Why Time Above Target Requires an Individual-Level Calculation
Some dose-selection targets specify that concentration must exceed a threshold for a required portion of a dosing interval. Whether an individual meets that target depends on the total time their concentration profile spends above the threshold, not merely on one concentration value or on the average concentration across the interval. A profile may cross the threshold several times, so the calculation must account for every above-target segment.
The interval used must match the estimand. For example, a calculation may use a steady-state interval or a first-dose interval, but these are different profile choices. Once the interval is specified, the calculation below applies to that interval. Concentrations and thresholds must use the same units, time must use a consistent unit, and the simulated profiles must correspond to the dose and regimen being evaluated.
The strict inequality \(C_i(t)>C_{\text{tar}}\) is a common definition of “above target.” A concentration that equals the threshold at one isolated crossing time contributes no duration to the integral, so using \(>\) or \(\ge\) makes no difference at such points. A profile that stays exactly at the threshold for a nonzero period is different: that period is excluded under \(>\) and included under \(\ge\). The convention should therefore be specified and applied consistently.
Finding the Above-Target Duration
For a continuous profile, crossing times divide the dosing interval into segments on which concentration is either above or below the target. Add the lengths of all above-target segments to obtain \(T_i\). If the concentration is available as a model function, crossing times can be found by solving \(C_i(t)=C_{\text{tar}}\). If it is available only at sampled times, an interpolation rule is needed to estimate where crossings occur.
With piecewise linear interpolation between two samples \((t_j,C_j)\) and \((t_{j+1},C_{j+1})\), a crossing between their concentrations occurs at:
This formula follows by setting the line through the two samples equal to \(C_{\text{tar}}\). It applies when \(C_{j+1}\ne C_j\) and the target lies between the endpoint concentrations. If the endpoints are on the same side of the threshold, there is no crossing within that linear segment. A segment that lies exactly at the threshold requires the strict-versus-inclusive convention described above.
After identifying all crossings, order them along with the interval boundaries. Determine which resulting segments are above target, then sum their durations. Do not assume there is only one crossing: absorption and elimination can produce an initial rise above target followed by a decline, and a more complex profile can move above and below the threshold several times.
Worked Example: One Above-Target Segment
Suppose a hypothetical individual has a 12-hour interval and a target of \(2\) mg/L. The model-predicted concentrations are \(1\) mg/L at 0 hours, \(3\) mg/L at 2 hours, \(4\) mg/L at 5 hours, \(1\) mg/L at 8 hours, and \(1\) mg/L at 12 hours. Use piecewise linear interpolation between samples and require at least half the interval above target, so \(q=0.50\).
On the first segment, concentration rises from 1 to 3 mg/L over 2 hours. The upward crossing is:
On the segment from 5 to 8 hours, concentration declines from 4 to 1 mg/L. The downward crossing is:
Under the piecewise linear profile, concentration is above target from 1 to 7 hours and below target elsewhere. Thus \(T_i=7-1=6\) hours, and \(F_i=6/12=0.50\). Since \(F_i\ge q\), this individual attains the target, exactly at the required fraction.
Worked Example: Multiple Threshold Crossings
Consider a hypothetical 12-hour profile with concentrations of 3, 1, 3, 1, 3, 1, and 1 mg/L at times 0, 2, 4, 6, 8, 10, and 12 hours, respectively. The target is \(2\) mg/L, interpolation is piecewise linear, and the required fraction is \(q=0.40\).
The profile starts above target. It crosses down between 0 and 2 hours at \(t=1\) hour. It crosses up between 2 and 4 hours at \(t=3\), then down between 4 and 6 hours at \(t=5\). It crosses up between 6 and 8 hours at \(t=7\), then down between 8 and 10 hours at \(t=9\). For example, the first crossing is \(0+(2-3)(2-0)/(1-3)=1\) hour; the other crossing times follow from the same interpolation formula.
The above-target segments are from 0 to 1 hour, 3 to 5 hours, and 7 to 9 hours. Their durations sum to \(1+2+2=5\) hours, so \(F_i=5/12=0.4167\). Because \(0.4167\ge0.40\), the individual attains the target. If the required fraction were \(0.50\), the same individual would not attain it.
From Individual Results to Population PTA
For each simulated individual, first apply the required-fraction rule to that individual’s profile. The population PTA is the proportion of simulated individuals who pass:
where \(N\) is the number of simulated individuals included in the defined population, and \(I(F_i\ge q)\) equals 1 when individual \(i\) attains the target and 0 otherwise. This is an estimate of the probability of attainment for the population and regimen represented by the simulations. The denominator must correspond to that defined population; excluding profiles because they fail the target would inflate the estimate.
