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The parents of two day old twin boys who died after being treated at Stafford Hospital say they're deeply upset and distressed.
Police say preliminary results suggest Alfie and Shaun McQuillin died because they were premature -not as a direct result of medical treatment.
It's as a public inquiry started this week into allegations of 'appalling care" at Stafford Hospital between 2005 and 2009.
A member of staff has been suspended and an independent paediatric doctor has been tasked with conducting an investigation into the circumstances.
In a statement released by Staffordshire Police, Phillip McQuillin and Ami Dean, from Stafford, said: "We are deeply upset and distressed by the deaths of Alfie and Harry.
"A full investigation into the circumstances has been launched, which should now be allowed to run its course.
"We would ask the media to respect our privacy as we try to come to terms with what has happened."
Maggie Oldham, Chief Operating Officer of Mid Staffordshire NHS Foundation Trust, said: "Our deepest sympathy and our thoughts continue to be with the twins' parents and their loved ones at this most difficult time.
"We have commissioned a full external investigation and this will be led by an independent paediatric doctor."
She added: "The public inquiry started this week and we would like to refer to Robert Francis QC's opening statement on Monday in which he said, 'It is inevitable there will continue to be incidents giving cause for concern, both at Stafford and in other hospitals as well. Such incidents do not of themselves show that there has been no improvement, any more than that the absence of such incidents would prove everything has been put right.'
"We at Mid Staffs have made significant improvements.
"At the time of the Healthcare Commission Review in 2008, this hospital's mortality rate was in the bottom 10% in the country and now our mortality rate is in the top 10%.
"This however has not made us complacent and we are working continually to improve our services and the care we provide for our patients and their loved ones."
Staffordhire Police said preliminary tests indicated that the boys, who were born at 27 weeks, died because they were so premature "rather than as a direct result of any medical treatment".
According to unconfirmed reports, health officials are examining whether the twins were given incorrect doses of a drug.
A police spokesman said: "Post-mortem examinations by a Home Office pathologist proved inconclusive and further tests will now be carried out.
"However, preliminary results suggest that the twins died because of their prematurity rather than as a direct result of any medical treatment."
The babies died at the University Hospital of North Staffordshire after being transferred from Stafford Hospital.
An inquest into the deaths was opened and adjourned by south Staffordshire coroner Andrew Haigh on Monday.