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Are you asking about the SMZ/TMP or something different now?
Your original question was about the antibiotic....
SMZ (sulfamethoxazole)/TMP (trimethoprim) is a Sulfonamide type antibiotic that treats a whole host of things. I will name a few....urinary tract infections, otitis media, acute and chronic prostatitis, shigellosis, traveler's diarrhea, chronic bronchitis, and Pneumocystis jiroveci pneumonitis. I would suggest calling the office tomorrow and find out what is going on.
Yes, infection can be detected through a blood test (CBC-complete blood count is the name of the test). If your white blood cell count was elevated, it could be indicitive of infection. A white blood count of 3.3 is rather low and typically would not suggest a systemic bacterial infection. The 3.3 reading is most likely regarding the Coumadin you are on.
Your question regarding your blood being too think or too thin when you are on Coumadin therapy is determined by a test called PT/INR. Patients typically should have an INR reading of 2.0 to 3.0 for basic "blood-thinning" needs. For some patients who have a high risk of clot formation (as would be determined by your Dr), the INR needs to be higher - about 2.5 to 3.5. Your doctor will use the INR to adjust your drug to get the PT into the range that is right for you. So, depending on what your Dr is wanting your level to be, the dosage of Coumadin will be adjusted accordingly.