Simulating Concentration-Time Distributions

Why Summarize Concentration-Time Distributions?

A simulation from a fitted nonlinear mixed-effects model can produce many concentration-time profiles, one for each simulated subject or subject-occasion. At a fixed time, these profiles may differ because of between-subject variability, between-occasion variability, residual variability, or parameter uncertainty, depending on how the simulation was constructed. A time-specific summary makes the distribution of concentrations at that time easier to describe and compare across the dosing interval.

The key is to define exactly which simulated values are being summarized. A distribution across new subjects at a specified occasion answers a different question from a distribution across all subject-occasion records. The resulting summaries describe the simulated ensemble and the simulation design; they are not automatically summaries of every possible source of uncertainty.

For \(S\) simulated units, write the concentrations at time \(t\) in increasing order as \(c_{(1)}(t),\ldots,c_{(S)}(t)\). The summaries are calculated separately at each selected time. For clarity in the calculations below, empirical quantiles use the linear-interpolation convention \(h=1+(S-1)p\): if \(j\) is the integer part of \(h\) and \(\gamma=h-j\), then \(Q_p=(1-\gamma)c_{(j)}+\gamma c_{(j+1)}\), with the endpoints used when \(h\) is an integer. This convention gives a reproducible finite-sample summary; software packages can use different quantile conventions, so the convention should be stated when exact agreement matters.

The median is \(Q_{0.50}\). The interquartile range (IQR) is \(Q_{0.75}-Q_{0.25}\), a measure of the spread of the central half of the simulated values. The sample standard deviation describes spread around the sample mean \(\bar{x}\). For \(S\geq 2\), it is \(s=\sqrt{\sum_{i=1}^{S}(c_i-\bar{x})^2/(S-1)}\); for \(S=1\), the sample standard deviation is undefined because its denominator is zero. All of these concentration summaries have the same units as concentration.

How the Summaries Describe a Distribution

At each time \(t\), the population distribution can be represented by its cumulative distribution function \(F_t(c)=P(C(t)\leq c)\). Its \(p\)th quantile is the concentration at or below which a proportion \(p\) of the distribution lies. In a simulation, the sorted values approximate that distribution, and the empirical quantile is a finite-sample summary of those values. Thus, if the simulated 90th percentile at a particular time is 12 mg/L, approximately 90% of the simulated units have concentrations at or below that value at that time, subject to simulation variability and the chosen quantile convention.

A brief justification explains why the median is often useful. For a proposed center \(m\), consider the total absolute deviation \(L(m)=\sum_i|c_i-m|\). Between successive ordered observations, increasing \(m\) by a small amount reduces the deviations from observations above \(m\) and increases the deviations from observations below \(m\). The net change is determined by the difference between the counts on either side. The loss is minimized where at least half the values lie on each side: at a middle observation when \(S\) is odd, or anywhere between the two middle observations when \(S\) is even. The interpolated median convention above selects the midpoint in the even case. This is why the median is less influenced by a few extreme values than the mean, though it does not make the median immune to unusual values or poor simulation design.

Summarize the Intended Simulation Units

The index \(i\) must match the scientific question. If the target is a new subject at a specified occasion, take one value per simulated subject for that occasion. If the target is a randomly selected subject-occasion, then pooling subject-occasion values may be appropriate, provided the occasions and their weights represent that target. If the target is a particular occasion, summarize that occasion separately rather than mixing it with other occasions.

Pooling can change the distribution, not merely increase its sample size. For example, suppose each of two subjects has a concentration on each of two occasions. A pooled summary gives four values equal weight, while a subject-level summary may first choose or average within subjects, depending on the estimand. Repeated values from one subject are also dependent. They should not be treated as additional independent subjects just because they appear as separate rows.

Summaries Across a Time Grid

A concentration-time distribution is often summarized at a prespecified set of times \(t_1,\ldots,t_J\). At each time, calculate the desired summaries across the same intended type of simulation unit. Reporting the median together with lower and upper quantiles can reveal changes in both the typical concentration and the distribution’s spread over time. A time grid should be sufficiently informative for the intended description; summaries at a few sparse times can miss rapid changes between them.

Diagnostics, Alternatives, and Limitations

A compact table of medians and quantiles is useful, but diagnostics help determine whether it represents the simulated profiles adequately. At selected times, inspect the underlying simulated values or a distribution plot, check for implausible concentrations, and verify that every value corresponds to the intended time and simulation unit. If concentrations must be nonnegative under the model, negative simulated concentrations may indicate a model or residual-error specification issue rather than a reason to silently truncate values. Also check that concentrations from different occasions or dosing histories have not been combined unintentionally.

• Few simulated units: Quantiles are coarse when \(S\) is small. With \(S=1\), a median can be reported as the single value, but an empirical spread cannot be assessed; the sample standard deviation is undefined.
• Extreme values or skewness: Compare the median and IQR with the mean and sample standard deviation, and inspect the values. A mean can be informative for an average-concentration question, while the median describes the center by rank. Neither should be substituted for the other without regard to the target.
• Time-grid problems: A summary only describes the simulated values at its stated time. Coarse time spacing may conceal a short-lived peak or a rapid change; denser sampling or a justified interpolation of each profile may be needed before summarizing at additional times.
• Changing simulation composition: If some subjects or occasions are absent at certain times, the summaries may be based on different units at different times. Check the counts and clarify whether the changing composition is intended.

The median and quantiles are robust descriptions of location and spread, while the mean and standard deviation provide summaries based on arithmetic deviations. A full distribution plot can show multimodality or tails that a few quantiles conceal. These approaches are complementary: select summaries to answer the intended descriptive question, and retain enough information to notice features the summaries omit.

A time-specific median curve is formed by joining medians calculated separately at each time. It need not correspond to any one simulated subject’s trajectory. Likewise, the pointwise 10th and 90th percentiles at successive times do not identify a single set of subjects who remain between those percentiles throughout the interval. They describe marginal distributions at each time, not simultaneous trajectory-level coverage. The relationship between measurements within a profile is preserved in the simulated data, but it is not fully represented by separate time-specific quantiles.