Do "dead" cancer cells produce PSA?

This might seem like the world's daftest question but could dead PCa cells release PSA? It is a protein, and dead cells are composed of various proteins. The reason I ask is that I just had a dream that someone told me this. (My husband's PSA recently went up from 0.3 to 1.9, and this has been on my mind a lot.) I was intrigued by the idea and so googled it to no avail other than a site selling supplements that claimed cancer markers could be higher when there is cancer cell die-off, and in literature from the author of Outsmart Your Cancer. Neither was backed up by any scientific data. Any biologists here to refute/explain? I know that the most likely cause is ADT resistance, but the PSA was falling and then rapidly went up - right after dosing heavily on some alternative treatments. Not the result I'd have expected. We have an appointment with the oncologist tomorrow and with the naturopathic oncologist next week.