Fig 1.
H&E (top) and anti-pimonidazole (bottom) stains of one of our study’s canonical tumor sections.
Fig 2.
Loci of single-bundle hypoxia gradients.
Circles (red) defined by the rm found by the Intensity-Sample-Ray-Bundles algorithm for each of the three centers we specified, corresponding to vessel locations in the registered H&E image. Here we show m = 1 sector (2π radians per sector) for each center. Sectors are labeled with red numbers, counterclockwise, just outside of the red sector contour.
Fig 3.
Intensity level analysis produced by the Intensity-Sample-Ray-Bundles algorithm for centers 1 (left 3 panels), 2 (middle 3 panels), and 3 (right 3 panels). Intensity-Sample-Ray-Bundles creates three plots of the data, where the horizontal axis denotes distance from the center (pixels), and the vertical axis denotes intensity level. The first panel shows every ray measurement (light gray), upon which (blue) and
(red) are overlaid; its title gives rm (pixels). The second panel shows
(blue) ±
(gray), overlaid with segmented least squares fits to
(black); its title gives the length (l, pixels), slope (s), and least squares error (e, pixels) for each fitted segment. The third panel shows
(red) ±
(gray), overlaid with segmented least squares fits to
(black); its title gives the length (l, pixels), slope (s), and least squares error (e, pixels) for each fitted segment. The segmented least square fits are given by a dynamic programming algorithm using a cost parameter C = 200.
Fig 4.
Tissue types by manual spatial partitioning.
Another (unsmoothed) canonical anti-pimonidazole image (top), its manual partitioning (middle), and its labeled partitions (bottom). Key: V = viable, N = necrotic; unlabeled, brown regions are hypoxic.
Table 1.
Otsu’s multithreshold segmentation of unsmoothed versus smoothed images over the total set of images (nT = 66), high anti-pimonidazole images (nH = 36), and low anti-pimonidazole images (nL = 30).
We report pixel areas as proportions of the entire set of pixels in the image (I); hence H:I, V:I, and N:I. We also report another proportion of interest, namely that of hypxic to viable cells in the image, H:V.
Fig 5.
Otsu segmentation and smoothing.
How Otsu’s multithreshold segmentation differs between unsmoothed gray (upper left) and smoothed gray (lower left) images. Corresponding images on the right show dark blue regions that denote hypoxic cells, light blue regions that denote viable cells, and yellow regions that denote necrotic cells.
Table 2.
1-segment radii in high anti-pimonidazole images.
The values of ng and ni report that the statistics are from a sample of 2 gradients found in 1 image.
Table 3.
2-segment radii in high anti-pimonidazole images.
The values of ng and ni report that the statistics are from a sample of 7 gradients found in 4 images.
Table 4.
3-segment radii in high anti-pimonidazole images.
The values of ng and ni report that the statistics are from a sample of 16 gradients found in 8 images.
Table 5.
1-segment radii in low anti-pimonidazole images.
The values of ng and ni report that the statistics are from a sample of 4 gradients found in 2 images.
Table 6.
2-segment radii in low anti-pimonidazole images.
The values of ng and ni report that the statistics are from a sample of 20 gradients found in 8 images.
Table 7.
3-segment radii in low anti-pimonidazole images.
The values of ng and ni report that the statistics are from a sample of 5 gradients found in 4 images.
Fig 6.
Inferred hypoxia gradients (gray) superimposed onto the canonical raw H&E (top) and anti-pimonidazole (bottom) images at half-opacity. Note that the positions of the gradient centers have been corrected as per our earlier observation regarding adjacent image registration (see S7 Fig).
Fig 7.
One way to verify that viable (V, tan) and necrotic (N, gray) regions are nowhere contiguous (i.e., they are everywhere separated by a hypoxic region (H, brown) is to follow arbitrary trajectories in the 2D or 3D space, each of which represents a spatiotemporal logical proposition, any number of whose results can be conjoined to obtain a system-wide propositional truth value. (A) The trajectory obtained by holding x2 fixed at some arbitrary value and allowing x1 to vary across an arbitrary extent, from some min1 to some max1. (B) The trajectory obtained by holding x1 fixed at some arbitrary value and allowing x2 to vary across an arbitrary extent, from some min2 to some max2. (C) A curvilinear trajectory, parameterized here by some arbitrary t, extending from some tmin to some tmax.
Fig 8.
Hypoxia gradients and segmented tissue areas.
Left panel: verifying the hypoxia gradient image feature against empirically measured and statistically bounded values. (A) The trajectory obtained by starting at in a necrotic (N) region, locating a hypoxic (H) region, and then descending the hypoxia (p = anti-pimonidazole) gradient into a viable (V) region to reach the vessel centroid at
, using the ∇−(x1, x2, p) gradient following function. (B) The trajectory obtained by starting at the vessel centroid at
in the viable (V) region, then ascending the hypoxia (p = anti-pimonidazole) gradient into a hypoxic (H) region until reaching a necrotic (N) region at
, using the ∇+(x1, x2, p) gradient following function. Right panel: verifying the viable-to-hypoxic area ratio (
) image feature against empirically measured and statistically bounded values.