1
Cutaneous Temperature Response after Infrared
Radiation
Ken Steffen Frahm, B.Sc.
Department of Health Science and Technology, Aalborg University, Denmark
Abstract—In human pain research lasers have often been used
for delivering transient-like noxious stimuli at the skin surface.
In this paper a model was developed which simulates the heat
transfer in the skin, into the layers where the heat sensitive
nociceptors are placed.
The model was based on the bio-heat equation (pure conduction).
For creating and solving the model, finite element modelling and
COMSOL multiphysics were used. For the validation a 100 W
CO2 laser was used and to monitor the surface temperature
an infrared (IR) camera was used. The laser was set to have
an output of 3.7 W and five stimuli durations were tested (150
ms, 200 ms, 250 ms, 300 ms, and 350 ms). The experiment was
conducted on a single male subject in the dorsal and volar left
forearm and left palm. A VAS score was registered to monitor
the degree of pain.
The experiments showed an agreement between the model and
the IR recordings. The best agreement was found in the longest
solution durations. Common for all the tested skin sites; the
dispersion of the maximum temperatures was greater in the
IR recordings than the theoretical. Typically the IR recordings
maximum temperatures at each recording site were spread across
approx. 20 ◦ C whereas the theoretical only was spread across
approx. 10 ◦ C. (IR: Min temp.: 42-51 ◦ C / Max temp.: 60-73
◦
C - Theoretical: Min temp.: 48-52 ◦ C / Max temp.: 56-61 ◦ C)
The spatial dispersions of the temperatures were greater in the
theoretical data than the IR recordings.
The model showed that due to different thermal properties of
the skin (different thickness of the epidermis) in the forearm and
palm, the heat transfer is different. This suggests that the reason
for the response latency of nociceptors in glabrous skin partly
can be found in the thermal properties of the skin.
Index Terms—CO2 laser, bio-heat equation, heat transfer,
nociceptors, pain, finite-element, COMSOL multiphysics.
I. I NTRODUCTION
T
HE CO2 laser has been used in clinical experiments for
some time. The ability of the laser to deliver a large
amount of energy in the most superficial layers of the skin
(penetration depth approx. 20 µm) is very useful in studies.
Furthermore the laser is not only able to deliver heat in the
most superficial layers it is also able to do so in a transient
like manner, which means that the effect from the laser
has a very short rise time. Lasers are also used in several
clinical applications such as dermatology and surgery, e.g.
retinal surgery, but this paper will focus on lasers in skin pain
research.
Receptors which sense noxious (painful) stimuli are called
nociceptors. Heat sensitive nociceptors are free nerve endings
placed in and close to the epidermal ridge located between the
epidermis and dermis, however, a few are placed some distance
from the epidermal ridge. [Bromm and Treede, 1983] [Tillman
et al., 1995] [Martini, 2004]
Treede et al. suggest that different types of heat sensitive
nociceptors exist, dependent on skin type. [Treede et al., 1995]
[Treede et al., 1998] Treede et al. 1998 divide the nociceptors
(A fibres) into two groups. Type I has high thermal threshold
(> 53 ◦ C), can be found in both glabrous and hairy skin,
has long response latency (5 ± 5.4 sec.), and do not adapt to
stimuli. Type II has a thermal threshold at 46 ◦ C, can only be
found in hairy skin, has short response latency (0.22 ± 0.18
sec.), and show some adaption to stimuli. [Treede et al., 1998]
Besides A fibre nociceptors, C fibres exist, these have lower
conduction velocities than A fibres and can be found in both
hairy and glabrous skin. [Treede et al., 1995] [Treede et al.,
1998] The C fibres in hairy skin had a response latency of 0.1
± 0.25 sec. [Treede et al., 1995] When Treede et al. ( [Treede
et al., 1995]) investigated A fibres in hairy and glabrous skin
they also investigated C to see if any different response latency
could occur to different skin thickness. They found that the
response latency of C fibres was a bit higher in glabrous skin
(0.1 sec. in hairy and 0.25 sec. in glabrous skin). [Treede et al.,
1995]
This paper will develop and validate a model based on the
bioheat equation for the skin heat transfer after laser irradiation. The model will be used for investigating the heat transfer
from the surface into the layers where nociceptors are found.
