Before getting into the specifics of nutritional supplementation I hope it is clear at this point that treatment plans need to be individualized. Holistic medicine works best when treating the whole body. If the primary problem is coming from the adrenal glands, or digestive system, you are not going to see much benefit from taking some "thyroid supplements."
In depth information of supplementing with iodine for hypothyroidism is found in section 6.
Iodine can be taken in pill form, and as a liquid. Lugol's iodine solution is probably the most well know. High amounts of iodine are in sea vegetables such as kelp, dulse, hijiki, nori, wakame and kombu.
Selenium is perhaps the most well thyroid supplement after iodine. It is essential for the proper conversion of thyroid hormone to T3. It has also been shown to decrease thyroid antibodies. Therefore selenium is often added to thyroid supplements, although by itself it is not a miracle cure for hypothyroidism. The thyroid gland uses selenium with iodine, so if iodine supplements are used, it may be best to include selenium as well.
Zinc is used in the manufacture of thyroid hormone, and the conversion from T4 to T3. 
A simple way to test for zinc deficiency is the zinc taste test. All you do is take about a teaspoon of zinc mixed into water, and taste it for about 10 seconds. If it has a strong metallic taste, you don't need zinc. If there is no taste at all, or it tastes good, then there is a need for zinc. I do this test on all my new clients. Low levels of zinc are also associated with poor wound healing.
Since the body uses zinc for hundreds of different reactions, including production of hydrochloric acid for digestion, I consider adequate zinc to be foundation for good health.
Zinc supplementation can potentially be harmful because zinc needs to be in proper balance with copper. Therefore it is only best to supplement with zinc if there is a need for it.
Vitamins A, D and essential fatty acids (such as the omega-3 fats in fish oil) plays an important part in in making T3 effective. Therefore, even with adequate thyroid hormone, there still may be symptoms of hypothyroidism if the body does not have the nutrients it needs to make T3 work. .
Vitamin A deficiency is associated with acne and lots of raised bumps, or "chicken skin," on the back of the upper arms. This is an called of hyperkeratosis. Vitamin A supplements can be used to treat this, but sometimes the body has enough vitamin A, it just can't use it. The liver uses zinc to convert much of dietary vitamin A (beta-carotene) to it's active form. So the real cause of vitamin A deficiency symptoms may really be zinc deficiency.
Hypothyroid patients may also have a yellow orange coloring to their skin, because without zinc, the liver can not convert beta-carotene into the active form of vitamin A. So the carotene goes to the skin and turns it yellow. 
Vitamin A is sometimes said to be dangerous due to potentially teratogenic (harmful to the developing fetus) effects in pregnant women. Therefore if a women has a chance of getting pregnant, she can not supplement with that much vitamin A.
Vitamin D is important for thyroid health in two ways. First, Vitamin D is needed to regulated the expression of T3. . Secondly, low levels of vitamin D are associated with autoimmune disease. Conventional lab ranges range from 30.0 to 74.0 (ng/mL).  However, many holistic practitioners like to see vitamin D at at least 60 to 80 (ng/mL). In autoimmune some practitioners get best results when vitamin D is up to 80 – 100 (ng/mL).
Vitamin D increases calcium absorption from the digestive system. At first this sounds like a good thing. But calcium depends on Vitamin K to work properly. Without vitamin K, calcium can get stuck in the wrong places, such as the arteries and contribute to arteriosclerosis.  Therefore, in my practice I only use combined vitamin D/K products. What it comes down to is the fat soluble vitamins: A, D and K all seem to work together. So a deficiency of one, can cause deficiency of all three. Since these vitamins are essential for proper thyroid hormone function, all three should be considered together. 
Vitamins E, B2, B3 and B6 are also used in the production of thyroid hormone. 
Tyrosine is an amino acid (a building block of protein) that along with iodine is what thyroid hormone is mostly made of. Tyrosine is included in many thyroid supplements. If someone if not consuming enough protein in their diet, then production of thyroid hormone can be affected (along with many other symptoms). However, there does not seem to be much evidence that supplementing with extra tyrosine, on top of a diet that supplies enough protein will help hypothyroidism.  However, as stated in previous sections, some hypothyroid patients may not be eating enough calories. If someone happens to be malnourished, and in need of protein, I would supplement with additional protein by using a protein powder, or amino acid supplement that contains all the essential amino acids. I would not only give tyrosine.
This is a supplement from the company standard process which is made from cow thyroid gland. It contains no thyroid hormone. I'm unsure exactly how it works, since it's a specialty product made by just one supplement company. I've heard that it acts as a decoy in auto immune disease so the immune system goes after the supplement and not the thyroid gland. I have also heard that it helps to actually repair the the thyroid. However, many practitioners have had good results using it on clients, not matter how exactly it does work. This is only available through health care professionals who use Standard Process supplements.
Proper supplementation for hypothyroidism needs to go beyond just giving "thyroid supplements." Other systems related to thyroid function such as the liver, digestive system, and adrenal glands must also be supported.
Since hyperthyroidism may also indicated deeper problems with auto immune disease and toxicity, other systems must also be supported with supplements. These would most likely be the liver, digestive system and detoxification in general. However, there are some supplements that are specific to hyperthyroidism.
Although these supplements may be helpful in controlling symptoms, and limiting need for medications, this would be only a small part, of a comprehensive plan for someone with graves disease. Refer to section 16 hyperthyroidism, for more detail on what a full plan does entail
Medical doctors use iodine in acute hyperthyroidism crisis. Iodine is not used in natural health for hyperthyroidism.
L-carnitine is a supplement usually used to improve energy. For hyperthyroidism, it has been shown to decrease symptoms, most notably nervousness and palpitations. The results where the same at 2 - 4 grams per day. It seems to do this by inhibiting the action of thyroid hormone on cell. It does not inhibit production of thyroid hormone. 
Lycopus virginicus (Bungleweed), Mellissa officinalis (lemon balm) and Iris versicolor (blue flag) have traditionally been used to treat hyperthyroidism. 
Selenium, and low dose lithium may be helpful. Often people are scarred of lithium, because of its reputation as a psychiatric drug. In psychiatry a starting dose of 600 is typical, compared to less than 5mg as a nutritional dose. The effects of a low, nutritional dose of lithium, therefore, can not be compared to that used in psychiatry. However, if someone still does not feel comfortable with lithium, carnitine and botanicals can be used instead.
Supplements are best used, as a way to control symptoms and lower need of medications, while other work is done on the underlining cause of graves disease. Supplements can be used as a long term substitute for drugs. However, natural health has a lot more to offer people with graves disease, by working on the root cause of immune system dysfunction.
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