A General Set of Bioheat Transfer Equations Based on the Volume Averaging Theory - Porous Media: Applications in Biological Systems and Biotechnology - page 14

We can easily see that the foregoing equation reduces to the Klinger equa-
tion when the ratio of vascular volume to total volume (i.e., porosity ε )is
suciently small. Since the porosity is generally less than 0.1, the foregoing
two equations are quite close to each other.
Another interpretation on the directional effect on the tissue temperature
field is possible. When the blood flow is strong enough to neglect the macro-
scopic diffusion, the energy equation (1.25) for the blood flow reduces to

a f h f

s

ρ f c p f ω

s (1.32)

∂
∂x j

f = f f

ρ f c pf u j T − T − T − T − T

Substitution of the foregoing equation into the energy equation for the
tissue (1.26) yields the Klinger equation (1.30). The assumption implicit here
is that the blood flow velocity is suciently high that the ratio of the bulk
convection heat transfer to conduction heat transfer, namely, the Peclet num-
ber, is much greater than unity. Thus, the Klinger model applies to the tissue
with comparatively large vessels.

1.4.5 Chen and Holmes Model
Chen and Holmes (1980) assumed that all tissue-arterial blood heat exchange
occurs along the circulatory network after the blood flows through the terminal
arteries and before it reaches the level of the arterioles, which prompted them
to propose the following bioheat transfer model:

ρc ∂T t
∂t

∂
∂x j

+ ρ f c pf u j T t

( εk f +(1 + ρ f c p f ω j ( T a −

∂
∂x j

ε ) k s ) ∂T t

∂x j + k p ∂T t

= − T t )+(1 − ε ) S m

∂x j

(1.33)

where

ρ = ερ f +(1 − ε ) ρ s (1.34a)

c =( ερ f c pf +(1 − ε ) ρ s c s ) /ρ (1.34b)

and

T t = ερ f c p f

s ρc f +(1

T − ε ) ρ s c s T (1.34c)

is the temperature of the continuum based on a volume average. Moreover,
ω j is the perfusion bleed-off to the tissue only from the microvessels past
the j th generation of branching, while T a is the blood temperature at the
j th generation of branching. Both ω j and T a require the anatomical data.
Chen and Holmes (1980) also took account of the “eddy” conduction due to
the random flow of blood, by introducing the thermal conductivity k p , which
corresponds to our dispersion thermal conductivity k dis . The energy equation