The Wulff bio-heat transfer model revisited: directional blood enthalpy transport, the biological Peclet number, and implications for laser-induced thermal therapy

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Original abstract

Bio-heat transfer models play a fundamental role in predicting temperature fields during laser-induced thermal therapy (LITT). Among continuum bio-heat transfer models, the Pennes equation remains the most widely adopted formulation. However, by representing blood perfusion as an isotropic volumetric source, it neglects the directional transport of thermal energy associated with blood flow. The Wulff model overcomes this limitation by incorporating blood averaged enthalpy transport directly into the heat flux. Despite its physical significance, the derivation of the Wulff formulation and the assumptions required to obtain its governing equation remain only briefly discussed in the original work. The aim of the present work is twofold. First, the physical formulation originally proposed by Wulff is revisited in order to clarify its derivation and the assumptions required to obtain the governing equation. In particular, the physical origin of the modified heat flux is investigated and an equivalent formulation with an independent metabolic source term is discussed. Second, the resulting model is assessed against the classical Pennes formulation through representative laser-induced thermal therapy benchmark problem. The numerical results show that accounting for blood flow directionality may substantially alter the predicted temperature field and the extent of thermal damage. A dimensionless analysis further leads to the definition of a biological Peclet number, which quantifies the magnitude of directional blood enthalpy transport relative to thermal diffusion. Finally, a simple extension accounting for venous stasis is proposed, mitigating the excessive temperature rise predicted by the original Wulff formulation

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