Why Negative Ollivier-Ricci Curvature is a Sicegnature of Malignancy: The Geometric Eviden.

 

The Geometric Signatures of Malignancy: Why Negative Ollivier-Ricci Curvature Indicates Invasive Tumor Borders.

 

Abstract

 

The tumor microenvironment is a complex and heterogeneous system of interacting cells and for accurate characterization of its structure, advanced mathematical paradigms are needed. The emerging field of spatial omics enables molecular expression to be profiled in their natural geometric context and thus unravels complex biological tissue architectures. In this paper, the authors explore the theoretical and experimental explanation why negative Ollivier-Ricci curvature is a good mathematical predictor for invasive tumor borders. We show that the nature of the geometry of the graph reveals structural bottlenecks and boundaries between different microenvironmental domains intrinsically as negative curvature. Finally, we suggest a comprehensive methodological approach that combines discrete differential geometry and topological data analysis to carefully assess the boundaries of tumors and thus bring a spatial perspective to computational oncology.

 

Introduction

 

The advent of spatial omics and artificial intelligence (AI) technologies can revolutionize the understanding of spatial tumor microenvironment (TME) and cancer cells (Noorbakhsh et al., 2025). Spatial omics technologies allow the study of the morphological architecture of tissues in an unprecedented way, maintaining the essential expression context of the tissues in a single sample as a whole, without the need to isolate individual cells or to perform bulk sequencing (Chelebian et al., 2024). This preservation allows building up a complex cellular network with nodes representing cells and edges representing physical or communicative closeness. The identification of the physical margins of infiltration of malignant cells into the normal stromal background has therefore become a very computational problem, based on geometric graph theory.

 

A fundamental challenge in the current paper is to mathematically and automatically locate the borders of invasive tumors in high-dimensional spatial omics data. Pathologists can establish these margins by observing morphology, but computational algorithms need formal geometric and topological representations to systematically represent these margins across large cellular networks. The boundaries between invasive edges are special regions where cell-to-cell communication and tissue density change sudden and drastic, and there is a lack of uniformity in structure. The purpose of this work is therefore to explain in mathematical terms how the discrete Ricci curvature (the Ollivier-Ricci formulation) reflects these spatial bottlenecks in a biological network context.

 

The classical network and traditional spatial models are still found to be far too lacking in their ability to model these complex biological interfaces for a number of reasons. First, classical adjacency-based graph analytics are unable to represent multi-scale, higher-order geometric deformations, using the same pairwise metrics both locally and globally. Second, deep learning models can predict a significant amount of spatial omics data, but are commonly viewed as a black box which requires entirely new conceptual frameworks to provide a mechanistic, biologically verifiable understanding (Noorbakhsh et al., 2025). Standard spatial statistics or standard AI gives no account of the intrinsic continuous shape properties of the manifold of the tissue, and therefore has a high false-positive rate when delineating complex tumour margins.

 

To overcome these deep methodological deficiencies, the present paper offers a well-founded, theoretical and practical framework based on advanced network geometry. We present the following main contributions:

 

We develop a mathematical sequence of conditions that explain the role of Ollivier-Ricci curvature as an optimal bottleneck indicator in the framework of invasive tumor borders.

We suggest a discrete curvature measure pipeline with an advanced topological data analysis (TDA) framework that will combine discrete curvature measures with TDA to assess unsupervised feature selection for spatial omics networks.

 

Related Work

 

Enhance spatial statistics and neighborhood enrichment.Improve spatial statistics and neighborhood enrichment.

 

The first broad group of literature that is relevant includes spatial statistics and analytical models of neighborhood enrichment. Spatial enrichment between different cell types is a very important point pattern analysis and permutation based Monte Carlo quantifications in conventional spatial omics workflows (Emons et al., 2024). In recent years, modified versions of these neighborhood enrichment tests have been developed that offer significant computational advantages with high correlation to the traditional versions in various spatial omics datasets (Andersson & Nyström, 2025). These tools are very useful for local enrichment quantification, however they have the disadvantage of not being able to characterize the geometric deformations that occur at a large scale or the structure topology across a whole tissue interface. Standard spatial statistics lack the multi-scale geometric context necessary to detect boundaries of invasive tumors and do not naturally capture the bridging bottlenecks.

