Age and Generation-Based Model of Metastatic Cancer: From Micrometastases to Macrometastases

Abstract

Metastasis remains the decisive event in most solid cancers, yet the mathematical tools we use to describe it are still largely tied to tumor size. In the curent work, we recast metastatic spread in terms of tumor age and generational ancestry, turning an implicit notion into the main organizing principle of the model. Building on the classical Iwata-Kawasaki-Shigesada (IKS) framework, we construct a transformation from size to age and derive a hierarchy of integral equations indexed by metastatic generation. Our reformulation yields closed-form expressions for first-generation metastases and simple one-dimensional recursive integrals for higher generations, avoiding direct numerical solution of the original size-structured partial differential equation (PDE) with its nonlocal boundary condition. Using Gompertzian growth and power-law metastatic emission, we show that the age-generation model reproduces IKS predictions over clinically relevant time scales while offering improved numerical stability and interpretability. The generational decomposition reveals a robust pattern: lesions seeded directly from the primary dominate early in the disease course, whereas successively younger generations, emit ted by existing metastases, come to dominate the total lesion count at small sizes, leaving older generations to occupy the macroscopic tail of the distribution. Introducing an explicit detection threshold naturally separates a small number of radiologically visible macrometastases from a much larger, unseen pool of micrometastases. Together, these results provide a transparent and computationally efficient framework that links primary-tumor growth, metastatic seeding across generations, and the hidden microscopic burden that underlies clinical presentation.