Worked Example: Estimating PTA Across Simulated Individuals
Suppose six hypothetical simulated individuals have fractions of their 12-hour intervals above target of \(0.75\), \(0.60\), \(0.50\), \(0.49\), \(0.33\), and \(0.20\). The required fraction is \(q=0.50\).
The first, second, and third individuals meet the rule \(F_i\ge0.50\); the remaining three do not. Therefore:
The estimated PTA is 50% for these simulated individuals. The equality case matters: the individual with \(F_i=0.50\) attains the target because the rule is “at least” \(q\), not strictly greater than \(q\). The corresponding above-target durations are 9, 7.2, 6, 5.88, 3.96, and 2.4 hours; comparing each with \(0.50\times12=6\) hours gives the same three successes.
Assumptions, Diagnostics, and Alternatives
The calculation is meaningful only when its inputs and definitions match the dose-selection question. Check that the concentration target, concentration units, interval length, and required fraction are specified before classifying profiles. Use a profile for the intended dosing condition and time window. If the estimand concerns a latent model-predicted concentration, calculate from that profile; if it instead concerns a concentration process that includes residual variability, that variability must be represented consistently in the simulated profiles.
A numerical concentration grid does not reveal the true profile between sample times by itself. Piecewise linear interpolation defines one approximation, not a guarantee about the underlying model curve. Sparse samples can miss a brief excursion above target or conceal two crossings within a single grid segment. When the model function is available, root finding on \(C_i(t)-C_{\text{tar}}\) can locate crossings more directly. Alternatively, a denser time grid can improve a numerical approximation, but convergence should be checked by repeating the calculation at finer resolution and verifying that crossing times, individual classifications, and PTA are stable.
Useful diagnostics include plotting profiles near the threshold, checking every crossing in time order, and verifying that \(0\le T_i\le\tau\) and \(0\le F_i\le1\). For a piecewise linear profile, inspect every adjacent sample pair that straddles the threshold. Compare the sum of above-target segment lengths with the interval boundaries, and confirm that an individual exactly at \(q\) is handled according to the chosen equality rule. Profiles with results near \(q\) deserve particular attention: a small interpolation error can change an individual’s pass/fail classification.
A common mistake is to average the individual fractions first and then compare that mean with \(q\). In the six-person example, the mean fraction is:
Comparing \(0.4783\) with \(0.50\) would suggest a different answer from the individual-level PTA of \(0.50\). It is the wrong calculation because PTA asks what proportion of individuals meet the criterion, not whether the average individual fraction meets it. Calculate each \(I(F_i\ge q)\) first, then average those indicators.
There are several ways to obtain the time above target, but they answer the same defined question only when they use compatible profiles and crossing rules. Solving for crossings from a model function can be more accurate when reliable root finding is available. Piecewise linear interpolation is transparent and often practical for sampled predictions, but depends on the sampling density and interpolation assumption. Counting sampled time points above target is simpler, but generally approximates duration poorly unless the grid is sufficiently fine and the treatment of interval endpoints and time weights is explicit. None of these methods removes uncertainty in the model, target, or simulated population; the result is specific to those choices.
When the Result Is Useful
Time-above-target PTA links the shape of each concentration-time profile to a regimen-specific success rule. It distinguishes profiles that may have similar peak or overall levels but spend different portions of the interval above the target. It also makes the order of operations explicit: calculate duration for each individual, compare that individual’s fraction with the requirement, and then summarize the fraction of individuals who attain it. This discipline prevents a population average from being mistaken for a probability of individual success.
The result should be interpreted as the estimated probability of meeting the stated time-above-target criterion under the simulated population, dose, regimen, interval, and profile definition. It does not by itself establish that the target is clinically appropriate, or that the model accurately describes the population. Those are separate assumptions that must be considered when using the estimate for dose selection.
Check Your Understanding
Answer each question using the stated time-above-target definition.
- A 10-hour profile is above target for a total of 4.5 hours, possibly in several segments. What is its fraction above target, and does it meet \(q=0.45\)?
- Two adjacent samples are 1 and 5 mg/L, 3 hours apart. Under linear interpolation, when does the profile cross a target of 3 mg/L?
- Why is comparing the mean of individuals’ fractions above target with \(q\) not a calculation of PTA?
- A profile equals the threshold continuously from hour 2 to hour 3. What duration does this segment contribute under the strict \(C_i(t)>C_{\text{tar}}\) convention?
Solutions. (1) The fraction is \(4.5/10=0.45\), so it meets \(q=0.45\) under the “at least” rule. (2) The crossing occurs at \(t=t_j+(3-1)(3)/(5-1)=t_j+1.5\) hours, or 1.5 hours after the first sample. (3) PTA is the proportion of individuals whose own fractions meet the criterion; averaging fractions before applying the criterion does not count those individuals. (4) The segment contributes 0 hours under the strict inequality, because its concentration equals, but does not exceed, the target throughout.