First remove it from the big tube. After that it is just a straight piece of hose that can be cleaned by anything that will go through it. The bottle brush will work. Really anything that will fit.
I got a pipe cleaning brush from my LFS for about £2. Long flexible handle/stem thing, then a smal brush on the end. Perfect for making sure that the water filter pipes are fully clean.
Pretty sure that's the kind of thing your on about.
A small screw or nail, a piece of string a couple of feet longer than the tubing, and a small rag. Tie the weight to one end, the rag to the other. Feed the weight through, grab the string, and pull the rag through.