Bicycle Mechanics - how can I extract a cable bead end from a STI shifter

I have Tiagra shifters on my commuter. The rear derailleur cable went slack and I found the cable had frayed and broken where the housing met the nipple on the shifter. When I shift the lever to the highest gear I can see what looks like the metal bead end of the cable. The clearance is so small all I can do is manipulate it a mm or two using a pin...zero chance of getting a needle nosed plier in there.
Any suggestions on how to extract it?

did you try removing the cable and then pushing it back through the brifter from the outside? That should do what you want i think?

Shift to the smallest cog position and then push the cable through the shifter, if possible.

Al

Yes, I tried to insert another cable through the nipple (opposite end) and push the broken length through but the cable inside the brifter didn't move. It may be so frayed that the pushing piece of cable isn't making good contact with it or there's so much grime in the brifter that it's resisting any movement. I was surprised by how much fine grime seems to have accumulated inside the mechanism over years of use.
I guess I can try shooting aerosol solvent into the brifter to see if that would free the broken length of cable enough to push it through.

I did get it out. The cable broke very nearly at the bead end itself. There were just a few strands of the cable left attached to the bead. I got them out of the housing with a pin and there was just enough of them to grab with a needle nose plier and wiggle out the bead end.
A good argument for replacing cables more frequently...these were the originals with more than 6 years of heavy use.

Common problem on all shimano shifters, My DA10s right shifter cables last about 5000 to 10000 miles...I've had 2 break. They shift funky for one day, then the cable breaks completely