Instantaneous Versus Cumulative Exposure Effects on Hazard

Two Different Meanings of Exposure

A hazard model can relate event intensity to the concentration present now, or to exposure accumulated over the period up to now. These are different scientific assumptions. Two individuals can have the same concentration at a particular time but different exposure histories; a current-concentration model treats them alike at that time, while a cumulative-exposure model can assign different hazards.

The distinction matters when a drug's effect is expected to track its current concentration closely versus when prior exposure is expected to contribute to the present event intensity. Neither model is automatically preferable: the choice should reflect the proposed mechanism and the information available in the observed concentration histories.

Definition: The hazard $\lambda(t)$ is the instantaneous event rate at time $t$, conditional on the individual remaining event-free just before $t$. A current-concentration model makes the hazard depend on $C(t)$, the concentration at time $t$. A cumulative-exposure model makes it depend on $A(t)$, the area under the concentration-time curve accumulated from treatment start to $t$: $$ A(t)=\int_0^t C(u)\,du. $$ If concentration is measured in mg/L and time in hours, then $A(t)$ has units mg·h/L.

A simple current-effect model and a simple cumulative-effect model can be written as

\lambda(t)=\lambda_0(t)\exp\{\beta_C C(t)\} \qquad\text{and}\qquad \lambda(t)=\lambda_0(t)\exp\{\beta_A A(t)\}.

Here $\lambda_0(t)$ is the baseline hazard, with units of inverse time, and the two expressions describe alternative models. The current-effect coefficient $\beta_C$ has units inverse to concentration; $\beta_A$ has units inverse to concentration-times-time. The exponential terms are dimensionless. A positive coefficient means that greater modeled exposure is associated with a greater hazard, conditional on the baseline hazard and any other covariates in the model. Such an association is not, by itself, proof of a causal drug effect.

These equations assume a multiplicative effect on the hazard scale and a linear effect of the stated exposure measure on the log-hazard scale. For the cumulative model as written, concentrations contribute equally per unit of concentration-time area, regardless of when that area occurred. The models also presume that the relevant concentration or exposure history is measured adequately and that the chosen baseline hazard and other covariates account appropriately for remaining systematic differences in event intensity.

Why Equal Current Concentrations Can Give Different Predictions

Suppose two exposure profiles are observed at the same time $t$, have the same baseline hazard, and share the same current concentration, but have different cumulative exposures. Under the current-concentration model, their hazards are equal because the concentration term is equal. Under the cumulative-exposure model, the ratio of their hazards depends on the difference in accumulated exposure:

\frac{\lambda_1(t)}{\lambda_2(t)} = \frac{\lambda_0(t)\exp\{\beta_A A_1(t)\}} {\lambda_0(t)\exp\{\beta_A A_2(t)\}} = \exp\{\beta_A[A_1(t)-A_2(t)]\}.

This is a direct consequence of the model equations: the common baseline hazard cancels, leaving the cumulative-exposure contrast. If $\beta_A=0$, or if the two cumulative exposures are equal, the cumulative model also predicts equal hazards. Otherwise, the predicted hazards differ even though current concentration is identical.

Worked Example: Same Current Concentration, Different Exposure Histories

Consider two hypothetical individuals assessed at 6 hours. Individual 1 has had a constant concentration of 2 mg/L for 6 hours. Individual 2 had 1 mg/L for the first 4 hours and 2 mg/L for the last 2 hours. Both have a current concentration of 2 mg/L. Assume a common constant baseline hazard of 0.0200 per hour.

For individual 1, accumulated exposure is

A_1(6)=2\ \text{mg/L}\times 6\ \text{h}=12\ \text{mg·h/L}.

For individual 2, it is

A_2(6)=(1\ \text{mg/L}\times4\ \text{h})+ (2\ \text{mg/L}\times2\ \text{h}) =8\ \text{mg·h/L}.

First use a current-concentration model with $\beta_C=0.100$ L/mg. For either individual,

\lambda(6)=0.0200\exp(0.100\times2) =0.0200\exp(0.200) =0.02443\ \text{per hour}.

The equal predictions follow from the equal current concentrations. Now use a cumulative-exposure model with $\beta_A=0.0200$ L/(mg·h). For individual 1,

\lambda_1(6)=0.0200\exp(0.0200\times12) =0.0200\exp(0.240) =0.02542\ \text{per hour}.

For individual 2,

\lambda_2(6)=0.0200\exp(0.0200\times8) =0.0200\exp(0.160) =0.02347\ \text{per hour}.

The cumulative model predicts a higher instantaneous hazard for individual 1 at 6 hours because that individual has greater accumulated exposure and $\beta_A$ is positive. These are hypothetical model predictions, not observed event rates. A hazard is an instantaneous rate conditional on being event-free up to that time; it is not the probability of an event over the entire 6-hour period.

Interpreting and Comparing the Models

In the current-concentration model, $\exp(\beta_C\Delta C)$ is the hazard ratio for a concentration difference $\Delta C$ at the same time, when other modeled factors are held fixed. In the cumulative model, $\exp(\beta_A\Delta A)$ is the hazard ratio for a cumulative-exposure difference $\Delta A$, under the same conditions. These interpretations concern the modeled hazard, not absolute event probability.

The cumulative model also has a specific memory assumption. Since $A(t)$ includes all concentration area since treatment start, a unit of exposure early in treatment contributes just as much to $A(t)$ as the same unit shortly before $t$. If the biological effect fades, is delayed, or depends on a more selective time window, this unweighted cumulative measure may not represent the relevant history. A current-concentration model has the opposite limitation: once current concentration is specified, the model ignores earlier concentration history.