Since the thermal conductivity is lower in epidermis than in the
dermis it is assumed that in skin with thick epidermis a noxious
stimulus will not be perceived as painful as in skin with thinner
epidermis [Gowrishankar et al., 2004] [Jiang et al., 2002]
[Wilson and Spence, 1988] [Seteikin and Krasnikov, 2006].
Furthermore will the perception of the noxious stimulus be
delayed due to the low thermal conductivity in the thicker
epidermis. Since the epidermis generally is thicker in glabrous
skin, the different response latencies (of A fibres) found in
glabrous and hairy skin might depend on the thickness of
epidermis.
The above leads to following aim
Aim
The aim for this paper is
• To develop and validate a heat transfer model for the
superficial skin layers after infrared radiation
• To test the following hypothesis
The different response latencies in heat sensitive nociceptors
in glabrous and hairy skin are due to different thermal
properties.
2
II. M ETHODS AND
MATERIALS
Laser
100 W Synrad 57-1 CO2 laser was used (10.6 µm).
The laser was controlled using the Synrad UC-1000 laser
control and a simple NI LabView application.
Experiments were conducted on six sites on the forearm and
palm to validate the model and test the hypothesis:
1) Dorsal side of the forearm, in the hairy skin, 6 cm from
the elbow
2) Dorsal side of the forearm, in the hairy skin, 6 cm from
the wrist.
3) Volar side of the forearm, in the hairy skin, only a few
hairs were present, placed 8 cm from the elbow.
4) Volar side of the forearm, in the hairy skin, only a few
hairs were present, placed 11 cm from the wrist.
5) Volar side of the forearm, in the hairy skin, only a
few hairs were present, placed 2 cm from the wrist.
Measurement deliberately placed on a site with a large
amount of underlying vessels.
6) Palm, glabrous skin at the root of the thumb.
7) Palm, glabrous skin, in the middle, in between the two
large lines.
Of the seven sites above, six were modelled and validated
(site 5 was left out).
All the sites and the model were tested during five stimulus
durations (150 ms, 200 ms, 250 ms, 300 ms, and 350 ms). The
order of shoot time was randomised using www.random.org.
[Haahr, 2007] Each experiment was randomised repeated at
least three times.
The target temperature at the skin surface was set to approx. 60
◦
C. This should ensure the temperature at the thermal nociceptors above the threshold for a noxious stimulus [Tillman et al.,
1995]. The skin temperature will only rise to this temperature
very briefly to prevent any tissue damage. [Arendt-Nielsen
and Chen, 2003] The output from the laser was set to 3.7
W which should ensure the target temperature being reached
within the maximum stimuli duration. To prevent any effects
from previous stimuli a cool time of 40 s inbetween stimuli
was chosen. An Ophir power meter was used before and after
each experiment to ensure that the power level was correct.
A
Infrared camera
The surface temperature was monitored using an Agema
Thermovision 900 infrared camera using the Agema900 kit
from Automation Technology GmbH upgrade package and
the ThermaCAM Researcher Pro software. The spectral wave
length of the IR camera is 2-5.4 µm which is different from
the 10.6 µm of the laser radiation. This ensures that the camera
can record during laser stimuli and will not be ’blinded’
from the laser radiation. [FLIR systems, 2005] The maximum
temperature was measured at the end of each stimulus.
Pain measurement
To see whether any nociceptors were activated a Visual
Analog Scale (VAS)-score was registered after each stimulus.
The VAS score was anchored at 0 - no sensation; 5 - pain
threshold; 10 - maximum pain.
Experimental setup
The laser beam from the CO2 laser is reflected in such a
manner that the beam is perpendicular to the object (forearm
and palm). The IR camera is installed such that the line of
sight is as close to perpendicular to objects surface, see figure
1. The subjects arm rested on a table to ensure little or no arm
movement during the experiments.
