Page 61 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

38 BIOMECHANICS OF THE HUMAN BODY

Supply Vein Transverse Venule

Q out,v Q in,v

Capillary

Q in,a Q out,a

Supply Artery
Transverse Arteriole

FIGURE 2.1 Schematic diagram of a tissue volume perfused by a single artery and vein pair.

Then the Pennes bioheat equation becomes
∂ T 2
T +
(
ρC t =∇ T + ωρ C T − ) q m (2.5)
k
b b
t
c
t
t
t ∂
This is a partial differential equation for the tissue temperature. As long as an appropriate initial con-
dition and boundary conditions are prescribed, the transient and steady-state temperature field in the
tissue can be determined.
The limitations of the Pennes equation come from the basic assumptions introduced in this
model. First, it is assumed that the temperature of the arterial blood does not change when it trav-
els from the heart to the capillary bed. As shown in Sec. 2.2, small temperature variations occur
only in blood vessels with a diameter larger than 300 μm. Another assumption is that the venous
blood temperature is approximated by the local tissue temperature. This is valid only for blood ves-
sels with a diameter smaller than 50 μm. Thus, without considering the thermal equilibration in the
artery and vein in different vessel generations, the Pennes perfusion source term obviously overes-
timates the effect of blood perfusion. To accurately model the effect of blood perfusion, the tem-
perature variation along the artery and the heat recaptured by the countercurrent vein must be taken
into consideration.
Despite the limitations of the Pennes bioheat equation, reasonable agreement between theory
and experiment has been obtained for the measured temperature profiles in perfused tissue subject
to various heating protocols. This equation is relatively easy to use, and it allows the manipulation
of two blood-related parameters, the volumetric perfusion rate and the local arterial temperature, to
modify the results. Pennes performed a series of experimental studies to validate his model. Over
the years, the validity of the Pennes bioheat equation has been largely based on macroscopic
thermal clearance measurements in which the adjustable free parameter in the theory, the blood per-
fusion rate (Xu and Anderson, 1999) was chosen to provide a reasonable agreement with experi-
ments for the temperature decay in the vicinity of a thermistor bead probe. Indeed, if the limitation
of Pennes bioheat equation is an inaccurate estimation of the strength of the perfusion source term,
an adjustable blood perfusion rate will overcome its limitations and provide a reasonable agreement
between experiment and theory.