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A short answer is malignancy (high grade) with 8-35% probability.
Cytology came back AUC. Non-superficial, non-degenerated urothelial cells with an increased nuclear to cytoplasmic ratio. Below lists explanation of each terminology.
Non-superficial: Superficial cells are cells which form the superficial (also called umbrella) layer of the lining of the bladder. Superficial layer protect urine from urine penetrating further inside the lining of the bladder. Superficial cells look flat. Superficial cells are ignored in cytology. So, the pathologist is saying the cells referred are not superficial cells.
non-degenerated urothelial cells : below superficial layer, there is intermediate layer which consists of 5-7 layers of cuboidal cells, then basement layer, which consists of a single layer of spherical cells. These cells are called urothelial cells. Urothelial cells often degenerate when they are exposed chemotherapy and BCG therapy. So, the pathologist is saying that the cells being referred did not look degenerated.
with an increased nuclear to cytoplasmic ratio (N/C):
A cell is filled with jell like liquid called cytoplasm. Inside cytoplasm, there is nucleus which contain DNAs in the form of chromosomes. A ratio of the size of the nuclear vs cytoplasm of healthy cell is about 30% of cytoplasm. The ratio becomes higher often in high grade tumor. The cytopathologist saw the ratio was higher than normal but high enough to say "High Grace UC" or "Suspicious for High Grade UC". So it was put in a category of Atypical UC. To be diagnosed as HG UC, at last 5-10 abnormal cells must be found and N/C ration must be 0.7 or greater. In analogy to our eye, the white part (sclera) of the eye is cytoplasm and lens (iris) is nucleus. The ration of the size of Iris vs the size of sclera is greater than .7, there is a concern of high grade UC.
AUC : Atypical Urothelial Carcinoma
Cells were not variant and they look typical urothelial cells.. This is good because most available treatments are for urothelial carcinoma. Major criterion to be diagnosed as Atypical UC, TPS defines Non-superficial and non-degenerated UCs with increased cytoplasmic (N/C) ration greater than 0.5. There are other criteria which defines ad Atypical UC, but the cytopathologist who prepared the report for you did not mention it.
The cytology report you have described indicates that the cytopathologist analyzed the sample and reported based upon The Paris Reporting System (TPS), which was established in 2013 at the international cytopathologist conference held in Paris, France. There, they decided cytology was to look for high risk (high grade / CIS) but not low grade as they had realized that it was challenging to accurately detect low grade, and typically low grade is not life threatening. Accordingly, under TPS, only when a specific feature (stem like structure) is found in the sample, the report will say low grade. The implementation of TPS also reduced significantly the number Atypical cases reported before TPS.
Categories in TPS and the likelihood of finding high grade tumors .
1. Nondiagnostic/Unsatisfactory - < 5- 10 %
2. Negative for high-grade urothelial carcinoma 0-10%
3. Atypical urothelial cells 8-35%
4. Suspicious for high grade urothelial carcinoma 50-90%
5. High grade urothelial carcinoma > 90%
6. Low-grade urothelial carcinoma ~ 10%
7. Others > 90%
Note that the range of likelihood is the result of multiple studies of TPS implementation among different cytopathology labs, e.g. for Atypical urothelial cells - among different labs which diagnosed urine the sample as Atypical, the lab which most correctly diagnosed as Atypical still found 8% were diagnosed as high grade by biopsy, and the lab which most poorly diagnosed as Atypical found 35% their diagnosis were actually high grade by biopsy.