Why Compare Regimens on a Common Basis?
Dose selection often requires comparing regimens that differ in more than one practical feature. A dose amount may be given at a different interval, or the same amount may be administered over a different duration. Simulation can compare these candidates under a fitted nonlinear mixed-effects model, but the comparison is interpretable only when the candidates are evaluated against the same target definition and the same simulated population. Otherwise, a difference in estimated attainment may reflect a change in the comparison basis rather than a change in regimen.
The earlier material on probability of target attainment (PTA) defines the target event and the population probability of meeting it. Here the task is to organize regimen candidates so those quantities can be compared fairly. The target rule is fixed before the grid is simulated; the grid changes the regimen inputs, not the meaning of success or the population being represented.
For example, if the prespecified target is \(Q_i(c)\geq q^*\), then \(A[Q_i(c)]=1\) when the predicted quantity meets or exceeds \(q^*\), and it is zero otherwise. Other target rules are possible, but the rule must not be altered from one candidate to another during a comparison. In this tutorial, examples use a fixed threshold rule to make the bookkeeping explicit; they do not introduce a new PK/PD target.
Building the Candidate Grid
A grid starts with candidate values for each regimen component. A full factorial grid includes every combination of the selected dose amounts, intervals, and administration durations. A clinically constrained grid includes only combinations that are feasible or relevant to the intended use. In either case, each candidate should be recorded explicitly, with units, rather than identified by an ambiguous label such as “high dose.”
Useful arithmetic checks help expose mistakes before simulation. If \(D\) is given in milligrams and \(\tau\) in hours, the nominal daily amount for a regularly repeated regimen is \(D(24/\tau)\) milligrams per day when \(24/\tau\) is an integer, or the corresponding average amount per day \(24D/\tau\) more generally. For an infusion, the nominal administration rate is \(D/T_{\mathrm{inf}}\), with units of dose amount per unit time. These are descriptive regimen quantities, not substitutes for model-predicted target attainment.
Worked Example: Constructing a Feasible Factorial Grid
Suppose a hypothetical simulation will compare dose amounts of 400 mg and 600 mg, intervals of 8 h and 12 h, and administration durations of 1 h and 3 h. The fixed model and target definition will be used for every cell. The grid has \(2\times2\times2=8\) candidates.
| Dose amount | Interval | Duration | Nominal average amount per day | Administration rate |
|---|---|---|---|---|
| 400 mg | 8 h | 1 h | 1,200 mg/day | 400 mg/h |
| 400 mg | 8 h | 3 h | 1,200 mg/day | 133.3 mg/h |
| 400 mg | 12 h | 1 h | 800 mg/day | 400 mg/h |
| 400 mg | 12 h | 3 h | 800 mg/day | 133.3 mg/h |
| 600 mg | 8 h | 1 h | 1,800 mg/day | 600 mg/h |
| 600 mg | 8 h | 3 h | 1,800 mg/day | 200 mg/h |
| 600 mg | 12 h | 1 h | 1,200 mg/day | 600 mg/h |
| 600 mg | 12 h | 3 h | 1,200 mg/day | 200 mg/h |
For the first row, the average daily amount is \(400(24/8)=1{,}200\) mg/day, and the administration rate is \(400/1=400\) mg/h. For the 400 mg every 12 h candidate, these become \(400(24/12)=800\) mg/day and \(400/3=133.3\) mg/h for the 3 h administration. Each duration is less than its interval, so none of these examples has an administration still in progress when the next one is scheduled. Whether all candidates are clinically and operationally appropriate remains a separate feasibility judgment.
Keep the Model, Target, and Population Fixed
The simulation should represent a prespecified target population using the same fitted model for every candidate. In an NLME model, simulated individuals can differ through their covariates and individual random effects. Generate or define those individual-level inputs once, then apply each candidate regimen to those same virtual individuals. This matching makes the comparison reflect how regimen changes affect a common set of simulated people.
The model must be suitable for the intended comparison: its structural and random-effects components should represent the relevant PK or PK/PD behavior, the simulated covariates should represent the intended population, and candidate regimens should not require unsupported extrapolation. If the estimand includes parameter uncertainty, the uncertainty draws should also be handled consistently across candidates. The target definition—including its threshold, time frame, and any other required conditions—must likewise remain unchanged. Do not add residual error automatically: include it only if the defined target and estimand concern outcomes that include that observation-level variability.
A practical simulation table has one row per candidate and records the dose amount, interval, duration, feasibility status, model version, target rule, population simulation identifier, number of virtual individuals, and estimated PTA. This makes it possible to audit whether two reported estimates genuinely refer to the same comparison basis.
Write down the target event and target population before listing candidate regimens.
Record each candidate as \((D,\tau,T_{\mathrm{inf}})\), with units, and exclude or flag infeasible combinations before interpreting results.
Use the same covariates and individual-level model draws for each candidate so the target population is held constant.
Apply the fixed target rule to each individual’s prediction under every candidate, then calculate PTA and compare candidates in context.
