I think its important to clear this up. There are no cysts to burst. There is not one shred of evidence that such a thing exists. To begin with "cyst" isn't very descriptive. There are two possible meanings in the medical world.
The first is used to describe any closed cavity or sac -- both normal and abnormal -- that is lined by epithelial cells, although in some locations, it may be lined by connective tissue or bone. Obviously THAT has nothing to do with Lyme
The second is used to describe a stage in the life cycle of certain parasites (e.g., Echinococcus granulosus) during which they are enclosed within a protective sac called a hydatid cyst. Some bacteria (Bacillus and Clostridia species) -- certainly not Borrelia burgdorferi -- form protective structures called spores; however, no bacteria form cysts.
The use of the term "cyst" with reference to B. burgdorferi or any other bacterium is incorrect. In most cases, the term is used to convey the false impression that, by forming "cysts", Borrelia are some how able to escape destruction by antibiotics and host immune defense mechanisms, so that they can establish a long-term persistent infection. or that they have some how built immunity to antibiotics. The fact is, if the bugs were encased in a "cyst," they could not POSSIBLY cause active disease.
The other possible use of the term use the term "cyst" could be to describe those structures (e.g., L-forms or "cell-wall deficient" variants) that are not part of the normal growth cycle of Borrelia, but which are formed after exposure to antibiotics that influence cell wall formation.
There are two types that differ only in the amount of residual cell wall material that they possess: spheroplasts, which still contain some remnants of cell wall material; and, protoplasts which are completely devoid of any cell wall material . Neither one would be antibiotic resistent.
These types can revert to the original form when placed in an antibiotic- free environment. If they revert, it happens quickly after antibiotic treatment stops. Neither is surrounded by a "cyst-like" protective structure. There is no reason to assume that they are any less permeable or susceptible to antibiotics than the original parental cell type, and their presence is EASILY determined by normal testing methods. The problem is those two types of cells have NEVER been found in humans, nor has any link been found to connect them to actual disease.
That's the basic science. It sadly has developed a life of its own by some practitioners. There are most certainly parts of the human body that it is more difficult to "reach" with antibiotics. The male prostate is the most difficult, the human eye and cerebral spinal fluid also being difficult. The idea, though, of a bacteria having its own immune system, the ability to build a fortress around itself (while still being able to venture out and do its deed) is really pretty far fetched.