Mark Lundy claims he was convicted of murder on the basis of unreliable science, becoming a "guinea pig" of untested methods.
Lundy's team, led by Jonathan Eaton QC, are in the Court of Appeal today fighting to overturn his convictions for murdering his wife and daughter in 2000.
Lundy is not in court for today's appeal.
Central to his legal case is the two spots on his polo shirt, which prosecutors argued was Lundy's wife Christine's brain tissue.
Eaton said the heart of their appeal was that the jury at the 2015 retrial should never have heard the mRNA evidence, as the analysis was too new and unproven to be relied upon.
He said Lundy had become a "guinea pig" for a method of analysis that had never been used anywhere in the world before, and had never been used to identify unknown material in the 16 years since the first trial.
"When you've got that level of complexity, this is an issue that should have been thrashed out by the experts, through peer review and conferences.
"Not in front of a jury, saying 'right well there's our conflicting views, you work out if it's reliable or not'."
Eaton said it was highly unusual to have a murder conviction resting on such controversial evidence, that had even been branded as unreliable in the 2015 retrial.
He argued there was a duty to make sure all evidence submitted in court could be relied upon.
"The courts are dropping the ball on the admissibility of new science," he said.
"And hence we have the UK Law Commission expressing their concerns in reports before you.
"We have the lengthy 2009 report from the Nation Research Council in the United States, which presents all sorts of concerns.
"And then in September last year, post Mr Lundy's conviction, we have the PCast report to Mr Obama regarding the presentation of science in criminal courts.
"And when one reads those reports and sees the climate of concern in relation to forensic science in criminal justice, it is instructive to this court in regards to what the standard ought to be in New Zealand.
"And it all comes back to the issue of scientific validity."
Eaton said part of the problem was the public perception of Lundy. He said there was an "elephant in the room" in terms of the effect of the widely seen video of Lundy at the funeral of Christine and Amber.
"What is it about Lundy that sets him apart from other cases?
"There is this view that he is a big fat so-and-so, who was with a prostitute on the night, and people saying 'did you see the unconvincing performance at the funeral? Of course he is guilty'.
"He has engendered no public sympathy."
The Crown legal team, led by Philip Morgan QC, has not yet presented its rebuttal arguments.
Today's hearing is the latest in a long-running legal saga, since August 2000 when Christine and Amber Lundy were found dead in their Palmerston North home.
Mark Lundy was first convicted of their murders in 2002, and his first appeal attempt actually resulted in the court increasing his prison sentence to 20 years.
After a long campaign, his conviction was quashed by the Privy Council in 2013, which ruled there were problems with the analysis of the brain tissue, as well as time of death.
Lundy and his lawyers tried to have the mRNA evidence of brain tissue on his shirt thrown out before his retrial, but his team failed to successfully challenge the evidence in both the Court of Appeal and Supreme Court.
In a 2015 retrial at the High Court in Wellington, Lundy was again convicted of the murders.