Smith sustained the injury during his side's defeat to the Pittsburgh Steelers after taking a hit, but practised earlier this week before the problem was diagnosed.
He told reporters: "It certainly wasn't something that prohibited me athletically, I didn't feel like it was muscular, just discomfort. Nausea coming and going, I knew a bug was going around, I thought maybe it was that. I practised both days, I felt great athletically running around and throwing.
"I had this constant deal going on and then doc urged me to go get the test. I was pretty reluctant yesterday. I said I was alright and they pushed me to get checked out and I got the call that I had the laceration on there and it was enlarged."
Chase Daniel will replace Smith at quarterback in a must-win game for the Chiefs.