The transport of thermal energy in living tissue is a complex process involving multiple phenomenological mechanisms including conduction, convection, radiation, metabolism, evaporation, and phase change. The study of tissue heating processes at high temperatures is relevant to therapeutic applications (such as RF, microwave, and laser ablation and hyperthermia) and food processing applications (such as baking and frying).
Pennes' bio-heat equation, based on the heat diffusion equation, is a much used approximation for heat transfer in biological tissue. Many publications have shown it is a valuable approximation, especially at hyperthermia temperatures. However, at the higher temperatures seen during ablation, the Pennes’ bio-heat equation does not incorporate all the physical processes affecting final tissue temperature. These processes include but are not limited to the effects of the movement and diffusion of tissue water due to temperature and changes in local water content due to heating, water evaporation at high temperatures, the diffusion of this generated water vapor, and its possible re-condensation.
To add to the complexity, the thermal and other physical properties are a function of temperature, water content and the changes in the mechanical stresses on the tissue. Attempting to model this complex physical system, which involves electromagnetic (EM), thermal and mass transfer modeling, mechanical stresses, etc, is challenging due to the interdependent nature of the physical properties.
Below is the Pennes’ bio-heat diffusion equation
Where r is density [kg/m3], C is specific heat [J/kgoC], T is temperature [oC], k is thermal conductivity [W/moC], Q is the microwave power density [W/m3], QB is a term which accounts for the effects of perfusion [W/m3], and A is the metabolic heat generation term [W/m3] which is considered insignificant with respect to the heating term.
Where rB is the blood mass density [kg/m3], CB is the blood specific heat [J/kgoC], wB is the blood perfusion rate [1/s], and TB is the ambient blood temperature [oC] before entering the ablation region. Note, all variables but are spatially dependent. For purposes of clarity the spatial dependence is left out of the equations and is to be implied.
The term QB models the perfusion heat loss. Vascular tissues generally experience increased perfusion as temperature increases. The above term is always considered in cases of tissues with a high degree of perfusion, such as liver. Regarding cardiac ablation, the perfusion heat loss is incorporated in some models, but is generally ignored since its effect is negligible.118 In general, ωb is assumed to be uniform throughout the tissue. However, its value may increase with heating time because of vasodilatation and capillary recruitment.
In order to build a theoretical model, the values of the basic physical characteristics have to be set for all the material of the model: mass density (ρ), specific heat (c), thermal conductivity (k), and electrical conductivity (σ). In general, it is difficult to measure tissue properties because they are spatially, temporally, and even temperature dependent. The above properties, however, are considered to be isotropic and their values are usually taken from the scientific literature as shown in Table I
One advantage of Pennes bio-heat equation is its simplicity. Given the relevant properties and perfusion rates, it becomes fairly easy to solve for tissue temperature as a function of spatial location and time. It is well known that the Pennes’ perfusion source term overestimates the actual blood perfusion effect in tissue in two ways:
· The first limitation is that it considers that all the heat leaving the artery is absorbed by the local tissue and there is no venous re-warming. A correction coefficient that is close to zero implies a significant countercurrent re-warming of the paired vein and a coefficient of unity implies no re-warming.
· A second limitation of the Pennes perfusion source term is that the arterial temperature is assumed to be equal to the body core temperature. The alternative to the Pennes equation is to employ a decidedly more complex model that explicitly describes heat exchange between vessel pairs.
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