A General Set of Bioheat Transfer Equations Based on the Volume Averaging Theory - Porous Media: Applications in Biological Systems and Biotechnology - page 4

numerically. The freezing front in a tumor during percutaneous cryoabla-
tion can be traced exploiting a bioheat equation. It will be shown that there
exists a limiting size of the tumor that one single cryoprobe can freeze at the
maximum. The freezing front moves radially outward from the cryoprobe and
reaches the end, where the heat from the surrounding tissue to the frozen
tissue balances with the heat being absorbed by the cryoprobe. An excellent
agreement between the analytical and numerical results is achieved for the
time required to freeze the tumor using the cryoprobe of one single needle.
The resulting analytical expression for estimating the limiting radius provides
useful information for cryotherapy treatment plans.

1.2 Volume Averaging Procedure
In an anatomical view, three compartments are identified in the biologi-
cal tissues, namely, blood vessels, cells and interstitium, as illustrated in
Figure 1.1. The interstitial space can be further divided into the extracel-
lular matrix and the interstitial fluid. However, for sake of simplicity, we
divide the biological tissue into two distinctive regions, namely, the vascu-
lar region and the extravascular region (i.e., cells and the interstitium) and
treat the whole anatomical structure as a fluid-saturated porous medium,
through which the blood infiltrates. The extravascular region is regarded as a
solid matrix (although the extravascular fluid is present), and will be simply
referred to as the “tissue” region to differentiate it from the “blood” region.
Thus, we shall try to apply the principle of heat and fluid flow in a fluid-
saturated porous medium to derive a set of the volume averaged governing
equations for the bioheat transfer and blood flow. For the volume averaging
(smoothing process) to be meaningful, we consider a control volume V in a
fluid-saturated porous medium, as shown in Figure 1.2, whose length scale
V 1 / 3 is much smaller than the macroscopic characteristic length V 1 / c , but,
at the same time, much greater than the microscopic (anatomical structure)
characteristic length (see e.g., Nakayama [1995]). Under this condition, the
volume average of a certain variable φ is defined as

1
V

φ ≡ φdV (1.1)

V f

Another average, namely, intrinsic average, is given by

1
V f

f

φ ≡ φdV (1.2)

V f

where V f is the volume space that the fluid (blood) occupies. Obviously, two
averages are related as

f

φ = ε φ (1.3)