Critical Evaluation of Current Skin Thermal Property Measurements
Anand Mani
University of Notre Dame
Department of Aerospace and Mechanical Engineering
May 8, 2013
Abstract
Burn treatment is an area of major importance in medicine, however, there is not currently an
analytic method to determine the depth and magnitude of burns. Complications, such as a relative
lack of knowledge of the specific thermal properties of skin and the relation between heat transfer
and biological processes, have prevented a comprehensive model to explain the field of bioheat
transfer. Research was conducted to estimate the relevant thermal and material properties in the
field of bioheat transfer. Subsequently, an experiment was proposed, and a mathematical model
was developed for the experiment, to allow for detection of burns through the use of heat transfer
analysis. Burned skin and healthy skin differ in that there is no blood flow and thus no directionality
to the heat transfer in burned skin. Thus, the experiment was designed to determine if there was
a directionality was present in the heat transfer in the skin. Using the estimated skin properties,
calculations were done to determine the viability of the proposed experiment. The experiment was
refined to account for the findings and modified to more accurately detect burns in human skin.
1
Introduction
The detection of burns in human skin is an important aspect of medicine. Currently, there are
no analytical methods that allow for the detection of burns, for two main reasons. The first is
a lack of knowledge about the specific properties and morphology of the skin. The thermal and
material properties of skin are not well measured, and many of the values that are used currently
are little more than estimates, based on knowledge of the composition of skin. Some properties
are unable to be directly measured due to their dependence on living tissue. For example, it is
impossible to directly measure the perfusive rate of blood, a measure of how much blood is flowing
in a given volume of tissue, because perfusion depends on blood flow in living tissue. Thus, in vitro
measurements are impossible because removal of flesh for study would stop any perfusion present
in the tissue. In addition, thermal energy generation, through methods such as metabolism, vary
greatly depending on location. For example, there is a much greater heat generation rate near the
brain as opposed to the skin in the arm. The second reason for the lack of a comprehensive model
is the complexity of the contribution of blood flow to heat transfer in skin. The different types of
blood vessels, ranging from the main veins and arteries to the capillaries requires knowledge of the
structure of the vessels to determine how each type contributes to the overall heat transfer.
2
Background
Burned skin and healthy skin have vastly different properties by which they can be characterized. One of the major differences between burned and healthy skin is that blood flow is not
present in burned skin. In the skin, there is both directional (anisotropic) and non-directional
1
(isotropic) heat transfer. The anisotropic heat transfer however, is dependent on blood flow, and
therefore should not be present in burned skin. Burned skin also lacks heat generation due to
metabolism, one of the major contributors to the overall heat transfer in skin. Our hope is that
by searching for anisotropies in the skin, we can determine at what depth blood starts flowing
again, and therefore, the depth of burns. If successful, this technique could be used to develop an
analytic procedure that can be used to determine burned areas in human skin. Thus, a greater
understanding and knowledge of the thermal and material properties of skin, and how they influence blood flow is required. To that end, a comprehensive analysis of currently available literature and studies was undertaken in an effort determine the necessary thermal and material
properties of skin. Using the currently available values to estimate the material and thermal
properties of skin allows for minimum and maximum possible values of the different heat transfer
rates to be calculated to determine whether the proposed method of burn determination is viable.
The goal was to find studies which determined the thermal or material properties of skin, either
through experimentation or computationally, to find values that could be used in calculations to
determine the viability of our proposed method of burn determination. In situations where the
available values were fairly close to each other, minimum and maximum values were taken to be
used in calculation. However, studies that provided greatly varying values of a single parameter were
critiqued to determine the cause of the differing values. Many of the studies that were evaluated used
pig skin tissue or other animal tissues in their determination of thermal and material properties. As
the proposed technique is to be utilized in human skin, only values which were derived for human
skin were accepted. In addition, a number of studies not dealing directly with burn determination,
such as studies regarding cancer treatment, were evaluated as some of them contained material
relating to determination of thermal properties or location of temperature variances in the skin.
