Inter-plaque distances

Over-segmentation assessment

Published

March 23, 2026

Problem

HALO image analysis software segments ThioS-positive objects by intensity thresholding and connected-pixel labelling and may subdivide one physical lesion into multiple adjacent objects. This high object counts would bias cell-to-plaque distance calculations: same cells or genes near a heavily fragmented vessel would be classified as “proximal” (< 50 µm) many times

Code 
source("R/setup.R")
library(patchwork)
library(igraph)

THRESHOLDS <- c(25, 50)   # µm — candidate merge thresholds

plaques <- thios_all |>
  mutate(
    x_plaque  = ((XMin + XMax) / 2) * 0.2125,
    y_plaque  = ((YMin + YMax) / 2) * 0.2125,
    bb_width  = (XMax - XMin) * 0.2125,
    bb_height = (YMax - YMin) * 0.2125,
    bb_diag   = sqrt(bb_width^2 + bb_height^2)
  )

Plaque size overview

To establish the physical scale of annotated objects, we first characterise their size distribution. Two size metrics are reported: the maximum diameter (the longest axis of the object as reported by HALO) and the bounding-box diagonal (the diagonal of the smallest rectangle enclosing the object, computed from pixel coordinates converted to µm at 0.2125 µm/pixel).

Both metrics provide an upper bound on object size and are used later to determine whether inter-object distances are plausibly smaller than the objects themselves.

Code 
plaques |>
  group_by(brain, plaque_type) |>
  summarise(
    n_objects       = n(),
    median_area     = median(Area),
    median_max_diam = median(`Maximum Diameter`),
    median_bb_diag  = median(bb_diag),
    .groups = "drop"
  ) |>
  kbl(
    digits = 1,
    col.names = c("Brain", "Type", "N objects", "Median area (µm²)",
                  "Median max diam (µm)", "Median BB diag (µm)")
  ) |>
  kable_styling(bootstrap_options = c("striped", "condensed"), full_width = FALSE)
Brain Type N objects Median area (µm²) Median max diam (µm) Median BB diag (µm)
Brain1 CAA 345 58.8 12.6 6.8
Brain1 Parenchymal 3961 49.8 10.7 5.9
Brain2 CAA 446 67.8 14.4 7.3
Brain2 Parenchymal 2681 105.2 15.3 8.3
Brain3 CAA 366 73.5 14.4 7.8
Brain3 Parenchymal 2760 51.8 11.4 6.2
brain4 CAA 389 61.5 13.8 7.0
brain4 Parenchymal 3559 54.0 11.4 6.3
brain5 CAA 370 81.1 15.2 8.0
brain5 Parenchymal 3187 67.2 13.1 7.2
brain6 CAA 284 57.1 12.5 6.7
brain6 Parenchymal 2743 61.5 12.8 7.0
Code 
p1 <- plaques |>
  ggplot(aes(`Maximum Diameter`, fill = plaque_type)) +
  geom_histogram(bins = 60, colour = NA, alpha = 0.8) +
  lims(x = c(0, 50))+
  facet_wrap(~plaque_type, scales = "free_y") +
  scale_fill_manual(values = c(CAA = "#d62728", Parenchymal = "#1f77b4"),
                    guide = "none") +
  labs(x = "Maximum diameter (µm)", y = "Count",
       title = "Maximum diameter distribution") +
  theme_minimal(base_size = 10)

p2 <- plaques |>
  ggplot(aes(bb_diag, fill = plaque_type)) +
  geom_histogram(bins = 60, colour = NA, alpha = 0.8) +
  lims(x = c(0, 30))+
  facet_wrap(~plaque_type, scales = "free_y") +
  scale_fill_manual(values = c(CAA = "#d62728", Parenchymal = "#1f77b4"),
                    guide = "none") +
  labs(x = "Bounding-box diagonal (µm)", y = "Count",
       title = "Bounding-box diagonal distribution") +
  theme_minimal(base_size = 10)

p1 + p2

Individual HALO objects are very small.

Across all 6 brains, the median maximum diameter is 12.5–15.2 µm for CAA objects and 10.7–15.3 µm for parenchymal objects, with bounding-box diagonals of 6.7–8.0 µm and 5.9–8.3 µm respectively. Both metrics are narrowly distributed, with the vast majority of objects falling below 50 µm. These size values serve as a reference frame for interpreting inter-object distances: any nearest-neighbour distance smaller than the median maximum diameter implies that two objects are closer to each other than their own width, making physical overlap, and thus over-segmentation, highly probable.

