Predicting Event Risk Under Repeated Dosing

From a Repeated-Dose Profile to Event Risk

A repeated-dose regimen produces a concentration-time profile, not a single exposure value. Peaks, troughs, accumulation, and the time spent at different concentrations can all matter when drug concentration changes the instantaneous event hazard. To predict risk over a follow-up period, the relevant quantity is the cumulative hazard obtained by integrating the hazard along the regimen’s concentration profile.

The calculations below use a concentration-dependent proportional-hazards model. The same integration principle applies to other specified hazard models, but each model must be evaluated using its own assumptions and hazard function.

Definition: Let $C(t)$ be the concentration at time $t$, and let $h_0(t)$ be the baseline hazard. Under the log-linear concentration-effect model, $$ h(t\mid C)=h_0(t)\exp\{\beta C(t)\}, $$ where $\beta$ is the change in log hazard per concentration unit. The cumulative hazard through time $T$ is $$ \Lambda(T)=\int_0^T h_0(t)\exp\{\beta C(t)\}\,dt. $$ For an individual with this hazard profile, survival through $T$ is $S(T)=\exp\{-\Lambda(T)\}$, and the probability of an event by $T$ is $F(T)=1-S(T)=1-\exp\{-\Lambda(T)\}$.

These predictions require a defined time origin and follow-up horizon, a specified concentration profile, and a hazard model appropriate for the population and context. In the examples, the baseline hazard is constant, concentration is treated as constant within each one-day interval, and there are no additional covariates. Those are simplifying assumptions for calculation, not general properties of repeated dosing. Concentration and time units must also match the units of $\beta$ and $h_0(t)$.

The predicted survival is conditional on the specified hazard profile and model parameters. It is not automatically a causal treatment effect: causal interpretation also depends on study design and control of relevant confounding. If the profile is based on a PK model, errors in that model can propagate into the event-risk prediction.

Calculating Risk Across Concentration Intervals

When both the baseline hazard and concentration are constant within interval $i$, of duration $\Delta t_i$, its cumulative-hazard contribution is $\lambda_{0,i}\exp(\beta C_i)\Delta t_i$. Summing contributions gives the total. For a smoothly changing profile, the corresponding integral can be evaluated analytically when possible or approximated numerically using sufficiently fine time steps.

Key result: For piecewise-constant baseline hazards and concentrations, $$ \Lambda(T)=\sum_{i=1}^{m}\lambda_{0,i}\exp(\beta C_i)\Delta t_i, \qquad F(T)=1-\exp\left\{-\sum_{i=1}^{m}\lambda_{0,i}\exp(\beta C_i)\Delta t_i\right\}. $$ If the concentration effect persists after the last dose, use the post-dose concentration profile during that period too. Stopping doses does not, by itself, imply that concentration or drug effect immediately becomes zero.

The result follows from the definition of hazard. Over a short interval of length $dt$, conditional survival is approximately $1-h(t\mid C)dt$. Multiplying these conditional survival factors over fine intervals and taking the limit gives $$ S(T)=\exp\left\{-\int_0^T h(t\mid C)\,dt\right\} =\exp\{-\Lambda(T)\}. $$ Thus, hazards accumulate additively on the cumulative-hazard scale, while survival probabilities combine multiplicatively. The event probability is obtained from survival as $1-S(T)$; cumulative hazard itself is not an event probability.

Worked Example: Equal Average Concentration, Different Profiles

Consider a hypothetical four-day comparison with constant baseline hazard $\lambda_0=0.020$ per day and $\beta=0.60$ L/mg. Concentration is expressed in mg/L and is constant within each one-day interval. Regimen A has $C=0.45$ mg/L on all four days. Regimen B has daily concentrations $0,\ 0.30,\ 0.60,\ 0.90$ mg/L. Both have a four-day average of $0.45$ mg/L. Calculate their predicted event risks by day 4.

For regimen A, the cumulative hazard is $$ \Lambda_A(4)=4(0.020)\exp(0.60\times0.45) =0.080\exp(0.27) =0.1048. $$ Therefore, $$ F_A(4)=1-\exp(-0.1048)\approx0.0995. $$ The predicted event probability is about 9.95%.

