Page 72 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

HEAT TRANSFER APPLICATIONS IN BIOLOGICAL SYSTEMS 49

where Q = absorbed radiant energy per unit area
ρ= mass density
C = specific heat
l = sample thickness
α= thermal diffusivity

The maximum temperature at the rear surface is determined by the volumetric heating as
T = T (l, 0) + Q/(ρCl) (2.16)
max
The thermal diffusivity in the direction of heat flow is usually calculated by the expression

l 2
α = 138 (2.17)
.
2
π t 12 /
where t 1/2 is the time required for the rear surface to reach half of its maximum temperature.
The simplicity of the method described above is often offset by the difficulty in satisfying the
required adiabatic boundary conditions. In order for this solution to be valid, the radiant energy incident
on the front surface is required to be uniform, and the duration of the flash must be sufficiently short
compared with the thermal characteristic time of the sample. In addition, it assumes that the sample
is homogeneous, isotropic, and opaque, and that the thermal properties of the sample do not vary
considerably with temperature.

Temperature Pulse Decay (TPD) Technique. Temperature pulse decay (TPD) technique is based
on the approach described and developed by Arkin, Chen, and Holmes (Arkin et al., 1986, 1987).
This method needs no insulation, in contrast to some of the methods described above, since testing
times are short, usually on the order of seconds. However, the determination of the thermal conduc-
tivity or the blood flow rate requires the solution of the transient bioheat transfer equation.
This technique employs a single thermistor serving as both a temperature sensor and a heater.
Typically in this technique, either a thermistor is inserted through the lumen of a hypodermic nee-
dle, which is in turn inserted into the tissue, or the thermistor is embedded in a glass-fiber-reinforced
epoxy shaft. Figure 2.7 shows the structure of a thermistor bead probe embedded in an epoxy shaft.
Each probe can consist of one or two small thermistor beads situated at the end or near the middle
of the epoxy shaft. The diameter of the finished probe is typically 0.3 mm, and the length can vary
as desired. Because the end can be sharpened to a point, it is capable of piercing most tissues with
very minimal trauma.
During the experiment, a short-heating pulse of approximately 3 seconds is delivered by the
thermistor bead. The pulse heating results in a temperature rise in the area near the tip of the
probe. After the pulse heating, the temperature of the probe will decrease. During the pulse heat-
ing and its subsequent temperature decay, the temperature at the tip of the probe is measured by
the thermistor bead. To determine the thermal conductivity or blood flow rate, a theoretical pre-
diction of the transient temperature profile is needed for the same tissue domain as in the exper-
imental study. Typically, the theoretically predicted temperature profile is obtained by solving a
bioheat transfer equation in which the blood flow rate and thermal conductivity have to be given
as input to the model. The predicted temperature profile is then compared with the experimental
measurements. The values of the blood flow rate and/or thermal conductivity will be adjusted to
minimize the square difference between the predicted temperature profile and the experimental
measurements using the linear-square residual fit. The values for the blood flow rate and thermal
conductivity that give the best fit of the experimentally measured temperature profile are the cal-
culated blood flow rate and thermal conductivity of the tissue sample.
Typically, the Pennes bioheat transfer equation is used to predict the temperature transient. It is
assumed that the thermistor bead is small enough to be considered a point source inserted into the
center of an infinitively large medium. The governing equation and initial condition for this thermal