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+Complete Reference: estimate_concentration_field()
+Physics, solvers, and +practical tuning
+Raredon +Laboratory
+ +2026-04-05
+ + Source:vignettes/complete-reference.Rmd
+ complete-reference.Rmd+
+
+
+Overview +
+Spatial transcriptomics data gives us discrete transcript or cell
+locations, but many biological processes — ligand gradients, cytokine
+fields, metabolite concentrations — are continuous.
+estimate_concentration_field() bridges this gap by
+computing the steady-state concentration field that a
+set of point sources (mRNA transcripts, detected proteins) would produce
+under a physically motivated diffusion-clearance model.
The result is a dense concentration map over the tissue, interpolated +from point source locations. This can be used to:
+-
+
- Identify where a signalling molecule is most concentrated +
- Compute the concentration seen by each cell +
- Create two-gene ratio maps (spatial domains of ligand/receptor +balance) +
- Visualise how diffusion length affects signal range +
+
+
+
+
+
+
+The physical model +
+The function solves the screened Poisson equation +(steady-state diffusion-clearance PDE):
+ +| Symbol | +Meaning | +Parameter | +
|---|---|---|
| + | Steady-state concentration at position x + | +(what we solve for) | +
| + | Diffusion coefficient (spatial spread rate) | +D |
+
| + | First-order clearance rate | +lambda |
+
| + | Source density (production rate per unit area) | +production_rate |
+
The key derived quantity is the diffusion +length:
+ ++sets the characteristic distance over which concentration decays from a +point source. It is the single most important parameter to tune. The +shape of the field depends only on +; +the amplitude scales as +.
+++Practical interpretation: + +is roughly the radius of influence of a single transcript. At distance + +from a source, the concentration is negligible. At +, +concentration is near its maximum.
+
+
+
+
+
+Shared helpers +
+
+# ---- Plotting theme ----
+theme_demo <- function(bs = 9.5)
+ theme_minimal(base_size = bs) +
+ theme(plot.title = element_text(face = "bold", size = bs + 0.5),
+ plot.subtitle = element_text(size = bs - 1, color = "grey40",
+ margin = margin(b = 4)),
+ panel.grid = element_line(color = "grey94"),
+ legend.key.width = unit(0.35, "cm"),
+ legend.title = element_text(size = bs - 1),
+ legend.text = element_text(size = bs - 1.5),
+ axis.text = element_text(size = bs - 2))
+
+# ---- Field visualiser ----
+plot_field <- function(result, transcripts = NULL,
+ title = "", subtitle = "",
+ palette = "magma", log_scale = FALSE,
+ show_pts = TRUE, show_contours = TRUE, n_contours = 6L,
+ pt_size = 0.35, pt_alpha = 0.45, pt_color = "#00e5ff",
+ fill_label = "Conc.", symmetric = FALSE) {
+ df <- field_to_df(result, inside_only = TRUE)
+ fill_col <- if (log_scale) { df$fv <- log1p(df$field); "fv" } else "field"
+ fill_lbl <- if (log_scale) paste0("log1p(", fill_label, ")") else fill_label
+ p <- ggplot(df, aes(x = x, y = y, fill = .data[[fill_col]])) +
+ geom_raster(interpolate = TRUE) + coord_equal() +
+ labs(title = title, subtitle = subtitle, x = "X", y = "Y") +
+ theme_demo()
+ if (symmetric) {
+ lim <- max(abs(df[[fill_col]]), na.rm = TRUE)
+ p <- p + scale_fill_distiller(palette = "RdBu", limits = c(-lim, lim),
+ name = fill_lbl, na.value = "transparent")
+ } else {
+ p <- p + scale_fill_viridis_c(option = palette, name = fill_lbl,
+ na.value = "transparent")
+ }
+ if (show_contours && any(!is.na(df$field)))
+ p <- p + geom_contour(aes(z = field), color = "white",
+ alpha = 0.35, linewidth = 0.25, bins = n_contours)
+ if (show_pts && !is.null(transcripts))
+ p <- p + geom_point(data = transcripts, aes(x = x, y = y),
+ inherit.aes = FALSE, color = pt_color,
+ size = pt_size, alpha = pt_alpha)
+ p
+}
+
+# ---- Sample points uniformly from inside a mask ----
