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One of Rostropovich's trainers, David Hayes, added: "He couldn't weight bear. But I think with a lot of painkillers he's relieved. It's a sad story anyway."
Racing Victoria issued a statement on Rostropovich's condition and said vets found the horse to be lame after the race.
"Racing Victoria (RV) stewards advise that the Hayes, Hayes and Dabernig-trained Rostropovich was inspected by RV veterinarians following the running of the Melbourne Cup at Flemington Racecourse today and was found to be lame," the statement read.
"After receiving veterinary care on the course, the horse has been transported to the University of Melbourne Equine Centre for further assessment."
The Animal Justice Party of NSW said in a tweet it heard Rostropovich had been "badly injured" while the RSPCA also reacted to the news.
There have been several deaths in the past five years of the glamour race.
In 2014, Admire Rakti, who collapsed and died in his stall after the race, and Araldo, who broke his leg and was euthanised, were casualties.
The following year fan favourite Red Cadeaux shattered a leg and was put down on the track.
In 2017, Regal Monarch died after a sickening fall in another race at Flemington on Melbourne Cup Day, before The Cliffsofmoher's fall early in last year's race.
Trainer and commentator Richard Freedman this week pointed to the foreign takeover of the event as a possible cause of the spike.
"Of the last six horses that have come to a catastrophic injury in a Melbourne Cup, all six of them were imported horses," Freedman said on Nine's Sports Sunday. "The last Australian horse to die as a result of the Melbourne Cup was in 1979."
Racing Victoria assesses each international horse and scratched last year's runner-up Marmelo and British horse Ispolini from this year's race after inspections by its veterinarians.
RECENT MELBOURNE CUP DEATHS
2013: Verema (France)
2014: Admire Rakti (Japan), Araldo (Great Britain)
2015: Red Cadeaux (Great Britain)
2017: Regal Monarch (Great Britain)
2018: The Cliffsofmoher (Ireland)