Differential co-localisation analysis of multi-sample and multi-condition experiments with spatialFDA

The paper introduces spatialFDA, an open-source Bioconductor R package that integrates spatial statistics and functional data analysis to accurately quantify and test for differences in cellular co-localization across multiple conditions in spatial omics data, demonstrating its effectiveness through simulations and biological applications in type-1 diabetes.

The Gist

Imagine you are a detective trying to solve a mystery in a bustling city. In this city, the "citizens" are not people, but cells inside your body. Some cells are immune soldiers, some are insulin factories, and others are structural bricks.

For a long time, scientists could only take a photo of this city and count how many of each type of citizen lived there. They knew, for example, "In a healthy city, there are 100 insulin factories. In a sick city, there are only 10." But they didn't know where those factories were standing or who they were hanging out with.

This is where Spatial Omics comes in. It's like getting a map that shows not just how many citizens there are, but exactly where they are standing.

The Mystery: Who is Hanging Out with Whom?

The big question scientists want to answer is: Do certain types of cells like to stand next to each other, or do they avoid each other?

• The Healthy City: Maybe the immune soldiers stand far away from the insulin factories, keeping a respectful distance.
• The Sick City: Maybe the immune soldiers are crowding right up against the factories, causing trouble.

This "hanging out" behavior is called Co-localisation. The paper introduces a new tool called spatialFDA to measure exactly how much these cells are clustering together, and whether that clustering changes when a person gets sick.

The Problem with Old Tools

Before this new tool, scientists tried to measure this "hanging out" by taking a snapshot of the whole city and calculating a single number, like an "average friendliness score."

The Flaw: Imagine two different cities.

1. City A: Everyone is standing in one giant, tight huddle.
2. City B: Everyone is standing in small, tight groups of three, scattered everywhere.

If you just calculate the "average friendliness," both cities might get the same score. But the pattern is totally different! Old tools were like a blunt instrument that missed these subtle, important differences. They also struggled when you had data from many different people (samples) and many different photos (fields of view) from the same person, often getting confused and giving false alarms.

The New Solution: spatialFDA

The authors created spatialFDA (Spatial Functional Data Analysis). Think of it as upgrading from a still camera to a high-definition movie camera.

Instead of squashing all the data into one single number, spatialFDA looks at the entire story of how cells interact at different distances.

• The Analogy of the Radius: Imagine you are standing in the middle of a crowd.
  • Step 1: You look at who is touching your shoulder (0–10 micrometers).
  • Step 2: You look at who is within arm's reach (10–50 micrometers).
  • Step 3: You look at who is in the same room (50–100 micrometers).

Old tools just gave you one number for the whole room. spatialFDA gives you a curve (a line on a graph) that shows exactly how the "hanging out" changes as you zoom out from shoulder-touching to room-wide.

How It Works (The "Magic" Part)

1. The Movie: It takes the map of cells and calculates a "friendship curve" for every photo.
2. The Comparison: It uses advanced math (called Functional Data Analysis) to compare the entire curves between healthy people and sick people.
3. The Detective Work: It can tell you: "In the sick people, the immune cells are crowding the insulin factories specifically within 20 micrometers, but they are fine at 50 micrometers."

This is crucial because it tells you where the problem is happening, not just that a problem exists.

The Real-World Test: Type 1 Diabetes

The authors tested their new tool on a real medical mystery: Type 1 Diabetes.

• The Background: In Type 1 Diabetes, the body's immune system attacks and destroys the insulin-producing cells in the pancreas.
• The Discovery: Using spatialFDA, they confirmed what we already knew: immune cells are attacking the insulin factories.
• The New Insight: But they found something new! They saw that this "attack" (clustering) was most intense in the early stages of the disease. In the late stages, the insulin factories were already gone, so the immune cells had nothing to crowd around.

Without spatialFDA, they might have just seen "immune cells are present" and missed the timing and the specific distance of the attack.

Why This Matters

Think of spatialFDA as a super-powered microscope for social behavior in your body.

• It doesn't just count the crowd; it understands the dance.
• It handles complex data (many patients, many photos) without getting confused.
• It helps doctors understand diseases like cancer or diabetes by seeing exactly how cells are rearranging themselves to cause trouble.

In short, this paper gives scientists a better way to read the "social map" of our bodies, helping us understand disease not just as a change in numbers, but as a change in relationships.

Technical Summary

1. Problem Statement

Spatial omics technologies (e.g., spatial transcriptomics, proteomics) generate high-dimensional data recording the spatial location of cells. A critical biological question is Cellular Co-localisation (CCoL): determining if specific cell types cluster together or are spaced apart more than expected by chance, and how this changes across experimental conditions (e.g., disease vs. healthy).

Existing challenges in differential CCoL analysis include:

• Loss of Information: Many current methods (e.g., spicyR, smoppix) reduce complex spatial metrics (functions of radius $r$) into a single scalar value (e.g., Area Between Curves - ABC). This compression can lead to a loss of sensitivity, as distinct spatial patterns may yield identical scalar summaries.
• Handling Variability: Complex experimental designs often involve multiple Fields of View (FOVs) per sample and multiple samples per condition. Standard statistical models often fail to account for this nested variability (non-independent FOVs), leading to inflated false discovery rates (FDR) or overly optimistic results.
• Existing Functional Methods: While Functional Data Analysis (FDA) offers tools to compare entire functions, existing tools like SpaceANOVA and mxfda lack the flexibility to model nested random effects or are restricted to specific regression setups (e.g., scalar-on-function).

