A two-equation coupled system for determination of liver tissue temperature during thermal ablation

Ablation techniques have become a widespread choice for the treatment of cancerous tumours, where surgical resection techniques have poor prognosis. However, the extent and the shape of the ablation zone can be significantly affected by heat loss due to blood perfusion, which is difficult both to measure and to model accurately. A two-equation coupled bio-heat model is thus presented here to model the heat exchange between blood flow and its surrounding biological tissue, by considering the vasculature as a porous medium. The cooling effects of different generations of the vasculature are examined separately in the model. It is shown both analytically and computationally that the model behaviour is dependent on a non-dimensional number, γ, so termed here the thermal significance coefficient: for vessels in the highest generation of the vasculature such as big arteries and veins, γ is found to be much larger than unity, indicating that the blood in these large vessels holds a constant temperature, in which case the two-equation coupled bio-heat model can be simplified into the Pennes model; on the other hand, for small vessels in the bottom generation of the vasculature such as arterioles, capillaries, venules, γ is far smaller than unity, suggesting that the blood in small vessels is in continuous equilibrium with tissue temperature, in which case the model is equivalent to the Klinger model. However, most of the temperature equilibration occurs as the blood travels via the middle generations of the vasculature such as terminal artery branches, for which γ is of order unity and hence the two-equation coupled model cannot be further simplified. The implications of this for practical implementation of the bio-heat model are discussed.