"There are some tumours that require radiotherapy to the whole brain and whole spinal cord," said Dr Yen-Ching Chang, a specialist in paediatric radiotherapy and lead on proton beam therapy at University College London hospitals, which will have one of the UK's two machines from 2018.
They put their case for proton beam therapy in a YouTube video that won them widespread support.
Paediatric intensive care consultant Dr Peter Wilson said the family put him and his colleagues in a difficult position, as Ashya had not been deemed suitable for proton beam therapy.
At the time, Ashya could not swallow and had to be fed through his nose, but the couple announced last month that he has now made a "miracle" recovery after receiving proton beam therapy in Prague.
That makes proton beam therapy really useful for attacking tumours that can be precisely delineated and targeted, to spare other parts of the brain.
While they did not want to discuss Ashya's individual case, a team of leading cancer specialists explained why the brain tumour Ashya has, medulloblastoma, is not on the list of cancers approved by the NHS specialist board for proton beam therapy.
If you shone a torch at the nearest star and simultaneously fired a Cern particle, the first torchlight would arrive in four years and the accelerated proton less than two seconds later.
A high court judge subsequently approved the move to take Ashya to Prague for proton beam therapy, which was claimed to be more effective than the conventional radiotherapy offered on the NHS.
Cancer doctors have to talk to parents who have heard that proton beam therapy is a better treatment and hope it may prove a magic bullet that will provide a happy ending to a terrible story.
The NHS will pay for children with certain specific cancers to have proton beam therapy abroad if it offers benefits over conventional radiotherapy.
Instead Lyn Evans, the physicist in charge of the collider, cautiously nursed the first proton beam around the circuit an eighth of the way at a time.
