Tumor Interaction Score (TIS)

• Tumor Interaction Score (TIS) is a quantitative biomarker that summarizes spatial interactions in tumor tissues by measuring cell mixing and network connectivity.
• It integrates cell-level marked point process models and graph-theoretic connectomics to distill complex tumor organization into a single predictive score.
• Implementing TIS requires precise cell or voxel segmentation followed by robust Bayesian inference or graph analysis to ensure accurate clinical interpretation.

The Tumor Interaction Score (TIS) is a quantitative biomarker designed to summarize cell–cell or region–region spatial interactions within tumor tissue, typically distilled to a single scalar per image or patient. TIS frameworks have been devised in both cell-level and network-level spatial modeling, most notably in point process models of pathology images and graph-theoretic models of imaging-derived tumor connectomes. In both contexts, the TIS aims to encapsulate clinically relevant aspects of tumor organization such as cell mixing, cross-compartment interaction, or changes in connectivity patterns, and has demonstrated utility for prognosis and treatment prediction.

1. TIS from Cell-Type Marked Point Process Models

A major implementation of TIS is grounded in spatial marked point process modeling of digital pathology images, as developed by Chen et al. (Li et al., 2018). Each cell is spatially indexed at coordinate $(x_i, y_i)$ within a normalized 2D region and labeled with a mark $z_i$ specifying its type among $$\{\mathrm{lymphocyte}, \mathrm{stromal}, \mathrm{tumor}\}$$, yielding a marked point pattern. The point process is modeled via a Gibbs energy functional that incorporates both first-order cell-type abundances and second-order pairwise interaction effects.

The energy function for a mark configuration $z$ is given by:

$$V(z\mid\omega,\Theta,\lambda) = \sum_{q=1}^Q \omega_q\sum_{i=1}^n \mathbf{1}\{z_i=q\} +\sum_{q=1}^Q\sum_{r=1}^Q \theta_{qr} \sum_{\substack{i<i'\ d_{ii'}\le c}} e^{-\lambda\,d_{ii'}} \mathbf{1}\{z_i=q,\,z_{i'}=r\},$$

where $\omega_q$ controls marginal abundance, $\theta_{qr}$ quantifies interaction strength between types $q$ and $r$, $\lambda$ governs spatial decay, and $d_{ii'}$ is Euclidean distance truncated beyond $c$.

The model’s posterior is sampled using double Metropolis–Hastings (DMH) due to the intractable normalization. The key fitted parameters $\hat\omega, \hat\Theta, \hat\lambda$ enable direct calculation of pairwise mark-interaction probabilities:

$$\hat\phi_{qr} = \frac{\exp(-\hat\theta_{qr})} {\sum_{s=1}^Q \exp(-\hat\theta_{sr})}.$$

TIS is constructed as $$\mathrm{TIS} = \hat\phi_{\mathrm{tum}, \mathrm{str}}$$, denoting the estimated probability that a cell immediately adjacent to a stromal cell is a tumor cell. Averaging over images yields a patient-level score.

2. TIS from Tumor Connectomics and Graph Theory

Complementary to point process modeling, TIS can also be defined on spatial graphs representing intra-tumoral interaction networks, as in the Tumor Connectomics Framework (TCF) of Parekh and Jacobs (Parekh et al., 2019). Here, spatially localized tissue regions (voxels) derived from dynamic contrast-enhanced MRI serve as nodes, with edges constructed from normalized geodesic distances in multi-dimensional feature space reflecting tissue characteristics.

Key graph metrics computed per tumor include:

• Degree centrality: $C_D(v) = \deg(v) = \sum_{j=1}^N a_{ij}$
• Average path length: $$APL(v_i) = \frac{1}{|R(v_i)|}\sum_{v_j \in R(v_i)} d_G(v_i, v_j)$$
• Clustering coefficient: $C_C(v_i) = \frac{2\,T(v_i)}{k_i (k_i - 1)}$ for $k_i \geq 2$

Percent change for these features between baseline and post-treatment is computed as $\Delta M = (M^1 - M^0)/M^0$.