2.1
Bioheat Transfer Models
Pennes Model
The Pennes model was the first model developed to examine the process of heat transfer in the
human skin. The model looks at metabolic heat generation and the effects of blood perfusion in
heat transfer. Pennes developed a thermal diffusion equation incorporating the two terms for perfusion and metabolism, which is written below:
ρc
∂T
= ∇· k∇T + (ρc)b ωb (Ta − T ) + qm
∂t
(1)
Thus, the term (ρc)b ωb (Ta − T ) is the heat transfer rate due to perfusion, qp , which accounts for
energy exchange between blood and the local tissue.
In developing his model, Pennes made some assumptions about the heat transfer process and properties of the human body:
• The metabolic heat rate is homogeneously distributed throughout the region.
• Blood Perfusion is homogeneous and isotropic
• Thermal equilibrium is reached in the microcirculatory bed
• No energy transfer occurs before or after the blood passes through the capillary bed
One of Pennes’ main assumptions, that thermal equilibrium is reached in the microcirculatory bed,
was later disproved by the work of Chen, Chato, Holmes, et al. as well as by Wienbaum et al. They
2
showed that thermal equilibriation actually occurs in the precapillary arterioles, not the capillary
bed. However, the Pennes model was found to be valid in the initial branchings of the largest microvessels, where the vasculature is comprised of many small, thermally significant vessels.
Continuum Model
The continuum model analyzes the larger vessels separately from the smaller vessels and tissue.
The later group is modeled as a continuum. In this analysis, the solid tissue is further separated
from the blood in an analysis of a differential element. Of all the continuum models, the Chen,
Chato and Holmes (CCH) Model is the most developed.
Their model replaced the perfusion term, (ρc)b ωb , with 3 terms, which represent the heat transfer
in: 1) non-equilibrated blood in thermally significant vessels, 2) blood equilibrated with tissue and
3) nearly-equilibrated blood. The equation then becomes:
ρc
∂T
= ∇· k∇T + (ρc)b ω ∗ (Ta∗ − T ) − (pc)b u· ∇T + ∇· kp ∇T + qm
∂t
(2)
where kp is the perfusion conductivity term.
Chen, Chato and Holmes also made some initial assumptions, namely:
• Neglecting the mass transfer between the vessel and tissue space
• Treating the thermal conductivity and temperature within the tissue-blood continuum as that
of solid tissue, since the volume of solid tissue is far more than the vascular volume
However, the CCH model requires much more detail than the Pennes model, which makes implementation difficult. Considering that, in the vast majority of situations, the CCH model yields similar
results as the Pennes model, and can be applied to the exact same regions as the Pennes model, in
most situations the Pennes model is better to use.
Wienbaum-Jiji-Lemons Model
The Wienbaum-Jiji-Lemons (WJL) Model is a vasculature-based model that looks at the countercurrent vasculature present in the skin. That is, it models artery-vein pairs as two parallel cylinders,
in which the heat is transferred due to advection and convection of the blood flow in the cylinders.
Metabolic heat transfer and collateral heat transfer are assumed to be normal to the plane of the
cylinders.
This model had one major assumption, namely heat transfer due to perfusion is treated as heat
transfer in a porous medium, and its direction is perpendicular to the artery or vein
Their analysis was performed on the different tissue layers (deep, intermediate and cutaneous) and
yielded the finding that the majority of the heat transfer is due to heat exchange between artery-vein
pairs. They developed their findings into one simple equation:
ρc
∂T
∂
∂T
=
(ke
) + qm
∂t
∂x
∂x
(3)
where ke is the effective conductivity term.Their model has one significant limitation though, which
is that it is only valid in situations where the ratio of the length of the vessel required for the blood
to reach thermal equilibrium with the surrounding tissue to the actual vessel length, is significantly
less than one. This condition is satisfied in analysis of heat transfer near the skin, where the ratio
3
is on the order of 10−4 .