Infrared camera
Scanner head
100W CO2 laser
6
2
7
5
4
1
3
Fig. 1. The figure illustrates the experimental setup used. The lasers scanner
head is set so the laser beam is perpendicular to the object (forearm and
palm). The infrared camera is installed such that its line of sight is as close
to perpendicular to the surface as possible. The numbers indicate where the
sites are placed.
High resolution ultrasound scanner
To create acurate models of the epidermis and dermis
high resolution ultrasound scans were conducted on the sites,
mentioned above. The scanner used was a Derma Scan, 50
Mhz probe with a resolution of 25 x 60 µm and a penetration
depth of 3 mm [Cortex technology, 2007]. These specifications
are sufficient for the use in this paper. The thickness of both
layers could be extracted from the data.
Modelling
A model to investigate skin heat transfer was developed.
The modelling was based on the bioheat equation:
ρc[
∂T (r, t)
] = ∇[k∇T (r, t)]+ρb cb wb [Tart (r, t)−T (r, t]+S(r, t)
∂t
(1)
where ρ is the tissue density, c is the specific heat, ρb is
the density of blood, cb is the specific heat of blood, wb is
the tissue average volumetric blood perfusion, T is the tissue
temperature, Tart is the temperature of the arterial blood, and
S is heat from the laser. [Welch and van Gemert, 1995a]
3
Some assumptions were made considering the modelling of the
heat transfer. First it was assumed that all of the heat transfer
was due to conduction. Hence, this paper only focuses on the
most superficial layers of the skin where no vessel are present,
the model does not include heat transfer by convection.
Secondly it was assumed that no heat was transferred due
to radiation, since the time durations were short. [Welch and
van Gemert, 1995a] Following these assumptions the bioheat
equation reduces into:
∂T
ρC
− ∇(k∇T ) = S(r, t)
(2)
∂t
The constants used in Eq. 2 can be seen in Table I
Constant
ρ
c
k
Epidermis
1200
3600
0.21
Dermis
1200
3800
0.58
finite element model of skin heat transfer was developed,
afterwards experiments using a CO2 laser and an infrared
camera were conducted to validate the model.
A
Experimental results
The laser outputs measured before and after the experiment
are displayed in table III. The power is averaged from measurements before and after the experiment.
Site
1
2
3
4
6
7
Unit
[kg/m3 ]
[J/(kg·K]
[W/(m·K)]
Laser power [W]
3.8
3.75
3.7
3.55
3.65
3.7
TABLE III
Measured laser output at each site
TABLE I
Constants used for modelling
To model the heat transfer, described by Eq. 2 a finite
element model was used. For the creation and computation of
the finite element model Matlab and COMSOL Multiphysics
3.4 was used. It was assumed that the skin profile was
symmetrical around the point of irradiation, and makes a 2D
axial model sufficient for modelling.
The laser beam had a Gaussian profile and the absorption
in the tissue can be expressed using Beers law. [Welch and
van Gemert, 1995b] The heat source is abbreviated Q and the
following expression was used to model the laser heat source
−r2
1
) [W/m3 ] (3)
Q = Pin ·µa ·exp(µa ·z)·( √ )2 ·exp(
2 · σ2
σ 2π
Where Pin is the power from the laser [W], µa is the
absorbtions coefficient [1/m], σ is the standard deviation of
the Gaussian function which describes the beam profile [m]
and r and z are the 2D axial coordinates [m] (radius and depth,
respectively). Note that at the surface z is zero and decreases
with depth.
The different constants used in Eq. 3 for the modelling can be
seen in Table II
Constant
Pin
µa
σ
Value
3.7
50000
0.0035
Unit
[W]
[1/m]
[m]
TABLE II
Constants used for modelling
The recordings made with the infrared camera and VASscore are displayed in table IV. The amount of energy is the
stimulus duration (150 ms - 350 ms) multiplied by the laser
output, see table III.
