Page 66 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

HEAT TRANSFER APPLICATIONS IN BIOLOGICAL SYSTEMS 43

Head

Internal
Organs Muscle

T = 25°C 1.8 m
air
2
h = 4.5 W/m °C

z
y x
FIGURE 2.3 Schematic diagram of the whole body geometry.

A recently developed whole body model by our group (Zhu et al., 2009) utilizes the simple rep-
resentation of the Pennes perfusion source term to assess the overall thermal interaction between the
tissue and blood in the human body. As shown in Fig. 2.3, a typical human body (male) has a body
3
weight of 81 kg and a volume of 0.074 m . The body consists of limbs, torso (internal organs and
muscle), neck, and head. The limbs and neck are modeled as cylinders consisting of muscle. Note
that the body geometry can be modeled more realistically if one includes a skin layer and a fat layer
in each compartment. However, since our objective is to illustrate the principle and feasibility of the
developed model, those details are neglected in the sample calculation. The simple geometry results
2
in a body surface area of 1.8 m . Applying the Pennes bioheat equation to the whole body yields
∂ T 2
ρ c t =∇ T + q + ρ c ω T − ) (2.8)
(
T
k
tt
t
t
a
m
b b
t
t ∂
where subscripts t and b refer to tissue and blood, respectively; T and T are body tissue temperature
t a
and blood temperature, respectively; ρ is density; c is specific heat; k is thermal conductivity of
t
3
tissue; q is the volumetric heat generation rate (W/m ) due to metabolism; and ω is the local blood
m
perfusion rate. The above governing equation can be solved once the boundary conditions and ini-
tial condition are prescribed. The boundary at the skin surface is modeled as a convection boundary
subject to an environment temperature of T and a convection coefficient of h.
air
Based on the Pennes bioheat equation, the rate of the total heat loss from the blood to tissue at
any time instant is
c ω(
)
Q blood-tissue = ∫∫∫ ρ bb T t − T dV body = ρ bb T t() − t T V) body (2.9)
c ω((
)
a a
a
t
body volume
where V body is the body volume, T is the blood temperature which may vary with time. Equation (2.9)
a
implies that both density ρ and specific heat c are constant. In Eq. (2.9), ω is the volumetric average
blood perfusion rate defined as
1
ω = ∫∫∫ ω dV body (2.10)
V
body body volume