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MI_spatialAnalysis

 MI_spatialAnalysis

Julia Wrobel

April 03, 2023
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  1. Julia Wrobel, PhD
    Department of Biostatistics and Informatics
    Spatial analysis of multiplex imaging data

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  2. • Spatial summary statistics are extracted to describe spatial relationship
    amongst cells in an image
    • Also often separated by tumor/stroma
    • Univariate spatial summary metrics
    • Clustering or dispersion of cells of one type
    • Bivariate spatial summary metrics
    • Co-expression or co-clustering of two cell types
    • Methods based on spatial point processes
    • Analyze number of neighbors: K-function, L-function
    • Analyze distance to nearest neighbor: G-function
    Spatial summary statistics
    2

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  3. • Locations of cells are considered random and
    to follow a point process
    • Marked point patterns have covariates (cell
    shape, area, expression of CD3) associated
    with each point
    • Multitype point patterns have multiple types
    of points
    • Gray are tumor cells, red are immune cells
    • Specifically interested in spatial arrangement of
    immune cells
    Spatial point patterns
    3

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  4. • Green cells are in tumor tissue
    area
    • Pink cells are in stromal tissue
    area
    • Black cells are macrophages
    • Quantify macrophage clustering
    in tumor and stromal areas
    Another example: macrophages in ovarian cancer
    4

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  5. • Capture important features of the point pattern
    • Assume location of points in point pattern are random variables
    • Typically number of points n is also a random variable
    • Often to detect deviations from complete spatial randomness (CSR)
    • Clustering or repulsion
    Spatial summary functions based on point processes
    5

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  6. • Homogeneity: 𝐸 𝑛(𝑋 ∩ 𝐴) = 𝜆|𝐴|
    • The expected number of points falling in
    a given region A should be proportional
    to the area of the region
    • Spatial independence
    • The counts in disjoint subregions of R are
    independent random variables
    Then, the number of points falling in R
    follows a Poisson distribution
    • Homogeneous Poisson point process
    • Deviations from Homogeneous Poisson
    point process represent spatial clustering
    Complete Spatial Randomness
    6
    𝑃 𝑛 𝑋 ∩ 𝐴 = 𝑘 = 𝑒!|#|
    𝜆 𝐴 $
    𝑘!
    𝜆 is the intensity, or expected number of
    random points per unit area

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  7. • K-function is a popular metric for analyzing spatial correlation in point patterns
    • Essentially the standardized average number of neighbors of a cell within radius r
    /
    𝐾 𝑟 =
    |𝐴|
    𝑛(𝑛 − 1)
    4
    %
    &
    4
    %'(
    𝐼 𝑑 𝑐%, 𝑐) ≤ 𝑟 𝑒%)
    • Assumes homogeneity
    • Has theoretical value under CSR
    • Compare observed to theoretical value to assess clustering
    𝐸!"#
    "
    𝐾 𝑟 = 𝜋𝑟$
    Ripley’s K-function
    7

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  8. Ripley’s K function for simulated cells
    8

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  9. Transformations of Ripley’s K
    9
    /
    𝐾 𝑟 =
    |𝐴|
    𝑛(𝑛 − 1)
    4
    %
    &
    4
    %'(
    𝐼 𝑑 𝑐%, 𝑐) ≤ 𝑟 𝑒%)
    L- function
    - 𝐿 𝑟 = * +
    ,
    Marcon’s M
    - * +
    ,+!

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  10. Can interpret as probability of a neighboring cell occurring within radius r
    • 𝑑!
    = 𝑚𝑖𝑛"#!
    ||𝑥"
    − 𝑥_𝑖|| finds the nearest neighbor
    • 𝐺 𝑟 = ℙ 𝑑 𝑢, 𝑿\u ≤ 𝑟|𝑿 ℎ𝑎𝑠 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑢 for any location u
    Nearest Neighbor G-function
    10

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  11. • Mark correlation function
    • Derivative of K function
    • Moran’s I
    • Can be used to quantify continuous marks
    • Local and global versions
    • Univariate and bivariate
    Other spatial summary statistics
    11

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  12. • Rare cell types: spatially summary metrics cannot be computed for
    images with no or few cells of a certain subtype
    • Makes it challenging to compare across images
    • Multiple images per subject
    • Needs to be accounted for in analysis
    • Violation of homogeneity assumptions bias estimation of K, G functions
    • Inhomogeneity can give appearance of spatial clustering
    Issues that arise in multiplex imaging data
    12

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  13. Homogenous vs. Inhomogeneous Poisson processes
    13

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  14. Ovarian tumor samples have inhomogeneity due to poor
    tissue quality
    14

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  15. • Cells are missing, resulting appearance of inhomogeneity of 𝜆 when
    underlying unobserved process may have been homogeneous
    • Violates homogeneity assumption of K-function!
    • Can no longer compare to theoretical K under CSR of 𝜋𝑟-
    • Wilson et. al, 2022 resolve this using a permutation approach
    • Calculates empirical estimate of K-function under CSR by randomly
    permuting cell labels
    • Then compare empirical K to permuted K rather than theoretical K
    • Effectively separates clustering from “patchy” effects
    Permutation solution to TMA inhomogeneity
    15

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  16. Permutation solution to TMA inhomogeneity
    16

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  17. Fridley Lab
    Moffitt Cancer Center
    spatialTIME package

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  18. • Visualize samples
    • Calculate spatial summary
    statistics across samples
    • Correct for spatial
    inhomogeneity using
    permutation approach
    • iTIME: Shiny interface
    What can we do with spatialTIME?
    18

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