Calculating Time Above and Below Exposure Limits

Why Calculate Time in an Exposure Region?

A concentration summary such as a peak, trough, or area under the curve compresses an entire profile into one number. That can conceal clinically relevant timing. Two simulated subjects can have similar AUC but spend different amounts of a dosing interval above an efficacy threshold. Likewise, a subject may have a high peak yet spend little time above the efficacy threshold, or spend a concerning duration above a safety limit.

For each simulated subject, the calculation begins with a concentration-time profile over a defined observation window. The window might be one dosing interval at steady state or a specified period after a dose; the interpretation depends on that choice. The calculation then measures the duration of the profile in each specified concentration region. Cohort time-based target attainment is the proportion of simulated subjects whose durations satisfy prespecified criteria.

The definitions use “at or above” for efficacy and “at or below” for safety. A protocol may instead specify strict inequalities; that choice should be made before analysis. For a smooth profile, a concentration that equals a threshold at only isolated crossing times contributes zero duration, so strict versus inclusive inequalities do not change the result. If a profile remains exactly at a threshold for a nonzero period, the chosen convention does matter.

The two durations are separate measures, not complementary parts of the interval. If \(E<S\), time between the thresholds is counted in both \(T_{E,i}\) and \(T_{S,i}\). The resulting targets might require at least a specified duration above \(E\) and no more than a specified duration above \(S\); that latter duration is \(\tau-T_{S,i}\) when equality occupies zero time. State precisely which time region and criterion are intended.

Locating Threshold Crossings

A simulator may provide concentrations at a finite set of times rather than a continuous formula. To calculate a duration, determine where the profile crosses each threshold, divide the interval at those crossing times, and add the lengths of the subintervals that meet the condition. The crossing estimate depends on how concentration is represented between simulated times. Use the simulator’s underlying model or a justified interpolation rule; do not treat a finite grid as though it were the complete continuous profile.

For example, under piecewise-linear interpolation between \((t_j,C_j)\) and \((t_{j+1},C_{j+1})\), a threshold \(K\) lying between the two concentrations is crossed at

This follows by writing the interpolated concentration as \(C(t)=C_j+(C_{j+1}-C_j)(t-t_j)/(t_{j+1}-t_j)\), setting \(C(t)=K\), and solving for \(t\). The formula applies when \(C_{j+1}\ne C_j\) and \(K\) lies between the endpoint concentrations. If both concentrations equal \(K\), the entire segment is at the threshold and the inequality convention determines how it is counted.

For a segment assumed to decline exponentially, a different interpolation is appropriate. If \(C_j>0\), \(C_{j+1}>0\), and \(C(t)=C_j\exp[-k(t-t_j)]\) on the segment, then \(k=\log(C_j/C_{j+1})/(t_{j+1}-t_j)\). Solving \(C(t_K)=K\) gives \(t_K=t_j+\log(C_j/K)/k\). This log-linear calculation is justified only when exponential decline is the chosen representation for that segment. A simulator that provides an analytic concentration function can instead be used to locate the roots directly.

Using a Log-Linear Decline

Concentrations often decline approximately exponentially over a terminal portion of a profile. When that is the model for a segment, log-linear interpolation uses the straight line between the log concentrations rather than between the concentrations themselves. It can yield a different crossing time from linear interpolation, particularly when concentrations change substantially across a long segment.

From Subject Durations to Cohort Attainment

A cohort target must be defined in terms of each subject’s calculated durations before the proportion is computed. For example, a subject might meet a target only if time above efficacy is at least \(L_E\) and time above the safety limit is no more than \(U_S\). Since \(T_{S,i}\) measures time at or below the safety limit, the latter condition can equivalently be written \(T_{S,i}\ge \tau-U_S\), provided the same window and threshold conventions are used.

For \(N\) simulated subjects, define \(I_i=1\) when subject \(i\) meets all required duration criteria and \(I_i=0\) otherwise. The cohort attainment estimate is the number meeting the criteria divided by the number simulated. This is an empirical proportion across the simulated cohort; it is not an average concentration or a fraction of clock time.

This is the time-based analogue of a subject-level attainment calculation: each simulated subject contributes one success or failure according to the prespecified rule. The calculation is meaningful only if the profiles cover the same relevant window, or if durations are appropriately expressed as fractions of each subject’s window. When windows differ, raw hours are not directly comparable without a clear target definition.

Assumptions, Diagnostics, and Limitations

The calculation requires a well-defined exposure measure, thresholds, and time window. Each simulated profile should represent the intended concentration quantity, such as the model-predicted plasma concentration, and should use consistent units. The window should match the clinical question: a single-dose period and a steady-state dosing interval are not interchangeable. If the question concerns a repeating steady-state interval, the profile must represent that interval rather than an arbitrary segment during accumulation.

A time grid approximates a continuous profile. Sparse output can miss a short excursion above a safety limit or a brief dip below an efficacy threshold, especially near a sharp peak or trough. Useful diagnostics include checking that all crossing times lie within the segments used to calculate them, confirming that each duration is between 0 and \(\tau\), and verifying that the counted segments cover the full interval without gaps or double counting. Refine the simulation time grid and repeat the calculation; material changes in crossing times or cohort attainment indicate that the original grid or interpolation is inadequate.

Also inspect the number and order of crossings. A profile may cross a threshold multiple times, so assuming one continuous above-threshold interval can be wrong. For piecewise-linear interpolation, a segment can have at most one crossing of a given threshold unless it lies exactly on that threshold. For a model-based profile, use all roots within the observation window and sort them before classifying the resulting intervals.

A frequent error is to count grid points above a threshold and multiply by the time step. That treats each point as a whole time bin and can bias the duration, particularly with irregular grids or when crossings occur between points. Instead, estimate crossing times under a stated profile representation and sum interval lengths. Another error is to report time below the safety limit as time above it; calculate the two regions explicitly, or use \(T_S=\tau-\text{time above }S\) only when the same complete window is covered.

Duration-based targets complement, rather than replace, other exposure summaries. Cmax, Cmin, and AUC answer different questions: for instance, Cmax does not specify how long concentration stays high, while AUC does not uniquely determine time above a threshold. A sampled-point proportion is a rough alternative when only discrete measurements are available, but it is not a continuous-time duration unless a time-bin interpretation is specified. Direct numerical root-finding on the simulated concentration function can be preferable to interpolation when that function is available; interpolation is practical when only a grid is retained, but its assumptions and resolution should be reported.

Finally, the cohort proportion depends on the simulated population, model, regimen, target definitions, and numerical calculation. It summarizes the simulated cohort under those conditions; it does not establish clinical efficacy or safety by itself. The duration target and the concentration threshold must have an appropriate clinical or pharmacological rationale.