Yes, you can use mirtazapine and Tapazole together with no problems.
Most hyperthyroid cats eat ravenously and still lose weight. So I'm not sure why Joe was on mirtazapine (i.e. an appetite stimulant with some anti-nausea effects), but we can't be sure he was eating very well before Tapazole only because of mirtazapine which may have been masking another issue. There are a few possibilities for why Joe may not be eating well and vomiting sometimes:
1. Intolerance for the drug Tapazole. Liver problems and/or bone marrow suppression can occur in a very small number of cats. Some cats experience only gastrointestinal signs that usually become obvious after using the drug for a week but I wouldn't rule this out just because Joe has been on the drug longer than that. Sometimes a switch from oral Tapazole to the transdermal gel form (absorbed through the skin) resolves the issue. If that doesn't work, then radioactive I131 is the next best option as it cures 98% hyperthyroid of cats.
2. Use of Tapazole may have lowered the metabolic rate which can unmask underlying kidney disease. This is one of the reasons for checking a full lab panel before starting methimazole and after using it for about 1 month. Sometimes if your cat's kidney values are borderline, then they will warn you about the possibility that symptoms of kidney disease can develop when the thyroid level normalizes. This could cause vomiting and inappetance. Usually you just try to keep the thyroid level at the upper end of normal rather than very low for these cats and then support them as needed for their kidney disease. It is advisable to treat hyperthyroid cats with Tapazole even if they have kidney disease.
3. There may be another problem that has yet to be identified which is the cause of Joe's poor appetite and vomiting. I would generally say this is the least likely of the possible scenarios but I don't know why Joe was taking an appetite stimulant before his hyperthyroidism was diagnosed. Maybe that was just to get him to eat Hill's y/d although his appetite for his usual food was great? (just guessing)
My advice would be to contact your veterinarian and let him know Joe's appetitie is poor and he is vomiting occasionally. He may be able to shed some light on the cause based on previous lab results. I suspect he will advise you to bring Joe in for repeat lab work ASAP. He may be able to do some of the lab work in house (i.e. check kidney values, liver values and a complete blood count which are all important to determine how the Tapazole is impacting Joe). Usually thyroid tests are sent to outside labs. Once the blood is drawn, then it is reasonable to stop the Tapazole and provide any other supportive care that Joe may need (nausea medication, fluids if he is dehydrated, etc.). If you stop the Tapazole before checking lab work, then it makes it a little confusing because we dont' know if Joe is having an adverse reaction to the medication (i.e. liver values elevated, bone marrow suppresion which are rare but possible), having gastrointestinal issues due to the tapazole OR if he is having kidney value elevation/symptoms after his thyroid value has come down.
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