+sample_in_mask <- function(mask_poly, n, max_tries = 20) {
+ bb <- sf::st_bbox(mask_poly); mu <- sf::st_union(mask_poly)
+ pts <- data.frame(x = numeric(0), y = numeric(0))
+ for (i in seq_len(max_tries)) {
+ if (nrow(pts) >= n) break
+ cnd <- data.frame(x = runif(n * 8, bb["xmin"], bb["xmax"]),
+ y = runif(n * 8, bb["ymin"], bb["ymax"]))
+ ok <- as.logical(sf::st_within(
+ sf::st_as_sf(cnd, coords = c("x","y")), mu, sparse = FALSE)[, 1L])
+ pts <- rbind(pts, cnd[ok, ])
+ }
+ pts[seq_len(min(n, nrow(pts))), ]
+}+
+
+
+Function reference +
+
+estimate_concentration_field(
+ mask, # sfc polygon -- output of fit_spatial_mask()
+ transcript_coords, # data.frame with columns x and y
+
+ # Physics
+ D = 1.0, # diffusion coefficient (arbitrary units)
+ lambda = 0.1, # first-order clearance rate
+ diffusion_length = NULL, # override: set L directly
+ production_rate = 1.0,
+
+ # Solver
+ method = "fd", # "fd" | "green" | "kde"
+ grid_resolution = 256L, # N x N grid
+ boundary_condition = "dirichlet", # "dirichlet" | "neumann" (fd only)
+ fd_solver = "direct", # "direct" | "iterative" (fd only)
+
+ # Weighting
+ weight_col = NULL, # column name for per-transcript weights (e.g. "umi")
+
+ # Optional
+ include_external = FALSE,
+ normalize = FALSE,
+ log_transform = FALSE,
+ clip_negative = TRUE,
+ n_cores = 1L,
+ plot = FALSE,
+ verbose = TRUE
+)
+
+
+
+
+Return value +
+| Element | +Contents | +
|---|---|
field |
+N x N matrix; NA outside the mask |
+
+x, y
+ |
+Grid centre coordinates (length N each) | +
+hx, hy
+ |
+Grid cell width and height | +
mask |
+N x N logical matrix (TRUE = inside mask) | +
sources |
+N x N matrix of binned source density | +
params |
+List of all input parameters | +
diagnostics |
+Timing, counts, etc. | +
Use field_to_df(result) to convert to a tidy data frame
+for ggplot2.
+
+
+
+
+
+
+Section 1 – Synthetic tissue and transcript layout +
+A kidney-bean-shaped tissue with two synthetic genes:
+-
+
- +Gene A: focal cluster at upper-left, plus sparse +background. High UMI count. +
- +Gene B: 500 transcripts with a right-biased spatial +distribution. Low UMI count. +
+in_kidney <- function(x, y, oa = 9, ob = 7, ncx = 5, nr = 3.8)
+ (x/oa)^2 + (y/ob)^2 < 1 & sqrt((x - ncx)^2 + y^2) >= nr
+
+cands <- data.frame(x = runif(18000, -11, 11), y = runif(18000, -9, 9))
+tissue_pts <- cands[in_kidney(cands$x, cands$y), ][1:3000, ]
+
+if (requireNamespace("TissueMask", quietly = TRUE)) {
+ tissue_mask <- TissueMask::fit_spatial_mask(
+ tissue_pts, method = "raster",
+ raster_resolution = 256L, raster_sigma = 0.7, raster_threshold = 0.15,
+ verbose = FALSE)
+} else {
+ pts_sf <- st_as_sf(tissue_pts, coords = c("x","y"))
+ tissue_mask <- st_convex_hull(st_union(pts_sf))
+}
+mu_t <- sf::st_union(tissue_mask)
+# Gene A: focal cluster at (-4.5, 3.8) + sparse background
+tc_A_cl <- data.frame(x = rnorm(350, -4.5, 1.2), y = rnorm(350, 3.8, 0.9))
+A_in <- as.logical(sf::st_within(
+ sf::st_as_sf(tc_A_cl, coords = c("x","y")), mu_t, sparse = FALSE)[, 1L])
+tc_A_cl <- tc_A_cl[A_in, ]
+tc_A_sc <- sample_in_mask(tissue_mask, 50)
+tc_A <- rbind(tc_A_cl, tc_A_sc)
+tc_A$umi <- c(rpois(nrow(tc_A_cl), 15), rpois(nrow(tc_A_sc), 2))
+
+# Gene B: right-biased random distribution
+tc_B_all <- sample_in_mask(tissue_mask, 2000)
+prob_B <- plogis(tc_B_all$x / 3)
+tc_B <- tc_B_all[runif(nrow(tc_B_all)) < prob_B, ][1:500, ]
+tc_B$umi <- rpois(nrow(tc_B), 5)
+tissue_sf <- sf::st_sf(geometry = tissue_mask)
+
+ggplot() +
+ geom_sf(data = tissue_sf, fill = "grey92", color = "grey60", linewidth = 0.4) +
+ geom_point(data = tc_A, aes(x = x, y = y, color = "Gene A", size = log1p(umi)),
+ alpha = 0.6) +
+ geom_point(data = tc_B, aes(x = x, y = y, color = "Gene B", size = log1p(umi)),
+ alpha = 0.5) +
+ scale_color_manual(values = c("Gene A" = "#e74c3c", "Gene B" = "#2980b9"),
+ name = NULL) +
+ scale_size_continuous(range = c(0.3, 2.5), name = "log(UMI+1)") +
+ coord_sf(datum = NA) + theme_demo() +
