I would wager to say that if it does, the magnitude is so so low that it is negligible.
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Think of it this way, there have been many studies on the benefits of fasting involving many people. If toothpaste DID adversely affect fasting, then all the people in those studies who brushed their teeth would not be receiving benefits, just the pigs who never brush their teeth while fasting would š Kind of an odd comparison but hope that kind of made sense lol
I think it might work like sugar free gum, which can trigger a very minimal spike in insulin, probably low enough that it doesnt really matter. And you swallow gum spit while you donāt swallow toothpaste spit, so that may also factor in. Also toothpaste is in our mouth for a much shorter duration.
In short I donāt think itās something to worry about.