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Table 1 Examples of accumulative risk functions

From: Comparative quantification of health risks: Conceptual framework and methodological issues

Accumulative Risk Function, ƒ(x) Interpretation Relative Risk, RR Example
Relative risk depends only on current exposure, with no contribution from past exposure. Instantaneous poisoning as a result of exposure to high levels of toxic chemicals; injuries or death in accidents due to binge drinking; infection with Hepatitis B or C as a result of an infected injection
ƒ(x) = 1 Relative risk depends on the accumulated exposure (or average exposure if normalized with respect to exposure time), without any effects from the temporal distribution of exposure. Cancer risk from life-time exposure to carcinogens which have no threshold level
Relative risk depends on current and past exposures. But the role of past exposure lasts for a limited time, K, and decline as a linear function of time.  
ƒ(x) = e a(t-T) Relative risk depends on current and past exposures. But the role of past exposure decays as an exponential function of time.  
  1. For simplicity of notation, in all these cases we assume that: 1) L = 0. Including a lag is straightforward and can be done by replacing t with (t - L) in the corresponding formulas; and 2) there is no threshold for exposure. Including the threshold level is also straightforward using the TRUE (x (t) ≥ X) function. In scenarios 1, 3, and 4, where the effects of past exposure declines over time, risk reversibility can take place if exposure is reduced or removed. In 1 there is immediate risk reversibility; in 3, there is full reversibility after time K; in 4, risk reversibility asymptotically reaches 100%. In scenario 2 there is no risk reversibility and the effects of past exposure remain for an indefinite period.