There is no indication that budget concerns played a role in the Southampton hospital’s decisions about Ashya. The health service has paid for proton beam therapy abroad for other patients (it is not yet available in Britain). The treatment is said to be more precise than conventional radiotherapy, with its effects focused within two millimeters, rather than more than a centimeter with conventional methods. Precision is particularly important in treating children, because damage to adjacent healthy tissue can leave lifelong effects.

Mr. King learned of the therapy on the Internet and suggested it to Ashya’s doctors, but they told the boy’s parents that they were not convinced that it would work better than traditional radiation. According to Cancer Research UK, a charity, proton beam therapy is recommended for just 1 percent of cancer patients in Britain.

About 400 patients, mostly children, have been sent abroad for proton beam therapy by the National Health Service since 2008, officials said. Matthew Watts, a spokesman for University Hospital Southampton, said that the hospital refers about five cases a year to centers in the United States.

One beneficiary was Ross Anderton, who was found to have a rare cancer, orbital rhabdomyosarcoma, six years ago when he was 16 months old. His mother, Lesley Anderton, said she was given two choices: doctors could remove Ross’s affected eye, or he could undergo radiotherapy, with a risk of damage to his pituitary gland, his hearing and parts of his brain. His doctors did not mention proton beam therapy until Mrs. Anderton found it on the Internet and suggested it.

Setting aside their initial hesitation over an unfamiliar new treatment, the doctors referred her son to a hospital in Florida, and the National Health Service covered the cost — around $325,000 for the eight-week treatment, Mrs. Anderton said — other than the travel expenses. The service has also paid for follow-up treatments for Ross, who is now 6.

The circumstances of the Ashya King case are different. Doctors removed his tumor, called a medulloblastoma, and recommended a combination of chemotherapy and radiotherapy after surgery to maximize his chances of surviving. The Kings disagreed, arguing that conventional radiation was unsuitable for a child as young as Ashya. “The hospital wanted to kill him, turn him into a vegetable,” Mr. King said at the Madrid news conference.

Proton beam therapy is generally not recommended for a medulloblastoma, a highly malignant cancer that emerges in the back and the bottom of the brain, cancer experts said. Roger Taylor, oncology vice president of the Royal College of Radiologists, said conventional radiotherapy is more effective when there is a risk that cancer cells have spread, such as through the cerebrospinal fluid in the brain and spine; that kind of spreading is common with medulloblastomas.