2. Methodology: spatialFDA

The authors introduce spatialFDA, an open-source Bioconductor R package that combines spatial statistics with Functional Data Analysis (FDA) to perform differential CCoL analysis.

Core Workflow:

1. Data Representation: Cells are approximated as points (centroids) with categorical marks (cell types).
2. Spatial Metric Calculation: For each FOV, spatial metrics are calculated as functions of the radius $r$. The package supports:
   • Correlation functions: Ripley's $K$ and Besag's $L$ functions (sensitive to large-scale interactions).
   • Spacing functions: Nearest-neighbour $G$ function (sensitive to short-scale interactions).
3. Statistical Modeling (fGAMM): Instead of summarizing the metric into a scalar, spatialFDA treats the spatial metric as a functional response. It employs Functional Generalized Additive Mixed Models (fGAMMs).
   • Model Structure: $E[y_i(r) | X_i, g(i)] = g^{-1}(\mu(r) + \beta_{g(i)}(r) + \sum f_j(X_{ji}, r))$
   • Random Effects: The model explicitly accounts for nested variability (e.g., FOVs within samples) using random intercepts, ensuring valid inference in multi-sample experiments.
   • Covariance Correction: To address autocorrelation and heteroscedasticity in the residuals along the domain $r$, the method uses cluster-robust covariance matrix estimators (sandwich estimators) post-hoc.
4. Inference:
   • Global Test: An omnibus $F$-test determines if the functional curve differs significantly between conditions across the entire domain.
   • Local Interpretation: Model coefficients are estimated as functions of $r$, allowing researchers to visualize where (at which spatial scale) the co-localization differs.

3. Key Contributions

• Framework Integration: First method to unify spatial point pattern analysis with functional mixed-effects models, preserving the full spatial scale information ($r$) rather than compressing it.
• Handling Nested Designs: Explicitly models the hierarchical structure of spatial omics data (multiple FOVs per sample), addressing a major source of statistical error in previous approaches.
• Robust Inference: Implements cluster-corrected sandwich covariance matrices to handle the non-independence of residuals along the functional domain.
• Software Implementation: Provides a user-friendly Bioconductor package (spatialFDA) that integrates with standard spatial analysis workflows (SpatialExperiment).

4. Results

Simulation Study 1 (Baker et al. framework):

• Setup: Simulated tissue scaffolds with varying cell type proportions and interaction probabilities. Compared spatialFDA against spicyR, smoppix, SpaceANOVA, mxfda, and a linear mixed model baseline.
• Findings:
  • Calibration: Methods ignoring nested variability (spicyR.LM, SpaceANOVA.Uni) were severely miscalibrated (high FDR).
  • Sensitivity vs. Specificity: spatialFDA (specifically the $L$-function variant, spatialFDA.L) demonstrated the best balance, maintaining conservative calibration (controlled FDR) while retaining high True Positive Rates (TPR).
  • Scalar Methods: Scalar mixed models (spicyR.MM, smoppix) controlled Type I errors better than non-mixed scalar methods but suffered from reduced statistical power (lower TPR).
  • Functional Competitors: SpaceANOVA showed inflated FDR; mxfda failed to detect effects in the tested setup.

Simulation Study 2 (Canete et al. framework):

• Setup: Simulated CCoL changes at specific length scales (10µm to 100µm).
• Findings:
  • spatialFDA.L showed stable performance across all length scales.
  • spatialFDA.G performed best for short-scale interactions (<50µm) but lost power at larger scales.
  • Scalar methods (spicyR) were highly dependent on the chosen integration range; if the range did not match the true biological scale, performance dropped.

Case Study: Type-1 Diabetes (T1D):

• Data: Imaging Mass Cytometry (IMC) data from 12 donors (Non-diabetic, Onset, Long-duration).
• Findings:
  • Validation: Successfully recovered known biological phenomena, such as the progressive loss of $\beta$-cells and the increased co-localization of immune cells (T-cells) with islet cells in the onset phase.
  • New Insight: Identified that the increased co-localization of Th cells and $\delta$-cells in onset T1D is most pronounced at short distances (10–50 µm), a detail that would be lost in scalar summaries.
  • Intensity vs. CCoL: Demonstrated that changes in cell intensity (proportion) and spatial arrangement (CCoL) are distinct; for example, $\delta$-cell intensity remained stable while their co-localization with Th cells changed significantly.

5. Significance

• Biological Insight: By retaining the spatial scale, spatialFDA allows researchers to pinpoint where biological interactions occur (e.g., immediate cell contact vs. broader tissue organization), offering deeper mechanistic insights than scalar methods.
• Statistical Rigor: It sets a new standard for analyzing multi-sample spatial omics data by properly accounting for experimental design complexity (nested FOVs), reducing false positives.
• Accessibility: As an open-source Bioconductor package, it lowers the barrier for the community to apply advanced functional data analysis to spatial biology, moving the field beyond simple point-pattern summaries.

In conclusion, spatialFDA represents a significant methodological advancement for spatial omics, offering a sensitive, calibrated, and interpretable framework for detecting differential cellular co-localization across complex experimental designs.