  Recent progress in microdissection and in DNA sequencing has enabled
subsampling of multi-focal cancers in organs such as the liver in several
hundred spots, helping to determine the pattern of mutations in each of these
spots. This has led to the construction of genealogies of the primary,
secondary, tertiary and so forth, foci of the tumor. These studies have led to
diverse conclusions concerning the Darwinian (selective) or neutral evolution
in cancer. Mathematical models of development of multifocal tumors have been
developed to support these claims. We report a model of development of a
multifocal tumor, which is a mathematically rigorous refinement of a model of
Ling et al. (2015). Guided by numerical studies and simulations, we show that
the rigorous model, in the form of an infinite-type branching process, displays
distributions of tumors size which have heavy tails and moments that become
infinite in finite time. To demonstrate these points, we obtain bounds on the
tails of the distributions of the process and infinite-series expression for
the first moments. In addition to its inherent mathematical interest, the model
is corroborated by recent reports of apparent super-exponential growth in
cancer metastases.
