The character has a high-pitched laugh and is unable to speak certain words due to his plastic surgery. He is also seen drinking from a surgical bag and reinflates his own face after being punched.
Brandt, who ran successful practices in Manhattan and Miami catering for the social elite, freely admitted to testing new treatments on himself.
Lesley Abravanel, a Miami Herald blogger, tweeted that Dr Brandt was "devastated" by Short's portrayal of him, which appeared in one episode.
She added: "Dr Brandt's publicist says he was suffering from an 'illness'. Sources close say that illness was depression." Abravanel tweeted that the cause of death was a suspected suicide.
Brandt's publicist Jacquie Trachtenberg told the New York Post: "The show definitely deeply hurt him, he was being made fun of because of the way he looks. It is mean and it was bullying. But the show was not the reason for his depression, and it was not the reason he would take his own life."
Trachtenberg would not confirm the cause of death but said in a statement: "He was suffering from an illness. Everyone who knows him is devastated. I worked with him for over 20 years and he was an amazing man, not only was he a brilliant doctor, but he was the kindest human being."
The Miami medical examiner performed an autopsy on Monday.
A New York Times profile last year described Dr Brandt, one of the early proponents of Botox and a prominent art collector, as "unnervingly ageless-looking". He presented a "mask of serene immobility, a face with a creaseless brow, a square firm jawline, lips feminine in their puffy fullness."
The doctor told the paper: "I've been kind of a pioneer in pushing the limits to see how things work and what the look would be. Would I change anything I've done? I might not have used as much Botox, because you don't want to look quite as frozen."
Fey declined to comment. In her new series Kimmy (Ellie Kemper) is rescued from an underground bunker after years as a cult leader's hostage. She is hired as a nanny by a socialite who visits Franff for "toenail resizing".
Where to get help:
• Lifeline: 0800 543 354 (available 24/7)
• Suicide Crisis Helpline: 0508 828 865 (0508 TAUTOKO) (available 24/7)
• Youth services: (06) 3555 906
• Youthline: 0800 376 633
• Kidsline: 0800 543 754 (4pm to 6pm weekdays)
• Whatsup: 0800 942 8787 (1pm to 11pm)
• The Word
• Depression helpline: 0800 111 757 (available 24/7)
• Rainbow Youth: (09) 376 4155
• CASPER Suicide Prevention
If it is an emergency and you feel like you or someone else is at risk, call 111.
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- Independent