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Neurontin is excreted whole and not metabolized by the liver.
So liver damage is not reported for it.

Many anti-seizure drugs do affect the hair, and typically the recommendation is to use zinc for that.
One OptiZinc a day should do the trick.

Also some people find MSM (a source of sulfur) helps.

Some women lose hair when taking birth control pills or hormone replacement drugs.

Poor protein intake can also lead to hair loss.

Sometimes the pituitary can malfunction as well, but a doctor would have to evaluate that. Hair loss is pretty common in stress situations, and prolonged illness.