A Mathematical Framework for Estimating Tumour Age Using Ki-67 and Mitotic Figures

Tumour pathology assessment has traditionally relied heavily on spatial information at a single time point, such as tumour size and morphology, while the temporal aspect of tumour growth has not been explicitly addressed. However, tumour progression is inherently a time-dependent biological process, and a conceptual framework is needed to infer tumour dynamics from routinely available morphological data. Ki-67 labelling index and mitotic counts, both widely used in diagnostic pathology, reflect cell cycle activity but are typically interpreted as independent static indicators. In this study, we propose a mathematical framework that integrates the ratio of Ki-67- positive cells to mitotic figures to estimate the elapsed time from a single transformed cell to the observed tumour size, referred to here as “tumour age.” Furthermore, by incorporating cell size and tumour cell area fraction (𝛼), the model accounts for differences in cellular packing density within tissue. A pilot application to representative tumour cases (breast carcinoma) suggested a consistent relationship between histological proliferative features and the estimated tumour age. This framework provides a conceptually simple and practically implementable approach for introducing a temporal dimension into routine pathological assessment. Future integration of additional factors, such as necrosis and fibrosis, may serve as a basis for a more dynamic understanding of tumour biology.