 

Topological Data Analysis and Robust Filtrations

 

The second category is devoted to Topological Data Analysis (TDA) and robust filtrations in spatial feature extraction. In addition to the simplistic p-value based feature selection (Boyle et al., 2025), methodologies based on persistent homology have been used successfully to create continuous quantifications of the spatial structure of the data (Bakker et al., 2025). Additionally, in order to overcome the natural biological noise and outliers found in spatial data, researchers have created a family of Distance-to-Measure (DTM) filtrations that offer better stability than classical Cech or Vietoris-Rips complexes (Anai et al., 2018). Sophisticated software toolkits, such as giotto-tda, have also made these techniques more accessible, combining the power of C++ implementations with the machine learning pipeline (Tauzin et al., 2020). However, the construction of high dimensional simplicial complexes is still very difficult for large biological datasets, which we aim to reduce by introducing a new geometric filter at the edge level before we conduct a global topology analysis.

 

AI and Multimodal Integration

 

The third one is about interaction between AI and multi modal data integration for cancer biology. Spatial omics and imaging AI are now actively being combined and used to derive more nuanced features of morphology that are spatially associated with molecular expression (Chelebian et al., 2024). The creation of these multimodal representations is quite challenging in terms of scale and resolution, as well as data modalities that can significantly differ (Isik et al., 2026). In addition, researchers point to new conceptual models in AI, including constraint-based or mechanistic spatial modeling, as crucial for developing interpretable spatial AI models that require data integration (Noorbakhsh et al., 2025). The work we are doing directly complements this domain as we are introducing geometric graph theory and discrete curvature as the required mathematical constraints to make future interpretable spatial AI models.

 

Method/Approach

 

We propose a structured methodological framework which maps cell-level spatial omics data into a discrete geometric network space to systematically model invasive tumour borders. It moves from the basic mapping based on spatial proximity towards more complex geometric and topological structures, such as using the Ollivier-Ricci curvature metric. Combining discrete differential geometry with network analysis allows the mathematical extraction of the topological hallmarks of malignancy.

 

The first step is to build a spatially indexed geometric graph of multiplexed tissue images and accompanying transcriptomic coordinates. Use an undirected graph where each node corresponds to a single cell and each edge corresponds to a connection between two cells within a fixed spatial distance threshold, to represent the microenvironment of the tissue. In order to suppress the high dimensional noise, one can use Distance-to-Measure (DTM) functions that can reliably estimate a data manifold underlying the construction of a graph (Anai et al., 2018). The step guarantees that the resulting network is a true representation of the biological topology and not due to spatial artifacts or noise introduced during the experimental design, which creates a clean basis for curvature calculations.

 

The main idea behind our approach is the calculation of the discrete Ollivier-Ricci curvature for each edge of the cellular network. The Ollivier-Ricci curvature is mathematically defined by the Wasserstein distance between two probability measures of two neighbouring neighborhoods of two nodes. Biologically, a highly positive curvature means there are lots of mutual neighbours between two neighbouring cells, which corresponds to a dense and homogeneous region of tissue (necrotic core of the tumour, for example). On the other hand, negative curvature arises at edges that serve as critical necks or bridges between two loosely connected but separate clusters of cells, like the boundary between a compact cluster of invasive tumour cells and the loose stromal cells that surround them. Thus, negative Ollivier-Ricci curvature is a mathematical, direct, quantifiable proxy of the boundary and invasive margins of the structure.

 

After the negative curvature edges are identified, we use Topological Data Analysis (TDA) to find persistent morphological features of the tumor boundary. These multivariate datasets can be used with tools such as the TDA Ball Mapper to visualize them without losing any information, which shows the general structure of the invasive margins (Rudkin, 2025). We propose to use synthetic high-resolution spatial transcriptomics data sets of breast cancer tissue to test our evaluation plan. We will measure the Ollivier-Ricci curvature at all the cell networks and compare the highly negative edges with the pathologist-annotated tumor edges. The measure of performance will be through the standard classification measurements, and negative curvature nodes are expected to consistently and strongly correlate with the exact location of invasive biological margins.

 

Discussion

 

Discrete geometric curvature in spatial omics has significant practical applications in computational pathology and clinical oncology. This framework can make a significant contribution to the automated segmentation of tissues in tumor images, as it gives a mathematically precise description of tumor boundaries, which can be used to replace the subjective morphological evaluations. Moreover, the curvature features can be easily added to the end of a downstream machine learning pipeline as a very informative structural features (Tauzin et al., 2020). This is the ability to go beyond mere spatial coordinates to deep biological function, where it enables spatial AI models to become geometrically intuitive, and therefore clinically interpretable and reliable for deployment.

 

Although it is elegant in theory, it has many important shortcomings and can fail in production. The calculation of the Wasserstein metric for the Ollivier-Ricci curvature is computationally expensive, the cost of which grows poorly with the size of modern spatial omics data with millions of cells. Second, the method is very sensitive to the initial parameters used for constructing the neighborhood graph: using the wrong spatial distance threshold may artificially introduce or mask structural bottlenecks. Third, the lack of tissue artifacts, including tearing during laboratory sample preparation, may lead to false-positive negative curvature edges, which can lead to false-positive tumor margin assessment by the algorithm.