+ labs(title = "Section 1: Synthetic kidney-bean tissue",
+ subtitle = "Gene A: focal upper-left | Gene B: broad right-biased",
+ x = "X", y = "Y")
+
+
+
+
+
+Section 2 – Method comparison: fd, green, kde +
+
+pb <- list(D = 1, lambda = 0.3, production_rate = 1,
+ grid_resolution = 256L, verbose = FALSE)
+
+res_fd <- do.call(estimate_concentration_field,
+ c(list(mask = tissue_mask, transcript_coords = tc_A,
+ method = "fd", fd_solver = "direct",
+ boundary_condition = "dirichlet"), pb))
+
+res_green <- do.call(estimate_concentration_field,
+ c(list(mask = tissue_mask, transcript_coords = tc_A,
+ method = "green"), pb))
+
+res_kde <- do.call(estimate_concentration_field,
+ c(list(mask = tissue_mask, transcript_coords = tc_A,
+ method = "kde", kde_bandwidth = sqrt(1/0.3)), pb))
+
+res_diff <- res_fd
+res_diff$field <- res_fd$field - res_green$field
+(plot_field(res_fd, tc_A, title = "2a. fd (Dirichlet)",
+ subtitle = "Exact BCs; recommended") +
+ plot_field(res_green, tc_A, title = "2b. green (FFT)",
+ subtitle = "Infinite domain; no boundary effects")) /
+(plot_field(res_kde, tc_A, title = "2c. kde",
+ subtitle = "Bandwidth = L; phenomenological") +
+ plot_field(res_diff, title = "2d. fd - green",
+ subtitle = "fd correctly suppresses near-edge concentration",
+ symmetric = TRUE, show_contours = FALSE, show_pts = FALSE)) +
+plot_annotation(
+ title = "Section 2: Solver Comparison (Gene A, D=1, lambda=0.3, L ~ 1.83)",
+ theme = theme(plot.title = element_text(face = "bold", size = 13)))
Key observations:
+-
+
-
+
fdandgreenagree well in the interior; +difference map (2d) showsfdcorrectly zeros concentration +at the tissue boundary (Dirichlet), whilegreenleaks +outward.
+ -
+
kdeis visually similar because the bandwidth was set +to +, +but it has no physical interpretation and no boundary conditions.
+
+
+
+
+
+
+
+
+Section 3 – Diffusion length sweep +
+The diffusion length + +is the primary tuning parameter.
+
+L_vals <- c(0.5, 1.5, 4.0, 10.0)
+
+sw3 <- lapply(L_vals, function(Lv) {
+ res <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_A,
+ D = 1, diffusion_length = Lv, production_rate = 1,
+ method = "fd", fd_solver = "direct", grid_resolution = 256L, verbose = FALSE)
+ f <- res$field
+ res$field <- (f - min(f, na.rm = TRUE)) / diff(range(f, na.rm = TRUE))
+ plot_field(res, tc_A,
+ title = sprintf("L = %.1f", Lv),
+ subtitle = sprintf("lambda ~ %.3f", 1/Lv^2),
+ fill_label = "Norm. Conc.", pt_size = 0.4) +
+ scale_fill_viridis_c(option = "magma", name = "Norm.\nConc.",
+ limits = c(0, 1), na.value = "transparent")
+})
+wrap_plots(sw3, nrow = 1) +
+ plot_annotation(
+ title = "Section 3: Diffusion Length Sweep (each panel normalised independently)",
+ subtitle = "Small L: tight peaks | Large L: tissue-wide field",
+ theme = theme(plot.title = element_text(face = "bold", size = 13),
+ plot.subtitle = element_text(size = 9, color = "grey40")))
+peaks <- sapply(L_vals, function(Lv) {
+ res <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_A,
+ D = 1, diffusion_length = Lv,
+ method = "fd", fd_solver = "direct", grid_resolution = 256L, verbose = FALSE)
+ max(res$field, na.rm = TRUE)
+})
+knitr::kable(data.frame(L = L_vals, peak_concentration = round(peaks, 5)))| L | +peak_concentration | +
|---|---|
| 0.5 | +11.13878 | +
| 1.5 | +36.13160 | +
| 4.0 | +58.53364 | +
| 10.0 | +67.03298 | +
+
+
+How to choose L in practice +
+-
+
- +What is the typical inter-cluster distance? + +should be 10–20% of this distance. +
- +Are you modelling a small diffusing molecule or a large +one? Small molecules have higher + +and thus larger +. +
- +Does your field look biologically reasonable? +Inspect the output and compare to known biology. +
+
+
+
+
+
+Section 4 – Fixed L, varying D and lambda +
+Field shape depends only on +. +Changing + +and + +while keeping + +constant produces identical spatial patterns, scaling amplitude as +.
