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Thermal models are used to analyze responses of muscle and lung tissue to transient (30-45 min) and chronic (4-7 week) heating in vivo. The general bioheat model, which describes one-dimensional temperature dynamics, incorporates heat conductance and perfusion. In general, perfusion changes with time and distance from a heated surface. One of the main objectives of this study was to analyze long-term perfusion change, which reflects tissue adaptation associated with angiogenesis. The database for these models was obtained using heated disks implanted in calves for up to seven weeks. Tissue temperature distributions were obtained repeatedly from thermistors protruding 1 to 10 mm from the heated disk surface. The perfusion parameter was estimated from the transient experiments at least several times each week by nonlinear, least-squares fitting of the model predicted temperature to the measured temperature response. Chronic heating at a heat flux 0.08 W/cm2 caused perfusion of muscle tissue to increase with postimplant day (PID). Under the same conditions, lung tissue perfusion increased with chronic heating from early to late PID, but less than that for muscle tissue. During chronic heating above 42 degrees C and below 50 degrees C, a decrease in tissue temperature is associated with higher perfusion that develops with time. Over seven weeks, perfusion of muscle tissue near the heated disk surface increased by about 70% at 0.08 W/cm2 and 40% at 0.06 W/cm2. Furthermore, the model can be used to predict tissue and perfusion changes continuously over weeks for heat fluxes around 0.08 W/cm2.

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For a spatially distributed model describing the transient temperature response of a thermistor-tissue system, Wei et al. [J. Biomech. Eng., 117:74-85, 1995] obtained an approximate transformation for fast analysis of the temperature response at the tissue surface. This approximate transformation reduces the model to a single ordinary differential equation. Here, we present an exact transformation that yields a single differential-integral equation. Numerical solutions from the approximate and exact transformations were compared to evaluate the differences with several sets of parameter values. The maximum difference between the exact and approximate solutions did not exceed 15 percent and occurred for only a short time interval. The root-mean-square error of the approximate solution was no more than 5 percent and within the level of experimental noise. Under the experimental conditions used by Wei et al., the approximate transformation is justified for estimating model parameters from transient thermal responses.