Nearest-neighbour distances between plaques

For each annotated object, we computed the distance to the closest other annotated object of the same type within the same brain, using a k-d tree (RANN::nn2). This nearest-neighbour (NN) distance is the primary metric for detecting over-segmentation: if two objects represent fragments of the same physical lesion, their centroids will be separated by a distance smaller than the lesion’s diameter, producing a NN distance close to zero. Distances are computed separately for CAA and parenchymal plaques to avoid cross-type interference.

Code 
compute_nn <- function(df) {
  df |>
    group_by(brain, plaque_type) |>
    group_modify(\(grp, key) {
      if (nrow(grp) < 2) return(mutate(grp, nn_dist = NA_real_))
      mat <- as.matrix(grp[, c("x_plaque", "y_plaque")])
      res <- nn2(data = mat, query = mat, k = 2)
      mutate(grp, nn_dist = res$nn.dist[, 2])
    }) |>
    ungroup()
}

plaques_nn <- compute_nn(plaques)

Distance distribution

The ECDF shows, for each distance value on the x-axis, what fraction of all annotated objects have a nearest neighbour closer than that distance. A curve that rises steeply near zero indicates that most objects are tightly packed. Vertical dashed lines mark the two candidate merge thresholds (25 µm and 50 µm).

Code 
plaques_nn |>
  filter(!is.na(nn_dist)) |>
  ggplot(aes(nn_dist, colour = brain)) +
  stat_ecdf(linewidth = 0.7) +
  geom_vline(xintercept = THRESHOLDS, linetype = "dashed", colour = "grey40") +
  annotate("text", x = THRESHOLDS[[1]], y = 0.02,
           label = paste0(THRESHOLDS[[1]], " µm"), hjust = -0.1, size = 4,
           colour = "grey30") +
  annotate("text", x = THRESHOLDS[[2]], y = 0.1,
           label = paste0(THRESHOLDS[[2]], " µm"), hjust = -0.1, size = 4,
           colour = "grey30") +
  facet_wrap(~plaque_type) +
  scale_x_continuous(limits = c(0, 500)) +
  labs(x = "NN distance (µm)", y = "Cumulative fraction",
       title = "ECDF of nearest-neighbour inter-plaque distances",
       colour = "Brain") +
  theme_minimal(base_size = 10)

The NN distance distributions differ in an informative way between the two plaque types. For CAA, the median NN distance ranges from 9.8 to 11.8 µm; for parenchymal plaques, from 11.1 to 20.1 µm. In both cases, median distances are equal to or smaller than the median maximum diameter of the objects themselves (11–15 µm), meaning the gap between adjacent annotated objects is on average no wider than the objects’ own diameter. The 10th percentile falls between 5.0 and 5.9 µm for CAA and 4.5 and 5.5 µm for parenchymal plaques — distances at which two centroids would overlap even with the smallest objects.

The log-scale histogram is plotted with a logarithmic x-axis to reveal structure at very small distances (1–10 µm) that would be invisible on a linear scale. A mode well below the 25 µm threshold is consistent with over-segmentation. A bimodal distribution would indicate two co-existing populations: over-segmented fragments (first mode, short distances) and genuinely distinct adjacent lesions (second mode, longer distances). The valley between the modes identifies the natural threshold that best separates the two populations.

Code 
plaques_nn |>
  filter(!is.na(nn_dist), nn_dist > 0) |>
  ggplot(aes(nn_dist, fill = plaque_type)) +
  geom_histogram(bins = 60, colour = NA, alpha = 0.8) +
  geom_vline(xintercept = THRESHOLDS, linetype = "dashed", colour = "grey30") +
  facet_wrap(~plaque_type, scales = "free_y") +
  scale_x_log10() +
  scale_fill_manual(values = c(CAA = "#d62728", Parenchymal = "#1f77b4"),
                    guide = "none") +
  labs(x = "NN distance (µm, log scale)", y = "Count",
       title = "Nearest-neighbour distance histogram (log-scale x)") +
  theme_minimal(base_size = 10)

Code 
plaques_nn |>
  filter(!is.na(nn_dist)) |>
  group_by(brain, plaque_type) |>
  summarise(
    n         = n(),
    p10       = quantile(nn_dist, 0.10),
    p25       = quantile(nn_dist, 0.25),
    median_nn = median(nn_dist),
    .groups = "drop"
  ) |>
  kbl(
    digits = 1,
    col.names = c("Brain", "Type", "N", "10th pct (µm)",
                  "25th pct (µm)", "Median NN (µm)")
  ) |>
  kable_styling(bootstrap_options = c("striped", "condensed"), full_width = FALSE)

The log-scale histogram reveals a qualitative difference between the two types. For CAA, the distribution is unimodal, with a single mode between approximately 5 and 20 µm and a monotonic decay beyond it. This is consistent with a population that is almost entirely composed of over-segmented fragments: there is no second peak at longer distances that would indicate a distinct population of well-separated CAA objects. For parenchymal plaques, the distribution is bimodal: a first peak at 5–20 µm (fragments of the same plaque deposit) and a second, broader peak centred around 50–200 µm (genuine distinct plaques in the tissue). The valley between the two modes falls around 25–50 µm and provides an empirical, data-driven justification for the merge threshold: a 25 µm cutoff sits in the trough between the two populations, capturing fragments while leaving the second-mode objects intact (actual distinct plaques).

