Why Presentation Is Part of Dose Justification
A dose-justification result is not fully interpretable from a single percentage. A statement such as “PTA was 80%” leaves open what exposure was simulated, which patients or covariates the simulation represents, what counted as attainment, and how much of the reported precision reflects finite simulation size. A useful presentation connects the exposure distribution to a clearly defined target and states the assumptions under which both were generated.
The earlier material on sensitivity analysis for target definitions, propagating parameter uncertainty into PTA, and diagnosing simulated exposure distributions provides the analytical foundations. Here the emphasis is on communicating results so that a reviewer can reconstruct the estimand, evaluate uncertainty, and see whether the recommendation follows from the displayed evidence.
Define the Target Before Showing the Results
A target must be stated in terms that determine exactly how each simulated individual is classified. Name the exposure metric (for example, AUC over a dosing interval or minimum concentration at steady state), its units and time window, and the inequality or range that defines success. For a joint target, specify whether all criteria must be met. “AUC and Cmin target” is insufficient unless the two limits and the rule for combining them are explicit.
For example, a hypothetical joint target might be \(AUC_{0-24,ss}\geq 100\ \text{mg·h/L}\) and \(C_{\min,ss}\geq 1.0\ \text{mg/L}\). An individual attains this target only if both conditions hold. The corresponding PTA is the proportion satisfying the conjunction, not the average of the two separate attainment percentages. If a target has both a lower and upper bound, define success as being within the range, rather than merely exceeding the lower bound.
Targets are decision criteria, not properties estimated from a simulation. State their source or rationale in the analysis report, and distinguish efficacy and safety criteria where relevant. A dose may have high attainment of an efficacy threshold while also producing concerning upper-tail exposure. The full exposure distribution helps reveal that pattern; one PTA value cannot communicate it by itself.
Worked Example: Summarizing a Simulated Exposure Distribution
Suppose a hypothetical simulation produces steady-state AUC values of 8, 9, 10, 11, 12, 13, 14, 16, 18, and 22 mg·h/L for ten simulated individuals. The analysis defines an efficacy target as AUC of at least 10 mg·h/L and a desirable range as 10 to 20 mg·h/L, inclusive. These ten values are illustrative; they are not patient data.
Distribution summary. The values are already sorted. The median for an even number of observations is the average of the fifth and sixth values:
Using the nearest-rank convention, the 10th percentile is the observation at rank \(\lceil 0.10\times10\rceil=1\), or 8 mg·h/L. The 90th percentile is at rank \(\lceil 0.90\times10\rceil=9\), or 18 mg·h/L. State the percentile convention because quantile algorithms can differ, especially in small samples.
Attainment. Eight of the ten values are at least 10 mg·h/L, so the estimate for the lower efficacy threshold is \(8/10=0.80\), or 80%. Seven values are between 10 and 20 mg·h/L inclusive, so attainment of the desirable range is \(7/10=0.70\), or 70%. The value 22 meets the lower threshold but exceeds the upper bound; that is why the two target definitions give different answers.
A presentation could show the points or a distribution plot alongside the target limits, and report the median and selected quantiles with units. With only ten simulated individuals, however, the apparent distribution is coarse and PTA is imprecise. This small example demonstrates the calculation, not a suitable simulation size for a dose recommendation.
Show Both the Distribution and the PTA Curve
An exposure-distribution plot should make the target visible. A histogram or density plot can show distribution shape; an empirical cumulative distribution function (ECDF) shows the fraction at or below each exposure value. Add vertical target limits and label the exposure metric, units, dose, regimen, population, and simulation condition. If comparing regimens, use consistent axes and target definitions so that visual differences are not artifacts of scaling or presentation.
A PTA curve places dose or regimen on the horizontal axis and estimated PTA on the vertical axis. Mark the prespecified PTA decision threshold, if one exists, and state whether the plotted points are based on the same simulated population and target at every dose. A curve displays how attainment changes across candidate regimens; it does not establish that any particular PTA threshold is universally acceptable. That threshold is a decision criterion which must be justified for the application.
