Regional Contributions to Cumulative Absorption

From Simulated Flux to Regional Contribution

A segmental absorption model can report an absorption flux for each gut region at every time point. A flux is an instantaneous rate, not an amount: a region with a large flux for a short interval may contribute less total absorbed drug than a region with a smaller flux that persists longer. To compare regions over a simulation, convert each flux into its cumulative absorbed amount, then express that amount relative to the total across regions.

The calculations describe absorption across the boundary represented by the model’s flux. They do not, by themselves, establish how much drug reaches systemic circulation: subsequent processes such as gut-wall metabolism or hepatic first-pass extraction are outside this regional accounting unless explicitly included in the flux definition.

Amounts have units of mass, such as mg, while fractions are dimensionless. At a fixed time \(T\), the regional contribution fractions sum to one, provided the listed regions exhaust the fluxes included in the total:

This fraction answers, “What share of the absorption represented in this model by time \(T\) came from region \(i\)?” It is not the same as the fraction of the administered dose absorbed. If the oral dose is \(D\), then \(A_i(T)/D\) is the fraction of that dose absorbed through region \(i\) by time \(T\), and \(A_{\mathrm{tot}}(T)/D\) is the total fraction absorbed by that time. These dose-based fractions sum to \(A_{\mathrm{tot}}(T)/D\), which may be below one.

Why Integration Gives Cumulative Absorption

The amount absorbed over a short interval is approximately flux multiplied by interval duration. Summing those contributions over time and taking the limit as the intervals become small gives the integral of flux. The result relies on the flux being integrable over the time interval; piecewise-continuous fluxes commonly used in simulations satisfy this condition.

The cumulative amount \(A_i(t)\) is absolutely continuous when \(J_i(t)\) is integrable. Its derivative equals \(J_i(t)\) almost everywhere, not necessarily at every time point. For a continuous flux, the derivative identity holds at every point of continuity. The integral identity for cumulative amount remains valid even when the flux has jumps.

This distinction matters when interpreting simulated profiles. If a flux jumps at a time boundary, the cumulative amount remains continuous, but its left and right slopes can differ. For example, if \(J_i(t)=3\) mg/h before 2 h and \(J_i(t)=1\) mg/h after 2 h, the cumulative amount has left slope 3 mg/h and right slope 1 mg/h at 2 h; its derivative at exactly 2 h is not defined. Integration across the intervals is still well-defined.

Calculating Amounts from Simulated Time Grids

Simulation output is often available at discrete time points rather than as a symbolic flux function. If \(J_i(t_k)\) is reported at times \(t_0,\ldots,t_K\), a common approximation is the composite trapezoidal rule. Each adjacent pair of flux values is treated as the endpoints of a straight line, and the area of the resulting trapezoid approximates absorption over that interval.

This approximation is most reliable when the time grid is fine enough to capture changes in flux. A large time step across a sharp peak, rapid transition, or discontinuity can miss substantial area. For a known piecewise-constant flux, integrating each constant interval directly is preferable to interpolating between output points as though the flux changed linearly.

Time Patterns and Regional Ranking

A regional contribution is time-specific. A region can dominate early absorption without supplying the largest amount by the end of the simulation. The cumulative amount curve records the area accumulated so far; the cumulative fraction curve compares that area with the total accumulated across regions at the same time. Both curves can be useful when the question concerns not only the final amount but also when each region contributes.

Checks, Comparisons, and Limitations

Several checks help identify errors before interpreting regional shares. First, confirm that flux and time units are compatible: mg/h multiplied by h gives mg. Second, verify that all regions are included exactly once in the total. Third, check that nonnegative regional fluxes yield nondecreasing cumulative amounts. Finally, check that the regional amounts sum to the integrated total flux and that contribution fractions sum to one, apart from rounding.

If the model represents only some regions, the computed fractions describe shares within that represented set, not necessarily shares of all intestinal absorption. If the fluxes are net fluxes and can be negative, cumulative amounts need not be nondecreasing, and the resulting ratios may not behave like shares of nonnegative absorption. In that case, the sign convention and scientific meaning of the net flux must be stated; do not label a signed ratio an absorption contribution without justification. Likewise, a total of zero makes \(F_i(T)\) undefined, not zero.

There are two common calculation approaches. Direct integration of the model’s continuous or piecewise-defined flux is appropriate when its functional form is available. Numerical quadrature, such as the trapezoidal rule, is practical for time-grid output but introduces discretization error. Comparing refined-grid estimates is a useful diagnostic: if the estimated amounts change materially as the grid is refined, the original grid may not resolve the flux profile. Comparing regional peak fluxes is not an alternative calculation of contribution; it answers a different question about instantaneous rates.

Regional fractions also depend on the chosen endpoint \(T\). Fractions at an early time describe the composition of absorption accumulated so far; they need not equal eventual fractions. Conclusions should therefore state the integration interval and the flux boundary being counted. Regional amounts from a simulation are model-based quantities: they describe the specified model and inputs, and alone do not establish that the regional pattern has been identified from observed data.