Site
1
2
3
4
6
7
Epidermal
thickness [µm]
105.67
105.67
105.67
105.67
105.67
86
86
86
86
86
46.33
46.33
46.33
46.33
46.33
82
82
82
82
82
209
209
209
209
209
109.67
109.67
109.67
109.67
109.67
Energy [J]
0.57
0.76
0.95
1.14
1.33
0.56
0.75
0.94
1.12
1.31
0.56
0.74
0.93
1.11
1.30
0.53
0.71
0.89
1.10
1.24
0.55
0.73
0.91
1.10
1.28
0.56
0.74
0.93
1.11
1.30
Mean temp [◦ C]
± SD
50.14 ± 1.26
54.92 ± 0.97
59.22 ± 1.98
63.46 ± 3.10
67.88 ± 3.77
47.02 ± 1.20
51.02 ± 0.90
54.68 ± 1.40
57.92 ± 0.95
61.5 ± 2.28
51.57 ± 0.83
56.2 ± 1.25
62.33 ± 2.10
67.67 ± 2.29
73.2 ± 1.21
46.58 ± 0.29
51.3 ± 0.87
55.92 ± 1.59
61.62 ± 1.41
63.22 ± 1.85
42.57 ± 2.31
46.47 ± 1.45
50.77 ± 1.99
56.37 ± 1.48
61.67 ± 1.60
46.3 ± 1.47
51.9 ± 1.25
57.53 ± 1.56
60.93 ± 0.59
65.33 ± 1.89
Mean VAS
± SD
3.1 ± 1.893
4.7 ± 1.5
6.2 ± 2
6.9 ± 0.29
6.8 ± 1.32
1.3 ± 2.11
3.4 ± 2.70
6.2 ± 0.27
6.8 ± 0.45
7.5 ± 0.5
4.33 ± 1.15
6.17 ± 0.29
6.5 ± 0.5
7.67 ± 0.58
8 ± 0.5
4.5 ± 1.73
6.6 ± 0.96
6.2 ± 1.10
8.6 ± 0.65
9.2 ± 0.45
1.8 ± 2.31
2.5 ± 3.02
2.9 ± 3.01
7 ± 1.62
7.3 ± 1.60
0.67 ± 0.29
3.83 ± 2.47
5.33 ± 1.15
6.83 ± 0.29
7.33 ± 0.76
The initial temperature for the skin surface was set to 34 ◦ C
in the forearm and 31 ◦ C in the palm, based on IR recordings.
TABLE IV
IR recordings and VAS scores from the six sites tested.
Data processing
After the IR recordings were finished the spatial temperature
profiles at each site were averaged and compared to theoretical
data.
As seen in table IV the temperatures after 150 ms stimulus
ranged from 42 to 51 ◦ C and the temperatures after 350 ms
stimulus ranged from 61 to 73 ◦ C. The difference between the
minimum and maximum temperature in each site ranged from
approx. 15 ◦ C to approx. 20 ◦ C.
The VAS scores ranged from 0.67 to 4.5 after 150 ms and
III. R ESULTS
4
from 6.8 to 9.2 after 350 ms. Of the VAS scores seen in table
IV only sites 1 and 3 were blinded during the experiment.
The spatial dispersion of the surface temperature at site 4 is
plotted in figure 2. At a distance of 7.5 cm from the centre
of the beam the temperature had declined approx. 25 ◦ C.
At this site there was approx. 17 ◦ C between the maximum
temperature after 150 ms stimulus and 350 ms stimulus.
Site 4 mean IR recordings
65
Temperature [ºC]
150 ms
60
200 ms
55
250 ms
300 ms
50
350 ms
45
40
35
30
0
0.2
0.4
0.6
r (radius) [cm]
Fig. 2. The figure illustrates the surface temperature at site 4 at the end of
different stimuli. Five different stimulus duration is plotted, see the legend.
The maximum temperature is spread across approx. 17 ◦ C.
On the filmstrip in figure 3 images from the IR camera is
shown. Each image was recorded at the end of stimulus. The
spatial surface dispersion of heat can be seen on the images
(the half circle in the top of the images is 1.5 cm in diameter).
