Structural Limits of Histopathological Detection: A Geometric Model of Lymphatic Invasion

In routine pathological practice, breast cancer specimens are examined microscopically to assess lymphatic vessel invasion (Ly). However, only a limited number of histological sections are evaluated, making it impossible to observe the entire tumour. As a result, the true extent of Ly may be underestimated, yet accurate detection is clinically critical given its strong association with sentinel lymph node metastasis. In this study, we propose a simplified spherical band model treating the tumour as a sphere and derive the probability of detecting Ly on a histological section. Our analysis revealed that current pathological practice allows observation of only approximately 0.051% of the tumour surface. This finding quantifies a structural limitation of histopathological diagnosis that has not previously been measured and provides a new perspective for the evaluation of Ly in clinical practice. Remarkably, despite such a low observable proportion, Ly remains strongly correlated with sentinel node metastasis, a paradox that suggests fundamentally non-linear behaviour in histopathological detection of lymphatic invasion.