Page 60 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

HEAT TRANSFER APPLICATIONS IN BIOLOGICAL SYSTEMS 37

therapeutic thermal exposure. In applications where point-to-point temperature nonuniformities
are important, vascular model has been proved to be necessary to predict accurately the tissue
temperature field (Zhu et al., 1996a). In recent years, with the breakthrough of advanced compu-
tational techniques and resources, vascular models (Raaymakers et al., 2000) for simulating vas-
cular networks have grown rapidly and already demonstrated its great potential in accurate and
point-to-point blood and tissue temperature mapping.

2.3.1 Continuum Models
In continuum models, blood vessels are not modeled individually. Instead, the traditional heat con-
duction equation for the tissue region is modified by either adding an additional term or altering
some of the key parameters. The modification is relatively simple and is closely related to the local
vasculature and blood perfusion. Even if the continuum models cannot describe the point-by-point
temperature variations in the vicinity of larger blood vessels, they are easy to use and allow the
manipulation of one or several free parameters. Thus, they have much wider applications than the
vascular models. In the following sections, some of the widely used continuum models are intro-
duced and their validity is evaluated on the basis of the fundamental heat transfer aspects.
Pennes Bioheat Transfer Model. It is known that one of the primary functions of blood flow in a
biological system is the ability to heat or cool the tissue, depending on the relative local tissue tem-
perature. The existence of a temperature difference between the blood and tissue is taken as evidence
of its function to remove or release heat. On the basis of this speculation, Pennes (1948) proposed
his famous heat transfer model, which is called Pennes bioheat equation. Pennes suggested that the
effect of blood flow in the tissue be modeled as a heat source or sink term added to the traditional
heat conduction equation. The Pennes bioheat equation is given by

∂T 2
ρC t =∇ T + q blood + q m (2.2)
k
t
t
∂t
where q is the metabolic heat generation in the tissue, and the second term (q ) on the right side
m blood
of the equation takes into account the contribution of blood flow to the local tissue temperature dis-
tribution. The strength of the perfusion source term can be derived as follows.
Figure 2.1 shows a schematic diagram of a small tissue volume perfused by a single artery and vein
pair. The tissue region is perfused via a capillary network bifurcating from the transverse arterioles,
and the blood is drained by the transverse venules. If one assumes that both the artery and vein keep
a constant temperature when they pass through this tissue region, the total heat released is equal to
the total amount of blood perfusing this tissue volume per second q multiplied by its density ρ ,
b
specific heat C , and the temperature difference between the artery and vein, and is given by
b
qρ C (T − T ) = (Q − Q ) ρ C (T − T ) (2.3)
b b a v in out b b a v
The volumetric heat generation rate q defined as the heat generation rate per unit tissue volume,
blood
is then derived as
q = [(Q − Q )/V]ρ C (T − T ) =ωρ C (T − T ) (2.4)
blood in out b b a v b b a v
where ω is defined as the amount of blood perfused per unit volume tissue per second.
Note that both T and T in Eq. (2.4) are unknown. Applying the analogy with gaseous exchange
a v
in living tissue, Pennes believed that heat transfer occurred in the capillaries because of their large
area for heat exchange. Thus, the local arterial temperature T could be assumed as a constant and
a
equal to the body core temperature T . As for the local venous blood, it seems reasonable to assume
c
that it equilibrates with the tissue in the capillary and enters the venules at the local tissue temperature.