Abstract
The McKendrick/Von Foerster equation is a transport equation with a non-local boundary condition that appearsfrequently in structured population models. A variant of this equation with a size structure has beenproposed as a metastatic growth model by Iwata et al.  Here we will show how a family of metastatic models with 1D or 2D structuring variables, based on the Iwata model, can bereformulated into an integral equation counterpart, a Volterra equation of convolution type, for which a rich numerical andanalytical theory exists. Furthermore, we will point out the potential of this reformulation by addressing questions coming up inthe modelling of metastatic tumour growth. We will show how this approach permits to reduce the computational costof the numerical resolution and to prove structural identifiability.
Highlights
The organism-scale nature of cancer is a major challenge for clinical oncology
Another major difficulty is that small metastases are often invisible on medical images, such that the metastatic state of a patient is only known with certainty once the metastatic growth has already advanced
For the description of metastatic growth, mathematical models have a potential to estimate the metastatic state in situations where it cannot be seen on medical images
Summary
The organism-scale nature of cancer is a major challenge for clinical oncology. As long as the disease is spatially confined, it can often be cured by a local intervention, but once a cancer metastasises, the prognostic deteriorates rapidly. The numerical resolution of these models is not without difficulties: several authors have described problems arising when using typical PDE schemes due to large scale differences in model dynamics for biologically relevant parameters [2, 10, 6]. It comes at a considerable computational cost, in the 2D case. In a recent work [16], the techniques described in this article are used for model building based on preclinical data and the adequacy of the modelling approach is discussed in detail
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