Repeated-Dose Steady-State Checks

Why Repeated-Dose Steady State Is Periodic

After a single dose, drug amounts and concentrations may rise and fall as absorption, distribution, and elimination proceed. With repeated dosing, the profile usually becomes similar from one dosing interval to the next. It does not become constant: concentrations can still fluctuate substantially within each interval. The relevant question is whether the same profile repeats at the same phases of successive intervals.

This distinction matters in PBPK models because a concentration measured just after a dose cannot be compared directly with one measured near the end of another interval. A valid comparison matches both the dosing phase and the time since dose. For a multicompartment model, the check should also consider whether the amounts in relevant tissues and input compartments repeat, rather than looking only at plasma.

Definition: For a dosing interval of length \(\tau\), repeated-dose steady state is a periodic solution in which the complete model state at time \(t+\tau\) equals the state at time \(t\), for every phase \(t\) in the interval. Thus \(x(t+\tau)=x(t)\). A concentration profile may vary within the interval while satisfying this condition.

Here \(x(t)\) denotes the vector of model states, such as drug amounts in the gut, blood, and tissues. Concentrations calculated from those states should also repeat at matching phases. This definition assumes the dosing schedule and model conditions remain unchanged, and that the model has reached a stable one-interval periodic response. In practice, a simulation approaches this response over successive intervals; its states need not match exactly at any finite interval.

Two Complementary Checks

The first check compares the profile in one interval with the profile at the same phases in the next. Select several phase points, including points where rapid absorption or distribution may occur, and compare concentrations and relevant amounts. A comparison of only pre-dose troughs can miss differences earlier in the interval. A comparison of only plasma can miss ongoing redistribution or drug accumulation in tissue.

One useful summary for a chosen state \(x_j\) is its relative discrepancy at a matched phase:

\delta_j(t)=\frac{|x_j(t+\tau)-x_j(t)|} {\max\{|x_j(t+\tau)|,\ |x_j(t)|,\ S_j\}}

Here \(S_j\) is a positive reference scale in the same units as \(x_j\), chosen to prevent an unstable ratio when the state is close to zero. For example, it might be a clinically meaningful concentration scale or a stated numerical scale appropriate to an amount. Report the scale and the largest discrepancy across the states and phases examined. There is no universal tolerance: it should be set in relation to the solver accuracy and the purpose of the simulation. A small discrepancy supports convergence; it does not establish that the model structure or inputs are correct.

The second check uses mass balance. Let \(A_{\mathrm{tot}}\) be the total amount of the tracked drug across the model states. During an interval, internal transfers between compartments cancel when the compartment balances are added. The change in total amount therefore reflects external inputs and external losses:

A_{\mathrm{tot}}(t+\tau)-A_{\mathrm{tot}}(t) = D_{\mathrm{in}}-A_{\mathrm{elim}}-A_{\mathrm{other}}

In this expression, \(D_{\mathrm{in}}\) is the total dose or other external drug input during the interval, \(A_{\mathrm{elim}}\) is the amount eliminated, and \(A_{\mathrm{other}}\) represents any other modeled external loss. At periodic steady state, the left side is zero. Consequently, interval input equals interval elimination plus other external losses. This condition applies to the defined model boundary: for an oral model, for example, the bookkeeping must include the dose entering the modeled gut state and any separately represented losses. A parent-drug mass balance may count metabolism as loss of parent even if metabolites are not included as states.

Key result: A convincing repeated-dose steady-state check combines phase-matched profile comparisons with interval mass balance. Profile repetition tests whether the modeled states recur; the balance check tests whether the interval’s net external input is consistent with no net change in tracked amount.

Worked Examples

Worked Example: Checking the One-Compartment Benchmark

Consider a hypothetical one-compartment model with IV bolus doses of 100 mg every 12 h, volume \(V=20\) L, and first-order elimination rate constant \(k=0.20\ \mathrm{h}^{-1}\). Check the expected post-dose steady-state concentration and trough, then compare the result with the first dose.

For a bolus, each dose raises concentration by \(D/V=100/20=5.00\) mg/L. Let \(C_n^+\) be the concentration just after dose \(n\). Over the next interval the concentration decays by the factor \(e^{-k\tau}\), after which the next bolus adds \(D/V\). Thus:

C_{n+1}^+=C_n^+e^{-k\tau}+\frac{D}{V}

At the repeating profile, \(C_{n+1}^+=C_n^+=C_{\mathrm{ss}}^+\). Solving the recurrence gives:

C_{\mathrm{ss}}^+=\frac{D/V}{1-e^{-k\tau}} =\frac{5.00}{1-e^{-(0.20)(12)}} =\frac{5.00}{1-0.09072} =5.499\ \mathrm{mg/L}

The corresponding trough is \(C_{\mathrm{ss}}^+e^{-k\tau}=(5.499)(0.09072)=0.4989\) mg/L. After only the first dose, the post-dose concentration is 5.00 mg/L and the first trough is \((5.00)(0.09072)=0.4536\) mg/L. The first interval has not yet reached the repeating profile. This calculation is a useful benchmark for a simulation of this same linear model; it is not a substitute for checking all states in a PBPK model.