A useful report pairs the curve with a table of selected regimens. Include the number simulated, number attaining, estimated PTA, and Monte Carlo standard error (MC SE). For exposure distributions, report a small set of interpretable summaries—often the median and lower and upper quantiles—rather than an excessive list of statistics. Include upper-tail summaries when safety limits matter. A plot and a table serve different purposes: the plot reveals shape, while the table provides exact values.
Worked Example: Reporting PTA and Monte Carlo Precision
In a hypothetical simulation of 1,000 independent individuals per regimen, the target is AUC of at least 100 mg·h/L. The simulated counts meeting that target are 620, 810, and 890 for doses of 100, 200, and 300 mg, respectively. For each dose, calculate PTA as the count divided by 1,000, then calculate MC SE using the binomial formula.
For 100 mg, \(\hat{p}=620/1000=0.620\). The MC SE in percentage points is:
For 200 mg, \(\hat{p}=810/1000=0.810\), and:
For 300 mg, \(\hat{p}=890/1000=0.890\), and:
| Dose | Meeting target | Estimated PTA | MC SE |
|---|---|---|---|
| 100 mg | 620 of 1,000 | 62.0% | 1.535 percentage points |
| 200 mg | 810 of 1,000 | 81.0% | 1.241 percentage points |
| 300 mg | 890 of 1,000 | 89.0% | 0.9895 percentage points |
For the 100 mg regimen, a rough normal-approximation interval for Monte Carlo error alone is \(62.0\%\pm1.96(1.535)\) percentage points, or approximately 58.99% to 65.01%. This approximation describes uncertainty from using 1,000 Monte Carlo draws to estimate the model-implied PTA; it does not include uncertainty about model parameters, model structure, or the target population. The dose table also does not by itself justify choosing 300 mg: that decision requires the target, safety considerations, and the intended decision rule.
Separate the Different Sources of Uncertainty
A report should label uncertainty according to what varies. Between-subject variability produces a distribution of exposures and therefore a distribution of target indicators. Monte Carlo error arises because a finite number of simulated individuals approximates the model-implied population PTA. Parameter uncertainty arises because the model parameters are estimated rather than known. These are related but not interchangeable quantities.
If parameter uncertainty is propagated, a typical presentation summarizes PTA across parameter draws. For each parameter draw, simulate the target population and calculate a conditional PTA; then describe the distribution of those PTA estimates across draws. Report the number of parameter draws and the simulation design within each draw. If the within-draw number of subjects is small, Monte Carlo noise can inflate the spread across draws. Conversely, a very large number of simulated subjects cannot remove uncertainty about the parameters or whether the model is appropriate.
Worked Example: Summarizing PTA Across Parameter Draws
Suppose, for illustration, five hypothetical parameter draws produce conditional PTA estimates of 76%, 80%, 83%, 78%, and 84% for the same dose and target. These values illustrate reporting calculations; five draws would generally be too few for a stable characterization of parameter uncertainty.
The mean across draws is:
The ordered results are 76%, 78%, 80%, 83%, and 84%; their median is 80%, and their observed range is 76% to 84%. For a measure of spread, the sample standard deviation across the five PTA estimates is:
Here each deviation is measured from the mean of 80.2%. The mean, median, range, and sample standard deviation describe these five simulated PTA values; they are not a patient-level confidence interval. In a substantive analysis, use enough parameter draws to estimate relevant quantiles reliably, report the chosen quantiles and method, and assess whether conditional Monte Carlo error is small relative to the across-draw variation.
Make the Simulation Auditable
A reader should be able to identify what the simulation represents and reproduce its logic. State the regimen (dose, interval, route, and duration as relevant), the population and covariate distribution, the exposure metric and time window, and whether results are for steady state or a specified treatment period. Describe how individual variability and parameter uncertainty were handled, the number of individuals simulated, and any assumptions about adherence, missed doses, or other design features that affect the simulated exposures.
