Spatially Limited Immune Access Creates Tumour Refugia in Hepatocellular Carcinoma

Abstract

Spatially restricted immune access can undermine tumour control even when total immune recruitment appears sufficient. We develop a nondimensional reaction--diffusion--chemotaxis model for hepatocellular carcinoma that couples tumour growth and immune-mediated killing with effector diffusion, saturating chemokine-dependent recruitment, and migration along an effective CXCL9/CXCL10/CXCL11--CXCR3 signal. We establish nonnegativity, uniform tumour and mass bounds, and global boundedness of classical solutions in one dimension and, in arbitrary dimensions, when chemokine production is tumour-driven. Analysis of the homogeneous dynamics shows that the threshold ${\sigma }_0>\delta$ is only local and does not preclude bistable tumour persistence. A mode-wise dispersion relation identifies stationary finite-wavelength and oscillatory instabilities, whose critical sensitivities, dominant modes, growth rates, and frequencies are quantitatively recovered by conservative finite-volume simulations. Using matched initial states and identical mean recruitment, we show that a well-mixed model may predict clearance while margin-limited recruitment preserves a stable interior tumour refuge. Chemotaxis improves interior effector access and reduces, but need not eliminate, this refuge. Spatial-overlap biomarkers further connect model-generated phenotypes to pathology and spatial-omics observables. These results distinguish immune abundance from effective spatial access and identify access-limited recruitment as a mechanism of incomplete tumour control.