Encoding depth heterogeneity into time: a finite-window power-series thermal response model for non-invasive tissue characterization and early tumor detection

Objective Depth-dependent heterogeneity in perfusion and thermophysical properties remains a major obstacle to accurate non-invasive measurement of tissue thermal parameters. The objective was to develop an analytical photothermal model that encodes such spatial heterogeneity into a time-dependent effective thermal response coefficient (ETRC) for tissue characterization and early tumor detection. Approach A positive-definite ETRC framework was formulated by using the scaling relationship between thermal diffusion depth and time. The ETRC was first represented by an exponential parent form, β(t)=βs eλt, and the experimentally fitted model used its finite-window first-order approximation, β(t)≈β0+β1t, where β0=βs and β1=βsλ. An explicit analytical solution was then derived for this finite-window approximation through variable substitution. The model was evaluated against zero-coefficient and constant-coefficient formulations through numerical validation and longitudinal pulsed photothermal experiments in a murine CT26 subcutaneous tumor model. Main results The proposed model achieved the lowest root-mean-square error in the cooling phase among the tested models. In vivo, the thermal diffusivity α in the peritumoral vascular region decreased progressively from day 3 after inoculation and differed significantly from the pre-inoculation baseline by day 6 (p<0.01), approximately 3-4 days before tumors became palpable. The area under the receiver operating characteristic curve reached 0.94 on day 12. Parametric maps of β₀ and β₁ provided complementary contrast. β₀ reflected the initial near-surface ETRC, whereas β₁ described the first-order temporal change in the ETRC within the fitted cooling window as the surface signal became sensitive to deeper tissue heterogeneity. Significance An interpretable and computationally efficient approach is provided for extracting depth-dependent functional information from surface temperature decay without requiring pre-assumed tissue layers. The proposed method may support longitudinal non-invasive thermal imaging, early tumor detection and functional monitoring of tumor development.