How to clean the grey drive cylinder the toothbrush snaps on to

Periodically, small bits of solid black debris come out of the drive mechanism that the toothbrush snaps onto. If you hold that end under running water and run the toothbrush, small tiny bits of black material will come out. I have found the best way to get the stuff to come out is it take a mouth full of water while the toothbrush is running and blow into the very end and the water will shoot out the two small holes at the base of the cylinder and lots of tiny black specs come out into the sink. I rinse the toothbrush drive mechanism every time I use it but still, over time every month or two there are more of the black flecks. Is there a better way of cleaning the gap between the silver center driveshaft and the surrounding gray collar? The gap is only a couple mm. You know it is clean if you hold the end straight vertically under running water while the toothbrush is running and water is pumped out the two side holes at the base of the drive mechanism.

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