Nansen, 25, had impressed in recent Guinness PRO14 games alongside Matthew Screech in the second row.
However, the big forward has now been ruled out of action following a scan on the injury on Monday.
“Brandon is believed to out for a number of weeks, although no surgery is required,” revealed skills coach Alan Kingsley.
“It’s a massive blow for Brandon because he came in last summer and was starting to find a run of form, we were starting to see the best of him.
“He has settled into the group, has been carrying the ball well and has been really positive in training, which was reflected in his performances.
“It’s a blow for him to have the injury at a time of the season when he was starting to feel comfortable in the squad and dominating in the areas that he can.”
Meanwhile, Dragons have revealed prop Aaron Jarvis will see a specialist after also suffering a hamstring injury at Parc y Scarlets.
Better news surrounds back row Ollie Griffiths who has returned to training this week.
Dragons remain without Ashton Hewitt (shoulder), Arwel Robson (hamstring), James Sheekey (hip), Brok Harris (knee), Leon Brown (thumb), Ryan Bevington (concussion), Ross Moriarty (concussion), Tiaan Loots (pectoral), Josh Reynolds (hamstring), Lennon Greggains (bicep), Max Williams (concussion), Dan Babos (bicep) and Evan Lloyd (concussion).