This is the follow up, four years later. I have pretty much extirpated the knotweed. There are are a few plants left, but they have been targeted for termination.
The imazapyr is an incredibly powerful herbicide. A single drop of it will kill any plant it happens to fall on. The main problem I found with injection is that the stalks of the plant are pretty tough, so you have to search around to find a place where you can get the needle into the pith. One constant problem was that pith can jam the needle. I used up a lot of needles during the operation.
There was some pretty heavy collateral damage. Two mature trees were killed during the operation. They were about 10-15 feet away from the stand of knotweed, but I guess their roots must have been in that area. Even though the herbicide was delivered specifically to each individual knotweed plant, someway or other the herbicide must have been transported to the roots of the trees, thereby killing them.
The Imazapyr does get down into the root system of the knotweed because re-growth has been very restrained.
The whole area where the knotweed was is kind of a wasteland now, with just a few weeds growing on it. I will probably have to re-sod it, or probably just build a shed there.