The HSE and Children's Health Ireland (CHI) have said they acknowledge the "deep distress experienced" by children with spina bifida and scoliosis and their families.
In a joint statement, the HSE and CHI apologised "very sincerely for the huge additions to this distress caused by the unacceptably long waiting lists for surgery, and the failings in the care we have provided to these children which have been the subject of recent reports."
It comes as two advocacy groups representing people with spina bifida and scoliosis said they will withhold patient medical record consent and boycott the third review into Temple Street Hospital.
The HSE has commissioned an external review by an independent expert from the UK, however, the Spina Bifida & Hydrocephalus Paediatric Advocacy Group and The Scoliosis Advocacy Network said they want the HSE and CHI to have no involvement on drafting the terms of reference for the external review.
"[The review] will not be conducted by the HSE or CHI but by Dr Selvadurai Nagayam, entirely independent of us," the HSE and CHI said in a joint statement.
"Its terms of reference are broad, and they allow for review of the governance of the Paediatric Orthopaedic Surgery Service, implications for service capacity and access, the delivery of the current agreed plans and other matters," the statement added.
The review's terms of reference were published on Monday on the HSE website when it announced the appointment of a UK expert to lead a review into aspects of orthopaedic surgery at Temple Street.
"We understand the advocacy groups’ deep frustration and anger and we understand that the Minister for Health has offered to meet them. If the groups would find it useful, the CEO of the HSE Mr Bernard Gloster would be very happy to meet them," the joint statement said.
The advocacy groups' statement was backed by Sinn Féin's spokesperson on health, David Cullinane, who said CHI should not play a part in conducting the review.
He told RTÉ's Drivetime: "I don't believe the advocate groups have a difficulty with the independent expert who has been earmarked to do the job, or his expertise is in question, it's more that there has to be a complete separation from the HSE and CHI to establish all of the facts.
"It makes perfect sense that advocate groups would have concerns."
He added that Minister for Health, Stephen Donnelly, should be engaging with advocate groups and "the people at the centre" of this issue to draw up the review’s terms of reference.
"I think the first mistake that was made was that there was no meaningful engagement from the Minister and those families," Mr Cullinane said.
"I would go back to the drawing board because I don’t believe there will be any credibility in the Government, or the Minister, for continuing with a review that doesn’t have the confidence of the families," he said.
On Monday the HSE ordered an external review into aspects of orthopaedic surgery at the hospital.
The executive said that it will be led by a UK expert "following a number of serious spinal surgical incidents there".
The HSE has emphasised that the doctor is not currently conducting surgeries and a referral has been made to the Irish Medical Council.
The HSE said that the review arises from very serious concerns identified by CHI since last year relating to poor surgical outcomes in spinal surgery at Temple Street, plus the use of a certain spinal technique and the use of unauthorised implantable devices.
The executive said that late last year, senior management at CHI were made aware of patient safety concerns in relation to the treatment of a small number of patients with spina bifida who had spinal surgery at Temple Street.
These concerns related to poor clinical outcomes of some complex spinal surgery, including a high incidence of post-operative complications and infections and two particularly serious surgical incidents, which occurred in July and September last year.
Following this, CHI commissioned two reviews, one internally and one external to CHI.