Even when considering its faults, the Pennes model is the most widely used model in bioheat transfer analysis today. There are two main reasons for this. The first is that subsequent models are
significantly more complex and thus require much greater knowledge of the geometry and properties of the tissue being analyzed. Thus, as the Pennes model is simpler, it is much easier to use.
Second, while subsequent models have developed and improved the Pennes model, these models are
generally not much more accurate than the Pennes model except for very specific circumstances[1].
Thus, for general use, the added complexity of using these later models does not yield a noticeably
more accurate result, and thus are not worth using in most circumstances.
3
Proposed Experimental Setup
The goal of the proposed experiment is to determine whether or not the skin is burned, and how deep
the burn is, using thermal imaging or a similar technique that graphically shows heat distribution
and flow. In burned skin, since there will be no blood flow, the heat flow will be non-directional. In
addition, there should not be any metabolic heat generation in burned skin either. Thus, the only
heat transfer that would occur is the ”passive” heat transfer due to conduction, as seen in Equation
1. The experimental setup would use heated pads to create a temperature gradient across the skin.
The directionality of this heat flow can be calculated by orienting the pads in different directions,
as shown in Figure ??, to calculate the heat flow.
Figure 1: Proposed Experiment Setup [15]
A thermal image (such as a thermograph) would show the heat distribution in the skin. Any
directional flow other than that being induced by the heating pads, would be due to blood flow, and
thus would imply that the skin at that point is healthy.
4
Critique of Measurement and Calculation Methods
Over the course of the literature review and analysis, there were some situations in which huge variations in the estimated values of certain parameters were encountered. These variations, sometimes
differences of several orders of magnitude, were reviewed, and the studies that yielded them carefully analyzed and the methods and assumptions that yielded these differences critiqued. In such
situations, there are multiple reasons that could have led to the studies yielding differing values.
The major causes for the differing values are identified and discussed below.
4
• The experimentation process introduces the first area that could cause differing values. In any
experiment, there are different types of data that are taken. The first classification is the type
of quantity being measured, or measurand. An experiment generally has a goal of obtaining
data about at least one measurand and its response to specific stimuli, which is determined by
adjusting the variables of the experiment. In addition to a primary measurand, there are often
also secondary measurands, which are measured quantities that vary in response to the changes
in the primary measurand. There are often quantities being calculated from the measurands
yielded during the experiment as well. These calculated quantities are termed results. These
results must be calculated from the data yielded in the measurands, as well as a supposed
relationship between the two quantities. The use of results calculated from an experiment
creates a couple areas that cause problems. For example, if the goal of an experiment is to
measure the heat rate change due to conduction in a given situation, and the method of doing
so is taking temperature readings at certain points and then using Fourier’s Law to calculate
the heat rate change, the primary measurand is the temperature that is being recorded. The
heat rate that is being calculated is the result. In the case of the example that relationship
that is assumed is that of Fourier conduction.
The assumptions made during the experimentation process is of the main areas that can cause
differences in the differing values. In every experiment, and the subsequent process of deriving
data from the experiment, there are a multitude of assumptions that are made during the
course of the experiment itself and the resulting calculations. These are listed and explained
below, and will the throughly examined during the course of the literature analysis.
– First, is the choice of relationship that is assumed to relate the quantities. In proposing
a relationship, knowledge of the subject matter is required. If the subject matter is well
understood, then then it is usually easy to find the correct relationship that is to be used.
In the case where the subject matter is not fully understood, assuming the relationship can
be a problem, because there is the added doubt about whether the proposed relationship
is the correct relationship. If an incorrect relationship is assumed, then the calculated
results may not be correct.
– Another area where assumptions can cause differing values is the assumptions made during
the design of the model that is to be used in the experiment. When an experiment is being
designed, a process muse be created to accurately model whatever phenomenon is being
investigated. During this process, assumptions are made about how the phenomenon
behaves. These assumptions range from simple ones assuming the validity of a given
equation, such as the Pennes Bioheat equation, to assumption of initial conditions and
other values, such as the density of blood.
