Hyperthyroidism in cats is almost always associated with cancer. Sometimes cats respond badly to the prescribed medication. Surgery I suppose can be dangerous because care must be taken not to damage or remove the parathyroid glands which regulate calcium metabolism. If these glands are removed, the cat will need supplemental thyroid for the remainder of his life.
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It is said that the cost of these three treatments are similar in the long term although a cat owner would probably believe that the medication administered orally would be cheaper but this may not be the case in the long term and is something that needs to be checked with the veterinarian.
Radioactive iodine does or is intended to cure the problem (as opposed to treating symptoms) because it destroys the cause of the hyperthyroidism. The cost of the treatment in the USA is around $1,300. To undergo this treatment cats have their hearts checked out by a echocardiogram because hyperthyroidism can damage the heart. As I understand it, the radioactive iodine is administered by a pill. Treatment can take about 2 weeks and it appears to be residential treatment meaning that your cat stays at the veterinary hospital during treatment.
The radioactivity that is administered to the cat is shed in the cat's urine and faeces which are checked until it is reduced to a safe level.
Cats treated this way may need supplemental thyroid for the rest of their lives.
Provided this condition is caught early and before the heart or kidney is damaged treatment can be successful.
One sort of cancer which causes hyperthyroidism, a malignant adenocarcinoma, is harder to treat and the prognosis is very poor because the cancer has spread to other parts of the body.