Fig. 4. The figure illustrates the surface temperature and the spatial dispersion
of heat after each of the five stimulus durations, see the legend. r (radius) is
the distance from the axis of symmetry.
At the centre of the laser beam (r = 0 in figure 4) the
minimum surface temperature was approx. 52 ◦ C (after 150
ms) and the maximum surface temperature was approx. 60 ◦ C
(after 350 ms). At a distance of 6 mm from the centre the
temperature had dropped below 40 ◦ C.
Fig. 3. The figure illustrates the data from the infrared camera at the end of
each stimulus, recorded at site 4. The images show the spatial dispersion of
heat at the surface. The half circle seen in the top of the images is 1.5 cm in
diameter.
Modelling results
The results from COMSOL multiphysics is presented in the
following. Only data from site 4 and 6 is presented. The spatial
surface dispersion of heat can be seen on figure 4.
Fig. 5. The figure illustrates the modelling of the depth temperature, along
the axis of symmetry, and the spatial dispersion of heat after each of the five
stimulus durations, see the legend. z is the distance from the surface.
5
The depth plot, figure 5, clearly shows the transition from
epidermis to dermis at approx. 80 µm where the thermal
conductivity is almost tripled [Gowrishankar et al., 2004]
[Jiang et al., 2002] [Wilson and Spence, 1988] [Seteikin and
Krasnikov, 2006] and the heat is conducted more rapidly. The
temperature gradient along the z-axis is much higher in the
epidermis than the dermis. The average temperature gradient in
the epidermis is approx. -0.125 ◦ C/µm (10 ◦ C/80 µm) and the
equivalent in the dermis is approx. -0.025 ◦ C/µm (15 ◦ C/600
µm).
To compare the heat transfer in hairy and glabrous skin, (e.g.
site 4 and 6, respectively), plots were extracted from COMSOL
multiphysics which display the temperature vs. time at the
epidermal ridge, after a 350 ms stimulus.
Figure 6 displays the temperature dispersion at the epidermal
ridge at site 4 (epidermal thickness 82 µm). The maximum
temperature was approx. 48 ◦ C and was reached after 375
ms.
Fig. 7. The figure illustrates the temperature vs. time at the epidermal ridge
(depth 209 µm) in glabrous skin (site 6) after a 350 ms stimulus.
IV. D ISCUSSION
I
Fig. 6. The figure illustrates the temperature vs. time at the epidermal ridge
(depth 82 µm) in hairy skin (site 4) after a 350 ms stimulus.
Figure 7 displays the temperature dispersion at the epidermal ridge at site 6 (epidermal thickness 209 µm). The
maximum temperature was approx. 40 ◦ C and was reached
after 700 ms.
N this paper a finite element model for the heat transfer
in the skin was developed - and a hypothesis considering
different latencies of nociceptors, dependent on skin type,
was evaluated. In vivo experiments using a CO2 laser was
conducted to validate the model.
There was found to be good agreement between the model
and the IR recordings - the model was validated.
However, due to some faulty equipment the results did not
conclusive validate the model, but by taking the status of the
equipment into consideration an agreement could be found.
When comparing the surface plot from the IR recordings and
the model, one can see that the spatial dispersion of heat
is greater in the model than the IR recordings, see figure 2
and 4, this tendency was persistent in all recordings. This
disagreement can partly be explained by the state of the
scanner head. The lens had a large crack in the centre (Ø
approx. 1 cm). This crack made the beam uncollimated and
gave it a focus point approx. 20 cm from the lens, instead of
at infinity. The highly irregular structure of the crack made it
difficult to model, since the beam no longer had a Gaussian
profile. Changing the dispersion σ, describing the spatial beam
profile, would not have been sufficient to solve the problem,
since the crack changed both the shape and the intensity locally
within the beam.
By comparing figure 2 and 4 again it is also clear that the
maximum temperature in IR recordings is spread across 17
◦
C where in the model they are spread across 8 ◦ C, again
one reason for this is the equipment used, i.e. the CO2 laser.
