When you do this you are essentially increasing the blood flow to this area. The ice decreases the blood flow by constricting the vessels through the cold. That is why your skin turns more white, with less blood, when ice is applied. The subsequent shower increases the blood flow by heating up the area that will respond with more blood flow anyway after icing. Here is video on lip injections
Swelling after lip augmentation injections are common, as Hyaluronic acid used for the purpose can cause swelling itself in tissues. Normally this type of swelling builds up after few hours after injection and settles after 24-72hrs. When anesthesia is worn out pain also can kick in.
Bruising and infection are another reasons for pain and swelling. Your doctor should be able to check you for that. Arnika, Vit K or Heparin cream can help for the former and antibiotics or antivirals (depending on reason for the infection) -for the latter.
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