A General Set of Bioheat Transfer Equations Based on the Volume Averaging Theory - Porous Media: Applications in Biological Systems and Biotechnology - page 29

experimental technique to create a two-dimensional freezing problem associ-
ated with prostate cryosurgery with urethral warming. However, none of them
considered the case of lung cancer nor were concerned with the effects of the
blood perfusion on the temporal evolution of ice formation, which leads to the
fact, namely, that there exists the limiting size of the tumor that one single
cryoprobe can freeze at the maximum. No attempts were made to estimate
the limiting radius for freezing tumors.
In this section, we shall appeal to the bioheat equation derived using the
volume averaging theory and solve it both numerically and analytically to
simulate the ice ball evolution and to locate the freezing front as time goes by.
The analytical results based on the integral method agree very well with the
numerical results based on the enthalpy method. Thus, the resulting analytical
expression may be exploited for estimating the time for freezing a cancer of a
given size. It will also be pointed out that there exists the limiting size of the
cancer that one single cryoprobe can freeze at the maximum. It is believed
that the present results lend quantitative support to the current empirical
standards for cryosurgical clinical applications.

1.7.2 Bioheat Equation for Cryoablation
The general bioheat equation (1.26) for the solid tissue phase may be used to
attack this problem. When the ratio of blood to total lung volume is small,
equation (1.26) reduces to
ρ s c s ∂T
∂t

k s ∂T
∂x j

+ ρ f c p f ω ( T f −

∂
∂x j

= T )+ a f h ( T f − T )+ S m (1.79)

where the second time and third term on the right-hand side correspond to the
blood perfusion to the tissue and the interfacial heat transfer from the blood
to tissue through the vessel wall, respectively. Similarity between our equation
and Pennes' equation is obvious as we rewrite the foregoing equation as
ρ s c s ∂T
∂t

k s ∂T
∂x j

+ ρ f c p f ω eff ( T f −

∂
∂x j

= T )+ S m (1.80)

where

a f h
ρ f c p f

ω eff = ω +

(1.81)

is the effective perfusion rate. However, ω eff conceptually differs from Pennes'
perfusion rate ω Pennes in the Pennes equation (1.27), which is purely empirical.
It should also be noted that T f in ( T f −

T ) is the local blood temperature,
whereas T a 0 in equation (1.27) is the mean brachial artery temperature. The
interfacial convective heat transfer between the blood and tissue can never be
insignificant for countercurrent bioheat transfer. Even when there is no per-
fusion, that is, ω Pennes = 0, the effective perfusion rate never vanishes since
ω eff = a f h/ρ f c p f . Thus, equation (1.81) must always be used for countercur-
rent bioheat transfer for the case of closely aligned pairs of vessels.