+
+DL_cases <- list(
+ list(D = 0.1, lam = 0.025, label = "D=0.1, lam=0.025"),
+ list(D = 1, lam = 0.25, label = "D=1, lam=0.25"),
+ list(D = 5, lam = 1.25, label = "D=5, lam=1.25"),
+ list(D = 20, lam = 5, label = "D=20, lam=5")
+)
+
+dl_pl <- lapply(DL_cases, function(co) {
+ res <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_A,
+ D = co$D, lambda = co$lam, production_rate = 1,
+ method = "fd", fd_solver = "direct", grid_resolution = 256L, verbose = FALSE)
+ pk <- round(max(res$field, na.rm = TRUE), 5)
+ plot_field(res, tc_A,
+ title = co$label,
+ subtitle = sprintf("L=2 fixed | peak=%.5f", pk),
+ pt_size = 0.3)
+})
+wrap_plots(dl_pl, nrow = 1) +
+ plot_annotation(
+ title = "Section 4: Fixed L=2, Varying D and lambda",
+ subtitle = "Shape IDENTICAL (same L). Amplitude proportional to 1/D.",
+ theme = theme(plot.title = element_text(face = "bold", size = 13),
+ plot.subtitle = element_text(size = 9, color = "grey40")))
+
+
+
+
+
+
+Section 5 – Boundary conditions +
+
+res_dir <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_A,
+ D = 1, lambda = 0.2, method = "fd", fd_solver = "direct",
+ boundary_condition = "dirichlet", grid_resolution = 256L, verbose = FALSE)
+
+res_neu <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_A,
+ D = 1, lambda = 0.2, method = "fd", fd_solver = "direct",
+ boundary_condition = "neumann", grid_resolution = 256L, verbose = FALSE)
+
+res_bcd <- res_dir
+res_bcd$field <- res_neu$field - res_dir$field
+sl <- function(res, lbl) {
+ df <- field_to_df(res)
+ s <- df[abs(df$y) < res$hy * 1.5 & !is.na(df$field), ]
+ s <- s[order(s$x), ]; s$BC <- lbl; s
+}
+sl_all <- rbind(sl(res_dir, "Dirichlet"), sl(res_neu, "Neumann"))
+
+p5d <- ggplot(sl_all, aes(x = x, y = field, color = BC)) +
+ geom_line(linewidth = 0.8) +
+ scale_color_manual(values = c("Dirichlet" = "#e74c3c", "Neumann" = "#2980b9")) +
+ labs(title = "5d. Cross-section at y = 0",
+ subtitle = "Dirichlet drops to 0; Neumann stays elevated",
+ x = "X", y = "Conc.", color = "BC") +
+ theme_demo()
+(plot_field(res_dir, tc_A,
+ title = "5a. Dirichlet (C=0 at boundary)",
+ subtitle = "D=1, lambda=0.2, L ~ 2.24") +
+ plot_field(res_neu, tc_A,
+ title = "5b. Neumann (dC/dn=0)",
+ subtitle = "Molecules reflected; higher near edges",
+ palette = "plasma")) /
+(plot_field(res_bcd,
+ title = "5c. Neumann - Dirichlet",
+ subtitle = "Neumann retains more concentration near boundary",
+ symmetric = TRUE, show_contours = FALSE, show_pts = FALSE) +
+ p5d) +
+plot_annotation(
+ title = "Section 5: Boundary Condition Comparison",
+ theme = theme(plot.title = element_text(face = "bold", size = 13)))
Recommendation: Use "dirichlet" for
+tissue sections. Neumann BCs are mainly useful when comparing to
+analytic solutions or modelling sealed compartments.
+
+
+
+
+
+Section 6 – UMI weighting vs. equal weighting +
+
+res_uw <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_A,
+ D = 1, lambda = 0.3, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, weight_col = NULL, verbose = FALSE)
+
+res_wt <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_A,
+ D = 1, lambda = 0.3, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, weight_col = "umi", verbose = FALSE)
+
+nrm <- function(r) {
+ f <- r$field
+ r$field <- (f - min(f, na.rm = TRUE)) / diff(range(f, na.rm = TRUE))
+ r
+}
+plot_field(nrm(res_uw), tc_A,
+ title = "6a. Unweighted",
+ subtitle = "All transcripts equal weight",
+ fill_label = "Norm. Conc.") +
+plot_field(nrm(res_wt), tc_A,
+ title = "6b. UMI-weighted",
+ subtitle = "High-UMI cluster source amplified",
+ palette = "plasma",
+ fill_label = "Norm. Conc.") +
+plot_annotation(
+ title = "Section 6: UMI Weighting",
+ theme = theme(plot.title = element_text(face = "bold", size = 13)))
+
+
+
+
+
+Section 7 – External transcripts +
+include_external = TRUE includes transcripts outside the
+mask as sources, useful when transcripts fall just outside the fitted
+mask boundary.
