Spatial Linear and Mixed-Effects Modelling with spicyR
29 October 2024
Abstract
Perform linear and mixed-effects models to assess and visualise changes in cell localisation across disease conditions.
Package
spicyR 1.18.0
1 Installation
if (!require("BiocManager")) {
install.packages("BiocManager")
}
BiocManager::install("spicyR")# load required packages
library(spicyR)
library(ggplot2)
library(SpatialExperiment)
library(SpatialDatasets)
library(imcRtools)
library(dplyr)
library(survival)2 Overview
This guide provides step-by-step instructions on how to apply a linear model to multiple segmented and labelled images to assess how the localisation of different cell types changes across different disease conditions.
3 Example data
We use the Keren et al. (2018) breast cancer dataset to compare the spatial distribution of immune cells in individuals with different levels of tumour infiltration (cold and compartmentalised).
The data is stored as a SpatialExperiment object and contains single-cell spatial data from 41 images.
kerenSPE <- SpatialDatasets::spe_Keren_2018()The cell types in this dataset includes 11 immune cell types (double negative CD3 T cells, CD4 T cells, B cells, monocytes, macrophages, CD8 T cells, neutrophils, natural killer cells, dendritic cells, regulatory T cells), 2 structural cell types (endothelial, mesenchymal), 2 tumour cell types (keratin+ tumour, tumour) and one unidentified category.
4 Linear modelling
To investigate changes in localisation between two different cell types, we measure the level of localisation between two cell types by modelling with the L-function. The L-function is a variance-stabilised K-function given by the equation
\[ \widehat{L_{ij}} (r) = \sqrt{\frac{\widehat{K_{ij}}(r)}{\pi}} \]
with \(\widehat{K_{ij}}\) defined as
\[ \widehat{K_{ij}} (r) = \frac{|W|}{n_i n_j} \sum_{n_i} \sum_{n_j} 1 \{d_{ij} \leq r \} e_{ij} (r) \]
where \(\widehat{K_{ij}}\) summarises the degree of co-localisation of cell type \(j\) with cell type \(i\), \(n_i\) and \(n_j\) are the number of cells of type \(i\) and \(j\), \(|W|\) is the image area, \(d_{ij}\) is the distance between two cells and \(e_{ij} (r)\) is an edge correcting factor.
Specifically, the mean difference between the experimental function and the theoretical function is used as a measure for the level of localisation, defined as
\[ u = \sum_{r' = r_{\text{min}}}^{r_{\text{max}}} \widehat L_{ij, \text{Experimental}} (r') - \widehat L_{ij, \text{Poisson}} (r') \]
where \(u\) is the sum is taken over a discrete range of \(r\) between \(r_{\text{min}}\) and \(r_{\text{max}}\). Differences of the statistic \(u\) between two conditions is modelled using a weighted linear model.
4.1 Test for change in localisation for a specific pair of cells
Firstly, we can test whether one cell type tends to be more localised with another cell type in one condition compared to the other. This can be done using the spicy() function, where we specify the condition parameter.
In this example, we want to see whether or not neutrophils (to) tend to be found around CD8 T cells (from) in compartmentalised tumours compared to cold tumours. Given that there are 3 conditions, we can specify the desired conditions by setting the order of our condition factor. spicy will choose the first level of the factor as the base condition and the second level as the comparison condition. spicy will also naturally coerce the condition column into a factor if it is not already a factor. The column containing cell type annotations can be specified using the cellTypeCol argument. By default, spicy uses the column named cellType in the SpatialExperiment object.
spicyTestPair <- spicy(
kerenSPE,
condition = "tumour_type",
from = "CD8_T_cell",
to = "Neutrophils"
)
topPairs(spicyTestPair)
#> intercept coefficient p.value adj.pvalue
#> CD8_T_cell__Neutrophils -109.081 112.0185 2.166646e-05 2.166646e-05
#> from to
#> CD8_T_cell__Neutrophils CD8_T_cell NeutrophilsWe obtain a spicy object which details the results of the
modelling performed. The topPairs() function can be used to obtain the associated coefficients and p-value.
