Which mentions that liver regeneration (which can happen quickly) is dependent on prostaglandins (fatty acids) in rats. According to Wikipedias entry on Prostaglandin, NSAIDs "inhibit cyclooxygenase and reduce prostaglandin synthesis"
So if the body is low in fatty acids, the regeneration of the liver would be slower/not happen and if NSAIDs use is high, maybe the regeneration is completely blocked (along with lower fatty acids from the diet).
The OO (Olive-Oil) from the flush potion also contains alot of fatty acids, which may help regeneration too, thus healing the liver.