Metastatic Site Prediction in Breast Cancer using Kirchhoff's Law and Omics Knowledge Graph

Predicting the anatomical site of metastasis from a primary tumour remains an unsolved problem in breast cancer (BRCA) and metastatic disease more broadly. The difficulty is structural: metastatic biology is multi-site (bone, lung, liver, brain), multi-omics (genomics, proteomics, methylomics, drug response), and multi-modal (CNV, gene expression, DNA methylation, pathways, clinical associations). Existing classifiers either collapse this heterogeneity into a single feature vector or rely on a single omics layer, both of which discard the mechanistic structure that drives metastatic tropism. We introduce Kirchhoff Knowledge Graphs (K-KG), a framework that imports the conservation laws of electrical-circuit theory into knowledge graph reasoning. Our contributions are: (1) a layered RDF Cancer Decision Network integrating 36 polyomics datasets across mutations, pathways, drugs, diseases, and reactions; (2) two novel conservation laws--the Knowledge-Graph Voltage Law (KGVL) and Knowledge-Graph Current Law (KGCL)--that govern information flow during traversal and yield a principled measure of graph completeness; (3) topological motif mining on the conserved graph, replacing expression-based feature selection by identifying triangular sub-structures whose rewiring marks metastatic transition; (4) a Graph Convolutional Neural Network whose hidden layers are the omics layers themselves, predicting site-specific metastasis as a continuous percentage rather than a binary label. On TCGA-BRCA training plus one validation and four independent test cohorts from GEO, K-KG achieves 83.8% AUC for relapse prediction and up to 0.87 AUC / 0.91 F1 for Brain-site-specific prediction, outperforming Random Forest, Neural Network, and SVM baselines by 8-20 AUC points. To our knowledge this is the first application of Kirchhoffs laws (1845, 1847) to graph-based machine learning, and the first metastasis predictor that returns a per-site contribution profile rather than a single label.