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Background  Current intra-tumoral heterogeneous feature extraction in radiology is limited to the use of a single slice or the region of interest within a few context-associated slices, and the decoding of intra-tumoral spatial heterogeneity using whole tumor samples is rare. We aim to propose a mathematical model of space-filling curve-based spatial correspondence mapping to interpret intra-tumoral spatial locality and heterogeneity.    Methods  A Hilbert curve-based approach was employed to decode and visualize intra-tumoral spatial heterogeneity by expanding the tumor volume to a two-dimensional (2D) matrix in voxels while preserving the spatial locality of the neighboring voxels. The proposed method was validated using three-dimensional (3D) volumes constructed from lung nodules from the LIDC-IDRI dataset, regular axial plane images, and 3D blocks.    Results  Dimensionality reduction of the Hilbert volume with a single regular axial plane image showed a sparse and scattered pixel distribution on the corresponding 2D matrix. However, for 3D blocks and lung tumor inside the volume, the dimensionality reduction to the 2D matrix indicated regular and concentrated squares and rectangles. For classification into benign and malignant masses using lung nodules from the LIDC-IDRI dataset, the Inception-V4 indicated that the Hilbert matrix images improved accuracy (85.54% vs. 73.22%,  p  < 0.001) compared to the original CT images of the test dataset.    Conclusions  Our study indicates that Hilbert curve-based spatial correspondence mapping is promising for decoding intra-tumoral spatial heterogeneity of partial or whole tumor samples on radiological images. This spatial-locality-preserving approach for voxel expansion enables existing radiomics and convolution neural networks to filter structured and spatially correlated high-dimensional intra-tumoral heterogeneity.    Supplementary Information  The online version contains supplementary material available at 10.1186/s13244-021-01100-8.

Decoding intra-tumoral spatial heterogeneity on radiological images using the Hilbert curve