Epidemiologic studies that collect detailed exposure histories often incorporate this information into a regression model through a time-dependent cumulative exposure metric. This summary metric obscures variations in exposure rates among people and within persons over time. To disentangle the effects of cumulative exposure and exposure rate, one standard approach is to simultaneously model both cumulative exposure and average exposure rate. We propose an alternative regression model that uses a person's detailed exposure history information to describe the effect of the history of exposure increments on the relative hazard function. We illustrate this approach using data from a cohort study of radon exposure and lung cancer mortality among uranium miners. Compared with a standard cumulative exposure-average exposure rate model, our proposed approach yielded somewhat stronger evidence that the radon-lung cancer mortality association is modified by exposure rate. At low exposure rates, the estimated excess relative hazard per 100 working-level months was 0.63 (95% confidence interval = 0.32-1.37) under the standard approach, whereas under the proposed approach it was 1.23 (0.53-3.76). The proposed approach may provide better understanding of relationships between a protracted exposure and disease and is readily implemented using existing statistical software.