Objective

Question of validity of using sections data as a representation of real tissue statistical properties. This validity is discussed and it is believed that 2D projections of the tissues do not provide with the accurate spatial distribution patterns

The aim of this pipeline is to compare spatial metrics both on true spatial data and on 2D sections, also to elaborate the method how to overcome the bias derived from two-dimensional sampling

Theoretical Part

Pair-Correlation Function

Quick Introductions

Pair-correlation function $g(r)$ measures the probability of finding a pair of objects separated by a distance r relative to what would be expected for a completely random (Poisson) distribution

Derivation

Given that space is isotropic let suggest that probability of observing a point around location x is $$p(x)=\lambda(x)dx$$ where $\lambda(x)$ is intensity, or density of the point process (like Poisson process) and $dV$ - small region of volume

Let $p(x,y)$ be a probability of observing 1 point in x and 1 point in y, then $$p(x,y)=\lambda(x)dx\lambda(y)dy\bullet g(x,y)$$ where $g(x,y)=\frac{p(x,y)}{\lambda^2dxdy}$ - pair-correlation function (in simple case)

Intuition

Layers of spheres get more diffuse, so for large distances, the probability of finding two spheres with a given separation is essentially constant > a more dense system has more spheres, this it's more likely to find two of them with a given distance

The PC function accounts for these factors by normalizing by the density; this at large values of r it goes to 1, uniform probability

Calculation in 3D Case

1. Pick a value of dr
2. Loop over all values of r
   1. Count all particles that are a distance between r and r+dr away from the particle you're considering
   2. Divide your total count by N > average local density
   3. Divide this number by $4\pi r^2dr$ (volume of the spherical shell)
   4. Divide this by the particle number density $\rho$ - ensures that $g(r)=1$

For 2D volume correction will be just $2\pi r dr$

Calculations

Pipeline: reshape > express > cluster > choose > slices > main (dependent of slices)

• reshape.py - takes stacks of images corresponding to the certain cell markers and cell masks ~ clusterization. This script creates cell datasets with coordinates cell_coordinates.csv and with intensities of certain cell markers - cell_intensities.csv
• express.r - creates expression matrices, both raw expression_annotated.csv and corrected expression_annotated_corrected.csv with compensation matrix compensationMatrix.csv
• cluster.py - performs clusterization and manual annotation of expression matrix expression_annotated_corrected.csv. Makes various plots for analysis
• choose.py - selects cell cluster obtain in the previous script > cell_coordinates_clusters.csv
• module.r - contains function to perform random sections based on the 3d dataset cell_coordinates. Compute Pair-correlation function for each of the slices
• slice.r - visualize both pcf of 3d initial dataset and of its 2d slices