q)a_2. $$ For physically interpretable dose fractions and availabilities, $0\leq q\leq 1$, $0\leq a_1\leq 1$, and $0\leq a_2\leq 1$. Thus $F_{\mathrm{tot}}$ is also between 0 and 1.

The corresponding administered amounts assigned to the pathways are $qD$ and $(1-q)D$. The amounts reaching systemic circulation are $qDa_1$ and $(1-q)Da_2$, respectively. Their sum is $DF_{\mathrm{tot}}$. In this parameterization, $a_j$ is conditional on the dose being assigned to pathway $j$, whereas $q$ is an allocation fraction.

This definition assumes that the pathway allocation is specified for the administered dose and that each pathway has a well-defined systemic availability. The availabilities include all losses between assignment to a pathway and entry into systemic circulation. They do not describe how quickly the systemic input occurs; absorption timing is represented separately by the pathway input functions.

Constraints and a Useful Bound

Because $F_{\mathrm{tot}}$ is a weighted average of $a_1$ and $a_2$, it lies between them. This follows directly from the nonnegative weights $q$ and $1-q$, which sum to one. If $a_1\leq a_2$, then $$ F_{\mathrm{tot}}-a_1=(1-q)(a_2-a_1)\geq 0 \quad\text{and}\quad a_2-F_{\mathrm{tot}}=q(a_2-a_1)\geq 0. $$ Therefore $a_1\leq F_{\mathrm{tot}}\leq a_2$. If $a_2\leq a_1$, the same argument with the labels reversed gives $a_2\leq F_{\mathrm{tot}}\leq a_1$. Since both pathway availabilities lie in $[0,1]$, the total does too.

The conditions $a_1,a_2\in[0,1]$ and $q\in[0,1]$ are sufficient for a valid total in this model. They are not interchangeable with a constraint on the sum of pathway allocation fractions: $q+(1-q)=1$ describes how the dose is divided, while $F_{\mathrm{tot}}$ describes the fraction of the dose reaching systemic circulation. In a model that instead defines $g_1$ and $g_2$ as fractions of the administered dose reaching circulation through each pathway, the appropriate constraints are $g_1\geq0$, $g_2\geq0$, and $g_1+g_2\leq1$. Those are systemic contribution fractions, not dose-allocation fractions.

When fitting a model, bounds can be built into parameters rather than checked only after estimation. For example, a logistic transformation maps an unconstrained value $z$ to $q=1/(1+\exp(-z))$, which is strictly between zero and one; the same construction can be used for $a_1$ and $a_2$. Exact boundary values require separate handling or a fitting method that permits boundaries. A constraint prevents impossible estimates, but it does not guarantee that the parameters can be estimated precisely from the data.

Worked Example: Converting Allocation Into Systemic Amounts

Suppose a hypothetical 500 mg dose is allocated 70% to pathway 1 and 30% to pathway 2. Assume conditional systemic availabilities of 0.80 and 0.40, respectively. Find the total bioavailability and the systemic amount contributed by each pathway.

The allocation fractions are $q=0.70$ and $1-q=0.30$. The amounts assigned are $500(0.70)=350$ mg and $500(0.30)=150$ mg. After applying the conditional availabilities, the systemic contributions are $350(0.80)=280$ mg and $150(0.40)=60$ mg.

Thus the total systemic amount is $280+60=340$ mg, and $$ F_{\mathrm{tot}}=\frac{340}{500} =0.70(0.80)+0.30(0.40) =0.56+0.12=0.68. $$ The total bioavailability is 0.68, or 68%. The 70% allocation to pathway 1 is not itself the total bioavailability.

Connection to Systemic Input and Exposure

To connect the fractions to a concentration-time model, let $h_1(t)$ and $h_2(t)$ be nonnegative pathway input-rate functions normalized so that each integrates to one over time. They describe the timing of input per unit of systemically available amount. The total systemic input rate is then

R_{\mathrm{sys}}(t) =D\left[q a_1 h_1(t)+(1-q)a_2 h_2(t)\right].