TIS is then defined as a weighted linear combination of normalized feature changes:

$$\mathrm{TIS} = w_1 \frac{\Delta C_D - \min \Delta C_D}{\max \Delta C_D - \min \Delta C_D} + w_2 \frac{\Delta \mathrm{APL} - \min \Delta \mathrm{APL}}{\max \Delta \mathrm{APL} - \min \Delta \mathrm{APL}} + w_3 \frac{\Delta C_C - \min \Delta C_C}{\max \Delta C_C - \min \Delta C_C}$$

with weights $w_i$ proportional to the discriminative ability (AUC) of each metric.

3. Methodological Implementation

For pathology-based TIS (Li et al., 2018), the methodological workflow is as follows:

1. Cell segmentation and mark assignment from digital histology.
2. Transformation to a marked point pattern $(x_i, y_i, z_i)$.
3. Model fitting using Bayesian inference with DMH to handle the intractable partition function.
4. Estimation of $\hat\phi_{qr}$ for all pairs; extraction of $$\mathrm{TIS} = \hat\phi_{\mathrm{tum},\mathrm{str}}$$.
5. Aggregation over multiple images for a patient-level summary.

For connectomics-based TIS (Parekh et al., 2019):

1. Feature extraction from DCE–MRI voxel signatures within tumor volume.
2. Graph construction using k-NN or geodesic distance thresholding.
3. Calculation of degree centrality, average path length, and clustering coefficient at baseline and post-treatment.
4. Metric normalization and weighted integration into TIS as outlined above.

Both approaches yield a scalar TIS per patient suitable for statistical association and predictive modeling.

4. Interpretation and Clinical Significance

The cell-based TIS captures the immediate affinity or mixing between tumor and stromal cells. High values indicate spatially intermixed tumor–stromal niches, while low values reflect sharply segregated cellular arrangements. In lung cancer pathology, empirical TIS values spanned approximately $0.01$ (well-separated compartmentalization) to $0.2$ (deep mixing), with higher TIS correlating significantly with worse overall survival; the Cox model coefficient for TIS was highly significant ($P=0.0073$) (Li et al., 2018).

In connectomics, TIS quantifies the global level of inter-voxel connectivity change—network “destruction” or “intensification”—subject to therapy. Negative changes in degree centrality, path length, and clustering coefficient collectively reflect breakdown of microenvironmental networks, which are predictive of treatment response in breast cancer. Composite TIS gives a robust, integrative measure (AUC${}_\mathrm{TIS} \approx 0.88$ for discrimination of responders vs non-responders) (Parekh et al., 2019).

5. Comparison of TIS Definitions Across Modalities

Context	Underlying Model	Biological Interpretation	Clinical Use
Pathology	Marked Gibbs point proc.	Tumor–stromal cell affinity	Prognosis
Imaging	Feature-space graph	Network integrity/change	Treatment response prediction

TIS in both modalities distills complex multivariate spatial patterns into a single number directly relatable to outcome or risk. The cell-based approach offers micro-scale resolution, while the graph-based TIS provides network-level descriptors.

6. Limitations and Considerations

While TIS frameworks offer interpretability and empirical validation, several limitations must be considered:

• In the point process model, the intractability of the partition function necessitates computationally intensive DMH sampling.
• TIS values are inherently dependent on the spatial scale (cell vs. voxel), feature choices, and operational definitions (e.g., thresholds in graph construction).
• Both methods assume the relevant information is encoded in either pairwise mark interactions (pathology) or network topology (imaging), and may not explicitly address higher-order correlations or tissue mechanical context.
• External validation across diverse tumor types, imaging modalities, and cell-type schemas remains limited.

A plausible implication is that integrating cell-based and network-based TIS approaches could yield synergistic information for future biomarker development.

7. Related Metrics and Future Directions

TIS relates closely to general spatial interaction scores, such as other mark–covariate interaction summaries in Gibbs processes or multiscale graph invariants in network biology. Recent developments in spatial omics and advanced multiplexed imaging may enable more granular and high-throughput extension of the TIS concept.

Standardization of TIS computation, further validation across larger cohorts, and incorporation of higher-dimensional data represent clear directions for advancing the clinical and biological interpretability of tumor interaction scores.