There are basically two ways to find out if something is carcinogenic (that I know) quicker than with a retrospective study in Humans: One is to use lab animals, mostly rodents, administer them that substance in a high enough dose and see if they develop tumors. Problem: They aren't Humans and they don't get very old.
The other is by trying to grow a bacterium on a growth medium with an antibacterial agent and the substance in question. The clue: The bacterium has a gene for resistance against that antibiotic, but it is disabled by exactly one point mutation. The rate at which there are colonies on the petri dish tells you how mutagenic the substance is. Problem: The substance isn't metabolized like in a body.
This is not popular news yet but most of the tests with lab mice may be invalid. [1] I suppose now we need a movement to get natural mice used in lab tests or something along that line.
The other is by trying to grow a bacterium on a growth medium with an antibacterial agent and the substance in question. The clue: The bacterium has a gene for resistance against that antibiotic, but it is disabled by exactly one point mutation. The rate at which there are colonies on the petri dish tells you how mutagenic the substance is. Problem: The substance isn't metabolized like in a body.