Ms Pentleton is monitored 24-hours a day by an array of equipment and a team of specialists.
She arrived on Thursday evening after Swedish doctors chartered a private jet to fly the two hours to Scotland to pick her up.
Crister Classon, a spokesman for the hospital, said: 'We are happy to help Britain or any other country if they run out of beds.
'It is a normal procedure to help other countries when they need it.
'We have only four beds and we currently have two swine flu patients in them, so there there are presently only two spare beds.'
It is thought a second British patient may be transferred to the unit.
The hospital's Dr Palle Palmer explained that the ECMO machine - similar to a heart and lung machine - was used to 'buy time' for patients. He said people could be kept on the machine for up to two months, but added that most patients did not need ECMO treatment for that long. He said: 'Normally it takes about two weeks, that's the normal treatment. But it is possible to run it for longer.
The Glenfield Hospital in Leicester, where doctors had hoped to treat Miss Pentleton, has the first designated ECMO unit for adults in the UK. But because it was full - with two of its five machines already being used by swine flu patients- medics turned to Stockholm.
ECMO treatment has only recently been accepted into mainstream NHS practice, being regarded as experimental in adults until the completion of a trial six months ago. Seriously sick children have been successfully treated for some time.
Best chance: Scottish health secretary Nicola Sturgeon said it was vital Miss Pentleton was transferred to Sweden
Consultant cardio-thoracic surgeon Mr Richard Firmin director of the ECMO unit in Leicester, said an average of 100 patients a year are treated there and beds could be expanded to 10 if absolutely necessary.
Patients are attached to an ECMO machine while their lungs recover from a variety of conditions, including viral infections and trauma. It involves circulating the patient's blood outside the body and adding oxygen to it artificially, Mr Firmin said 'The circuit is basically an external lung. Anybody who ends up with ECMO is somebody who is at the very severest end of lung failure.'
Patients may need treatment for two to eight weeks, at a cost of £55,000 to £105,000 per patient. Professor David Menon, an intensive care specialist at Cambridge University, said a small minority of swine flu victims who need intensive care have suffered a direct viral attack on their lungs, rather than a secondary infection. The condition called pneumonitis involves destruction of lung tissue.