– A third type of assumption that could be of concern is in the use of mathematical analysis
to calculate a parameter as opposed to experimentation. Using a mathematical calculation
to determine the various parameter values is extremely dependent on the assumptions
made in creating the model and initial values used in the calculations. On the other
hand, experimental determinations are subject to variations in the surroundings and
ambient conditions which could affect the resulting values. Our literature critique will
address these concerns where possible and attempt to determine what effect, if any, these
conditions and assumptions had on the determination of parameters.
• Another area that could explain the differing values is uncertainty in experimental measurements. In any experiment where data is taken, instrument uncertainty is present. Every
measuring instrument has an uncertainty associated with it. This uncertainty is caused by the
5
degree of precision that the instrument is able to record data to. In experiments with multiple
pieces of equipment, such as the majority of the experiments that this analysis focuses on, the
uncertainty associated with all of the measuring instruments must be considered.
• In addition to previously noted causes of the differing values found during the course of the
literature analysis, there are a few complexities inherent to dealing with bioheat transfer. The
first, and arguably one of the biggest issues in the area of bioheat transfer, is that of finding
an appropriate model to explain bioheat transfer in humans. Heat transfer in tissues is a combined effect of conduction, convection, rediation, metabolic generation, evaporation and phase
change [1]. To accurately model the total effect of heat transfer in human tissue, it is necessary
to decouple and separately explain each component. This leads to the problem of developing
an accurate model to use. As discussed previously, the Pennes bioheat equation is the most
commonly used equation that separates the total effect of heat transfer into three distinct components: a conductive term, a metabolic generation term and a perfussive term [2]. The Pennes
equation has been improved by many groups, (Chen, Chato and Holmes, et al. and Weinbaum
and Jiji et al. for example) however, the original Pennes equation is still the most widely used
equation due to its simplicity and the relatively small impacts that further development yields.
• Another area that causes problems is the difference between in vivo and in vitro measurements.
In vitro measurements have a few inherent problems. First, the problem of certain effects,
such as perfusion, only appearing in vivo was discussed earlier. Second, in vitro measurements
made using tissue must be done very carefully due to the sensitivity of tissue to ambient its
surroundings. Improper handling of tissues can greatly affect the tissue, rendering unsuitable
for use in experimental parameter determination [1]. Additionally, in in vitro experimentation,
it is very hard to experimentally control and determine fluid flows and local metabolism in the
tissue [1].
• Locational differences in tissue structure and therefore the material and thermal properties
that accompany it also can cause problems. For example, in different areas of the body, the
tissue structure varies to suit the purposes of that organ or area. The area that a tissue is
from also influences the properties and processes that take place in that tissue, making specific
determination of properties that can be generalized throughout the body difficult.
• Finally, one last issue in experimental determinations of tissue properties is the sensitivity
of the measuring instruments to the surrounding conditions. For example, in many cases,
temperature measurements are made using thermocouple probes inserted into the skin. These
probes are very sensitive to surrounding conditions. Therefore, the mechanical trauma that
generally accompanies the insertion of the probe into the tissue can cause major variations in
the surrounding area, such as blood pooling, and can significantly alter the results [1].
The literature analysis will attempt to comment on and critique the effects that these previously
stated factors may have had on the estimation of parameter values.
5
Literature Analysis
A comparison of the values that were gathered from the various studies and papers is presented
below in Table 1 in Section 5.1. Many of the values that are presented, such as the values for the
6
density and specific heat of blood and the skin conductivity coefficient are fairly standard values
that are agreed upon in the field of bioheat transfer. These values are presented to give a representative sampling of the range of accepted values. Of note however, is the fact that while most
000
values are in a similar range, and seem to agree with each other, there are two quantities, qmet
and
ωb for which greatly differing values were observed. In the cases where the studies yielded values
for these quantities, a special interest is taken to understand how these values were derived and any
assumptions behind them.
