Page 40 - Computational Modeling in Biomedical Engineering and Medical Physics

26 Computational Modeling in Biomedical Engineering and Medical Physics

a modified relationship for calculating the blood perfusion term, and accounts for the
“eddy” conduction due to the random flow of blood, by introducing a modified ther-
mal conductivity. However, CH BHT model is difficult to implement because it
requires detailed knowledge of the vascular network and blood perfusion. Chato
(1980) introduced a countercurrent, two-temperature model. He obtained the tem-
perature profiles along the arterial and venous vessels and evidenced the heat transfer
enhancement by the perfusion “bleed-off” between the vessels as compared with con-
stant mass flow rates models (Nakayama and Kuwahara, 2008). To account for the
perfusion heat sources, which might be important particularly in the extremities
(Roetzel and Xuan, 1998) replaced the convection perfusion parameters with interfa-
cial convection heat transfer coefficients.
Three-temperature energy equations bioheat model proposed by Weinbaum et al.
(1984) account for the countercurrent blood flow effect, mainly applicable for the
intermediate tissue of the skin (Minkowycz et al., 2009) where small arteries and veins
are parallel, and their flow directions are countercurrent. Weinbaum and Jiji (1985)
evidenced three vascular layers—deep, intermediate, and cutaneous—in the outer
1 cm tissue layer in a study performed on rabbit limbs.
Bioheat transfer models, such as in the studies by Pennes (1948), Wulff (1974),
Klinger (1974), Chen and Holmes (1980), Weinbaum et al. (1984), and Nakayama
and Kuwahara (2008) identified the blood perfusion with a heat source term in the
energy equation, and use homogenization techniques to replace the complex anatomic
media with equivalent continuum media.
As seen, progressively, one-, two- and three-energy equations (temperatures), iso-
tropic and anisotropic models, were introduced, each of them with merits and short-
comings. Pennes one-temperature model is the simplest of them and despites its
limitations it is still used intensively. Two-temperature models are used to investigate
the countercurrent heat exchange between the arterial and venous blood vessels in the
circulatory system for idealized one-dimensional cases. Three-energy equations may be
more general since in its multidimensional and anisotropic form can be applied to all
regions peripheral heat transfer from the extremity to the surroundings (Nakayama
and Kuwahara, 2008). This model provides control volume-based recipes to calculate
the mechanical permeability, volume fractions, interfacial heat transfer coefficients, and
perfusion rates. Coupled with the continuity and Darcy’s laws, it results in a mathe-
matical model that may be solved to find both velocity and temperature fields.
However, despite these progresses, it is yet much to do (Nakayama and Kuwahara,
2008; Roetzel and Xuan, 1998). For instance, there is a need for the clarification of
some vasoconstrictor and vasodilator mechanisms, model “constants,” physiological
parameters (porosity, specific surface area), which depend on factors such as the body
temperature and its interaction with the environment, and have to be determined
experimentally.