+tc_ext <- data.frame(x = rnorm(80, 6.5, 0.4),
+ y = rnorm(80, 0, 0.5),
+ umi = rpois(80, 10))
+ext_in <- as.logical(sf::st_within(
+ sf::st_as_sf(tc_ext, coords = c("x","y")), mu_t, sparse = FALSE)[, 1L])
+tc_ext <- tc_ext[!ext_in, ]
+tc_Ax <- rbind(tc_A, tc_ext)
+
+res_ne <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_Ax,
+ D = 1, lambda = 0.15, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, include_external = FALSE, verbose = FALSE)
+
+res_we <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_Ax,
+ D = 1, lambda = 0.15, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, include_external = TRUE, verbose = FALSE)
+
+ext_pt_layer <- geom_point(data = tc_ext, aes(x = x, y = y),
+ color = "orange", size = 0.9, alpha = 0.8,
+ inherit.aes = FALSE)
+(plot_field(res_ne, tc_A, title = "7a. include_external = FALSE") +
+ ext_pt_layer) +
+(plot_field(res_we, tc_A, title = "7b. include_external = TRUE",
+ palette = "plasma") +
+ ext_pt_layer) +
+plot_annotation(
+ title = "Section 7: External Transcripts (orange = outside mask)",
+ theme = theme(plot.title = element_text(face = "bold", size = 13)))
+
+
+
+
+
+
+Section 8 – Holey mask: vascular clearance +
+A key strength of method = "fd" is correct handling of
+holes (vessel lumens) in the mask. The green method ignores
+topology.
+vd <- list(list(cx = -2, cy = 1.5, r = 1.2),
+ list(cx = 2.5, cy = -2, r = 1.0),
+ list(cx = -4.5, cy = -2.5, r = 0.8))
+in_v <- function(x, y) Reduce(`|`, lapply(vd, function(v)
+ sqrt((x - v$cx)^2 + (y - v$cy)^2) < v$r))
+
+n_h <- 3000
+cnd_h <- data.frame(x = runif(n_h * 8, -11, 11),
+ y = runif(n_h * 8, -9, 9))
+kp_h <- in_kidney(cnd_h$x, cnd_h$y) & !in_v(cnd_h$x, cnd_h$y)
+pts_h <- cnd_h[kp_h, ][1:n_h, ]
+
+if (requireNamespace("TissueMask", quietly = TRUE)) {
+ holey_mask <- TissueMask::fit_spatial_mask(
+ pts_h, method = "raster",
+ raster_resolution = 256L, raster_sigma = 0.4,
+ raster_threshold = 0.2, verbose = FALSE)
+} else {
+ pts_hf <- st_as_sf(pts_h, coords = c("x","y"))
+ holey_mask <- st_convex_hull(st_union(pts_hf))
+}
+tc_h <- sample_in_mask(holey_mask, 400)
+res_hfd <- estimate_concentration_field(
+ mask = holey_mask, transcript_coords = tc_h,
+ D = 1, lambda = 0.15, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, verbose = FALSE)
+
+res_htight <- estimate_concentration_field(
+ mask = holey_mask, transcript_coords = tc_h,
+ D = 1, lambda = 1.0, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, verbose = FALSE)
+
+res_hgrn <- estimate_concentration_field(
+ mask = holey_mask, transcript_coords = tc_h,
+ D = 1, lambda = 0.15, method = "green",
+ grid_resolution = 256L, verbose = FALSE)
+
+res_hdiff <- res_hfd
+res_hdiff$field <- res_hfd$field - res_hgrn$field
+(plot_field(res_hfd, tc_h,
+ title = "8a. fd | L ~ 2.58",
+ subtitle = "Vessel lumens correctly zeroed",
+ pt_size = 0.3) +
+ plot_field(res_htight, tc_h,
+ title = "8b. fd | L = 1 (tight)",
+ subtitle = "Steeper depletion near vessels",
+ palette = "inferno", pt_size = 0.3)) /
+(plot_field(res_hgrn, tc_h,
+ title = "8c. green -- topology blind",
+ subtitle = "FFT ignores holes entirely",
+ palette = "plasma", pt_size = 0.3) +
+ plot_field(res_hdiff,
+ title = "8d. fd - green",
+ subtitle = "fd zeros lumens; green does not",
+ symmetric = TRUE, show_contours = FALSE, show_pts = FALSE)) +
+plot_annotation(
+ title = "Section 8: Holey Mask -- Vascular Clearance",
+ subtitle = "Use method='fd' whenever the mask has holes",
+ theme = theme(plot.title = element_text(face = "bold", size = 13),
+ plot.subtitle = element_text(size = 9, color = "grey40")))
++Important: If your tissue mask has holes (vessel +lumens, necrotic cores), always use
+method = "fd". The +greenandkdemethods are unaware of domain +topology.
+
+
+
+
+
+Section 9 – Wide dynamic range and log-scale visualisation +
+When transcript counts span several orders of magnitude, use
+log_scale = TRUE in plot_field() or
+log_transform = TRUE in the solver call to reveal
+background structure.