Henriques and Moritz
Henriques and Moritz published a series of papers titled Studies of Thermal Injury[6] in 1947 . This
series was the one of the first to to investigate burns in depth. In the study, the authors applied
flowing water, heated to different temperatures, over porcine (pig) skin. Their goal was to measure
the exposure time necessary to cause burns. From their work, Henriques and Moritz developed a
single value, Ω, calculated through an Arrhenius integral, to quantify the damage done based on the
temperature and exposure time, and categorize the different level of burns. While the study did not
yield any data or values that were used in this report, Studies of Thermal Injury was a milestone in
research relating to burns and tissue damage, and the Arrhenius integral and accompanying damage
parameter is the basis of much of work that is being done today [8] [9] [10] .
Jones
The study by Jones [7] investigated the use of thermal imaging in medicine, including in an efforts to
detect burns. Their primary goal was to asses what differences in emissivity, if any, existed between
healthy and burnt skin, using infrared thermal imaging. Using thermographs of healthy and burnt
skin, the author was unable to discern an innate difference in emissivity between the two types of
skin. However, the author did find that there were other differences observed between healthy and
burnt skin, and thus suggested that thermal imaging would still be useful for other uses regarding
the investigation and treatment of burns. In addition, during the course of explaining some of the
aspects of bioheat transfer that were discussed earlier, the study presented a range of values for the
diameter of capillary vessels, which is presented in Table 1.
Ng
The study by Ng, et. al. [8] used heating pad analysis to determine the required temperature/time
combination to necessary to cause certain types of burns. Our proposed study used the same type of
heating pads, however, our goal is simply to detect burns, while their study was focused on finding
the appropriate combination of factors to prevent certain types of damage. However, their study
used many properties which were taken from the study by Diller and Hayes [14].
Diller and Hayes
The study by Diller and Hayes from 1983, titled A Finite Element Model of Burn Injury in Blood
Perfused Skin [14], is one of the most referenced studies in the area of bioheat transfer in the skin.
Their study used a finite element model to numerically solve for the Arrhenius integral to determine
tissue damage due to burns. In their study, they compiled a list of the quantities necessary to solve
the Pennes equation and their values. While they do not provide the model or details on how they
derived these values, their paper is often cited [5] [8] [11] qand the values that they used are accepted
as the standard values for these quantities. Their data is shown Table 1 below.
Upreti
The paper by Upreti and Jeje proposes and evaluates a method to determine the blood flow rate
7
Table 1: Values taken from Literature
Parameter [Units]
W
kd [ mK
]
kg
ρb [ m3 ]
J
cb [ kgK
]
ωb [s−1 ]
Ta [K]
Tt [K]
W
000
qmet
[m
3]
D [m]
W
]
kep [ mK
Jones
Upreti
0.3
1040
3300
2.175(10−4 )
310
Wang
1050
3800
36.97
Yang
0.5*
1000
4200
2.6(10−3 )
310
306
2148
Diller
1100
3300
4(10−4 )
310
307
100-300
3 − 10(10−6 )
0.21
*: Combined Conductivity for dermal and epidermal layers
and heat generation the arm. Essentially, they solved the improved Pennes equation developed by
Chen, Chato and Holmes for radial coordinates (Equation 2) over an entire limb, where they assumed the tissue was homogeneous throughout the limb and the entire limb could be modeled as a
cylindrical control volume. They used the condition that the conductive portion of heat loss ∇· k∇T
is completely due to convective heat loss over the surface of the limb h(T − T∞ ) where was known
and T and T∞ could be measured. For the material and thermal properties of blood and skin, they
compiled a list of the values from various sources. These values are well within the accepted range
of sources, and thus were deemed acceptable to be used. Using this approach, they found values for
000
and ωb , using an iterative numerical integration of the differential equation. These values are
qmet
shown in Table 1.
Wang
The paper by Wang, et. al.[12] deals with the cooling of brain temperature through the use of a
colling device situated on the carotid artery. While the content of the study is not important to this
analysis, their paper used a couple values, which they had compiled from other papers values shown
in Table 1.
