During the past week, Carter's parents have experienced both "highs and lows".
They have been encouraged by the support people are giving them, both in the hospital, across the region and around the world.
Sarah Cookson told ITV News:
In the last week, doctors had to temporarily paralyse Carter's body to avoid fatal bleeding. His dad, Chris Cookson, said this was particularly difficult.
For Carter to receive a new heart, it would mean another family have to lose their baby.
NHS Blood and Transplant's Lead Nurse, Angie Scales, told ITV News that although it is a difficult decision to make, donating the organs of your child can often benefit the parents during the grieving process.
Carter's parents know only too well that they, themselves, may have to make that very difficult decision if a heart cannot be found for their son.
They said they would have no qualms about donating Carter's organs in those circumstances.