  Primary tumors often emerge within genetically altered fields of premalignant
cells that appear histologically normal but have a high chance of progression
to malignancy. Clinical observations have suggested that these premalignant
fields pose high risks for emergence of secondary recurrent tumors if left
behind after surgical removal of the primary tumor. In this work, we develop a
spatio-temporal stochastic model of epithelial carcinogenesis, combining
cellular reproduction and death dynamics with a general framework for
multi-stage genetic progression to cancer. Using this model, we investigate how
macroscopic features (e.g. size and geometry of premalignant fields) depend on
microscopic cellular properties of the tissue (e.g.\ tissue renewal rate,
mutation rate, selection advantages conferred by genetic events leading to
cancer, etc). We develop methods to characterize how clinically relevant
quantities such as waiting time until emergence of second field tumors and
recurrence risk after tumor resection. We also study the clonal relatedness of
recurrent tumors to primary tumors, and analyze how these phenomena depend upon
specific characteristics of the tissue and cancer type. This study contributes
to a growing literature seeking to obtain a quantitative understanding of the
spatial dynamics in cancer initiation.
