A Brisbane man wants a public inquest into the death of his nine-year-old daughter, saying she died prematurely because she was repeatedly denied access to medicinal cannabis.

Steve Peek's daughter Suli had a complex neurological disorder and cerebral palsy.

She could communicate only by blinking her eyes.

Her condition resulted in life-threatening seizures, sometimes 200 times a day.

She was on epilepsy medication, but Mr Peek said it was largely ineffective and had horrendous side-effects, while the seizures continued.

"It's terrible, heartbreaking," he said.

In May 2015, after Suli suffered bleeding stomach ulcers, Steve and his wife Joyce knew they had to try something else.

"She wasn't in a very good place so we gave her the first dose of cannabis oil and within half an hour we saw an immediate, and I mean immediate, change," Mr Peek said.

Mr Peek says cannabis oil helped control his daughter Suli's seizures. ( ABC News: Andree Withey )

He said Suli's seizures dropped by 90 per cent and they weaned her off the epilepsy medication.

"She didn't go to hospital for epilepsy for over a year," Mr Peek said.

"She was meeting all of her goals for the first time at school."

Medicinal cannabis was not legal in Queensland then, but Mr Peek continued giving it to her.

Suli's GP Dr Roger Egerton also noticed the difference.

"I got the impression that she was enjoying a better quality of life," he said.

Mr Peek gave Suli cannabis oil before it was legal. ( Supplied: Steve Peek )

"Generally, she seemed more comfortable, more responsive, generally more alert.

"I think it's likely that the medicinal cannabis reduced the frequency and severity of her seizures."

Just over a year ago, new laws were passed to allow GPs to apply to Queensland Health for permission to prescribe medicinal cannabis, providing they had the support and advice of a relevant specialist.

Certain specialists such as oncologists, and paediatric neurologists could prescribe it without approval.

Suli's GP Dr Egerton was willing to apply, but no specialist would agree.

"I think the medical profession generally is cautious about prescribing medicinal cannabis, for good reason," Dr Egerton said.

"It's a medication that hasn't got approval from the TGA [Therapeutic Goods Administration]."

He said regulations and red tape also made applying for it a "complicated, onerous and time-consuming process".

"It would be helpful for specialists to be prepared to prescribe for carefully selected patients, and I certainly think more clinical trials are necessary to establish safety and efficacy of cannabis products."

Mr Peek said he believed specialists were reluctant because Queensland Health had issued clinical guidance urging caution when prescribing medicinal cannabis containing the compound THC.

The guidelines state THC products are not recommended for patients under 25.

Mr Peek said he was giving Suli an unregulated type of medicinal cannabis which was closely related to THC, but did not produce the cannabis high.

While Suli's epilepsy improved, she was admitted to the Lady Cilento Children's Hospital in June 2016 with complications.

Suli Peek in hospital. ( Supplied: Steve Peek )

Initially staff turned a blind eye and he was able to continue giving her the medicinal cannabis, but that stopped weeks later when a staff member threatened to call the police.

Although Mr Peek continued to give his daughter the treatment at home, he was no longer able to administer the cannabis oil when she was in hospital.

"I pleaded with them at one stage: 'Please, please prescribe it. There's nothing left for you to do for her, why can't you prescribe it?'," Mr Peek said.

The hospital offered Suli a different type of medicinal cannabis known as Epidiolex.

But Mr Peek said they declined because he feared it would be less effective, and required a "washout" period of one month before using it.

This meant Suli would be off all medication — a risk her family deemed too dangerous.

Lawyer Teresa Nicoletti, who is working pro-bono for the family, said she believed the hospital was blocking access to medicinal cannabis.

"That washout period … would have put [Suli's] life at risk," she said.

"For the parents, no, it wasn't an option, and would any parent consider that to be an acceptable option?"

Ms Nicoletti has written to Federal Health Minister Greg Hunt concerned at the "systematic and profound failure" of the regulatory framework to supply medicinal cannabis.

"The complexities of lawful access to medicinal cannabis in Australia remain nothing short of prohibitive," she stated.

She said the state and territory regimes "effectively neutered" the Commonwealth approval and unnecessarily duplicated federal schemes, doubling regulation and "maximising the hurdles" that patients and their prescribers needed to overcome to access medicinal cannabis.

"We have tried to assist not just the Peek family but other families in Queensland, and they are facing the same issues of having their access blocked," she told the ABC.

Certain specialists can prescribe medicinal cannabis. ( ABC Rural: Marty McCarthy )

In a statement, a Queensland Health spokesman said the department had an obligation to ensure medicinal cannabis was provided in a safe and appropriate way.

"The introduction of medicinal cannabis is still relatively new in Australia and state and Commonwealth departments continue to work together to streamline access to medicinal cannabis for patients," he stated.

Children's Health Queensland (CHQ) medical services executive director Andrew Hallahan said the organisation supported the prescription of legal medicinal cannabis where paediatric specialists judged it to benefit the patient.

"Illicit cannabis preparations are known to be highly variable and often have high levels of psychoactive THC which is unsafe for children and young people," Dr Hallahan said in a statement.

"The use of illegal cannabis is not supported by CHQ as it is contrary to state and federal law."

Suli Peek was admitted to the Lady Cilento Children's Hospital one last time in October 2017.

She had been having stomach problems, and died suddenly.

Suli's family wants a public inquest to look into procedures in the hospital. ( Supplied: Steve Peek )

The cause of death is unknown and the coroner is investigating.

A Queensland Health spokesman said the department had done everything in its power to achieve the best possible outcome for Suli.

"We offer our sincere condolences to Mr Peek and his family," he said.

But the Peek family wants a public inquest to look into procedures in the hospital.

"The fact is one week they let us give it to her and turned a blind eye, and three weeks later they say no … so there's this big inconsistency," Mr Peek said.

Suli was offered a different type of cannabis by the hospital, but the family declined. ( ABC News: Kathy Lord )

He said it was particularly frustrating no specialist would prescribe the THC drug because of concerns about long-term effects on the brain, when Suli had a regressive brain disorder and faced a limited life expectancy.

"I think they failed her," he said.

"We knew [Suli's death] was inevitable one day, but it was premature, and we honestly believe that the refusal for Suli to be prescribed the cannabis product we wanted contributed to her premature death."