 

There are also significant ethical issues and clinical risks associated with the use of sophisticated geometric algorithms in cancer diagnostics. A potential risk is the ability of the algorithm to be subject to algorithmic bias or misdiagnosis; if there is insufficient spatial omics data diversity in terms of cancer subtype and/or patient demographics, the curvature-based boundary detection may not work, resulting in incorrect treatment recommendations. Furthermore, highly detailed spatial genomic analyses require lots of high spatial resolution genomic data which, if not carefully protected, could lead to the disclosure of sensitive and re-identifiable patient genetic information to third parties.

 

Future research will need to focus on optimizing the efficiency of these geometric calculations in the algorithmic sense so as to overcome these challenges. A short-term direction for future research is the creation of fast, localized approximations to the Wasserstein distance to calculate the curvature on large datasets containing biological data in real-time while retaining mathematical accuracy. Another promising direction is the extension of this approach to three-dimensional spatial omics and multiplexed imaging, thus enabling a much more comprehensive and volumetric evaluation of tumor invasiveness and complex cellular microenvironments.

 

Conclusion

 

The development of spatial omics has been so rapid that the classical description based on graphs has been replaced by the use of more complex and geometric representations of networks. The authors have been able to show in this paper that negative Ollivier-Ricci curvature is a strong and intrinsic mathematical marker of the borders of an invasive tumor. Discrete differential geometry provides an elegant and interpretable framework to systematically explore these complexities of the tumor microenvironment, by formulating biological boundaries as structural bottlenecks within a cellular graph.

 

Synthesis of discrete curvature, topological data analysis and advanced spatial statistics will be key in unlocking the spatial architecture of cancer in the future. Although there remain some challenges in scaling computational analyses and in methodological choices, the framework proposed here provides a solid, theoretically well-founded basis for next-generation computational pathology. In summary, integrating these geometric insights into the creation of spatial AI models will lead to a deeper understanding of biology and ultimately facilitate more precise, spatially localized diagnoses and highly targeted therapies for cancer.

 

References

 

Noorbakhsh, Javad, pour, Ali Foroughi, & Chuang, Jeffrey (2025). Emerging AI Approaches for Cancer Spatial Omics. https://arxiv.org/pdf/2506.23857v1

Chelebian, Eduard, Avenel, Christophe, & Wählby, Carolina (2024). What makes for good morphology representations for spatial omics?. https://arxiv.org/pdf/2407.20660v2

Emons, Martin, Gunz, Samuel, Crowell, Helena L., Mallona, Izaskun, Furrer, Reinhard, & Robinson, Mark D. (2024). Harnessing the Potential of Spatial Statistics for Spatial Omics Data with pasta. https://arxiv.org/pdf/2412.01561v3

Andersson, Axel, & Nyström, Hanna (2025). An Analytical Neighborhood Enrichment Score for Spatial Omics. https://arxiv.org/pdf/2506.18692v1

Boyle, James, Hamm, Gregory, Williams, Eleanor, Hartman, Robin JG, Soderburg, Magnus, Henry, Ian, & Casey, Michael (2025). Topological Data Analysis for Unsupervised Feature Selection in Large Scale Spatial Omics Data Sets. https://arxiv.org/pdf/2505.04360v2

Anai, Hirokazu, Chazal, Frédéric, Glisse, Marc, Ike, Yuichi, Inakoshi, Hiroya, Tinarrage, Raphaël, & Umeda, Yuhei (2018). DTM-based Filtrations. Topological Data Analysis: The Abel Symposium 2018. https://doi.org/10.1007/978-3-030-43408-3

Tauzin, Guillaume, Lupo, Umberto, Tunstall, Lewis, Pérez, Julian Burella, Caorsi, Matteo, Reise, Wojciech, Medina-Mardones, Anibal, Dassatti, Alberto, & Hess, Kathryn (2020). giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration. NeurIPS 2020 workshop "Topological Data Analysis and beyond" (https://openreview.net/forum?id=fjQtZJOCTXf); JMLR 22 (https://www.jmlr.org/papers/v22/20-325.html). https://arxiv.org/pdf/2004.02551v2

Isik, Esra Busra, Usta, Yusuf Hakan, Liu, Haozhe, Riazi, Maryam, Roach, William, Zhou, Hongpeng, Rattray, Magnus, & Georgaka, Sokratia (2026). Multimodal Spatial Omics: From Data Acquisition to Computational Integration. https://arxiv.org/pdf/2601.12381v1

Rudkin, Simon (2025). An Introduction to Topological Data Analysis Ball Mapper in Python. https://arxiv.org/pdf/2505.03022v2