For regimen B, add the four one-day hazard contributions: $$ \Lambda_B(4)=0.020\left[\exp(0)+\exp(0.18)+\exp(0.36)+\exp(0.54)\right]. $$ Using $\exp(0)=1$, $\exp(0.18)\approx1.1972$, $\exp(0.36)\approx1.4333$, and $\exp(0.54)\approx1.7160$, $$ \Lambda_B(4)=0.020(5.3466)=0.1069. $$ Thus $F_B(4)=1-\exp(-0.1069)\approx0.1014$, or about 10.14%.

Under this model, the fluctuating profile gives a slightly higher predicted risk despite the same average concentration. This difference arises because hazard depends on the exponential of concentration, and the exponential function is convex. The example does not establish that fluctuation always increases risk under every hazard model; it shows why substituting an average concentration into a nonlinear model need not reproduce the integrated prediction.

Duration and Exposure After Dosing Stops

A risk prediction depends on the entire hazard history up to the stated horizon. Extending follow-up adds cumulative hazard for the extra time, and therefore cannot reduce cumulative event risk under a fixed model. The added hazard may reflect ongoing treatment, residual concentrations after discontinuation, or baseline risk alone. The appropriate choice depends on the specified concentration and hazard profiles.

Worked Example: Extending Follow-Up Beyond a Four-Day Profile

Use the same hypothetical $\lambda_0=0.020$ per day and $\beta=0.60$ L/mg. For days 1 through 4, let the daily concentrations be $0,\ 0.84,\ 1.84,\ 1.84$ mg/L. For this calculation, assume the concentration effect is negligible after day 4, so the hazard during days 5 through 8 is the baseline hazard alone. Find the event probabilities by days 4 and 8.

For the first four days, $$ \Lambda(4)=0.020\left[1+\exp(0.504)+2\exp(1.104)\right]. $$ Since $\exp(0.504)\approx1.6553$ and $\exp(1.104)\approx3.0162$, $$ \Lambda(4)=0.020(1+1.6553+6.0324) =0.1738. $$ The predicted risk by day 4 is $$ F(4)=1-\exp(-0.1738)\approx0.1595, $$ or 15.95%.

Under the stated assumption, the four additional days contribute only baseline cumulative hazard: $\Lambda_{5\text{–}8}=4(0.020)=0.0800$. Therefore, $$ \Lambda(8)=0.1738+0.0800=0.2538, \qquad F(8)=1-\exp(-0.2538)\approx0.2242. $$ The predicted probability of an event by day 8 is about 22.42%. This extension is valid only under the assumption that drug-related hazard has disappeared after day 4. If concentration declines gradually, the post-dose concentration and its corresponding hazard must instead be integrated over days 5 through 8.

Comparing Dose Levels Under a Repeated Schedule

For an alternative dose level or dosing interval, first obtain the concentration profile associated with that regimen, then evaluate the hazard over the same prediction horizon. The profile may come from observed concentrations or a PK model; the hazard calculation does not itself establish that the PK profile is accurate. Comparisons are most interpretable when the baseline hazard, follow-up period, covariates, and other modeling assumptions are held constant.

Worked Example: Comparing Two Hypothetical Dose Levels

Suppose two hypothetical repeated-dose regimens each have four one-day concentration intervals. The lower-dose regimen has concentrations $0.20,\ 0.40,\ 0.20,\ 0.40$ mg/L; the higher-dose regimen has concentrations $0.30,\ 0.60,\ 0.30,\ 0.60$ mg/L. Let $\lambda_0=0.020$ per day and $\beta=0.60$ L/mg for both. Calculate and compare their four-day predicted risks.

For the lower-dose regimen, $$ \Lambda_L(4)=0.020\left[2\exp(0.12)+2\exp(0.24)\right] =0.020\left[2(1.1275)+2(1.2712)\right] =0.09595. $$ Hence $F_L(4)=1-\exp(-0.09595)\approx0.0915$, or 9.15%.

