Clustering of murine lung metastases reflects fractal nonuniformity in regional lung blood flow.

In the experimental metastasis assay certain animals, from groups of similarly treated animals, develop more lung metastases than expected from random chance alone. This clustering of metastases is characterized by a power function relationship, sigma(2) = amu(b), between the variance, sigma(2), and mean, mu, of the numbers of lung metastases per animal (a and b are constants). To determine whether this clustering could be an artifact of experimental metastasis, whether it could be influenced by different experimental conditions, and to attempt to clarify its cause, 22 published data sets from experimental metastasis utilizing 2,145 mice, as well as 8 data sets from spontaneous metastasis utilizing 1,020 mice were analyzed. In these experiments cell cloning, cell-cell fusion, treatment with a protein kinase C inhibitor, treatment with cell adhesion compounds, and transfection with either the ras oncogene, the sialidase gene, or the urokinase sense and antisense genes were used to influence metastasis. They employed 14 different cell lines and 6 different strains of inbred mice. Clustering of metastasis was evident in animals from the spontaneous metastasis assays as well as from the experimental metastasis assays. It was apparent whether mice were injected with tumor cells derived from clones or from cell lines. Clustering was demonstrated within each data set, regardless of the experimental conditions employed. A single variance to mean power function (with a = 2.2 and b = 1.51) characterized the clustering in the 30 data sets. The regional distribution of blood flow through lungs and other organs is nonuniform, exhibiting a fractal symmetry on change of scale. This symmetry implies that the variance of a region's blood flow is related to its mean by the same power function as was observed with metastasis. Indeed, measurements of blood flow from isolated canine lungs yield b = 1.56, similar to the corresponding figure from murine lung metastasis. These findings lend support to the hypothesis that the observed clustering of metastases is a consequence of fractal variations in lung blood flow.