Brake Cable Hazard

I just bought a new rubber interior for my bike tires because I felt it was time to replace them having been vulcanized a few times. However, just recently, I was upset to find out another flat tire when I was about to bike down the city. I thought I was just too unlucky to hit another sharp object and pierced my tire.
Upon checking at the vulcanizing shop nearby, It was shocking to find out that the rubber interior had four pin-sized holes in it at different spots. Checking to see what the cause was, the attendant removed a very thin (hairlike) but sturdy wire about an inch long that got inside the tire. At first, I wondered where it came from and couldn't figure out at all. I just went home after it was vulcanized.
Later in the day, when I checked my brakes, I noticed some of the wires from the cable stuck out from the brake cable's bundle. From there, I concluded, that was where the wire that pierced my tire came from.
It has been months since I last checked on the brake system. Some of the wires hanged loose from the connection and one of them have certainly touched the surface of the tire after some time. With the wheels on motion it was easy to penetrate the rubber surface with that sturdy thin wire. Since then, I needed to check the cables often, of course not just for the tire, but also for my legs and my safety lest it finds another way to get me into trouble.