Yang
The study by Yang and Liu [13] focused on the effects of plaque in the carotid artery. While the
subject is not relevant to this analysis, the study is important because, in their initial calculations,
000
they calculated values for qmet
and ωb . Their approach to calculating these values was to solve the
Pennes equation using the standard values for the thermal properties of skin, which are shown in
Table 1. They assumed that the tissue was composed of large blood vessels, yielding the results for
000
qmet
and ωb shown in Table 1.
5.1
6
Parameter Values taken from the literature
Analysis
In this section, three sets of calculations are explained and performed. First, The heat loss along a
single capillary is analyzed in an effort to prove that blood flow creates a directionality of heat flow.
Second, each of the terms in the Pennes equation are calculated to find their impact in the overall
heat transfer in tissue. Finally, an example of the proposed experiment is shown and analyzed.
8
6.1
Heat Flow along a Vessel
To prove that blood flow creates a directionality in the heat flow, the amount of heat lost between
blood entering and exiting the capillary due to advection was examined. The control volume of a
single capillary was modeled as a pipe, as shown in Figure 2.
blood flow
D
T1
T2
dx
Figure 2: Control Volume for blood flow through a capillary
qint = ṁc(Tout − Ti )
(4)
Clearly, the outlet temperature of blood is necessary to find this value. The outlet temperature, To
can be derived using two equations, the simplified steady flow equation[3]:
∆u = ṁcdT
(5)
and the convective heat transfer equation[3]:
dq = hP (Ts − T )dx
(6)
where ∆u is the change in internal energy in a control volume, ṁ is the mass flow rate of the blood, c
and h the heat capacity and convection coefficient of blood, respectively, Ts the surface temperature
of the capillary and P the perimeter of the control volume. Using the relation
∆u = −dq
(7)
ṁcdT = −hP (Ts − T )dx
(8)
or
2
and making the substitutions ṁ = ρπD4 U where D is the vessel diameter and U is the velocity of
blood, and P = πD, yields the equation:
dT
h
=−
dx
TS − T
ρcU D
(9)
Solving the differential equation with the initial condition of T (0) = Ta , or the arterial temperature
of blood, and the assumption that the surface temperature of the vessel is the tissue temperature,
Tt , yields:
−4hx
T (x) = Tt + (Ta − Tt ) exp
(10)
ρcU D
9
Thus, the outlet temperature of the capillary, at x = Lcap is:
Tout = T (Lcap ) = Tt + (Ta − Tt ) exp
−4hLcap
ρcU D
(11)
Using this term in Equation 4, and making the same substitutions used earlier, namely for ṁ and
Ti , yields:
−4hLcap
ρπD2 U
c[Tt + (Ta − Tt ) exp
− Ta ]
(12)
qint =
4
ρcU D
which can be rearranged to
qint = (
ρπD2 U c
−4hLcap
)(Tt − Ta )(1 − exp
)
4
ρcU D
(13)
to yield the final expression for the amount of heat lost along a blood vessel.
6.2
Pennes Flow Heat Transfer Terms
The Pennes Equation, Equation 1, has three terms, which represent heat transfer due to conduction,
blood perfusion and metabolism. The sum of these three terms is the total amount of heat transfer
in a given volume of tissue.
6.2.1
Conductive
The first value calculated was the heat transfer due to conduction in a control volume of skin, as seen
in Figure 3. The control volume in this case is a cube of side length L with a thermal conductivity
of k, which represents a layer of dermal tissue.
T2
T1
k
L
Figure 3: Control Volume for a Layer of Dermal Tissue
The equation for this form of heat transfer is:
qcond = V ∗ (∇(k∇(∆T ))
10
(14)
where T1 and T2 are the temperatures at the entrance and exit of the control volume respectively,
V is the volume of the control, and k is the thermal conductivity in watts per meter kelvin. This
equation can be approximated to:
k
(15)
qcond = V ( 2 (∆T ))
L
6.2.2
Perfusive
The second mode of heat transfer that was examined was the amount of heat transfer due to
perfusion. This represents the amount of heat transfer due to blood and nutrient flow out of the
capillaries and into the capillary bed and skin. The control volume investigated was identical to
that of the previous section, namely a cube with sides of length, L.
qperf = V ((ρc)b ωb (Ta − Tt ))
(16)
where ωb is the blood perfusion rate for the tissue, ρ and c are the density and heat capacity,
respectively, Ta is the arterial temperature of the blood and Tt is is the temperature of the local
tissue.
