The blood service primarily works to boost blood supplies

Children like her need to find bone marrow that closely matches their own to stop their body rejecting it, and this where bone marrow registries - vast collections of samples from across the world - come in.

The Maddocks family is frantically working to increase the number of potential donors coming forward to be tested as a potential match for Alice.

In this case, it is the way bone marrow donation is organised in the UK that has caused them confusion and distress.

It is important to point out that funding is not the only issue in bone marrow donor recruitment

Anne Parish, Anthony Nolan Bone Marrow Trust

When they launched a website encouraging people to join the register through the National Blood Service - the arm of the NHS which recruits blood donors - they received a surprising reply from bosses there.

It said that appeals such as these were placing a huge strain on staff at the service - some of whom were cancelling holidays and working overtime to process the surge in donors produced.

In her letter, Professor Marcela Contreras, who runs the section of the service which tests potential bone marrow donors, said: "The laboratories need a steady influx of bone marrow volunteers throughout the year since there is a limit to the workload that our staff can reasonably cope with."

The blood service is investing in new machines to help test blood samples, but it would take a massive cash injection to radically increase the amount of bone marrow work that they can undertake.

And that is what Carol Maddocks demanded from the PM.

"Will you make that commitment and help to save her life?" she asked.

There are some within this part of the medical community who would question whether more money was the simple answer.

In the UK, the blood service is just one of three organisations which contributes names to an international bone marrow register numbering more than 6m people.

The biggest - by far - is the Anthony Nolan Bone Marrow Trust which currently holds more than 300,000 names.

Split priorities

A charity, independent from the NHS, it concentrates on bone marrow and nothing else.

The National Blood Service's main job is to handle blood donations - and its work on bone marrow an added extra, born from the recognition that it had access to a high number of public spirited individuals who might be willing to do both.

It currently approaches regular, reliable donors to ask about bone marrow, and provides leaflets for others explaining the position.

However, some experts say that it would be better to exploit the experience of the Anthony Nolan Trust than build new systems which duplicate their efforts at the blood service.

Anne Parish, from the Anthony Nolan Trust, told BBC News Online, said: "It is this expertise which has placed us in the unique position of carrying out a core health service, a position held in high esteem by transplant centre in this country and abroad.

"While some may find it strange that such a crucial service is organised and managed by a charitable trust, those in the field of bone marrow transplantation understand that our independence allows us to offer the best service to those patients who need this vital service.

Hope against hope

"It is important to point out that funding is not the only issue in bone marrow donor recruitment.

"To allow the public to believe that recruiting more donors would result in finding a match for every patient is misleading.

"A very large Register doesn't mean that patients will find donors - it is not, unfortunately, so simple."

If the blood service were to double the number of people who registered as bone marrow donors with them, it would still be no more than a drop in the ocean internationally.

Even a multi-million pound revolution in bone marrow registration in this country may not produce benefits quickly enough to help Alice Maddocks.

But every parent will understand why even a fraction of hope is enough to inspire her parents to take on the prime minister.

For more details about how to become a bone marrow donor, follow the link to the Anthony Nolan Bone Marrow Trust on the right hand side of the page, or ring the hotline on (0901) 8822234.