Geometric Limits of Histopathological Sampling: A Quantitative Framework Based on Spherical Band and Volumetric Models

Background In histopathological diagnosis, three-dimensional (3D) tissue structures are routinely inferred from thin two-dimensional (2D) tissue sections. However, this process inherently involves sampling error, and a mathematical framework quantitatively defining the theoretically observable fraction of tissue structures under standard histological sectioning conditions has not yet been established. Methods In this study, we constructed a mathematical model based on Archimedes’ hat-box theorem to quantify the geometric limitations of histopathological sampling. The tumor was approximated as a sphere, and superficial lesions were modeled using a spherical band framework. In addition, a separate volumetric distribution model was developed for internal lesions, and the theoretical observability under single or serial 2D sections was analyzed. For each model, the theoretical observation rate and detection probability were mathematically evaluated. Furthermore, the volumetric sampling fraction was analyzed and compared with the superficial lesion model to determine whether similar geometric constraints arise despite differences in lesion distribution. Results The present model demonstrated that the observable region under standard histopathological sampling conditions is geometrically extremely limited. In the superficial lesion model, the observation rate was shown to be independent of position on the spherical surface and was determined primarily by section thickness and slicing geometry. In the volumetric distribution model, observability depended on sectioning position in single-section sampling, peaking near the equatorial plane; however, under serial sectioning conditions, the effective sampling fraction was likewise governed primarily by section thickness and slicing interval. These findings indicate that thin-section sampling constraints are determined predominantly by slicing geometry rather than lesion distribution. Consequently, the model suggests that a high false-negative rate for microscopic lesions may be structurally unavoidable under routine sampling conditions. Conclusion This study redefines the observational limitations of histopathological diagnosis not as empirical or observer-dependent issues, but as predictable structural constraints governed by fundamental geometric principles. The present model provides a novel theoretical framework for quantitatively characterizing the intrinsic limitations of histological tissue sampling.