+tc_w <- rbind(
+ data.frame(x = rnorm(20, -5, 0.3), y = rnorm(20, 4, 0.3), umi = rpois(20, 200)),
+ data.frame(x = rnorm(300, 0, 4), y = rnorm(300, 0, 3), umi = rpois(300, 1)))
+tc_w_in <- as.logical(sf::st_within(
+ sf::st_as_sf(tc_w[, c("x","y")], coords = c("x","y")),
+ mu_t, sparse = FALSE)[, 1L])
+tc_w <- tc_w[tc_w_in, ]
+
+res_w <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_w,
+ D = 1, lambda = 0.5, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, weight_col = "umi", verbose = FALSE)
+plot_field(res_w, tc_w,
+ title = "9a. Linear scale",
+ subtitle = "Background invisible; cluster dominates",
+ pt_color = "white") +
+plot_field(res_w, tc_w,
+ log_scale = TRUE,
+ title = "9b. log1p scale",
+ subtitle = "Background and cluster both visible",
+ palette = "viridis", pt_color = "white") +
+plot_annotation(
+ title = "Section 9: Wide Dynamic Range",
+ subtitle = "Use log_scale=TRUE when max/min ratio > ~20",
+ theme = theme(plot.title = element_text(face = "bold", size = 13),
+ plot.subtitle = element_text(size = 9, color = "grey40")))
+
+
+
+
+
+Section 10 – Two-gene ratio map +
+A log2(A/B) ratio map shows spatial domains of
+ligand/receptor balance.
+ph <- list(D = 1, lambda = 0.25, production_rate = 1,
+ method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, verbose = FALSE)
+
+res_gA <- do.call(estimate_concentration_field,
+ c(list(mask = tissue_mask, transcript_coords = tc_A), ph))
+res_gB <- do.call(estimate_concentration_field,
+ c(list(mask = tissue_mask, transcript_coords = tc_B), ph))
+
+res_rt <- res_gA
+res_rt$field <- log2((res_gA$field + 1e-6) / (res_gB$field + 1e-6))
+p_ratio <- plot_field(res_rt,
+ title = "10c. log2(A/B) ratio",
+ subtitle = "Red=A territory | Blue=B territory | White=balanced",
+ symmetric = TRUE, show_contours = TRUE, n_contours = 5,
+ show_pts = FALSE, fill_label = "log2(A/B)") +
+ geom_point(data = tc_A, aes(x = x, y = y),
+ color = "#e74c3c", size = 0.3, alpha = 0.4,
+ inherit.aes = FALSE) +
+ geom_point(data = tc_B, aes(x = x, y = y),
+ color = "#27ae60", size = 0.3, alpha = 0.4,
+ inherit.aes = FALSE)
+
+p_dens <- ggplot(field_to_df(res_rt), aes(x = field,
+ fill = ifelse(field < 0, "B dominant", "A dominant"))) +
+ geom_density(alpha = 0.55) +
+ geom_vline(xintercept = 0, linetype = "dashed", color = "grey40") +
+ scale_fill_manual(values = c("B dominant" = "#2980b9",
+ "A dominant" = "#c0392b"), name = NULL) +
+ labs(title = "10d. Distribution of log2(A/B)",
+ subtitle = "Fraction of tissue in each territory",
+ x = "log2(A/B)", y = "Density") +
+ theme_demo()
+
+(plot_field(res_gA, tc_A, title = "10a. Gene A field") +
+ plot_field(res_gB, tc_B, title = "10b. Gene B field", palette = "viridis")) /
+(p_ratio + p_dens) +
+plot_annotation(
+ title = "Section 10: Two-Gene Ratio Map",
+ theme = theme(plot.title = element_text(face = "bold", size = 13)))
+
+
+
+
+
+
+
+
+
+
+
+Section 11 – 100k transcripts and solver timing +
+
+tc_100k <- sample_in_mask(tissue_mask, 100000)
+cat(sprintf("Sampled %d transcripts.\n", nrow(tc_100k)))## Sampled 100000 transcripts.
+t1 <- system.time(r1 <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_100k,
+ D = 1, lambda = 0.3, method = "fd", fd_solver = "direct",
+ grid_resolution = 256L, verbose = TRUE))## ============================================================
+## estimate_concentration_field
+## ============================================================
+## Method: fd | D: 1 | lambda: 0.3 | L: 1.82574 | N: 256 x 256
+## BC: dirichlet | solver: direct
+## ------------------------------------------------------------
+## Transcripts: 100000 used (0 external excluded)
+## Rasterizing 256x256 grid...
+## Inside-mask cells: 45594 (69.6%)
+## Sparse system: 45594x45594, 227002 nnz
+## Solve: 0.75s | Total: 1.27s | Range: [23.11, 1596]
+## ============================================================
+t2 <- system.time(r2 <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_100k,
+ D = 1, lambda = 0.3, method = "fd", fd_solver = "iterative",
+ sor_omega = 1.75, grid_resolution = 256L, verbose = TRUE))## ============================================================
+## estimate_concentration_field
+## ============================================================
+## Method: fd | D: 1 | lambda: 0.3 | L: 1.82574 | N: 256 x 256
+## BC: dirichlet | solver: iterative
+## ------------------------------------------------------------
+## Transcripts: 100000 used (0 external excluded)
+## Rasterizing 256x256 grid...
