Route Input Functions and Depot Mass Balance

From Administered Dose to Systemic Input

A dose is not the same thing as the amount that reaches systemic blood. After administration by a non-intravenous route, some drug may fail to enter the modeled route compartment, remain in a depot, be lost locally, or be absorbed but fail to reach systemic circulation. PBPK models need to keep these amounts distinct: otherwise, a model can appear to fit concentration data while violating mass balance or counting bioavailability twice.

The route-specific input function describes the rate at which drug crosses into systemic blood. It is a mass-per-time quantity, not a concentration and not necessarily the rate at which drug leaves the administration site. A route model can therefore have a large local release or absorption rate but a smaller systemic input rate.

For a general route, let \(D_{\mathrm{admin}}\) be the administered dose and \(D_{\mathrm{dep}}\) the amount initially placed in the modeled depot. If only a fraction \(f_{\mathrm{dep}}\) enters that depot, then \(D_{\mathrm{dep}}=f_{\mathrm{dep}}D_{\mathrm{admin}}\); the remainder is accounted for separately. Let \(A_{\mathrm{dep}}(t)\) denote the mass remaining in the depot, \(R_{\mathrm{abs}}(t)\) the rate at which drug leaves it by absorption, and \(A_{\mathrm{abs}}(t)\) the cumulative amount absorbed. If systemic availability is a constant fraction and absorption is the only route from depot to the absorbed side, then:

This proportional relationship assumes that the same fraction applies to each increment absorbed, with no delay or separate storage between absorption and systemic entry. If presystemic passage is delayed, or systemic availability varies over time, the rate relationship must represent that process rather than apply a single constant multiplier.

A Depot Balance and Its Conservation Identity

Consider a depot with first-order absorption and first-order local loss. First-order here means that each process rate is proportional to the amount currently in the depot. Let \(k_a\) be the absorption rate constant and \(k_d\) the local-loss rate constant, both in inverse time units. The depot balance is the incoming amount at time zero minus cumulative absorption and cumulative local loss.

With \(A_{\mathrm{dep}}(0)=D_{\mathrm{dep}}\), the solution is \(A_{\mathrm{dep}}(t)=D_{\mathrm{dep}}e^{-(k_a+k_d)t}\). Integrating the two outflow rates gives the cumulative absorbed amount and cumulative local loss:

The expressions follow directly by integrating \(k_aD_{\mathrm{dep}}e^{-(k_a+k_d)u}\) and \(k_dD_{\mathrm{dep}}e^{-(k_a+k_d)u}\) from \(u=0\) to \(u=t\). Adding the remaining depot amount and both cumulative outflows yields:

This identity is a useful diagnostic: at every time, the depot’s initial mass must be either still in the depot, absorbed, or locally lost. At infinite time, the absorbed fraction of depot mass is \(k_a/(k_a+k_d)\), not necessarily one. Systemic input is then only the systemically available fraction of that absorbed amount.

Keeping Absorption and Systemic Availability Separate

Absorbed amount describes transfer out of the route depot; systemic input describes subsequent entry into systemic blood. With constant \(F_{\mathrm{sys}}\), the absorbed amount is divided into systemic input and presystemic loss. It is incorrect to call the absorbed amount itself “systemic input” unless systemic availability is one. It is also incorrect to multiply a dose by availability and then apply the same availability factor again to the resulting systemic input.

A complete mass balance should count drug once, according to its current location or cumulative loss. For a model with no additional doses, the administered dose is partitioned into material currently remaining in modeled compartments plus cumulative losses, including losses in the route and after systemic entry. Cumulative systemic input is a transfer into systemic circulation, not a final disposition category to add alongside systemic elimination.

Coupling Route Input to a Central Compartment

Once drug enters systemic blood, a central-compartment balance describes what happens next. In a simplified one-compartment model with volume \(V_c\), concentration \(C_c=A_c/V_c\), and systemic clearance \(CL\), elimination occurs at rate \(CL\,C_c\). The central amount therefore obeys:

This simple form assumes constant \(V_c\) and \(CL\), and no explicit distribution to or from other compartments. In a multi-compartment PBPK model, the central balance also includes distribution transfers; those are internal transfers, not losses from the body. The systemic input function is the same conceptual boundary input, even when the downstream model is more detailed.

Assumptions, Diagnostics, and Modeling Choices

The first-order depot equations assume a well-defined depot amount, constant rate constants, and rates proportional to that amount. They also assume that absorption and local loss are the only depot exits. A route with dissolution limits, finite surface permeability, changing physiology, or multiple sequential route compartments may need a different structure. The balance principle still applies: each change in compartment amount must equal its inflows minus its outflows.

Useful diagnostics are direct consequences of the balances. Check that \(A_{\mathrm{dep}}(t)\) is nonnegative and does not increase in the no-input depot model; that cumulative absorption and cumulative loss do not decrease; and that cumulative absorption plus cumulative loss plus depot amount equals the initial depot amount. Also check that cumulative systemic input does not exceed cumulative absorption when \(0\leq F_{\mathrm{sys}}\leq1\). A failure may indicate an incorrect sign, unit mismatch, duplicated availability factor, or a missing pathway.

A prescribed systemic input function can be useful when the route kinetics are already characterized or when only the systemic boundary input is needed. A mechanistic depot model is preferable when the objective requires separate predictions for depot persistence, absorption, or local loss. A single availability multiplier is compact when presystemic processing can reasonably be treated as an instantaneous fraction; explicit presystemic compartments are more appropriate when delay or changing metabolism matters. These are different levels of representation, not interchangeable meanings of “dose.”

A common mistake is to treat “absorbed” as synonymous with “systemically available.” For example, multiplying the administered dose by the fraction entering a depot and then calling that whole amount systemic input ignores both incomplete absorption and presystemic loss. The correction is to label each transfer: administered dose to depot, depot to absorbed amount, absorbed amount to systemic input, and systemic input to central amount or elimination.