The laser had an unexpected long rise time, probably due
to the general state of the laser, with a shorter rise time the
dispersion should decrease. It is difficult to evaluate whether
other factors could play a part in these disagreements, this
should be investigated when the errors, mentioned above, are
corrected.
6
The VAS scores do not conclusive demonstrate an agreement
between the VAS score and the thickness of epidermis. It
was expected that the thicker the epidermis, the lower the
VAS score. The VAS score in the glabrous skin (site 6 &
7) was not significantly lower than in the hairy skin. It was
expected that site 1 & 2 had higher VAS scores, than was
the case. One reason for this might be that the hair density
in site 1 & 2 (dorsal forearm) is higher compared to site 3
& 4 (volar forearm). However, the VAS score is a highly
subjective measure, and the experiments were only blinded
in site 1 and 3. Furthermore only a single subject was used
in the experiments. For allowing more precise interpretation
of the VAS scores, more subjects should be tested, and their
scores averaged. An even better solution would be to conduct
in vivo measurements of action potentials from nociceptors,
yet very difficult to do.
During the modelling some assumptions were made (rotation symmetry, no convection or radiation of heat, and the
epidermis and dermis can be assumed as a homogeneous
matter). The assumptions considering convection and radiation
is acceptable, due to short stimulus duration. [Vendri and Vos,
1957] Based on the ultra sound scans, the assumed rotation
symmetry in skin sites is considered acceptable. The scan
did not indicate large variations within short distances from
the scan site (<1 cm) as long as the skin type was the
same. The assumption considering the epidermis homogenity is considered being acceptable, even though in glabrous
skin the thicker epidermis is partly due to a thicker stratum
corneum. The stratum corneum, which has a low content of
water (because the stratum corneum is made up of dead cells)
compared to the rest of the epidermis, will not absorb the
radiation from a CO2 (10.6 µm) laser as well as in the rest of
the epidermis where the water content is higher. This might
increase the penetration depth of the CO2 laser, and should be
taken into consideration when conducting further experiments.
The thermal conductivity will also depend on the amount of
water in the tissue. [Cohen, 1977]
The values for the thermal conductivity of epidermis and dermis suggested by the litterature have some variation. [Wilson
and Spence, 1988] [Gowrishankar et al., 2004] [Jiang et al.,
2002] [Wilson and Spence, 1988] [Seteikin and Krasnikov,
2006] The used values for the thermal conductivity for epidermis (0.21 W/(m·K)) and dermis (0.58 W/(m·K)), from Wilson
and Spence ( [Wilson and Spence, 1988]) is quite far apart,
some litterature suggest the conductivities to be more similar
(higher for the epidermis and lower for the dermis). [Wilson
and Spence, 1988] [Gowrishankar et al., 2004] [Jiang et al.,
2002] [Wilson and Spence, 1988] [Seteikin and Krasnikov,
2006] Choosing other values will change the modelling result
and should be investigated when the laser have been repaired.
Based on the experiments the hypothesis cannot be rejected.
First of all because no response latency was measured. Yet
the subject reported some pain latency in glabrous skin, but
since these experiments were not blinded no conclusions can
be drawn from this. However, the model was validated and
by using the model to calculate the temperature profile over
time at the epidermal ridge in hairy and glabrous skin, some
comparison can be made. The results of this technique clearly
show a difference between the temperature profile of hairy
and glabrous skin, see figure 6 and 7. In site 4 (hairy skin)
the peak temperature was reached almost at the end of the 350
ms stimulus (375 ms), where the peak in site 6 (glabrous skin)
was reached after approx. 700 ms, a 425 ms difference. This
clearly suggest that different thermal properties in glabrous
and hairy skin can be part of the reason for any difference in
response latencies seen.
Treede et al. 1995 found a difference in response latencies of
C fibres in hairy and glabrous skin of approx. 150 ms. [Treede
et al., 1995] However, the results found in this study suggests
it to be longer (425 ms) and could further substantiate that
some of the difference seen in A fibres response latencies are
due to different thermal properties.
