To clarify how polyunsaturated fats are involved with iodine, here is a clarification:
Iodine can attach to a fat molecule (lactone). And there is plenty of polyunsaturated fat (arachidonic acid) found in breast tumors, as evidenced by elevated levels of prostaglandins that are produced from arachidonic acid by the enzyme cyclooxygenase
(Arachidonic acid is a polyunsaturated fatty acid that is present in the phospholipids.)
When Iodine attaches to a fat molecule, iodolactones (6-IL) are formed
Iodine is able to bind to fat molecules, forming iodolactones (6-IL), that regulate apoptosis, and can even more powerfully effect apoptosis in Breast Cancer cells since lower concentrations are needed.