Prop Moray Low, meanwhile, has sustained medial ligament damage to a knee, and is expected to be unavailable for selection for eight to ten weeks. The tight-head does not, however, require surgery.
Both players have now begun their rehabilitation under the care of the Warriors medical team.
Head coach Gregor Townsend told www.glasgowwarriors.org: "DTH and Moray are both important players for us, so it's clearly disappointing to have them unavailable over the next few weeks.
"The news isn't as bad as it could have been, though, and I know both guys will be working extremely hard under the care of our medical team to return to the field and continue making a significant contribution to our season."
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