+## Inside-mask cells: 45594 (69.6%)## Solve: 6.05s | Total: 6.57s | Range: [23.11, 1596]
+## ============================================================
+t3 <- system.time(r3 <- estimate_concentration_field(
+ mask = tissue_mask, transcript_coords = tc_100k,
+ D = 1, lambda = 0.3, method = "green",
+ grid_resolution = 256L, verbose = TRUE))## ============================================================
+## estimate_concentration_field
+## ============================================================
+## Method: green | D: 1 | lambda: 0.3 | L: 1.82574 | N: 256 x 256
+## ------------------------------------------------------------
+## Transcripts: 100000 used (0 external excluded)
+## Rasterizing 256x256 grid...
+## Inside-mask cells: 45594 (69.6%)
+## Solve: 0.08s | Total: 0.62s | Range: [701.5, 1651]
+## ============================================================
+knitr::kable(data.frame(
+ Solver = c("fd/direct (N=256)", "fd/SOR (N=256)", "green/FFT (N=256)"),
+ Time_s = round(c(t1["elapsed"], t2["elapsed"], t3["elapsed"]), 2),
+ Has_BCs = c("Yes", "Yes (Dirichlet only)", "No"),
+ Handles_holes = c("Yes", "Yes", "No"),
+ Notes = c("Sparse LU; recommended up to N=400",
+ "Iterative; use for N > 400",
+ "Fastest; infinite domain only")
+))| Solver | +Time_s | +Has_BCs | +Handles_holes | +Notes | +
|---|---|---|---|---|
| fd/direct (N=256) | +1.27 | +Yes | +Yes | +Sparse LU; recommended up to N=400 | +
| fd/SOR (N=256) | +6.57 | +Yes (Dirichlet only) | +Yes | +Iterative; use for N > 400 | +
| green/FFT (N=256) | +0.62 | +No | +No | +Fastest; infinite domain only | +
+sub <- tc_100k[sample(nrow(tc_100k), 2000), ]
+plot_field(r1, sub,
+ title = sprintf("11a. fd/direct (%.1f s)", t1["elapsed"]),
+ subtitle = "N=256 | sparse LU",
+ pt_size = 0.15, pt_alpha = 0.2) +
+plot_field(r2, sub,
+ title = sprintf("11b. fd/SOR (%.1f s)", t2["elapsed"]),
+ subtitle = "N=256 | iterative",
+ palette = "plasma", pt_size = 0.15, pt_alpha = 0.2) +
+plot_annotation(
+ title = "Section 11: 100k Transcripts -- Solver Comparison",
+ subtitle = "2,000 of 100,000 points shown",
+ theme = theme(plot.title = element_text(face = "bold", size = 13),
+ plot.subtitle = element_text(size = 9, color = "grey40")))
+
+
+
+
+
+
+
+
+
+
+
+Section 12 – Quantitative field extraction +
+
+res_q <- res_fd
+
+interpolate_field <- function(result, query_pts) {
+ xg <- result$x; yg <- result$y; f <- result$field
+ vapply(seq_len(nrow(query_pts)), function(i) {
+ qx <- query_pts$x[i]; qy <- query_pts$y[i]
+ xi <- findInterval(qx, xg); yi <- findInterval(qy, yg)
+ if (xi < 1 || xi >= length(xg) || yi < 1 || yi >= length(yg))
+ return(NA_real_)
+ wx <- (qx - xg[xi]) / (xg[xi+1] - xg[xi])
+ wy <- (qy - yg[yi]) / (yg[yi+1] - yg[yi])
+ (1-wx)*(1-wy)*f[xi,yi] + wx*(1-wy)*f[xi+1,yi] +
+ (1-wx)*wy *f[xi,yi+1] + wx*wy *f[xi+1,yi+1]
+ }, numeric(1))
+}
+queries <- data.frame(
+ label = c("Cluster core", "Cluster edge", "Tissue centre",
+ "Far right", "Near notch"),
+ x = c(-4.5, -3.0, 0.0, 5.0, 3.0),
+ y = c( 3.8, 2.5, 0.0, 0.0, 0.0))
+queries$concentration <- interpolate_field(res_q, queries)
+knitr::kable(queries, digits = 5)| label | +x | +y | +concentration | +
|---|---|---|---|
| Cluster core | +-4.5 | +3.8 | +40.94720 | +
| Cluster edge | +-3.0 | +2.5 | +21.12701 | +
| Tissue centre | +0.0 | +0.0 | +2.15650 | +
| Far right | +5.0 | +0.0 | +0.75415 | +
| Near notch | +3.0 | +0.0 | +0.97991 | +
+df_q <- field_to_df(res_q)
+
+cat(sprintf("Mean concentration:\n Left half (x < 0): %.5f\n Right half (x >= 0): %.5f\n",
+ mean(df_q$field[df_q$x < 0 & !is.na(df_q$field)]),
+ mean(df_q$field[df_q$x >= 0 & !is.na(df_q$field)])))## Mean concentration:
+## Left half (x < 0): 6.73849
+## Right half (x >= 0): 0.80629
+pk <- which.max(res_q$field)
+N <- length(res_q$x)
+cat(sprintf("\nPeak: x=%.2f, y=%.2f, value=%.6f\n",
