Density case control studies
In cohort studies, incidence rates are sometimes called incidence density rates. By similarity we will call "density based sampling" a sampling method in which the sample used as controls will represent the person time experience of exposed (Pte) and unexposed (PTu) cohorts in the source population.  Thus the probability of any person from the source population to be selected in the sample is proportional to his/her person-time contribution to the denominators of the incidence rates in exposed and unexposed cohorts.
Incidence (density) rates in exposed and unexposed cohorts of the source population can be expressed as follows:
Where "a" is the number of cases exposed, "b" the number of cases unexposed, "PTe" the total person-time accumulated by exposed persons and "PTu" the total person-time accumulated by the unexposed group.
If instead of studying the entire denominators of person time being exposed and unexposed we were sampling them (let's say 10%) we would have the following table:
||Sample from source population
Obviously from the above table the inidence rate of disease cannot be computed since person time denominators sampled from exposed and unexposed are only a sampling fraction of these two populations. However, if incidence rate can no longer be computed for exposed and unexposed, the rate ratio remains the same. If in the rate ratio calculation we replace the person time denominators by the 10% samples representing them, we obtain the same value for the rate ratio.
In this study design the sample (control group) is randomly selected from the person time experience of the source population. As a consequence the rate ratio computed using this sample is equal to the rate ratio computed within the cohort study done with the entire person time denominators of the source population.
The next issue is obviously about how to select a sample and make sure it represents the person-time experience of the exposed and unexposed cohorts in the source population. It is in fact quite simple. Each time a case occurs, an individual (or several) is randomly selected from the source population which is still free of the disease at the time of the case onset. This is sometimes called prospective case control study. A mathematical explanation of this rational can be found in Rodrigues et al. 
For each person contributing time in the source population experience, the time that this person is eligible to be selected in the sample is the same time during which she is also eligible to become a case if the disease should occur. Selecting an individual at the time of disease onset in a case leads us to select a sample among people still at risk and therefore proportionally to their time participation so far in the study. People who have left, are dead or who are already cases cannot be selected from that time on. This is also meaning that a selected individual who is still at risk of disease can later become a case in the study.
Let us suppose that Boise and Monson  had decided to do a density case control study instead of a person-time cohort study. They would have identified the 56 cases that occurred in the two cohorts and selected a sample series of 470 women. The sample series group should be sampled so that the person time distribution of the sample mirrors the person time distribution of the source population. If randomly selected and unbiased, this would give us 280 exposed and 190 unexposed in the sample (59.6 % of the sample is exposed which is equal to the proportion of exposed person time in the source population, 28010 / 47027).
Cases and sample selected from breast cancer cases and person-years of observation for women with tuberculosis repeatedly exposed to multiple x-ray fluoroscopies and unexposed women with tuberculosis
||Rate ratio source population
||Rate ratio sample
Source: Boice & Monson 
The controls represent person years at risk experience among exposed and unexposed. Controls are selected concurrently from those still at risk when a case occur. A person selected as a control can later become a case, the opposite is not possible since a case is no longer at risk of developing disease (with a non recurrent disease). A control which later becomes a case is kept in both groups.
Using density sampling allows us to compute a rate ratio which is equal to the rate ratio we would have obtained if a cohort study had been conducted to compare rates between exposed and unexposed cohorts in the source population. Density case control studies require an analysis matched on time of disease onset and control selection.
When to conduct a density case control study?
Density case control studies are suited for estimation of rate ratios (incidence density rate ratios). They are simple to conduct. In fact, to select a control among persons still free of disease, at the time a case occurs, is common practice frequently called prospective case control study. It provides a good estimate of the rate ratio. Density case control studies are suited when unequal length of follow up occurs for study members.
1. Rothman KJ. Epidemiology. An Introduction. New York. Oxford University Press 2002
2. Rodrigues L. Kirkwood BR. Case-control designs in the study of common diseases: updates on the demise of the rare disease assumption and the choice of sampling scheme for controls. Int J Epidemiol 1990 Mar:19(1):205-13
3. Boice, J. D., and R. R. Monson. *** cancer in women after repeated fluoroscopic examinations of the chest. Journal of the National Cancer Institute 1977 59: 823–832.