Virtual Spectral Decomposition with Dendritic Tile Selection: An Explainable AI Framework for Multimodal Tissue Composition Analysis and Immune Phenotyping Across Pancreatic, Lung, and Breast Cancer

Background: Current deep learning models in computational pathology, radiology, and digital pathology produce opaque predictions that lack the explainable artificial intelligence (xAI) capabilities required for clinical adoption. Despite achieving radiologist-level performance in tasks from whole-slide image (WSI) classification to mammographic screening, these models function as black boxes: clinicians cannot trace predictions to specific biological features, verify outputs against established morphological criteria, or integrate AI reasoning into precision oncology workflows and tumor board decision-making. Methods: We present Virtual Spectral Decomposition (VSD), a modality-agnostic, interpretable-by-design framework that decomposes medical images into six biologically interpretable tissue composition channels using sigmoid threshold functions - the same mathematical structure as CT windowing. Unlike post-hoc xAI methods (Grad-CAM, SHAP, LIME) applied to black-box deep learning models, VSD channels have pre-defined biological meanings derived from tissue physics, providing inherent explainability without sacrificing quantitative rigor. For whole-slide image (WSI) analysis in digital pathology, we introduce the dendritic tile selection algorithm, a biologically-inspired hierarchical architecture achieving 70-80% computational reduction while preferentially sampling the tumor immune microenvironment. VSD is validated across three cancer types and imaging modalities: pancreatic ductal adenocarcinoma (PDAC) on CT imaging, lung adenocarcinoma (LUAD) on H&E-stained pathology slides using TCGA data, and breast cancer on screening mammography. Composition entropy of the six-channel vector is computed as a visual Biological Entropy Index (vBEI) - an imaging biomarker quantifying the diversity of active biological defense systems. Results: In pancreatic cancer, the fat-to-stroma ratio (a novel CT-derived radiomics biomarker) declines from >5.0 (normal) to <0.5 (advanced PDAC), enabling early detection of desmoplastic invasion before mass formation on standard imaging. In lung cancer, composition entropy from H&E whole-slide images correlates with tumor immune microenvironment markers from RNA-seq (CD3: rho=+0.57, p=0.009; CD8: rho=+0.54, p=0.015; PD-1: rho=+0.54, p=0.013) and predicts overall survival (low entropy immune-desert phenotype: 71% mortality vs 29%, p=0.032; n=20 TCGA-LUAD), providing immune phenotyping for checkpoint immunotherapy patient selection from a $5 H&E slide without molecular assays. In breast cancer, each lesion type produces a characteristic six-channel fingerprint functioning as an interpretable computer-aided diagnosis (CAD) system for quantitative BI-RADS assessment and subtype classification (IDC vs ILC vs DCIS vs IBC). A five-level xAI audit trail provides complete traceability from clinical decision support output to specific biological structures visible on the original images. Conclusion: VSD establishes a unified, interpretable-by-design mathematical framework for explainable tissue composition analysis across imaging modalities and cancer types. Unlike black-box deep learning and post-hoc xAI approaches, VSD provides inherently interpretable, clinically verifiable cancer detection and immune phenotyping from standard clinical imaging at existing costs - without requiring foundation model infrastructure, specialized hardware, or molecular assays. The open-source pipeline (Google Colab, Supplementary Material) enables immediate reproducibility and extension to additional cancer types across the pan-cancer TCGA atlas.