Threshold impact

To quantify the practical consequence of over-segmentation, we report, for each brain and plaque type, the number and proportion of objects whose nearest same-type neighbour falls within each candidate merge threshold. An object “within threshold” has at least one nearby neighbour with which it would be merged under the corresponding merging rule. High percentages indicate that the majority of HALO objects are candidates for consolidation.

Code 
map_dfr(THRESHOLDS, \(thr)
  plaques_nn |>
    group_by(brain, plaque_type) |>
    summarise(
      threshold_um = thr,
      n_total      = n(),
      n_within_thr = sum(nn_dist <= thr, na.rm = TRUE),
      pct_within   = mean(nn_dist <= thr, na.rm = TRUE) * 100,
      .groups = "drop"
    )
) |>
  arrange(plaque_type, threshold_um, brain) |>
  kbl(
    digits = 1,
    col.names = c("Brain", "Type", "Threshold (µm)", "N total",
                  "N within threshold", "% within threshold")
  ) |>
  kable_styling(bootstrap_options = c("striped", "condensed"), full_width = FALSE) |>
  collapse_rows(columns = 1:2, valign = "top")
Brain Type Threshold (µm) N total N within threshold % within threshold
Brain1 CAA 25 345 304 88.1
Brain2 25 446 334 74.9
Brain3 25 366 290 79.2
brain4 25 389 328 84.3
brain5 25 370 304 82.2
brain6 25 284 231 81.3
Brain1 50 345 315 91.3
Brain2 50 446 379 85.0
Brain3 50 366 323 88.3
brain4 50 389 350 90.0
brain5 50 370 328 88.6
brain6 50 284 256 90.1
Brain1 Parenchymal 25 3961 3077 77.7
Brain2 25 2681 1509 56.3
Brain3 25 2760 1947 70.5
brain4 25 3559 2606 73.2
brain5 25 3187 2146 67.3
brain6 25 2743 1939 70.7
Brain1 50 3961 3654 92.2
Brain2 50 2681 2093 78.1
Brain3 50 2760 2383 86.3
brain4 50 3559 3159 88.8
brain5 50 3187 2698 84.7
brain6 50 2743 2327 84.8

At the 25 µm threshold, 74.9–88.1% of CAA objects and 56.3–77.7% of parenchymal objects have at least one neighbour of the same type within that distance.

At the less stringent 50 µm threshold, these proportions rise to 85.0–91.3% and 67.4–92.2%, respectively. These values are not driven by a subset of densely annotated regions: the proportions are consistently high across all six brains and both slides, indicating pervasive over-segmentation rather than a localised artefact. The smaller percentage for parenchymal plaques relative to CAA at 25 µm reflects that parenchymal plaques, while still overwhelmingly clustered, are distributed across a wider tissue area and thus have a slightly lower packing density than CAA objects confined to vessel trajectories.

Connected components

The threshold impact analysis counts objects with nearby neighbours but does not account for “chain-like clustering”: an object may be within 25 µm of one neighbour, which is itself within 25 µm of a further neighbour, and so on. To quantify the number of distinct lesions that would remain after merging all transitively connected objects, we construct a graph in which each HALO object is a node and edges connect pairs of objects within the threshold distance.

The number of connected components (disconnected subgraphs) equals the number of merged lesions. Singletons are components containing a single object (i.e. isolated objects with no neighbour within the threshold). The % reduction is the percentage decrease in object count relative to the original, equivalent to the fraction of objects that would be absorbed into multi-object clusters.

