A subsequent X-ray identified a fracture that required a surgical repair.
He has now undergone surgery and is expected to be unavailable for around six weeks.
Glasgow Warriors head coach Gregor Townsend said: "Pat has featured regularly for us this season and has been in good form, while also stepping up as a leader, so it's disappointing to lose him for the next few weeks.
"He is due to be discharged home today and we look forward to him re-joining the squad when he has fully recovered."