The spatial composition and cellular heterogeneity of the tumor microenvironment plays a critical role in cancer development and progression. High-definition pathology imaging of tumor biopsies provide a high-resolution view of the spatial organization of different types of cells. This allows for systematic assessment of intra- and inter-patient spatial cellular interactions and heterogeneity by integrating accompanying patient-level genomics data. However, joint modeling across tumor biopsies presents unique challenges due to non-conformability (lack of a common spatial domain across biopsies) as well as high-dimensionality. To address this problem, we propose the Dual random effect and main effect selection model for Spatially structured regression model (DreameSpase). DreameSpase employs a Bayesian variable selection framework that facilitates the assessment of spatial heterogeneity with respect to covariates both within (through fixed effects) and between spaces (through spatial random effects) for non-conformable spatial domains. We demonstrate the efficacy of DreameSpase via simulations and integrative analyses of pathology imaging and gene expression data obtained from $335$ melanoma biopsies. Our findings confirm several existing relationships, e.g. neutrophil genes being associated with both inter- and intra-patient spatial heterogeneity, as well as discovering novel associations. We also provide freely available and computationally efficient software for implementing DreameSpase.

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