It's probobly best demonstrated by a video, but I'm not sure I'm motivated enough to make one.
It did take some trial and error, making a few messes. but I'll give you a few tips that may help.
-I use the cap from the "push" end of the insulin syringe as a place to stick the plunger. It won't "stand up", but its a sterile place to put the rubber plunger into and lay it down.
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-I use a 1-1.5" large gauge needle to avoid creating bubbles, stuck deeply into the syringe barrel, angled against the barrel wall, without completely blocking off the open end of the insulin syringe. This allows the liquid to flow to the bottom of the insulin syringe easily, without getting "caught", above a bubble, as you've likely experienced.
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-Once finished filing, holding both syringes in one hand I recap the 3ml filtered syringe and put it down.
-Then I stick the insulin syringe plunger in *just barely". It's not even in enough to stay in on its own. It only closes off the open end and must be held there. Obviously you don't want to push out any liquid.
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-Flip the insulin syringe needle side up (still capped btw), hold the barely in plunger to keep the open end closed.
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-I flick the barrel with my fingernail to move the air bubble to the top. Once it's at the top, I can push the plunger in, pushing out the air (you can still leave the needle end cap on), and set down the perfectly dosed syringe for later or immediate use.
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It's easier than it sounds once you practice a few times, Even small doses are no issue, though you should really be diluting to get more than .10ml volume for any peptide.
Maybe BD will sponser a video, lol.
Also, I realize this isn't "perfect", no technique is. I am improving this, eliminating the need for backfilling for per dose filtration, supplies are more expensive using this somewhat better technique. Another thing you should aim for, at least with the 3ml syringe that'll hold peptides for an extended time, is a silicone free syringe. Silicone, though common as a syringe lubricant, in increasingly understood to be bad to inject generally, but it has a negative effect on proteins/peptides in particular.
The other possibility, if you have really good control, is just to stick a zero dead space 31g needle on the end of the filtered 3ml syringe, then remove and dispose the needle after injecting. Like a "manual" injection pen. This adds a few complications too, but nothing insurmountable. It does add the risk of overdosing though and requires a very steady hand.