Now, let's see, a lack of oxygen might mean the are lesions in the lungs. A possibility then is TB caused by Mycobacterium tuberculosis causing TB. There's two Mycobacteria of note, M. tuberculosis and M. leprae (causing leprosy).
Today, M. leprae is not a problem anymore because it is quite sensitive to antibiotics. M. tuberculosis is huge problem, especially in Developing countries. Often it presents in people that have AIDS. It's identification is as follows: Bloody sputum passed early in the morning is ideal. One places the sputum on a slide and stains it with a Ziehl-Neelsen stain. One heats the stain on the slide for a few minutes, then you wash it off with a weak acid. M. tuberculosis is a small gram negative bacilli (reddish under the microscope). It is acid-fast, so it will absorb the ZN stain, whereas the acid will wash the rest of the slide. Mouth epithelial cells stain blue.
One can also stain TB bacilli with a fluorescent stain. But for that you need a fluorescent microscope. Under it, the small TB bacilli fluoresces and look like bright little stars against a black backdrop. Even if you do not see the bacilli, but suspect TB, you do a TB culture. This is done aerobically in bottles on a greenish egg medium. This takes approximately five weeks. The TB colonies present yellow on the green egg medium. All positive TB sputums are cultivated in this way. Afterwards, in specialized laboratories, antibiotic discs are used to check sensitivity (not sure whether they are using antibiotic discs or dilutions - we always sent the bottles away to a specialized lab to estimate antibiotic sensitivity).
The problem with M. tuberculosis is that it is highly contagious and very dangerous. Laboratory personel use a reverse vapour cuboard, which is sealed and sucks out the air away from the person working with the sputums. Recently resistant strains have developed. All antibiotic treatment fails. The organism systematically destroys the host. I haven't been in a lab for a long time, so I wouldn't know whether above techniques have been changed or improved.