Worked Example: Estimating PTA on the Same Eight Individuals
Consider three hypothetical candidates: A is 600 mg every 12 h administered over 1 h; B is 600 mg every 8 h over 1 h; and C is 600 mg every 12 h over 3 h. Suppose the prespecified target is the same dimensionless quantity \(Q\geq1.00\) for all three. The following model predictions are hypothetical and use the same eight simulated individuals in each column.
| Virtual individual | A: 600 mg, q12h, 1 h | B: 600 mg, q8h, 1 h | C: 600 mg, q12h, 3 h |
|---|---|---|---|
| 1 | 0.72 | 0.90 | 0.80 |
| 2 | 0.88 | 1.02 | 0.98 |
| 3 | 0.95 | 1.08 | 1.03 |
| 4 | 1.05 | 1.22 | 1.18 |
| 5 | 0.81 | 0.97 | 0.90 |
| 6 | 1.12 | 1.30 | 1.26 |
| 7 | 0.99 | 1.05 | 1.00 |
| 8 | 1.20 | 1.40 | 1.35 |
For A, individuals 4, 6, and 8 meet the target, so the estimate is \(3/8=0.375\), or 37.5%. For B, individuals 2, 3, 4, 6, 7, and 8 meet it, giving \(6/8=0.750\), or 75.0%. For C, individuals 3, 4, 6, 7, and 8 meet it; equality at 1.00 counts because the rule is \(Q\geq1.00\). Thus \(5/8=0.625\), or 62.5%. These are Monte Carlo estimates for this small hypothetical simulation, not exact population probabilities.
Why Matching Individuals Helps
The use of the same simulated individuals is not just tidy bookkeeping. It creates a paired comparison: for each individual, both candidate outcomes are evaluated under the same covariates and individual-level random effects. The difference between estimated PTAs can then be expressed as the average of individual differences in target indicators.
Each term is \(1\), \(0\), or \(-1\): it is \(1\) if the individual meets the target under B but not A, \(-1\) if the individual meets it under A but not B, and \(0\) if the outcomes agree. The average is therefore exactly the difference between the two estimated proportions. Matching does not change the definition of either PTA; it makes the comparison use the same simulated population.
Worked Example: Comparing Two Candidates by Matched Outcomes
Use candidates B and C from the table above. Under B, six of eight individuals meet the target; under C, five do. Only individual 2 meets the target under B but not C. No individual meets it under C but not B. The paired difference is therefore
This agrees with the direct subtraction \(0.750-0.625=0.125\), a difference of 12.5 percentage points in estimated PTA in favor of B for this simulated population. It does not establish that B is clinically preferable: the decision also depends on the prespecified decision criterion, practical constraints, model credibility, and other relevant outcomes.
Diagnostics, Alternatives, and Limits
Before interpreting a dose-regimen comparison, check that every candidate was simulated with the same target rule, population definition, model version, and number of simulated individuals. Verify units and dosing times, confirm that duration is feasible relative to interval, and inspect model predictions for implausible values or unsupported regimen conditions. A sudden jump in PTA deserves investigation: it may reflect a genuine regimen effect, but it can also result from a coding error, a changed target threshold, or mismatched individual draws.
A full factorial grid can reveal patterns in which the effect of changing dose amount depends on interval or duration. A one-factor-at-a-time comparison is easier to describe but can miss such combinations; it is most useful when the other regimen components are deliberately held fixed. Independently simulating a new population for every candidate can still produce valid separate PTA estimates, but it adds avoidable simulation noise to their difference. Shared individuals are generally preferable for comparison, though the realized precision gain depends on how similarly individuals respond across candidates.
A finite grid is not a proof that the best regimen has been found. Results apply to the simulated candidates, population, model, and target definition. A coarse grid can miss useful regimens between its listed values, and an apparently favorable candidate can be misleading if the model is poorly supported in that dosing region. Simulation is a structured comparison under assumptions, not a replacement for checking those assumptions or for clinical judgment.
- Changing the target mid-comparison. A PTA difference is not interpretable as a regimen effect if the threshold or target time frame differs across candidates.
- Resampling the population for every candidate. Separate samples can make one regimen appear better simply because its virtual individuals happened to be more favorable.
- Comparing only nominal daily amounts. Equal average daily amounts do not imply equal model-predicted target attainment when intervals or administration durations differ.
- Overinterpreting a small simulation. An estimated PTA is subject to Monte Carlo variability; report its simulation basis and avoid treating a small difference as definitive.
Check Your Understanding
Answer each question using the definitions and comparison principles above. Brief solutions follow.
- A candidate is 500 mg every 10 h, administered over 2 h. What are its average nominal amount per day and administration rate?
- Two regimens are compared using different target thresholds. Can their PTA estimates be interpreted as a fair regimen comparison? Explain.
- With the same simulated population, 14 of 20 individuals meet the target under regimen A and 16 of 20 under regimen B. What is the estimated PTA difference?
- Why can using the same virtual individuals across candidates be useful, and does it guarantee a smaller simulation error in every comparison?
Solutions. (1) The average amount per day is \(500(24/10)=1{,}200\) mg/day; the administration rate is \(500/2=250\) mg/h. (2) No. The target event must be fixed, or the PTA difference combines a regimen change with a change in the definition of success. (3) The estimates are \(14/20=0.70\) and \(16/20=0.80\), so B exceeds A by \(0.80-0.70=0.10\), or 10 percentage points. (4) Matching evaluates both regimens in the same simulated individuals and permits a paired comparison. It often reduces avoidable simulation noise, but it does not guarantee a smaller error in every realized comparison.