+ res_q$x[((pk-1L) %% N) + 1L],
+ res_q$y[((pk-1L) %/% N) + 1L],
+ res_q$field[pk]))##
+## Peak: x=-4.25, y=3.86, value=41.440719
+total <- sum(res_q$field[res_q$mask], na.rm = TRUE) * res_q$hx * res_q$hy
+theory <- nrow(tc_A) * 1.0 / 0.3
+cat(sprintf("\nField integral: %.4f\nTheory (n * r / lambda): %.4f\nAgreement: %.1f%%\n",
+ total, theory, 100 * (1 - abs(total - theory) / theory)))##
+## Field integral: 737.6707
+## Theory (n * r / lambda): 1300.0000
+## Agreement: 56.7%
+prof <- df_q[abs(df_q$y) < res_q$hy * 1.5 & !is.na(df_q$field), ]
+ggplot(prof[order(prof$x), ], aes(x = x, y = field)) +
+ geom_line(color = "#e74c3c", linewidth = 0.8) +
+ geom_area(fill = "#e74c3c", alpha = 0.15) +
+ geom_vline(xintercept = 0, linetype = "dashed", color = "grey50") +
+ labs(title = "12e. Concentration profile along y = 0",
+ subtitle = "Cluster peak at x ~ -4.5; near-zero at right edge",
+ x = "X", y = "Concentration") +
+ theme_demo()
+
+
+
+Parameter quick-reference +
+estimate_concentration_field() -- parameter guide
+--------------------------------------------------------------------
+
+PHYSICS
+ diffusion_length Primary tuning knob. Set directly: L = 0.5 to 10
+ Rule: L ~ 10-20% of inter-cluster distance
+ D Affects amplitude only (amplitude proportional to 1/D)
+ lambda Affects amplitude only; lambda = D / L^2
+ production_rate Global amplitude scaling; use weight_col for per-cell
+
+SOLVER
+ method = "fd" Default. Use whenever you need accurate BCs or holes.
+ method = "green" Faster for sweeps; ignores domain shape.
+ method = "kde" Quick baseline; no physics.
+
+ fd_solver = "direct" N <= 400 (sparse LU decomposition)
+ fd_solver = "iterative" N > 400 (red-black SOR)
+
+ boundary_condition = "dirichlet" C=0 at boundary. Default.
+ boundary_condition = "neumann" dC/dn=0. Sealed system.
+
+ grid_resolution = 256 Explore. 512 for publication quality.
+
+WEIGHTS
+ weight_col = "umi" Use UMI counts as source strength.
+
+POST-PROCESSING
+ log_transform = TRUE Apply log1p() after solving.
+ normalize = TRUE Rescale to [0,1] for comparison.
+ clip_negative = TRUE Remove small numerical negatives. Default on.
+--------------------------------------------------------------------+
+
+
+ Session information +
+
+sessionInfo()## R version 4.5.2 (2025-10-31)
+## Platform: aarch64-apple-darwin20
+## Running under: macOS Sonoma 14.6.1
+##
+## Matrix products: default
+## BLAS: /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
+## LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.1
+##
+## locale:
+## [1] en_US/en_US/en_US/C/en_US/en_US
+##
+## time zone: America/New_York
+## tzcode source: internal
+##
+## attached base packages:
+## [1] stats graphics grDevices utils datasets methods base
+##
+## other attached packages:
+## [1] patchwork_1.3.2 ggplot2_4.0.1 sf_1.0-21 TissueField_0.1.0
+##
+## loaded via a namespace (and not attached):
+## [1] sass_0.4.10 generics_0.1.4 class_7.3-23 KernSmooth_2.23-26
+## [5] lattice_0.22-7 digest_0.6.39 magrittr_2.0.4 evaluate_1.0.5
+## [9] grid_4.5.2 RColorBrewer_1.1-3 fastmap_1.2.0 Matrix_1.7-4
+## [13] jsonlite_2.0.0 e1071_1.7-16 DBI_1.2.3 viridisLite_0.4.2
+## [17] scales_1.4.0 isoband_0.3.0 textshaping_1.0.1 jquerylib_0.1.4
+## [21] cli_3.6.5 rlang_1.1.7 units_0.8-7 withr_3.0.2
+## [25] cachem_1.1.0 yaml_2.3.12 otel_0.2.0 tools_4.5.2
+## [29] dplyr_1.1.4 vctrs_0.7.1 R6_2.6.1 proxy_0.4-27
+## [33] lifecycle_1.0.5 classInt_0.4-11 fs_1.6.6 htmlwidgets_1.6.4
+## [37] ragg_1.4.0 pkgconfig_2.0.3 desc_1.4.3 pkgdown_2.1.3
+## [41] bslib_0.10.0 pillar_1.11.1 gtable_0.3.6 glue_1.8.0
+## [45] Rcpp_1.1.1 systemfonts_1.2.3 xfun_0.56 tibble_3.3.1
+## [49] tidyselect_1.2.1 knitr_1.51 dichromat_2.0-0.1 farver_2.1.2
+## [53] htmltools_0.5.9 rmarkdown_2.30 labeling_0.4.3 compiler_4.5.2
+## [57] S7_0.2.1