Choosing Intervals for Peak-Trough Constraints

Why Compare Both Peak and Trough?

Some concentration targets define a therapeutic window: concentrations should be high enough at the end of an interval to satisfy a lower limit, but not so high after a dose that they exceed an upper limit. The interval between doses affects both parts of this requirement. A schedule with an acceptable peak can still have a trough below the target; a schedule with a satisfactory trough can still produce an excessive peak.

The decision is therefore about joint attainment in the population: what proportion of individuals meet the upper peak limit and the lower trough limit under the same regimen? Simulating each candidate interval with the same population model gives a direct way to compare that proportion while retaining the relationship between each individual's peak and trough.

Definition: For a specified regimen and a specified steady-state dosing interval, let $C_{\max,ss,i}$ be individual $i$'s modeled maximum concentration during the interval and $C_{\min,ss,i}$ the minimum concentration during that interval. Given a lower limit $C_L$ and an upper limit $C_U$, individual $i$ meets the peak–trough constraints if $$ C_{\min,ss,i} \ge C_L \quad\text{and}\quad C_{\max,ss,i} \le C_U. $$ The joint peak–trough probability of target attainment (PTA) is the probability that both inequalities hold for an individual drawn from the target population under the specified regimen.

The limits must be defined for the clinical question at hand, with concentration units and the relevant time point or interval made clear. In this tutorial, “peak” and “trough” mean the modeled maximum and minimum at steady state over one dosing interval. If the target instead concerns a first dose, a sample at a particular time, or a different concentration summary, the simulation must calculate that quantity instead.

A Simple Steady-State Derivation

A one-compartment model with instantaneous bolus doses and first-order elimination provides a useful check on the calculations. Assume a constant dose $D$ is given every $\tau$ time units, volume $V$ and elimination rate constant $k$ are fixed for an individual, and elimination is linear. The concentration contribution from a dose declines exponentially. At steady state, the post-dose peak is the sum of the current dose's contribution and all remaining contributions from earlier doses.

C_{\max,ss} = \frac{D}{V}\sum_{j=0}^{\infty} e^{-jk\tau} = \frac{D/V}{1-e^{-k\tau}}, \qquad C_{\min,ss} = C_{\max,ss}e^{-k\tau} = \frac{D}{V}\frac{e^{-k\tau}}{1-e^{-k\tau}}.

The sum is a geometric series with ratio $e^{-k\tau}$, which lies between 0 and 1 when $k>0$ and $\tau>0$. Its sum is $1/(1-e^{-k\tau})$. Multiplying the post-dose concentration by $e^{-k\tau}$ gives the concentration just before the next dose. This derivation is a check for this simple model, not a substitute for simulating the model actually used for dose selection. In a nonlinear pharmacokinetic model, or with a different administration type, the peak and trough should be obtained from that model's predicted concentration-time profile.

Worked Example: Checking Three Intervals for One Hypothetical Individual

Suppose a hypothetical individual's one-compartment parameters give $D/V=5$ mg/L per dose and $k=0.14$ h$^{-1}$. Consider fixed doses every 8, 12, or 24 hours. The hypothetical concentration limits are $C_L=1.2$ mg/L and $C_U=8$ mg/L.

For $\tau=8$ h, $e^{-k\tau}=e^{-1.12}\approx0.32628$. Thus $$ C_{\max,ss}=\frac{5}{1-0.32628} =\frac{5}{0.67372} \approx 7.421\text{ mg/L}, \qquad C_{\min,ss}=7.421(0.32628)\approx2.421\text{ mg/L}. $$ The peak is below 8 mg/L and the trough is above 1.2 mg/L, so this individual meets both limits at the 8-hour interval.

For $\tau=12$ h, $e^{-k\tau}=e^{-1.68}\approx0.186374$. Therefore $$ C_{\min,ss} = 5\frac{0.186374}{1-0.186374} = \frac{0.931870}{0.813626} \approx 1.145\text{ mg/L}. $$ The corresponding peak is $5/0.813626\approx6.145$ mg/L. Although that peak is below the upper limit, the trough is below 1.2 mg/L, so the individual fails the joint constraint.

For $\tau=24$ h, $e^{-3.36}\approx0.0347353$. The peak is $5/(1-0.0347353)\approx5.180$ mg/L, and the trough is $5(0.0347353)/(1-0.0347353)\approx0.180$ mg/L. The peak passes, but the trough fails. This example demonstrates how a regimen can satisfy the upper constraint while missing the lower one.

From Individual Profiles to a Population Decision

In a nonlinear mixed-effects model, individuals can have different parameter values because of between-individual variability and covariates. A population simulation draws individual parameter vectors from the model's population distribution, calculates a concentration-time profile for each simulated individual under each candidate regimen, and records that individual's peak and trough. This yields simulated distributions of both quantities as well as their joint behavior.

For $N$ simulated individuals, let $I_i=1$ if individual $i$ meets both constraints and $I_i=0$ otherwise. The simulated joint PTA is the fraction meeting both:

\widehat{\mathrm{PTA}}_{\mathrm{joint}} = \frac{1}{N}\sum_{i=1}^{N} I_i = \frac{\#\{i:C_{\min,ss,i}\ge C_L,\ C_{\max,ss,i}\le C_U\}}{N}.

The candidate regimen must be specified completely: dose, interval, administration method, and the population and model used to generate profiles. If comparing intervals while holding the dose per administration fixed, state that choice. If the practical comparison holds total daily dose fixed or changes the dose along with the interval, simulate those actual regimens instead. The resulting comparison answers different dosing questions.

