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

Objective&#xD;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.&#xD;Approach&#xD;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.&#xD;Main results&#xD;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.&#xD;Significance&#xD;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.