I am writing an academic paper and I do not know whether my expression is formal. The full sentence is
There is a threshold (we call it take-off threshold) of ρc above which the proportion of the successful diffusion increases dramatically (from 0% all the way up to 100%) as ρc increases.
What I want to highlight is that the proportion explodes from 0% to 100% in a very short period. I do not know whether all the way up is a formal expression and whether it can highlight the explosion process.