Two women have challenged a High Court order to have their children vaccinated with the controversial MMR jab.
The case centres on whether the girls should have the MMR vaccine
The women took their case to the Court of Appeal after Mr Justice Sumner ruled in June that the girls, aged five and 10, should be given the three-in-one vaccine against their wishes.
The case was brought by the girls' fathers who want their children vaccinated against mumps, measles and rubella.
Lords Justices Thorpe and Sedley and Sir Anthony Evans heard evidence from both sides on Thursday.
They reserved judgement and are expected to give their decision in the coming days.
Effect on families
Elizabeth-Anne Gumbel, representing the mothers who cannot be named to protect
the identities of the children, told the court that Mr Justice Sumner had not
taken sufficient notice of the mothers' wishes and the effect the ruling would
have on the families.
It is understood that in both of these cases the mothers are against their daughters being given the three-in-one vaccine because of concerns over its safety.
Studies have suggested the jab may be linked to autism and bowel disease. However, these claims have been dismissed by doctors and the government, who all insist it is safe.
Mr Justice Sumner ruled that it was in the girls' best interests to be given the three-in-one jab.
He rejected the idea of giving the girls separate vaccinations against mumps, measles or rubella, saying that the gap between jabs could put them at risk of getting these diseases.
Both girls in this case live alone with their mothers. Their parents are either divorced or separated.
None of the people involved in the case can be identified for legal reasons.
Mr Justice Sumner said his decision was influenced by evidence given by medical experts.
The ruling sparked controversy not least because it could have implications for many parents in similar situations across the country.
One of the mothers issued a statement after last month's High Court ruling criticising the decision.
She said: "It's outrageous that, in a free society, a judge could make such a decision."
The British Medical Association has backed the High Court ruling. However, it has been criticised by anti-MMR campaigners, including JABS.