Worked Example: Comparing Three Candidate Intervals in a Hypothetical Population

Suppose 1,000 hypothetical virtual individuals are simulated at steady state for each candidate interval using the same fitted population model and the same dose per administration. The hypothetical limits are $C_{\min,ss}\ge1.2$ mg/L and $C_{\max,ss}\le8$ mg/L. The simulated counts are:

Interval	Peak at or below 8 mg/L	Trough at or above 1.2 mg/L	Both constraints met
8 h	850	980	840
12 h	960	850	820
24 h	995	420	415

For the 8-hour interval, the joint PTA is $840/1000=0.840$, or 84.0%. For 12 hours it is $820/1000=0.820$, or 82.0%. For 24 hours it is $415/1000=0.415$, or 41.5%. Each joint count is no greater than either of its marginal pass counts, as required.

Suppose the decision rule, specified before comparing the results, is to choose the longest interval whose estimated joint PTA is at least 80%. The 12-hour interval is the longest candidate that meets this rule at 82.0%; the 24-hour interval does not. This conclusion is conditional on the hypothetical model, population, dose comparison, constraints, and decision rule. Because 82.0% is relatively close to the 80% threshold, it would be sensible to check that Monte Carlo sampling variation is not materially affecting the decision.

Why Separate Attainment Rates Are Not Enough

The peak and trough are measured on the same individual's profile, so their attainment events are generally dependent. Individuals with particular combinations of clearance, volume, or other model parameters can have systematically different peaks and troughs. Knowing the proportion that passes the peak limit and the proportion that passes the trough limit does not by itself identify the proportion passing both.

Worked Example: Marginal Pass Rates Can Mislead

In a separate hypothetical simulation of 100 individuals under one regimen, 90 individuals meet the peak limit and 90 meet the trough limit, but only 70 meet both. The joint PTA is $70/100=70\%$. Each marginal pass rate is $90/100=90\%$, yet the regimen does not attain an 80% joint-PTA decision threshold.

Multiplying the two marginal rates gives $0.90\times0.90=0.81$, or 81%. That multiplication would represent the joint probability only if peak and trough attainment were independent. No such independence assumption is justified here, and the simulated joint count directly shows that the actual joint fraction is 70%. Count both conditions for each individual rather than combining marginal percentages.

A Practical Comparison and Its Checks

A disciplined comparison starts by defining the clinical constraints and the decision rule, then calculates both concentration summaries from each simulated profile. The following sequence keeps the comparison tied to the intended population and regimen:

Specify the question.
State the lower trough and upper peak limits, their units, whether the targets apply at steady state, the candidate regimens, and the rule used to choose among them.

Generate comparable profiles.
Use the same target population and population model for each regimen. Calculate the modeled maximum and minimum over the relevant interval for every simulated individual.

Count joint attainment.
For each individual, check both inequalities on that individual's profile. Divide the number passing both by the number simulated.

Inspect and compare.
Review the peak and trough distributions, the joint pass counts, and the simulated concentration-time profiles. Apply the stated selection rule only after checking that the simulation and model predictions are plausible.

For a simple linear model, an analytic calculation such as the geometric-series derivation can be a useful validation check, and individual parameter values may allow direct comparison of candidate intervals. For a population with random effects, covariates, or nonlinear pharmacokinetics, simulation is often more practical because it produces joint peak–trough outcomes across the population. Separate percentile comparisons are not a replacement for joint counting: two marginal summaries do not show how many individuals meet both conditions.

Diagnostics should include checking that simulated concentrations are in the intended units, peak and trough are taken over the correct interval, and the simulated state corresponds to the intended dosing history. Inspect representative profiles for implausible shapes, unexpected accumulation, or a peak or trough occurring at an unexpected time. Check that each joint count is no larger than either marginal count. If an estimated PTA is near the decision threshold, increase the number of simulated individuals and see whether the decision is stable; Monte Carlo error can affect the estimate even when the underlying model is held fixed.

Common pitfall: It is wrong to choose an interval because its median peak and median trough both meet their limits, or because the two marginal PTA estimates look adequate. The medians may describe different individuals, and marginal pass rates do not determine joint attainment. Instead, evaluate both inequalities for every simulated individual and report the fraction passing both.

A simulated PTA is conditional on the specified model and population distribution; it does not establish that the model is correct or that the regimen will achieve the same performance in every setting. Results can depend on covariate representation, the fitted pharmacokinetic model, and how the simulated population reflects the population for which the regimen is intended. A regimen should therefore be described with its assumptions and target population, not as universally satisfactory. This comparison focuses on the two specified concentration constraints; other clinical endpoints or broader benefit–risk decisions require additional criteria.

Check Your Understanding

Answer each question using the definitions and decision logic in this tutorial.

In a simulation of 500 individuals, 430 meet the peak limit, 400 meet the trough limit, and 350 meet both. What is the joint PTA?
Why is it not generally valid to multiply the peak pass rate by the trough pass rate to obtain joint PTA?
For a simple repeated-bolus model with $k>0$, what happens to the peak and trough formulas if $e^{-k\tau}$ is replaced by its value for a longer interval?
A candidate interval has estimated joint PTA of 78%, while the prespecified threshold is 80%. The peak and trough marginal pass rates are both above 80%. Does this candidate meet the stated joint decision rule?

Solutions. (1) The joint PTA is $350/500=0.700$, or 70.0%. (2) Peak and trough attainment are outcomes from the same individual profile and may be dependent; multiplying marginal probabilities assumes independence. (3) A longer interval makes $e^{-k\tau}$ smaller. In the stated fixed-dose linear model, the denominator $1-e^{-k\tau}$ increases, so the peak $ (D/V)/(1-e^{-k\tau}) $ decreases, and the trough $ (D/V)e^{-k\tau}/(1-e^{-k\tau}) $ also decreases. (4) No. The decision rule concerns the joint PTA, which is 78% and falls below 80%, regardless of the two marginal pass rates.