As the coefficient in spicyTestPair is positive, we find that neutrophils are significantly more likely to be found near CD8 T cells in the compartmentalised tumours group compared to the cold tumour group.
4.2 Test for change in localisation for all pairwise cell combinations
We can perform what we did above for all pairwise combinations of cell
types by excluding the from and to parameters in spicy().
spicyTest <- spicy(
kerenSPE,
condition = "tumour_type"
)
topPairs(spicyTest)
#> intercept coefficient p.value adj.pvalue
#> Macrophages__dn_T_CD3 56.446064 -50.08474 1.080273e-07 3.035568e-05
#> dn_T_CD3__Macrophages 54.987151 -48.38664 2.194018e-07 3.082595e-05
#> Macrophages__DC_or_Mono 73.239404 -59.90361 5.224660e-06 4.893765e-04
#> DC_or_Mono__Macrophages 71.777087 -58.46833 7.431172e-06 5.220399e-04
#> dn_T_CD3__dn_T_CD3 -63.786032 100.61010 2.878804e-05 1.208706e-03
#> Neutrophils__dn_T_CD3 -63.141840 69.64356 2.891872e-05 1.208706e-03
#> dn_T_CD3__Neutrophils -63.133725 70.15508 3.011012e-05 1.208706e-03
#> DC__Macrophages 96.893239 -92.55112 1.801300e-04 5.758129e-03
#> Macrophages__DC 96.896215 -93.25194 1.844241e-04 5.758129e-03
#> CD4_T_cell__Keratin_Tumour -4.845037 -22.14995 2.834659e-04 7.409016e-03
#> from to
#> Macrophages__dn_T_CD3 Macrophages dn_T_CD3
#> dn_T_CD3__Macrophages dn_T_CD3 Macrophages
#> Macrophages__DC_or_Mono Macrophages DC_or_Mono
#> DC_or_Mono__Macrophages DC_or_Mono Macrophages
#> dn_T_CD3__dn_T_CD3 dn_T_CD3 dn_T_CD3
#> Neutrophils__dn_T_CD3 Neutrophils dn_T_CD3
#> dn_T_CD3__Neutrophils dn_T_CD3 Neutrophils
#> DC__Macrophages DC Macrophages
#> Macrophages__DC Macrophages DC
#> CD4_T_cell__Keratin_Tumour CD4_T_cell Keratin_TumourAgain, we obtain a spicy object which outlines the result of the linear
models performed for each pairwise combination of cell types.
We can also examine the L-function metrics of individual images by using the
convenient bind() function on our spicyTest results object.
bind(spicyTest)[1:5, 1:5]
#> imageID condition Keratin_Tumour__Keratin_Tumour
#> 1 1 mixed -2.300602
#> 2 2 mixed -1.989699
#> 3 3 compartmentalised 11.373530
#> 4 4 compartmentalised 33.931133
#> 5 5 compartmentalised 17.922818
#> dn_T_CD3__Keratin_Tumour B_cell__Keratin_Tumour
#> 1 -5.298543 -20.827279
#> 2 -16.020022 3.025815
#> 3 -21.857447 -24.962913
#> 4 -36.438476 -40.470221
#> 5 -20.816783 -38.138076The results can be represented as a bubble plot using the signifPlot() function.
signifPlot(
spicyTest,
breaks = c(-3, 3, 1),
marksToPlot = c("Macrophages", "DC_or_Mono", "dn_T_CD3", "Neutrophils",
"CD8_T_cell", "Keratin_Tumour")
)
Here, we can observe that the most significant relationships occur between macrophages and double negative CD3 T cells, suggesting that the two cell types are far more dispersed in compartmentalised tumours compared to cold tumours.
