Yukio is trying to ignore the pain of his injuries, but then West tells him off and ignores him. There isn't time for this, there isn't time to wait for anesthesia to kick in and Rin to pull and pick and irrigate the gravel out of his injuries. He grits his teeth and turns to Rin, clearly angry and in pain.

"Nii-san, do you have full control of your flames?" Without waiting for an answer, he launches into his plan, with a gesture at his side. It's not any worse than what Rin did to him, when he walked into the med tent and said 'I want you to take these crappy blind eyes out of my head'. "I need you to burn this clean. Just the contaminants. Do it quickly so I can go help Dr. West."

Rin can probably do this, he's done it before, just burned away clothes and left flesh untouched.