A Predictive Framework for Detecting Non-Independent Tumor Dynamics in Longitudinal MR-Guided Radiotherapy

Abstract

Mean apparent diffusion coefficient (ADC) is the dominant quantitative endpoint in MR-guided radiotherapy response assessment. We prove that for any symmetric local coupling between tumour sub-regions, the coupling term cancels exactly from the volume mean. Mean ADC is therefore structurally blind to spatial interaction, not merely insensitive to it. We define the Tumour Coupling Index (TCI) as the normalised reduction in one-step voxel prediction error obtained by adding neighbourhood information. We derive a closed form, TCI = D²V ⁄ (D²V + σ²), where D is coupling strength and V is the conditional heterogeneity of the neighbourhood given the voxel's own value. Simulation confirms the expression to within 0.002 over 45 parameter combinations. At stationarity TCI depends only on D and the response rate, and is invariant to process-noise amplitude. Combining TCI with the autoregressive slope recovers D itself to within 0.004, so coupling is identifiable without solving a partial-differential-equation inverse problem. In 200 simulated ADC series with a full imaging chain, thirteen established metrics reach at most AUC 0.666 for detecting coupling; TCI reaches 0.749, raises recovered variance in D from R² = 0.078 to 0.389, and retains partial correlation 0.523 after all established metrics are removed. No surrogate null is zero-centred under the hypothesis of no coupling, because measurement noise makes neighbours predictive in its absence. We identify one dominant cause as attenuation of the autoregressive slope, which inflates the estimated response rate from a true 0.060 to 0.329 at 8% voxel noise, and derive an errors-in-variables correction that removes the noise dependence of the bias. A residual remains. Across geometries the bias at zero coupling is +0.017 with a standard deviation of 0.040, comparable to the coupling signal itself, and three independently constructed nulls fail to remove it. We conclude that an absolute null is not identifiable from a single patient's images, and that TCI must be used as a within-patient or matched-reference quantity rather than an absolute one. Detection in the clinical range is sequence-dependent. At 3.2% voxel-level ADC noise, corresponding to SPLICE or turbo-spin-echo diffusion on a 1.5 T MR-Linac, AUROC is 0.99 at D = 0.2 over 15 fractions. At 6.6%, corresponding to echo-planar imaging, detection fails at chance and requires 2×2 spatial binning to recover. Detectability is non-monotone in series length, peaking near 15 fractions and falling by 30 as coupling homogenises the tumour. The usable envelope is therefore: 4% voxel noise or binning, roughly 15 time points, noise correlation length below 3 voxels, and residual registration error below 1 mm.