To examine a specific cell type-cell type relationship in more detail, we can use spicyBoxplot() and specify either from = "Macrophages" and to = "dn_T_CD3" or rank = 1.
spicyBoxPlot(results = spicyTest,
# from = "Macrophages",
# to = "dn_T_CD3"
rank = 1)
5 Linear modelling for custom metrics
spicyR can also be applied to custom distance or abundance metrics. A kNN interactions graph can be generated with the function buildSpatialGraph from the imcRtools package. This generates a colPairs object inside of the SpatialExperiment object.
spicyR provides the function convPairs for converting a colPairs object into an abundance matrix by calculating the average number of nearby cells types for every cell type for a given k. For example, if there exists on average 5 neutrophils for every macrophage in image 1, the column Neutrophil__Macrophage would have a value of 5 for image 1.
kerenSPE <- imcRtools::buildSpatialGraph(kerenSPE,
img_id = "imageID",
type = "knn", k = 20,
coords = c("x", "y"))
pairAbundances <- convPairs(kerenSPE,
colPair = "knn_interaction_graph")
head(pairAbundances["B_cell__B_cell"])
#> B_cell__B_cell
#> 1 12.7349608
#> 10 0.2777778
#> 11 0.0000000
#> 12 1.3333333
#> 13 1.2200957
#> 14 0.0000000The custom distance or abundance metrics can then be included in the analysis with the alternateResult parameter. The Statial package contains other custom distance metrics which can be used with spicy.
spicyTestColPairs <- spicy(
kerenSPE,
condition = "tumour_type",
alternateResult = pairAbundances,
weights = FALSE
)
topPairs(spicyTestColPairs)
#> intercept coefficient p.value adj.pvalue
#> CD8_T_cell__Neutrophils 0.833333333 -0.7592968 0.002645466 0.3291833
#> B_cell__Tumour 0.001937984 0.2602822 0.004872664 0.3291833
#> Other_Immune__NK 0.012698413 0.2612881 0.005673068 0.3291833
#> Unidentified__CD8_T_cell 0.106626794 0.6387339 0.005906526 0.3291833
#> dn_T_CD3__NK 0.004242424 0.2148797 0.006317829 0.3291833
#> CD4_T_cell__Neutrophils 0.036213602 0.2947696 0.007902670 0.3291833
#> Tregs__CD4_T_cell 0.128876212 0.5726201 0.010207087 0.3291833
#> Endothelial__DC 0.008771930 0.3008523 0.011189533 0.3291833
#> Tumour__Neutrophils 0.021638939 0.2529045 0.011388850 0.3291833
#> Mesenchymal__Neutrophils 0.004504505 0.2494301 0.012761315 0.3291833
#> from to
#> CD8_T_cell__Neutrophils CD8_T_cell Neutrophils
#> B_cell__Tumour B_cell Tumour
#> Other_Immune__NK Other_Immune NK
#> Unidentified__CD8_T_cell Unidentified CD8_T_cell
#> dn_T_CD3__NK dn_T_CD3 NK
#> CD4_T_cell__Neutrophils CD4_T_cell Neutrophils
#> Tregs__CD4_T_cell Tregs CD4_T_cell
#> Endothelial__DC Endothelial DC
#> Tumour__Neutrophils Tumour Neutrophils
#> Mesenchymal__Neutrophils Mesenchymal NeutrophilssignifPlot(
spicyTestColPairs,
breaks = c(-3, 3, 1),
marksToPlot = c("Macrophages", "dn_T_CD3", "CD4_T_cell",
"B_cell", "DC_or_Mono", "Neutrophils", "CD8_T_cell")
)
6 Performing survival analysis
spicy can also be used to perform survival analysis to asses whether changes in co-localisation between cell types are associated with survival probability. spicy requires the SingleCellExperiment object being used to contain a column called survival as a Surv object.
kerenSPE$event = 1 - kerenSPE$Censored
kerenSPE$survival = Surv(kerenSPE$`Survival_days_capped*`, kerenSPE$event)We can then perform survival analysis using the spicy function by specifying condition = "survival". We can then access the corresponding coefficients and p-values by accessing the survivalResults slot in the spicy results object.
