With any premium barrel that has been finish lapped such as your Krieger Barrel, the lay or direction of the finish is in the direction of the bullet travel, so fouling is minimal. This is true of any properly finish-lapped barrel regardless of how it is rifled. If it is not finish-lapped, there will be reamer marks left in the bore that are directly across the direction of the bullet travel. This occurs even in a button-rifled barrel as the button cannot completely iron out these reamer marks.
Because the lay of the finish is in the direction of the bullet travel, very little is done to the bore during break-in, but the throat is another story. When your barrel is chambered, by necessity there are reamer marks left in the throat that are across the lands, i.e. across the direction of the bullet travel. In a new barrel they are very distinct; much like the teeth on a very fine file. When the bullet is forced into the throat, copper dust is released into the gas which at this temperature and pressure is actually a plasma. The copper dust is vaporized in this gas and is carried down the barrel. As the gas expands and cools, the copper comes out of suspension and is deposited in the bore. This makes it appear as if the source of the fouling is the bore when it is actually for the most part the new throat. If this copper is allowed to stay in the bore, and subsequent bullets and deposits are fired over it; copper which adheres well to itself, will build up quickly and may be difficult to remove later. So when we break in a barrel, our goal is to get the throat polished without allowing copper to build up in the bore. This is the reasoning for the "fire-one-shot-and-clean" procedure.
Barrels will vary slightly in how many rounds they take to break in because of things like slightly different machinability of the steel, or steel chemistry, or the condition of the chambering reamer, etc. . . For example a chrome moly barrel may take longer to break in than stainless steel because it is more abrasion resistant even though it is the same hardness. Also chrome moly has a little more of an affinity for copper than stainless steel so it will usually show a little more "color" if you are using a chemical cleaner. (Chrome moly and stainless steel are different materials with some things in common and others different.) Rim Fire barrels can take an extremely long time to break in -- sometimes requiring several hundred rounds or more. But cleaning can be lengthened to every 25-50 rounds. The break-in procedure and the clearing procedure are really the same except for the frequency. Remember the goal is to get or keep the barrel clean while polishing out the throat.
Finally, the best way to break-in the barrel is to observe when the barrel is broken in; i.e. when the fouling is reduced. This is better than some set number of cycles of "shoot and clean" as many owners report practically no fouling after the first few shots, and more break-in would be pointless. Conversely, if more is required, a set number would not address that either. Besides, cleaning is not a completely benign procedure so it should be done carefully and no more than necessary.