Integrating over time gives $$ \int_0^\infty R_{\mathrm{sys}}(t)\,dt =D\left[q a_1\int_0^\infty h_1(t)\,dt +(1-q)a_2\int_0^\infty h_2(t)\,dt\right] =DF_{\mathrm{tot}}. $$ This is a mass-balance justification for the weighted-sum formula: the pathway-specific time courses can differ, but their total integrated systemic input is determined by their systemic amounts.

Under linear disposition with clearance $CL$, complete exposure observation, and an appropriate intravenous reference, dose-normalized absolute bioavailability is related to AUC by $$ \mathrm{AUC}_{\mathrm{oral}}=\frac{DF_{\mathrm{tot}}}{CL}. $$ Here AUC has units of concentration times time, $D$ has units of amount, and $CL$ has units of volume per time. This relation assumes dose-proportional systemic disposition over the relevant range. If disposition is nonlinear, or AUC is not adequately captured, this simple exposure relation need not hold.

Worked Example: Comparing Dose Share With Systemic Contribution

For a hypothetical 500 mg dose, suppose $q=0.60$, $a_1=0.75$, and $a_2=0.50$. Calculate the dose assigned to each pathway, the systemic amount from each, and total bioavailability.

Pathway 1 receives $500(0.60)=300$ mg, of which $300(0.75)=225$ mg reaches systemic circulation. Pathway 2 receives $500(0.40)=200$ mg, of which $200(0.50)=100$ mg reaches systemic circulation.

The total systemic amount is $225+100=325$ mg, so $$ F_{\mathrm{tot}} =0.60(0.75)+0.40(0.50) =0.45+0.20=0.65. $$ Total bioavailability is 0.65, or 65%. Pathway 1 receives 60% of the dose but contributes $225/325\approx0.6923$, or about 69.2%, of the systemic amount. These percentages answer different questions: the first concerns dose allocation, and the second concerns the composition of systemic input.

Worked Example: Predicting AUC From Two Pathways

Consider a hypothetical 400 mg dose with $q=0.25$, $a_1=0.80$, and $a_2=0.60$. Suppose pathway 1 has first-order input rate $k_1=0.50\ \mathrm{h}^{-1}$, pathway 2 has rate $k_2=0.10\ \mathrm{h}^{-1}$, and systemic clearance is $CL=8.0\ \mathrm{L/h}$. For this example, let $h_j(t)=k_j\exp(-k_jt)$, which integrates to one for $k_j>0$. Find the total systemic amount, the initial input rate, and the AUC under linear disposition.

The systemic amount from pathway 1 is $400(0.25)(0.80)=80$ mg. The amount from pathway 2 is $400(0.75)(0.60)=180$ mg. Therefore, total systemic input is $260$ mg and $F_{\mathrm{tot}}=260/400=0.65$.

At $t=0$, each exponential input function has value $h_j(0)=k_j$. The total input rate is therefore $$ R_{\mathrm{sys}}(0) =80(0.50\ \mathrm{h}^{-1})+180(0.10\ \mathrm{h}^{-1}) =40+18=58\ \mathrm{mg/h}. $$ The units are amount per time. Under the stated linear-disposition assumptions, the AUC is $$ \mathrm{AUC}=\frac{260\ \mathrm{mg}}{8.0\ \mathrm{L/h}} =32.5\ \mathrm{mg\cdot h/L}. $$ The different input rates affect the concentration-time profile, but not this integrated AUC when the total systemic amount and clearance are fixed.

What the Data Can and Cannot Separate

A model with $q$, $a_1$, and $a_2$ has three quantities controlling systemic contributions, but an AUC measurement alone supplies only their weighted sum $F_{\mathrm{tot}}$. For example, if $q=0.50$, then $a_1=0.40$, $a_2=0.80$ gives $F_{\mathrm{tot}}=0.60$; so does $a_1=0.70$, $a_2=0.50$. The same total exposure does not identify which pathway had higher availability.