For the higher-dose regimen, $$ \Lambda_H(4)=0.020\left[2\exp(0.18)+2\exp(0.36)\right] =0.020\left[2(1.1972)+2(1.4333)\right] =0.1052. $$ Hence $F_H(4)=1-\exp(-0.1052)\approx0.0999$, or 9.99%. Under these assumptions, the higher-dose profile has about $0.84$ percentage points more predicted event risk by day 4. This is a model-based comparison, not evidence that the dose change causes that difference in a real population.

Assumptions, Diagnostics, and Alternatives

The calculation is only as reliable as its input profiles and model. Before interpreting a prediction, check that the hazard function is defined for all concentrations and times used, that concentration and parameter units agree, and that the prediction horizon matches the clinical question. If a fitted concentration-effect model is being used, predictions at concentrations well beyond those supporting the fit are extrapolations and should be treated cautiously.

Inspect the full concentration profile. Confirm that the dose times, accumulation, and post-dose decline are represented. A profile formed by coarse daily averages may miss short high peaks; refine the time grid if those peaks materially affect hazard.
Check the baseline hazard over the full horizon. Assuming a constant baseline hazard is inappropriate if the relevant hazard changes with time and that change is not represented in the model.
Separate prediction from parameter uncertainty. The examples treat $\lambda_0$ and $\beta$ as known. In practice they are estimated, so uncertainty in those estimates and in the PK profile can make the predicted risk uncertain.
Check the model’s concentration-effect form. A log-linear effect implies a multiplicative hazard change of $\exp(\beta\Delta C)$ for a concentration change $\Delta C$. If that form does not describe the data or is implausible over the relevant range, the resulting regimen predictions may be unreliable.
State what happens after treatment ends. Setting concentration to zero at the final dose is not justified when drug or effect persists. Use the assumed post-dose PK and PD behavior explicitly.

A useful alternative is to replace the varying concentration by one average value and calculate risk from that constant. As the first worked example demonstrates, this generally differs from integrating a nonlinear concentration-dependent hazard. It may be a reasonable approximation when concentrations vary little or the concentration effect is nearly linear over the observed range, but it should be checked against profile-based integration. Another alternative is a discrete-time event model with interval-specific event probabilities; that model requires its own link and interval assumptions and is not obtained by treating a hazard as a probability.

The central limitation is that a precise integration cannot correct a misspecified hazard model, inaccurate PK profile, or unsupported extrapolation. A predicted risk is a conditional consequence of the model and inputs, not a guarantee for an individual or a substitute for validation against relevant data.

Common pitfall: Using $1-\exp\{-h(T)T\}$ with one hazard value evaluated at the average concentration silently assumes that hazard is constant over the entire period. When concentration varies and the hazard depends nonlinearly on it, calculate $\Lambda(T)=\int_0^T h(t\mid C)\,dt$ from the time-varying profile first, then transform that cumulative hazard into risk.

Check Your Understanding

Answer each question using the stated assumptions. Brief solutions follow the questions.

If $\Lambda(T)=0.20$, what is the event probability by $T$, and why is it not 20%?
With constant baseline hazard and a log-linear concentration effect, why might two profiles with the same average concentration produce different risks?
A concentration effect persists for two days after the last dose. What should be included when predicting risk through those two days?
For a fixed hazard model, can extending the prediction horizon reduce cumulative event risk? Explain.
Name two checks needed before interpreting a regimen comparison as a useful prediction.

Solutions. (1) $F(T)=1-\exp(-0.20)\approx0.1813$, or 18.13%; cumulative hazard is not itself a probability. (2) The hazard depends on $\exp\{\beta C(t)\}$, so integrating the transformed concentrations is generally not equivalent to transforming their average. (3) Include the post-dose concentration profile and integrate its hazard contribution; do not set concentration to zero solely because dosing stopped. (4) No. Added time contributes nonnegative hazard, so $\Lambda(T)$ cannot decrease and $1-\exp\{-\Lambda(T)\}$ cannot decrease. (5) Examples include checking profile accuracy and dose timing, confirming units and the baseline hazard over the horizon, and avoiding unsupported concentration extrapolation.