Poisson regression for continuous variables? I understand that Poisson regression is used for count data because, among other reasons, it accommodates the count variable's mode being zero and the fact that it cannot be negative. What about continuous variables, for example C-reactive protein, in clinical contexts? It is heavily positive-skewed, cannot be negative, and can have mode of 0 in many scenarios (lower limit of detection can be, for example <3, and considered 0 for analysis). Transformations are inadequate. This situation is true for many biomarkers.
How should one approach regression with these variables, can Poisson regression be used?
 A: The Poisson model is not just skew and non-negative; for example it has a particular variance specification (the variance is equal to the mean). This will almost never be true for variables that are not counts (what happens if you change units? You change the shape of your Poisson model! That makes no sense).
While it would in some situations be possible to consider a quasi-Poisson model (variance proportional to the mean) that would still not usually be a suitable model for non-negative measurements of physical quantities. 
You would instead typically expect them to have standard deviation proportional to mean - you don't expect the distributional characteristics, aside from a known change to a scale parameter, to change when you change scale. 
If you're looking for a GLM, the obvious candidate with the characteristics non-negative, right skew, standard deviation proportional to mean is the gamma. (There are other commonly used standard-deviation-proportional-to-mean models -- the lognormal, the Weibull and so forth).
However, if you have exact zeros you might consider a zero-inflated or a hurdle model (gamma with a proportion of zeros).
On the other hand, if the low end is really censored (as with an instrument than cannot detect below some threshold and just records 0 there), then you may be better off to explicitly deal with that censoring. Since censored data is standard in survival models, programs for modelling survival often have what you need built right in (and offer multiple choices for survival time distributions, which you can parlay into models for C-reactive protein levels by simply putting that where survival time would go in the left-censored survival model)
