Naturopathic doctors are among those who are typically more open to diagnosing and treat hypothyroidism. clinically instead of just based on TSH. Good luck with your doctor. Please let us know how it goes for you.
Using TSH as the primary diagnostic overlooks most hypothyroidism because TSH has only a fair correlation with the actual thyroid hormones, and a negligible correlation with symptoms typical of hypothyroidism. So TSH is useful as a diagnostic only when at extreme levels typical of overt primary hypothyroidism.
Otherwise, your thyroid status is best determined by evaluation for symptoms typical of hypothyroidism, along with tests for both thyroid hormones: Free T4 and Free T3. You have multiple symptoms of hypothyroidism. You should always make sure they test for both FT4 and FT3. Free T3 is the biologically active thyroid hormone that largely regulates metabolism throughout the body. Hypothyroidism is best defined as "inadequate T3 genomic effect in tissue throughout the body, due to insufficient supply of, or response to, thyroid hormone". Unfortunately most doctors don't bother testing Free T3 on the false assumption that T4 always converts to T3 as needed.
Think of it this way. Patients don't go to their doctor complaining that their TSH is off. Patients go to the doctor because of unwanted symptoms. So they should be treated to relieve those symptoms. Doctors will say that symptoms are non-specific. That is true for some symptoms; however, when there is multiple symptoms typical of hypothyroidism, along with FT4 and FT3 in the lower part of their ranges, that is strong evidence for the need for treatment.
So the first thing I suggest is to go to this link and make a copy of the symptoms and mark up the ones you have. Then give the list to your doctor and emphasize that you need to be tested for the biologically active thyroid hormone Free T3, along with Free T4. It is also important for a hypothyroid patient to make sure Vitamin D is at least 50 ng/ml, B12 in the upper part of its range, and ferritin should be at least 100. So you need to test for those and supplement as needed to optimize. By the way, hair loss is frequently related to being hypothyroid and also having low ferritin ( a storage form of iron that is a precursor to serum iron levels).
If you want to confirm all of this, read my paper in the following link.
I also suggest giving a copy of the paper to your doctor and ask to be treated clinically, for symptoms, by testing and adjusting Free T4, Free T3, Vitamin A, B12 and ferritin as needed to relieve your symptoms. If the doctor refuses to do so, then you will need to find a doctor who will treat clinically as described.
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I am sending you a PM with some info that might help find a good thyroid doctor. They are hard to find. Just click on your name and then go to messages.