Biomedical Engineering Reference
In-Depth Information
Equation 1.95 is written to describe the thermal effects of
blood flow through a local region of tissue having a temperature
T
. It contains the familiar energy storage and conduction terms,
plus terms to account for convection with perfused blood and
metabolic heat generation. The middle term on the right side of
the equation corresponds to the enthalpy term in Equation 1.1
that accounts for the effects of mass entering and leaving a sys-
tem influencing the stored energy. The temperature of entering
perfused blood into a tissue region is that of the arterial supply,
T
a
, and the leaving temperature is
T
because the relatively small
volume of flowing blood completely equilibrates with the sur-
rounding tissue via the very large surface area to volume ratio
of the microvasculature through which it flows. Indeed, the level
of the vasculature at which thermal equilibration is achieved
between perfused blood and the surrounding tissue has been a
topic of considerable interest and importance for many applica-
tions of bioheat transfer (Chato 1980; Chen and Holmes 1980;
Shrivastava and Roemer 2006). There is little doubt that blood
comes to the temperature of the tissue through which it is flow-
ing within the arteriolar network long before the capillaries are
reached (in stark contrast to mass transport, which is focused in
the capillaries).
The Pennes model contains no specific information about the
morphology of the vasculature through which the blood flows.
The somewhat simple assumption is that the fraction of blood
flowing through a tissue that is diverted through the microvas-
culature comes to thermal equilibration with the local tissue
as it passes to the venous return vessels. A major advantage of
the Pennes model is that the added term to account for per-
fusion heat transfer is linear in temperature, which facilitates
the solution of Equation 1.95. Since the publication of this
work, the Pennes model has been adapted by many research-
ers for the analysis of a variety of bioheat transfer phenomena.
These applications vary in physiological complexity from a
simple homogeneous volume of tissue to thermal regulation of
the entire human body. As more scientists have evaluated the
Pennes model for application in specific physiological systems,
it has become increasingly clear that some of the assumptions
foundational to the model are not valid for some vascular geo-
metries that vary greatly among the various tissues and organs
of the body (Charney 1992).
Given that the validity of the Pennes model has been ques-
tioned for many applications, Wissler (1998) has revisited and
reanalyzed Pennes's original data. Given the hindsight of five
decades of advances in bioheat transfer plus greatly improved
computational tools and better constitutive property data,
Wissler's analysis pointed out flaws in Pennes's work that had
not been appreciated previously. However, he also showed that
much of the criticism that has been directed toward the Pennes
model is not justified, in that his improved computations with
the model demonstrated a good standard of agreement with
the experimental data. Thus, Wissler's conclusion is that “those
who base their theoretical calculations on the Pennes model
can be somewhat more confident that their starting equations
are valid” (Wissler, 1998).
Another important issue relating to convective heat trans-
fer between blood and tissue is the regulation of local perfusion
rate. The flow of blood to specific regions of the body, organs,
and tissues is a function of many variables. These include main-
tenance of thermogenesis, thermoregulatory function, type and
level of physical activity, existence of a febrile state, and others.
The regulation and distribution of blood flow involves a complex,
nonlinear feedback process that involves a combination of local
and central control inputs. Many existing models include these
various inputs on a summative basis. In contrast, Wissler (2008)
has recently introduced a multiplicative model that provides an
improved simulation of physiological performance. In summary,
quantitative analysis of the effects of blood perfusion on the inter-
nal temperature distribution in living tissue remains a topic of
active research after a half century of study.
1.3.2 thermal properties of Living tissues
Compilation of tables of the thermal properties of tissues has
lagged behind that of properties for inanimate materials. One
of the major challenges faced in measuring tissue properties is
the fact that inserting a measurement probe into a live speci-
men will alter its state by causing trauma and modifying local
blood perfusion. Also, there are large variations among different
individuals, and physical access to internal organs is difficult.
Nonetheless, there are increasing broad compilations of tissue
thermal properties such as that prepared by Ken Holmes in
Diller et al. (2005).
Thermal probe techniques are used frequently to determine
the thermal conductivity and the thermal diffusivity of bio-
materials (Balasubramaniam and Bowman 1977; Chato 1968;
Valvano et al. 1984). Common to these techniques is the use of a
thermistor bead either as a heat source or a temperature sensor.
Various thermal diffusion probe techniques (Valvano 1992) have
been developed from Chato's first practical use of the thermal
probe (Chato 1968). Physically, for all of these techniques, heat
is introduced to the tissue at a specific location and is dissipated
by conduction through the tissue and by convection with blood
perfusion.
Thermal probes are constructed by placing a miniature
thermistor at the tip of a plastic catheter. The volume of tissue
over which the measurement occurs depends on the surface area
of the thermistor. Electrical power is delivered simultaneously
to a spherical thermistor positioned invasively within the tissue
of interest. The tissue is assumed to be homogeneous within the
mL surrounding the probe. The electrical power and the result-
ing temperature rise are measured by a microcomputer-based
instrument. When the tissue is perfused by blood, the thermis-
tor heat is removed both by conduction and by heat transfer
due to blood flow near the probe.
In vivo
, the instrument mea-
sures effective thermal properties that are the combination of
conductive and convective heat transfer. Thermal properties
are derived from temperature and power measurements using
equations that describe heat transfer in the integrated probe/
tissue system.