Worked Example: A Positive Cumulative Effect After Concentration Falls

At a particular time, suppose the concentration has returned to 0 mg/L, but the hypothetical accumulated exposure is 20 mg·h/L. Let the baseline hazard be 0.0150 per hour. Under a current-concentration model with $\beta_C=0.500$ L/mg, the predicted hazard is

\lambda(t)=0.0150\exp(0.500\times0) =0.0150\ \text{per hour}.

Under a cumulative model with $\beta_A=0.0100$ L/(mg·h), the prediction instead is

\lambda(t)=0.0150\exp(0.0100\times20) =0.0150\exp(0.200) =0.01832\ \text{per hour}.

The difference is not a computational inconsistency. The current model says that the drug-related contribution at this time is determined by the current concentration, which is zero. The cumulative model retains an association with prior exposure. Whether that persistence is scientifically plausible is a modeling question, not something settled by the arithmetic.

The main alternative is not necessarily to choose one exposure measure and discard the other. A model may include both $C(t)$ and $A(t)$, if the data and scientific question support separate contributions. In that case, the coefficients describe conditional associations: the current-concentration effect is interpreted holding cumulative exposure fixed, and vice versa. Because current concentration and cumulative exposure may be related, estimates can be unstable or difficult to interpret when the observed profiles do not provide enough contrast. A more flexible delayed-effect model is another possibility when the timing of prior exposure matters, but it makes a different assumption about how history contributes.

Worked Example: Quantifying a Cumulative-Exposure Contrast

At the same time point, two hypothetical individuals have cumulative exposures of 6 and 15 mg·h/L, respectively. Suppose the baseline hazard is 0.0100 per hour and the cumulative-exposure coefficient is $\beta_A=0.0300$ L/(mg·h). The hazard ratio comparing the higher cumulative exposure with the lower is

\frac{\lambda_{15}}{\lambda_6} =\exp\{0.0300(15-6)\} =\exp(0.270) =1.310.

The corresponding hazards are

\lambda_6=0.0100\exp(0.0300\times6) =0.0100\exp(0.180) =0.01197\ \text{per hour},

\lambda_{15}=0.0100\exp(0.0300\times15) =0.0100\exp(0.450) =0.01568\ \text{per hour}.

Thus, in this model, the hazard at 15 mg·h/L is 1.310 times the hazard at 6 mg·h/L at that time, conditional on the same baseline hazard and other modeled factors. This is a relative instantaneous rate comparison, not a 31.0 percentage-point difference in event probability.

Assumptions, Diagnostics, and Common Pitfalls

A useful comparison begins by specifying exactly what “current” and “cumulative” mean in the analysis. The concentration $C(t)$ must refer to the same time scale as the event process. The cumulative quantity $A(t)$ must specify its starting point and exposure window. If concentration measurements are sparse, the calculated area depends on how values between observations are represented; that uncertainty should not be mistaken for precise knowledge of the exposure history.

Diagnostics should ask whether the model's chosen exposure summary captures the pattern in the data. Compare fitted hazards or model-based summaries across individuals with similar current concentrations but different histories, and across individuals with similar cumulative exposure but different current concentrations. If the model predicts no distinction within a group, but systematic discrepancies remain, its exposure summary may be inadequate. Also examine whether conclusions change with reasonable choices about the cumulative window or concentration interpolation. Such checks can reveal model mismatch, but they do not by themselves establish a causal mechanism.

A common mistake is to say, “The concentrations are equal now, so the hazards must be equal.” That conclusion follows only for a current-concentration model with the same baseline hazard and other covariates. It does not follow for a cumulative model, because the exposure histories can give different values of $A(t)$. The converse mistake is to treat equal cumulative exposure as proof of equal hazard under every model: equal $A(t)$ guarantees equal exposure terms in the stated cumulative model, but not in a current-concentration model if the current concentrations differ.

Neither formulation automatically represents every pharmacologic memory process. The current model can miss persistent effects of earlier exposure; the unweighted cumulative model can overstate persistence when old exposure should matter less. Adding exposure summaries can increase complexity and may not resolve these limitations if the data do not distinguish their contributions. Model choice therefore combines biological reasoning with assessment of fit and predictive behavior.

Key takeaway: At a fixed time and with a common baseline hazard, a current-concentration model compares individuals through $C(t)$, whereas a cumulative-exposure model compares them through $A(t)$. Equal current concentrations imply equal hazards only under the current-effect formulation; equal cumulative exposures imply equal hazards only under the cumulative-effect formulation.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

What are the units of $\beta_A$ if concentration is measured in mg/L and time in hours?
Two individuals have the same current concentration and baseline hazard but different cumulative exposures. Which model necessarily predicts equal hazards for them?
Under a cumulative model, what hazard ratio corresponds to a cumulative-exposure difference of 5 mg·h/L when $\beta_A=0.0200$ L/(mg·h)?
Why can an unweighted cumulative exposure model be unsuitable when the effect of past exposure is expected to fade?

Solutions. (1) Since $\beta_A A(t)$ must be dimensionless and $A(t)$ has units mg·h/L, $\beta_A$ has units L/(mg·h). (2) The current-concentration model predicts equal hazards, because $C(t)$, the baseline hazard, and other modeled factors are the same. The cumulative model can predict different hazards. (3) The ratio is $\exp(0.0200\times5)=\exp(0.100)=1.105$, so the higher cumulative exposure corresponds to a 1.105 hazard ratio under the stated model. (4) The unweighted integral gives equal contribution to equal concentration-time area regardless of when it occurred, so it does not make older exposure count less than recent exposure.

Instantaneous Versus Cumulative Exposure Effects on Hazard

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