Code 
count_components <- function(df, thr) {
  mat <- as.matrix(df[, c("x_plaque", "y_plaque")])
  k   <- min(nrow(mat), 50L)
  res <- nn2(data = mat, query = mat, k = k)
  within <- res$nn.dist <= thr & res$nn.idx != row(res$nn.idx)
  from   <- row(res$nn.idx)[within]
  to     <- res$nn.idx[within]
  edges  <- tibble(from, to) |> filter(from < to)
  g      <- graph_from_data_frame(edges, directed = FALSE,
                                  vertices = seq_len(nrow(mat)))
  comps  <- components(g)
  tibble(
    n_objects     = nrow(mat),
    n_components  = comps$no,
    n_singletons  = sum(comps$csize == 1),
    pct_reduction = (1 - comps$no / nrow(mat)) * 100,
    threshold_um  = thr
  )
}

component_results <- plaques_nn |>
  group_by(brain, plaque_type) |>
  group_modify(\(grp, key) {
    if (nrow(grp) < 2) return(tibble())
    map_dfr(THRESHOLDS, \(thr) count_components(grp, thr))
  }) |>
  ungroup()

component_results |>
  arrange(plaque_type, threshold_um, brain) |>
  kbl(
    digits = 1,
    col.names = c("Brain", "Type", "N objects", "N components",
                  "N singletons", "% reduction", "Threshold (µm)")
  ) |>
  kable_styling(bootstrap_options = c("striped", "condensed"), full_width = FALSE) |>
  collapse_rows(columns = 1:2, valign = "top")
Brain Type N objects N components N singletons % reduction Threshold (µm)
Brain1 CAA 345 113 41 67.2 25
Brain2 446 204 112 54.3 25
Brain3 366 153 76 58.2 25
brain4 389 148 61 62.0 25
brain5 370 142 66 61.6 25
brain6 284 119 53 58.1 25
Brain1 345 93 30 73.0 50
Brain2 446 153 67 65.7 50
Brain3 366 115 43 68.6 50
brain4 389 113 39 71.0 50
brain5 370 102 42 72.4 50
brain6 284 90 28 68.3 50
Brain1 Parenchymal 3961 1678 884 57.6 25
Brain2 2681 1709 1172 36.3 25
Brain3 2760 1380 813 50.0 25
brain4 3559 1737 953 51.2 25
brain5 3187 1679 1041 47.3 25
brain6 2743 1410 804 48.6 25
Brain1 3961 769 307 80.6 50
Brain2 2681 1112 588 58.5 50
Brain3 2760 845 377 69.4 50
brain4 3559 929 400 73.9 50
brain5 3187 1014 489 68.2 50
brain6 2743 894 416 67.4 50

At 25 µm, merging connected objects would reduce CAA counts from 284–446 objects to 90–204 components per brain (a reduction of 58.1–67.2%). Of these surviving components, 28–112 are singletons, meaning between one-quarter and one-half of merged lesions have no neighbours within 25 µm and would be unaffected by merging.

Applying the same threshold to parenchymal plaques would reduce counts from 2,681–3,961 to 1,380–1,709 components (a 36.3–57.6% reduction) with 804–1,172 singletons per brain. Increasing the threshold to 50 µm yields an additional reduction: CAA to 90–153 components (65.7–73.0% reduction) and parenchymal to 769–1,014 components (58.5–80.6% reduction), but the incremental gain over 25 µm is modest relative to the additional risk of merging genuinely distinct lesions.

Spatial maps

To determine whether clustering is spatially coherent, consistent with biological structure (vessel trajectories for CAA, plaque fields for parenchymal) rather than random annotation noise, we map each object’s position coloured by its NN distance. Objects coloured red have a neighbour within 25 µm; grey objects have neighbours at approximately the midpoint distance (25 µm); navy objects are well-isolated (> 100 µm to nearest neighbour). Spatial coherence of red objects would confirm that clusters represent real lesion structures rather than scattered annotation errors.

Code 
make_map <- function(type_label) {
  plaques_nn |>
    filter(plaque_type == type_label) |>
    ggplot(aes(x_plaque, y_plaque, colour = nn_dist)) +
    geom_point(size = 0.8, alpha = 0.5) +
    scale_colour_gradient2(
      low = "red", mid = "grey80", high = "navy",
      midpoint = 60, name = "NN dist (µm)",
      limits = c(0, 200), oob = scales::squish
    ) +
    facet_wrap(~brain, scales = "free") +
    theme_minimal(base_size = 10) +
    labs(title = type_label, x = "x (µm)", y = "y (µm)")
}

wrap_plots(make_map("CAA"), make_map("Parenchymal"), ncol = 1)

CAA objects are almost uniformly red across all six brains and form the characteristic linear, curvilinear, and branching patterns expected of cortical vessel distributions. The rare navy points represent larger, isolated CAA deposits without nearby same-type objects.

For parenchymal plaques, the maps show dense with blue/navy objects occurring only at the periphery of annotated regions. The spatial coherence of the red clusters in both types confirms that over-segmentation reflects the underlying anatomy: HALO has systematically fragmented continuous structures (vessel segments, plaque fields) rather than produced random annotation artefacts.