The model must be used in a way consistent with its intended structure. For example, specify whether the reported exposure is a latent model-predicted exposure or includes residual observation error. Do not add residual error merely because concentration measurements in the original analysis were noisy: the target may concern underlying exposure rather than a noisy measured concentration. Conversely, if the decision criterion is explicitly based on measured concentrations, explain how measurement variability is represented. State any exclusions, truncations, or constraints on simulated covariates or parameters, since these can change the population distribution and PTA.
Useful checks include confirming that simulated concentration-time profiles behave as expected for the regimen, that exposure summaries are plausible, that target indicators implement the stated inequalities, and that increasing the simulation size stabilizes PTA to the expected degree. When using parameter draws, examine conditional PTA estimates and Monte Carlo errors rather than presenting a single uncertainty band without its construction. If diagnostics reveal implausible distributions or instability, investigate the simulation and assumptions before polishing the plot.
| Presentation approach | Strength | Limitation | Use in dose justification |
|---|---|---|---|
| Exposure histogram or density plot | Shows distribution shape and possible tails | Bin or smoothing choices can affect appearance | Pair with target limits and numerical quantiles |
| ECDF | Shows the fraction at or below each exposure | Can be less familiar and visually dense across many regimens | Useful for reading proportions relative to exposure thresholds |
| PTA curve | Compares attainment across doses or regimens | Hides distribution shape and depends on target definition | Pair with an explicit target and exposure summaries |
| Single median exposure | Compact summary of central tendency | Does not reveal tail behavior or target attainment | Use as a supplement, not a substitute for PTA or distribution plots |
Common Mistakes and Practical Limits
- Reporting PTA without the target. A percentage has no stable meaning unless the exposure metric, time window, threshold or range, and joint-target rule are given.
- Calling MC SE between-subject variability. MC SE describes finite simulation precision. The spread of individual exposures is a different feature of the simulated population.
- Treating parameter-draw quantiles as a confidence interval for patients. Such quantiles summarize results across parameter uncertainty under the specified model and assumptions; they are not a statement that a specified fraction of individual patients has an exposure in that range.
- Using a median as evidence of high PTA. A median locates the center of a distribution. It cannot show how many individuals meet a particular target, especially when the target is joint or includes an upper limit.
- Presenting an attractive curve without its assumptions. A smooth curve does not validate the model, target, population, or simulation design. State these explicitly and include diagnostics.
All summaries remain conditional on the simulated population, model, and target definition. A distribution plot can reveal tails but cannot establish that the model predicts those tails correctly. PTA is a model-based decision aid, not a guarantee of clinical outcome. The strongest presentation makes these limits visible while still giving decision makers enough detail to compare candidate regimens.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A joint target requires AUC at least 100 mg·h/L and Cmin at least 1.0 mg/L. Of 500 simulated individuals, 350 meet the AUC condition, 300 meet the Cmin condition, and 240 meet both. What is joint PTA?
- A simulation estimates PTA as 75% from 2,500 independent simulated individuals. Calculate the approximate Monte Carlo standard error in percentage points.
- Why does a median exposure above a lower target not establish that a large majority of individuals attain that target?
- What should be reported to distinguish uncertainty across parameter draws from finite-simulation error?
Solutions. (1) Joint attainment requires both criteria, so PTA is \(240/500=0.48\), or 48%; the separate attainment counts cannot be averaged. (2) \(100\sqrt{0.75(1-0.75)/2500}=100\sqrt{0.000075}=0.8660\) percentage points. (3) The median describes the center, not the fraction above a specified limit; distribution shape and the proportion meeting the target are needed. (4) Report the number of parameter draws, the conditional simulation size and MC error within draws, and the summary method and quantiles across parameter draws. Those summaries describe different sources of uncertainty.