Brain Metastases: Insights from Statistical Modeling of Size Distribution

Brain metastases are a common finding on brain MRI. However, the factors that dictate their size and distribution are incompletely understood. Our aim was to discover a statistical model that can account for the size distribution of parenchymal metastases in the brain as measured on contrast-enhanced MR imaging.

Tumor volumes were calculated on the basis of measured tumor diameters from contrast-enhanced T1-weighted spoiled gradient-echo images in 68 patients with untreated parenchymal metastatic disease. Tumor volumes were then placed in rank-order distributions and compared with 11 different statistical curve types. The resultant R ² values to assess goodness of fit were calculated. The top 2 distributions were then compared using the likelihood ratio test, with resultant R values demonstrating the relative likelihood of these distributions accounting for the observed data.

Thirty-nine of 68 cases best fit a power distribution (mean R ² = 0.938 ± 0.050), 20 cases best fit an exponential distribution (mean R ² = 0.957 ± 0.050), and the remaining cases were scattered among the remaining distributions. Likelihood ratio analysis revealed that 66 of 68 cases had a positive mean R value (1.596 ± 1.316), skewing toward a power law distribution.

The size distributions of untreated brain metastases favor a power law distribution. This finding suggests that metastases do not exist in isolation, but rather as part of a complex system. Furthermore, these results suggest that there may be a relatively small number of underlying variables that substantially influence the behavior of these systems. The identification of these variables could have a profound effect on our understanding of these lesions and our ability to treat them.