Thermal ablation — heating a tumor until it dies — works differently in different tissues. The standard explanation is vascularity: blood carries heat away, so well-perfused tumors resist ablation. But even controlling for blood flow, liver metastases respond worse than primary liver tumors. Something else is happening.

The missing variable is geometry. Tumor tissue has fractal structure — irregular branching vasculature, tortuous interstitial spaces, heterogeneous cell packing. The fractal dimension and spectral dimension of this geometry jointly control how heat propagates. A tumor with higher fractal dimension has more surface area for heat to dissipate through. A tumor with lower spectral dimension has slower diffusion across its network, trapping heat in some regions while leaving others undertreated.

The coagulation zone — the volume of destroyed tissue — is controlled by the topology, not just the temperature. Two tumors at the same temperature can have different kill zones because the heat walks differently through their respective geometries.

This is a measurement problem dressed as a treatment problem. The ablation probe doesn’t know the fractal dimension of the tissue it’s heating. It delivers a fixed power and assumes uniform diffusion. The clinical variability isn’t noise — it’s signal from geometric structure the probe can’t see.