Page 39 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 25

the anatomical tissues with complex structures, heat transfer properties, and heat flow
mechanisms. Heat transfer in living tissues is a multiscale process involving metabolic
heat production, thermal conduction in tissues, heat convection in larger blood vessels,
and heat carried by perfused tissues, capillaries, microvasculature. Mathematical models
based on continuum media homogenization hypotheses are used for heat transfer anal-
ysis of the body heat balance, in thermoregulation, heat transfer in muscle and tissues,
skin burns, surgical procedures, for example, hyperthermia and hypothermia cancer
treatment, laser surgery, cryosurgery, cryopreservation of organs for transplant, resusci-
tation, thermal comfort, and extracorporeal equipment.
Pennes introduced for the first time a simple model, the bioheat equation (Pennes,
1948), by adopting the energy equation to yield

2
ρ @T tissue 5 kr T tissue 2 ρ C b ω b T b 2 T tissue Þ 1 qw: ð1:34Þ
ð
tissue C tissue b m
@t
Here ω b [s 21 ] is the blood perfusion rate, T b , ρ b , and C b are the temperature, mass
density, and specific heat of the arterial blood, T tissue , ρ tissue , and C tissue are the tempera-
3
ture, mass density and specific heat of the of the tissue, and qw [W/m ] is the meta-
m
bolic heat rate.
This model was and still is widely used to describe the heat transfer within living
tissues. Although attractive for its simplicity, it bears a number of shortcomings, which
prone its predictions to error. For instance, it assumes only the venous blood flow as
the fluid stream equilibrated with the tissue—tissue blood local thermal equilibrium.
Moreover it assumes uniform perfusion, neglecting the directionality of the blood
flow and the important anatomical features of the circulatory network system such as
countercurrent arrangement of the vascular system, the different sizes of the vessels,
from micrometers to millimeters, and the pending different rheological models for
blood, vascular geometry, the transvascular heat and mass transfer, the sharp spatial var-
iations of the material properties, the necrosis that might accompany the heat genera-
tion, to name some.
Aiming to overcome these difficulties several continuum models were proposed. Wulff
(1974) and Nakayama and Kuwahara (2008) assumed that the blood temperature is that of
the tissue temperature within a tissue control volume and not only the local tissue temper-
ature gradient. Klinger (1974) and later Cho (1992) assumed that the heat transfer between
the hemodynamic flowing and the irrigated tissue is proportional to the temperature dif-
ference between these two media with nonuniform velocity field in space and time, and
metabolic heat source (Zolfaghari, and Maerefat, 2010).
Chen and Holmes (1980) (CH BHT) showed that the major heat transfer pro-
cesses occur in the 50 μm to 500 μm diameter vessels and proposed that larger vessels
be modeled separately from smaller vessels and the embedding tissue. Their model
subdivides the tissue control volume into solid and bloodstream subvolumes, introduce