Edwards noted that stem-cell work uses blastocysts containing clumpsof 16 or 32 cells that were collected by fertility clinics but are no longer needed and would otherwise be thrown away.
It sounds like you're describing tonsillar stones (also called tonsilloliths), which are clumpsof food particles, mucus, dead tissue cells and more that can collect in the crypts found on the surface of the tonsils.
Dirks showed that some of the immature-looking cells from his patients could form tumorlike clumps in the test tube, unlike most other cells from the tumors, but he didn't have a method to pluck out the stem cells.