Checking Steady State by Repeated-Dose Convergence

Why Check Repeated-Dose Convergence?

With repeated dosing, a PBPK model may take multiple dosing intervals to settle into a repeating concentration pattern. The first few intervals can differ substantially from later ones because drug is still accumulating in tissues, being eliminated from the body, or moving between compartments. A simulation that simply runs for a chosen number of doses does not, by itself, demonstrate that the predicted pattern has stabilized.

A practical check is to compare the same quantities in successive dosing intervals: for example, the concentration peak and trough in each interval. The comparison must use the same dose schedule, model inputs, units, and phase of the interval. A quantitative tolerance then makes “close enough” explicit. This is a numerical adequacy check for the specified outputs and purpose; it does not establish that the PBPK model is physiologically correct.

For a bolus dose, the concentration may jump at the dose time, so the peak convention must say whether the post-dose value is included. A trough is often the predose value, but the convention should match the intended use. For an infusion, a peak may occur at infusion end or at another time; use the maximum over the interval rather than assuming its timing. The earlier material on extracting predictions at observation times is useful for recording values at fixed phases. A true maximum or minimum may require additional evaluation between observation times.

A Quantitative Cycle-to-Cycle Criterion

A relative difference alone is unstable when a concentration is near zero. An absolute difference alone can be too permissive for a large concentration. A mixed absolute-relative criterion addresses both scales. For any quantity \(q\), define the change from interval \(k-1\) to interval \(k\) as acceptable when

Here \(\mathrm{atol}_q\) is an absolute tolerance in the units of \(q\), and \(\mathrm{rtol}_q\) is a dimensionless relative tolerance. Both are selected before reviewing the result, based on the intended precision and use. For peak and trough concentrations, they may share a concentration scale, but the absolute tolerance should still be appropriate when a trough is close to zero. Passing this test means the cycle-to-cycle change is small by the stated rule; it does not mean the change is exactly zero.

A useful operational rule is to require both peak and trough criteria to pass for two consecutive comparisons. This reduces the chance that one unusually small difference is mistaken for stable behavior. The number of consecutive passes is a reporting choice, not a universal mathematical guarantee. For models where internal distribution matters, also compare relevant PBPK states at a common phase, such as immediately before each dose.

Why the Comparison Works—and What It Does Not Prove

Represent the PBPK model state by a vector \(x(t)\) containing amounts or concentrations in its compartments. Between dose events, the state evolves according to the model’s differential equations. At a discrete dose, the state may change according to the specified dose event. If identical doses are administered at regular intervals, the forcing pattern repeats. At an exact periodic steady state, the state at a given phase of one interval equals the state at that same phase in the next interval, and the concentration trajectory repeats over the full interval.

This gives the justification for comparing corresponding peaks and troughs: at an exact repeating trajectory, the same interval-specific maximum and minimum recur. Conversely, small differences in those extrema provide evidence of practical stability for those reported outputs. They are not, on their own, a proof that every component of \(x(t)\) has stabilized. A slowly changing tissue amount might have little visible effect on plasma peaks over the intervals examined.

A stronger supporting check compares the full state at a common phase. For example, let \(x_k^-\) be the state immediately before dose \(k\). For state component \(j\), a mixed criterion can be applied as

The component-specific absolute tolerance carries the units of that state. This check can reveal changes hidden by the selected concentration outputs. It is particularly relevant when the intended prediction depends on tissue distribution, organ exposure, or a state that accumulates slowly.

Implementation, Diagnostics, and Alternatives

Several patterns indicate that the check may be misleading or the computation may need attention:

• Peaks pass but troughs fail. The chosen peak alone does not represent the whole interval; continue the simulation or examine the trough-specific behavior.
• Outputs pass while a slow tissue state changes. Compare internal states relevant to the scientific question, not just plasma extrema.
• Extrema are recorded only at a coarse grid of times. A grid can miss a true maximum or minimum. Refine evaluation around the extremum or use a reliable event or optimization procedure consistent with the model.
• Concentrations jump at doses or infusion events. An inconsistent pre-event/post-event convention can create a false difference. Apply the same event-side convention in every interval.
• Differences fluctuate rather than shrink. Check dose timing, interval indexing, event handling, solver settings, and whether the trajectory is genuinely approaching a repeatable pattern. A convergence criterion is not a substitute for diagnosing numerical behavior.
• Tolerances are chosen after seeing the values. This makes the decision difficult to interpret. Choose tolerances based on reporting precision or decision needs before assessing convergence.

An alternative is to simulate a fixed number of doses and assume that number is sufficient. This is simple, but may stop too early for slowly equilibrating systems or spend unnecessary computation on rapidly stabilizing ones. Comparing peaks and troughs adapts the stopping decision to the simulated behavior, but only for the chosen outputs. Comparing states at a common phase is more informative about internal distribution, although it requires defensible tolerances for every state and can be difficult when states have very different scales.

The convergence check is also subject to numerical error. If cycle-to-cycle differences are comparable to integration error, apparent stability may reflect solver accuracy rather than biological dynamics. The solver tolerances and event handling should therefore be adequate for the differences being interpreted. A separate numerical convergence assessment can help distinguish these sources; passing a repeated-dose criterion does not itself establish numerical accuracy.

Common Mistake and Practical Limits

A common mistake is to say, “The peak changed by less than one percent, so the model is at steady state.” That reasoning is incomplete: the trough may still be changing, a hidden compartment may still be accumulating, the peak may have been missed by sparse evaluation, or the criterion may be smaller than the numerical error. The correction is to state exactly which outputs were compared, use a prespecified absolute-relative criterion, check more than one successive comparison, and inspect relevant internal states when the application depends on them.

No finite simulation can demonstrate exact equality of all future states from a finite set of numerical comparisons. The defensible result is therefore practical: the specified quantities changed by less than stated tolerances over the examined intervals. The tolerances should reflect the intended use. A difference acceptable for a rounded exposure summary may be unacceptable when a small change affects a dose decision.