6.2.3
Generated Heat
The generated heat is the amount of heat generated through natural body processes. In this case,
the focus is on the heat generated by metabolism, qmet .
000
qmet = V qmet
(17)
000
qmet
is the basal metabolic volumetric heat generation rate and the control volume is taken
where
to be the same as in Sections 6.2.1 and 6.2.2.
6.3
Proposed Experimental Setup
The heat transfer associated with the proposed experiment setup is shown below.
The heat transfer due to conduction in a 2-Dimensional surface is given by the equation
qpad = Sk k(Tp − Ts )
(18)
where Tp is the temperature of the pad, k is the thermal conductivity of the mater the pad is on,
and Sk is the shape factor for the given geometry. In this case, for a circular pad on a semi-infinite
medium, as the epidermis is being modeled, as shown in Figure 4, is Sk = 2d where d is the diameter
of the pad. Assuming the pad is 0.05 m in diameter and Ts = Tt , the equation is:
Th
Tc
q in
q out
Figure 4: Picture of the Proposed Thermal Pad Setup
qpad = (2)(0.05m)k(Tp − Tt )
11
(19)
Table 2: Blood Vessel and Blood Material Properties
Parameter
D
U
ρ
c
Value
8 × 10−6
0.025 m
s
kg
1060 m
3
J
3770 kgK
Table 3: Assumed Quantities
Parameter
L
∆T
h
Lcap
Dp
7
Value
0.005m
1K
1400 mW
2K
0.005m
0.05m
Results
Section 7.1 shows the other parameter values used in this analysis. In addition to the previously
stated values derived from the literature, there were other values that were collected from various
sources. Some of the parameters that were necessary had multiple values found for that quantity.
In those cases, Table 4 shows the minimum and maximum values found for those parameters.
Once values for the various quantities were determined, minimum and maximum values for the heat
rate were calculated using the minimum and maximum parameter values
7.1
Values Used in Calculations
In addition to the values previously noted, a few values for the properties of blood and structure of
blood vessels were taken from a Human Physiology Textbook to use as a general reference. Those
values are shown in Table 2. Some values were also assumed, such as the chosen control volume and
vessel dimensions. These values, shown in Table 3 are primarily useful only for estimates, and thus
in this case, there is no need for exact values. For quantities with more than one value presented
in Tables 1, 2 and 3, the minimum and maximum values were compiled into a single table, shown
below.
7.2
Heat Flow along a Vessel
To find the minimum and maximum values for qint , the values from the previous section were used
in Equation 13. This yielded:
min
qint
= 2.33 × 10−3 mW
(20)
and
max
qint
= 2.72 × 10−2 mW
12
(21)
Table 4: Minimum and Maximum Parameter Values
Parameter [Units]
W
]
kd [ mK
kg
ρb [ m 3 ]
J
cb [ kgK
]
−1
ωb [s ]
Ta [K]
Tt [K]
W
000
qmet
[m
3]
D [m]
W
kep [ mK
]
7.3
7.3.1
Jones
Upreti
0.3
1040
3300
2.175(10−4 )
310
Wang
1050
3800
36.97
Yang
0.5*
1000
4200
2.6(10−3 )
310
306
2148
Diller
1100
3300
4(10−4 )
310
307
100-300
3 − 10(10−6 )
0.21
Pennes Flow Heat Transfer Terms
Conductive
Using the values from the previous section in Equation 15 yields
min
qcond
= 1.5mW
(22)
max
qcond
= 2.5mW
(23)
and
for the minimum and maximum values of the conductive component of the Pennes equation.
