Fractal and Spectral Dimensions as Determinants of Thermal Ablation Outcomes in Cancer Tissues

Abstract

Clinical thermal ablation outcomes display significant variability that classical bio-heat models cannot fully explain. One reason may lie in the fractal architecture of biological tissues, which has been identified as a robust biomarker directly correlated with cancer grades. This structural heterogeneity, together with memory effects (e.g., thermotolerance), causes heat transfer in living tissues to differ from Fourier diffusion, resulting in anomalous biological transport. In this work, we implemented a realistic fractal-fractional bio-heat model, with non-linear perfusion and PI-controlled power delivery, to quantify the role of tissue fractality in ablation outcomes. Our results reveal that the expansion of coagulation zones is jointly controlled by fractal geometry and its associated topological connectivity. These findings highlight spectral dimension as a key driver of clinical variability, successfully reproducing the reduced ablative efficacy in liver metastases compared to primary carcinomas, and provide evidence for topologically informed treatment strategies for the thermal ablation of malignant neoplasms.

Figures (7)

• Figure 1: Temporal evolution and PI control at the heating centre. The temperature history at the application point ($r=0$) is shown for varying fractal geometries and fractional orders: (a) $\alpha=0.8$, (b) $\alpha=1.4$, and (c) $\alpha=1.8$. The feedback loop effectively constrains the maximum temperature to the target setpoint $T_{lim}=90^\circ$C, ensuring a controlled ablative plateau.
• Figure 2: Log-log plot of coagulation radius evolution over time. The coagulation radius ($\Omega=4.6$) is shown for three fractional orders: (a) $\alpha=0.8$, (b) $\alpha=1.4$, and (c) $\alpha=1.8$. Curves in each panel correspond to different fractal parameter sets {$D_f, d_s$}. The black dotted line indicates the end of heating ($t_{on}=600$ s). Additionally, thin vertical lines, coloured to match their corresponding curves, indicate the time at which the PI controller starts modulating the heat source.
• Figure 3: Effect of fractional order $\alpha$ and fractal parameters ($D_f, d_s$) on spatial temperature distribution. Heatmaps of temperature $T(x,y)$ at the end of the heating phase ($t_{on} = 600$ s). Each panel (a-d) corresponds to a different fractional order $\alpha$, as indicated. To visualise the effect of fractal and spectral dimensions, each panel is a composite of three distinct simulations: Region (i) [left] corresponds to ($D_f=1.7, d_s=1.5$); Region (ii) [top-right] corresponds to ($D_f=1.7, d_s=1.4$); Region (iii) [bottom-right] corresponds to ($D_f=1.6, d_s=1.5$). Overlaid white lines show the simulated ablation contours for coagulation ($\Omega=4.6$, solid) and periablation ($\Omega=2.1$, dashed; $\Omega=0.6$, dotted). Contours were computed after a variable cooling period ($20-100$ min) adjusted for each $\alpha$ to ensure the complete stabilisation of the accumulated thermal damage. The central 3 mm $\emptyset$ applicator is shown in gray. Legends defining regions, contours, and the applicator are embedded in panels (a) and (b).
• Figure 4: Impact of topological uncertainty on ablation predictability. The floating bars quantify the theoretical variability range of the final coagulation radius ($r_c$) resulting from the indeterminacy of the spectral dimension ($d_s$), which is varied across a broad range of physically plausible values and strictly bounded by $D_f$ ($1.05 \leq d_s \le D_f - 0.05$). The white horizontal marker indicates the response at the mid-point of the parameter range. (a) Variability across fractional diffusion regimes ($\alpha$) assuming a fixed malignant tumour geometry ($D_f=1.7$). (b) Variability across tissue fractal dimensions ($D_f$) for a fixed fractional order ($\alpha=1.8$). The cases represent healthy tissue ($D_f=1.5$), stage II-III malignant tumour ($D_f=1.7$), and theoretical saturation ($D_f=1.9$) Elkington2022.
• Figure 5: Comparative local sensitivity of ablation size to geometric and topological variations. The normalised local sensitivity index ($S = (p/r_c)|\partial r_c/\partial p|$) is presented for the fractal dimension ($p=D_f$, geometry) and the spectral dimension ($p=d_s$, topology). (a) Sensitivity analysis across different fractional diffusive regimes ($\alpha$) for a fixed malignant tumour baseline ($D_f=1.7$, $d_s=1.5$). (b) Sensitivity analysis across three stages of tumour progression Elkington2022 with fixed fractional order ($\alpha=1.8$): healthy tissue ($D_f=1.5$, $d_s=1.3$), stage II-III malignant tumour ($D_f=1.7$, $d_s=1.5$), and theoretical saturation ($D_f=1.9$, $d_s=1.7$).