Concentration-time data may contain additional information when the pathways have meaningfully different timing profiles. Even then, estimates can be weakly determined if the input profiles are similar, observations miss the time periods that distinguish them, or other model parameters can compensate for changes in pathway contributions. A mathematically valid constraint is not evidence that the data identify every constrained parameter.

Three modeling choices are useful to distinguish. A shared availability model sets $a_1=a_2=a$, so $F_{\mathrm{tot}}=a$ regardless of allocation. A pathway-specific availability model estimates $a_1$ and $a_2$, with the total given by the weighted sum. A lumped availability model estimates only $F_{\mathrm{tot}}$ and does not claim to resolve pathway contributions. The shared model is simpler but assumes equal availability; the pathway-specific model is more flexible but needs enough information to support the extra parameters; the lumped model targets total systemic input without attributing it to pathways.

Diagnostics, Limitations, and a Common Mistake

Check both the parameter values and the implications of the fitted model. Allocation fractions and conditional availabilities should be in $[0,1]$, and the calculated total should satisfy $0\leq F_{\mathrm{tot}}\leq1$. Recalculate the systemic amount by adding the pathway contributions and confirm it equals $DF_{\mathrm{tot}}$. When estimates approach boundaries, or change substantially under reasonable alternative initial values or input assumptions, the data may provide limited information about the separate parameters. Such behavior warrants checking the fit and the identifiability of the parameterization, not simply removing the bounds.

A further limitation is that $a_j$ summarizes net systemic availability for a pathway; it does not, by itself, distinguish incomplete absorption from presystemic loss. Nor does a good fit establish that the assumed pathway decomposition is uniquely correct. Interpretation depends on the model definition and the information available in the observed data.

Wrong reasoning: “Sixty percent of the dose is assigned to pathway 1, so total bioavailability is 60%.”
Correction: Allocation is not availability. Multiply each allocation fraction by its conditional availability, then add the pathway contributions: $F_{\mathrm{tot}}=q a_1+(1-q)a_2$.
Wrong reasoning: “The pathway dose fractions and the pathway bioavailabilities must add to one.”
Correction: The two allocation fractions add to one. The conditional availabilities are each bounded by zero and one; their weighted sum gives total bioavailability.

Key takeaway: In a parallel absorption model, dose allocation determines how much is sent to each pathway, and conditional pathway availability determines how much of that allocation reaches systemic circulation. Total bioavailability is the allocation-weighted sum, not the allocation fraction for either pathway.

Check Your Understanding

Answer each question, then compare with the solutions below.

A 200 mg dose has $q=0.30$, $a_1=0.50$, and $a_2=0.90$. Find total bioavailability and the total systemic amount.
If $a_1=0.40$, $a_2=0.80$, and $q=0.25$, show that total bioavailability lies between the two pathway availabilities.
Two models have the same $F_{\mathrm{tot}}$ and the same clearance. Under linear disposition, what can be concluded about their complete AUCs, and what need not be the same?
Why can an AUC-only analysis fail to determine $a_1$ and $a_2$ separately?

Solutions. (1) $F_{\mathrm{tot}}=0.30(0.50)+0.70(0.90)=0.15+0.63=0.78$. The systemic amount is $200(0.78)=156$ mg. (2) $F_{\mathrm{tot}}=0.25(0.40)+0.75(0.80)=0.10+0.60=0.70$, which is between 0.40 and 0.80. (3) Their complete AUCs are equal because each equals $DF_{\mathrm{tot}}/CL$ for the same dose and clearance; their concentration-time profiles need not match if pathway timing differs. (4) AUC determines the weighted sum $q a_1+(1-q)a_2$, not generally the two conditional availabilities separately; different pairs can produce the same sum.