7.3.2
Perfusive
The minimum and maximum values of the perfusive term of the Pennes equation is based on Equation
16, and are:
min
qperf
= 0.66mW
(24)
and
max
qperf
= 4.5mW
7.3.3
(25)
Metabolic
The metabolic heat generation rate of human tissue is given by Equation 17. Using the values found
previously, the minimum and maximum values for qmet are:
min
qmet
= 4.62 × 10−3 mW
(26)
max
qmet
= 0.269mW
(27)
and
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7.4
Proposed Experimental Setup
Using the proposed experimental setup, as described in Section 6.3, the heat transfer rate is given
by Equation 18. Assuming a 5◦ C temperature difference, the minimum and maximum heat rates
are given by:
min
qpad
= 105mW
(28)
and
max
qcond
= 126mW
(29)
For easy reading, the values in Sections 7.2-7.4 have been compiled into one table, Table 5, below:
Table 5: Summary of Minimum and Maximum Calculated Heat Rates
Quantity
Symbol Minimum [mW] Maximum [mW]
Internal Flow along a Vessel
qint
2.33 × 10−3
2.72 × 10−2
Pennes Conductive
qcond
1.5
2.5
Pennes Perfussive
qperf
0.66
4.5
Pennes Metabolic
qmet
4.62 × 10−3
0.269
Proposed Setup
qpad
105
126
8
Conclusion
The goal of this analysis was to determine if it is possible to detect burned skin through thermal
imaging by detecting anisotropies in the skin. The viability of this proposal depends on the quantities of qperf and qpad , the heat rate due to blood perfusion and the conductive heat rate in the
skin when the temperature is increased using heated pads. If there is a measurable difference in
the total heat rate with and without the perfussive heat rate, then this proposal should be successful.
The first step in determining whether the proposed experiment would be viable was to find values for the various quantities necessary in calculation of the heat rates. This was done through
a literature analysis and evaluation of the values found, to explain any discrepancies from various
studies. Once suitable values were found, the minimum and maximum possible heat rates were found
for the various modes of heat transfer. To find best-case scenario, the maximum possible perfusive
heat rate was compared to the minimum possible heat rate for the proposed setup. The ratio of
max
min
qperf
yields a value of 0.043, or a 4.3% difference. Similarly, the worst-case scenario is given by
/qpad
the ratio of the minimum possible perfussive heat rate to the maximum proposed experiment heat
min
max
rate , or qperf
/qpad
. This yields a value of 0.0052 or 0.52%.
While we feel that this field merits further research, and would like to begin testing our proposed
idea, the low values of the ratio of perfusive heat rate to the proposed experiment heat rate does
pose a problem. The low values, at best about a 5% difference, at worst 0.5%, means that any slight
variations in the testing procedure or uncertainty in the heat rate measurement could skew the results of the test. We think that this problem could be solved by refining the proposed experiment to
measure temperatures rather than heat rates. We propose to use two pads, that could be alternately
heated or cooled, attached to thermocouples. The pads would be alternately heated or cooled for
a given period of time, with one pad being hot while the other is cold, and vice-versa. Due to the
temperature gradient created by the two pads, heat should flow from the hot one to the cold one.
In the case of isotropic heat flow, the heat flow rate from one pad to the other should be the same
in one direction as another. Thus, if there is no blood flowing, and thus no directional heat transfer,
14
the pads will heat and cool at the same rate. However, if blood is flowing, there will be added heat
flow in a given direction, and thus there will be greater resistance of conductive heat flow in that
direction. In other words, if blood is flowing, implying healthy skin, one pad will cool faster than
the other when they are both exposed to an alternating heating and cooling cycle.
Thus, we believe that the information from this shows that it should be possible to detect burns
through analyzing heat transfer and in particular, detecting anisotropies. We believe that with our
refined proposed experiment, we should be able to detect the directionality of heat transfer, and
thus blood flow in human skin, which would indicate whether the skin is burned or healthy.
15
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