Fig. 2: Occupancy, fractal dimensions, and FD differences indicate that stromal B cells are more spread out in good outcome.

a–c Occupancy: a Example image showing how occupancy is computed. A grid of squares placed over the image. (Inset) 1’s (0’s) correspond to yes (no) answers to a binary question asked of each square, e.g., “Is there at least one B cell in the square?” Occupancy is fraction of squares with 1’s. b Plot of CD20⁺ B cell occupancy in stroma vs. square size for good clinical outcome (blue), poor clinical outcome (red), normal breast tissue (black) and points randomly distributed according to a uniform Poisson process with a density of 3 × 10² points/mm² (cyan). Error bars indicate 95% confidence intervals. Good (blue) outcome curve is consistently higher than poor (red) outcome curve. c Plots showing occupancy area under the curve (AUC) for CD20⁺ B cells in stroma is higher for good (n = 24) vs. poor (n = 12) outcome (p = 0.05). Center line represents the median, box limits represent the upper and lower quartiles, whiskers indicate maximum and minimum values within 1.5 interquartile ranges of the upper and lower quartiles; individual points are outliers. d Fractal dimension determined from slope of the log-log plot of the number of squares with at least one stromal CD20⁺ B cell vs. the inverse box size (Logarithms are base e). The fractal dimension is larger for good clinical outcome vs. poor outcome at large length scales, but this trend is reversed at small length scales. At long length scales (200–600 μm on the left side of the plot), mean fractal dimension s (slope) is 1.3 for good outcome (blue), 1.11 for poor outcome (red), 0.92 for normal tissue (gray), and 2.08 for Poisson (black). The p value for good vs. poor outcome is 0.08 at long length scales. At short length scales (10–40 μm on the right side of the plot), the mean fractal dimension is 0.58 for good outcome, 0.65 for poor outcome, 0.30 for normal tissue, and 0.15 for Poisson. The p value for good vs. poor outcome is 0.5 at short length scales. Black dashed lines show least-squares linear regression fit at long and short length scales. Because different images have different overall areas, fractal dimension is calculated by normalizing the number n(L) of boxes with 1’s with the total number N(L) of boxes with cells. Thus the y-axis values are negative. The error bars correspond to 95% confidence intervals. e Example illustrating how FD difference indicates whether cells are spread out or clustered. The left image with four quadrants shows four blue points that are spatially spread out in the upper two quadrants while the right image with four quadrants shows four clustered red points in the upper two quadrants. At long length scales (big boxes in left 2 quadrants), the big squares in the lower left quadrant have more 1 s in the left image than in the lower left quadrant of the right image because the blue points in the upper left quadrant are more spread out compared to the red points in the upper left quadrant of the right image, so the fractal dimension is larger in the left half of left image than in the left half of the right image. At small length scales (small boxes in 2 right quadrants), the upper right quadrants of both images have the same number of points and occupy the same number of squares; hence, the blue and red points have the same fractal dimension at small length scales. So the difference in fractal dimension between large and small length scales will be larger for the spatially dispersed blue points than for the clustered red points. f Plot showing that the FD difference for stromal CD20⁺ B cells is significantly higher in good outcome (p = 3 × 10⁻³, ROC AUC = 0.77), indicating that B cells are more spatially dispersed in the stroma in good outcome. Center line represents the median, box limits represent the upper and lower quartiles, whiskers indicate maximum and minimum values within 1.5 interquartile ranges of the upper and lower quartiles; individual points are outliers. g RFS plot showing the clinical significance of the FD difference for stromal CD20⁺ B cells with a threshold value = 0.564 (see “Methods”). For all FD differences calculated in this paper, the large length scale range is 200–600 μm and the small length scale range is 10–40 μm. Patient cohorts for all plots in this figure are n = 24 for good outcome (blue) and n = 12 for poor outcome (red). P-values in box plots are determined by unpaired, two-tailed Student t-tests. ns: not significant; *p < 0.05; **p < 0.01; ***p < 0.001; ****p < 1 × 10⁻⁴.