Some bodybuilders use aspirin because it lowers cortisol effects (not production, just effects). But it also knocks t and GH levels at the same time so the whole thing is kinda redundant. The effect on protein synthesis relates to this hormonal profile change too I believe. Unless you have just been chased by a dinosaur and you're worried about failing your upcoming cortisol level test aspirin supplementation is not for you
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If you have to use aspirin or similar regularly for something just follow the instructions on the box. They're conservative for a reason. I take aspirin once a week on average when my patellar tendons flare up no worries. The levels used to elicit a hormonal response in studies are pretty big. If you're gobbling that much over the counter pain med regularly you best get some professional help.