Worked Example: Comparing Matching Phases Across PBPK Intervals

Suppose a hypothetical PBPK simulation uses doses every 12 h. The following values are recorded in two late intervals, at matching times since dose. Concentrations are in mg/L.

State and phase	Interval 5	Interval 6
Plasma, 1 h after dose	1.240	1.246
Plasma, 6 h after dose	0.720	0.722
Plasma, just before next dose	0.410	0.411
Liver, 6 h after dose	0.980	1.010

Compare the values phase by phase. For plasma 1 h after dose, the absolute difference is \(|1.246-1.240|=0.006\) mg/L. Using the larger value as the denominator gives a relative discrepancy of \(0.006/1.246=0.004815\), or 0.482%. At the 6 h plasma phase it is \(0.002/0.722=0.002770\), or 0.277%; at the trough it is \(0.001/0.411=0.002433\), or 0.243%.

The liver difference is \(|1.010-0.980|=0.030\) mg/L, and its relative discrepancy is \(0.030/1.010=0.02970\), or 2.970%. The largest discrepancy in this table is therefore in liver, not plasma. The plasma profile appears close to repeating, but the larger liver difference warrants checking whether liver amounts and other tissue states continue to rise. These selected concentrations alone do not prove that the complete model state has converged.

Worked Example: Using Interval Mass Balance to Detect Residual Accumulation

A hypothetical PBPK model receives 100 mg by IV input during each interval. In one interval, the integrated amount eliminated is 99.60 mg, with no other external losses. The total tracked amount at the interval end is 0.40 mg greater than at the start. Is the interval consistent with steady state?

The mass-balance residual, defined as the observed change in total amount minus the change predicted by external flows, is:

r=\Delta A_{\mathrm{tot}}-(D_{\mathrm{in}}-A_{\mathrm{elim}}) =0.40-(100.00-99.60)=0.00\ \mathrm{mg}

The accounting is internally consistent, but the interval is not at steady state: \(\Delta A_{\mathrm{tot}}=100.00-99.60=0.40\) mg, rather than zero. The model is still accumulating drug. In a later interval, suppose elimination is 99.98 mg and the total amount rises by 0.02 mg. The mass-balance residual is again \(0.02-(100.00-99.98)=0.00\) mg, so the accounting still closes, but the nonzero change shows that steady state has not been reached exactly. If the change becomes negligible relative to an explicitly chosen scale and the matched profiles also repeat within the justified tolerance, the evidence for practical convergence is stronger.

Practical Checking and Failure Modes

Define the interval and model boundary.
Record the dose schedule, interval length \(\tau\), tracked drug states, and all external inputs and losses included in the balance.

Compare matching phases.
Choose times since dose across the interval, including peaks, transition periods, and troughs where relevant. Compare plasma and important tissue or input states.

Check interval amounts.
Calculate the total amount at both interval boundaries and compare its change with input minus elimination and other external losses.

Assess sensitivity of the check.
Examine whether the conclusion changes with a denser set of phase points or a tighter numerical tolerance. State the scales and tolerances used.

A common mistake is to compare a peak in one interval with a trough in the next and call the difference evidence against steady state. Those values are at different phases, so the comparison does not test repetition. Compare peak with peak, trough with trough, and intermediate points with the same time since dose.

Another pitfall is to infer steady state because plasma troughs look stable. A slowly equilibrating tissue or residual drug in an absorption compartment may still be accumulating. Conversely, a small nonzero mass-balance residual can reflect numerical integration error rather than meaningful accumulation. Compare the residual with the scale of the dose and tracked amounts, and distinguish a balance residual from the actual change in total amount.

Profile comparison and mass balance answer related but different questions. A profile comparison can reveal where in the interval states differ, while interval balance summarizes net external input and loss. An analytical accumulation calculation, such as the one-compartment example, provides a benchmark when its assumptions apply. It does not generally capture nonlinear or multicompartment PBPK behavior. None of these checks alone establishes that physiology, parameter values, or model structure are appropriate; they verify properties of the simulated repeated-dose behavior.

Key takeaway: Repeated-dose steady state means a repeating trajectory, not a flat concentration. Match interval phases, include states that could still accumulate, and verify that the interval change in total tracked amount is negligible and consistent with external input and loss.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A concentration at 2 h after dose \(n\) is compared with a concentration at 8 h after dose \(n+1\). Why is this not a phase-matched steady-state check?
During a 24 h interval, 250 mg enters the model, 245 mg is eliminated, and there are no other losses. What change in total tracked amount does mass balance predict? Is this interval at steady state?
Why can a repeated plasma trough fail to demonstrate whole-model steady state?
In the one-compartment bolus example, explain why the steady-state post-dose concentration is greater than the concentration after the first dose.

Solutions. (1) The observations are at different times since dose, so they may differ simply because the within-interval profile changes. Compare the same phase, such as 2 h after each dose. (2) The predicted change is \(250-245=5\) mg. Since the change is not zero, the interval is not at steady state. (3) A tissue or input compartment may continue accumulating or redistributing drug even while the plasma trough changes little. Check other relevant states and interval mass balance. (4) Residual drug remains when the next dose is given; the recurrence adds the new dose-related concentration increment to the residual from the preceding interval.

Repeated-Dose Steady-State Checks

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