| Abstract: |
| \begin{abstract}
We revisit the classical Iwata--Kawasaki--Shigesada model of metastatic dissemination by reformulating it in terms of lesion age rather than tumor size. Whereas the original framework classifies metastases according to size, size alone does not fully capture the biological history of a lesion: tumors of similar size may differ markedly in their age, growth trajectory, and capacity to generate further metastatic spread. By taking lesion age as the principal structuring variable, we obtain a formulation that is both mathematically more tractable and biologically more informative.
This shift in perspective transforms the original size-structured transport equation into an age-structured model with a simpler and more transparent structure. The resulting formulation preserves the behavior of the classical model over clinically relevant time scales, while improving computational efficiency and interpretability. In particular, it clarifies how metastatic burden accumulates over time, how newly seeded lesions contribute to the evolving population, and how a relatively small number of clinically visible metastases may coexist with a much larger, occult population of microscopic disease.
We further prove that the age-structured formulation is exactly equivalent to a renewal equation governing the production of new metastases. This equivalence makes the system amenable to direct analytical treatment. In the case of Gompertzian growth and power-law metastatic emission, the model yields explicit results for the asymptotic expansion of the metastatic population and provides a natural framework for the analysis of long-time behavior and detection thresholds. Overall, the age-based reformulation offers a transparent, biologically meaningful, and computationally efficient framework linking primary tumor growth, metastatic seeding, and the emergence of clinically detectable disease.
\end{abstract} |
|