# Running survival analysis
spicySurvival = spicy(kerenSPE,
condition = "survival")
# top 10 significant pairs
head(spicySurvival$survivalResults, 10)
#> # A tibble: 10 × 4
#> test coef se.coef p.value
#> <chr> <dbl> <dbl> <dbl>
#> 1 Other_Immune__Tregs 0.0236 0.00866 0.00000893
#> 2 CD4_T_cell__Tregs 0.0177 0.00685 0.0000124
#> 3 Tregs__Other_Immune 0.0237 0.00873 0.0000126
#> 4 Tregs__CD4_T_cell 0.0171 0.00676 0.0000285
#> 5 CD8_T_cell__CD8_T_cell 0.00605 0.00272 0.000332
#> 6 Tumour__CD8_T_cell -0.0305 0.0114 0.000617
#> 7 CD8_T_cell__Tumour -0.0305 0.0116 0.000721
#> 8 CD4_T_cell__dn_T_CD3 0.00845 0.00353 0.000794
#> 9 dn_T_CD3__CD4_T_cell 0.00840 0.00353 0.000937
#> 10 DC__Other_Immune -0.0289 0.0123 0.001037 Accounting for tissue inhomogeneity
The spicy function can also account for tissue inhomogeneity to avoid false positives or negatives. This can be done by setting the sigma = parameter within the spicy function. By default, sigma is set to NULL, and spicy assumes a homogeneous tissue structure.
For example, when we examine the L-function for Keratin_Tumour__Neutrophils when sigma = NULL and Rs = 100, the value is positive, indicating attraction between the two cell types.
# filter SPE object to obtain image 24 data
kerenSubset = kerenSPE[, colData(kerenSPE)$imageID == "24"]
pairwiseAssoc = getPairwise(kerenSubset,
sigma = NULL,
Rs = 100) |>
as.data.frame()
pairwiseAssoc[["Keratin_Tumour__Neutrophils"]]
#> [1] 10.88892When we specify sigma = 20 and re-calculate the L-function, it indicates that there is no relationship between Keratin_Tumour and Neutrophils, i.e., there is no major attraction or dispersion, as it now takes into account tissue inhomogeneity.
pairwiseAssoc = getPairwise(kerenSubset,
sigma = 20,
Rs = 100) |>
as.data.frame()
pairwiseAssoc[["Keratin_Tumour__Neutrophils"]]
#> [1] 0.9024836# obtain colData for image 24
cData = colData(kerenSPE) |> as.data.frame() |>
dplyr::filter(imageID == "24")
# obtain cells present in image 24
coords = spatialCoords(kerenSPE) |> as.data.frame()
coords$cellID = rownames(coords)
coords = coords |> dplyr::filter(cellID %in% cData$CellID)
cData$X = coords$x
cData$Y = coords$y
cData = cData |>
dplyr::mutate(cellTypeNew = ifelse(cellType %in% c("Keratin_Tumour", "Neutrophils"),
cellType, "Other"))
pal = setNames(c("#d6b11c", "#850f07"),
c("Keratin_Tumour", "Neutrophils"))
ggplot() +
stat_density_2d(data = cData, aes(x = X, y = Y, fill = after_stat(density)),
geom = "raster",
contour = FALSE) +
geom_point(data = cData |> filter(cellType != "Other"),
aes(x = X, y = Y, colour = cellTypeNew), size = 1) +
scale_color_manual(values = pal) +
scale_fill_distiller(palette = "Blues", direction = 1) +
theme_classic() +
labs(title = "image ID: 24")
Plotting image 24 shows that the supposed co-localisation occurs due to the dense cluster of cells near the top of the image.
8 Mixed effects modelling
spicyR supports mixed effects modelling when multiple images are obtained for each subject. In this case, subject is treated as a random effect and condition is treated as a fixed effect. To perform mixed effects modelling, we can specify the subject parameter in the spicy function.
spicyMixedTest <- spicy(
diabetesData,
condition = "stage",
subject = "case"
)9 References
Keren, L, M Bosse, D Marquez, R Angoshtari, S Jain, S Varma, S Yang, et al. 2018. “A Structured Tumor-Immune Microenvironment in Triple Negative Breast Cancer Revealed by Multiplexed Ion Beam Imaging.” Cell 174 (6): 1373–1387.e19.
Appendix
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