In future experiments more subjects should be tested using
more reliable equipment. Furthermore the hypothesis should
be tested using in vivo animal trials, e.g. rats or monkeys.
[Treede et al., 1998] [Treede et al., 1995] [Tillman et al., 1995]
In these experiments the action potential from the nociceptors
should be registered, if possible.
Future steps include testing the hypothesis using a laser with
higher penetration depth, e.g. a diode laser. The energy from
this type of laser will be absorbed deeper in the tissue, closer
to the area where the nociceptors can be found (epidermal
ridge). If there is less or no response latency using this type
of laser this will provide further evidence that the different
response latencies seen in hairy and glabrous skin occur due
to the thermal properties.
V. ACKNOWLEDGEMENT
T
HE author would like to thank Jørgen Serup MD, DMSc
and Kenichiro Ogoshi MD, PhD at the Department of
Dermatology at Bispebjerg hospital, Copenhagen, Denmark
for their aid in conducting high-resolution ultrasound measures
of the skin profile at several sites in the dorsal and volar
forearm, and palm. These measurements were a great aid in
constructing the heat transfer model of the skin.
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Journal of Physiology (1995), 485.3, pp. 753–765.
[Treede et al., 1998] Treede, R.-D.; Meyer, R. A.; and Campbell, J. N.
(1998). ”Myelinated Mechanically Insensitive Afferents From Monkey
Hairy Skin: Heat-Response Properties”. The American Physiological
Society, pages 1082–1093.
[Treede et al., 1995] Treede, R.-D.; Meyer, R. A.; Raja, S. N.; and Campbell,
J. N. (1995). ”Evidence for two different heat transduction mechanisms
in nociceptors primary afferent innervating monkey skin”. Journal of
Physiology (1995), 483.3, pp. 747–758.
[Vendri and Vos, 1957] Vendri, A. and Vos, J. (1957). ”A method for the
measurement of the thermal conductivity of human skin”. Journal of
applied physiology, 2, pp. 211–215.
[Welch and van Gemert, 1995a] Welch, A. J. and van Gemert, M. J. (1995a).
Optical-thermal response of laser-irridiated tissue, pages 367–384. Plenum
Press, first edition.
[Welch and van Gemert, 1995b] Welch, A. J. and van Gemert, M. J. (1995b).
Optical-thermal response of laser-irridiated tissue, pages 15–46. Plenum
Press, first edition.
[Wilson and Spence, 1988] Wilson, S. B. and Spence, V. A. (1988). ”A
tissue heat transfer model for relating dynamic skin temperature changes
to physiological parameters”. Phys. Med. Biol., 33, pp. 895–912.
Worksheets supporting the article
Cutaneous Temperature Response after
Infrared Radiation
Ken Steffen Frahm
Group 705E, December 2007
Department of Health Science and Technology,
Aalborg University, Denmark
The Faculty of Engineering, Science and Medicine
Aalborg University, Denmark
Title:
Cutaneous Temperature Response after Infrared Radiation
Abstract:
Project period:
7th semester
September 3rd - December 19th, 2007
Project group:
705E
Group member:
Ken Steffen Frahm
Supervisors:
Carsten Dahl Mørch
Lars Arendt-Nielsen
Numbers of copies: 3
Numbers of pages: 50
Enclosures: 2 DVDs
In human pain research lasers have often been used for delivering transient-like noxious stimuli at the skin surface. In
this paper a model was developed which simulates the heat
transfer in the skin, into the layers where the heat sensitive
nociceptors are placed.
The model was based on the bio-heat equation (pure conduction). For creating and solving the model, finite element
modelling and COMSOL multiphysics was used. For the
validation a 100 W CO2 laser was used and to monitor the
surface temperature an infrared (IR) camera was used. The
laser was set to a have an output of 3.7 W and five stimuli
durations were tested (150 ms, 200 ms, 250 ms, 300 ms, and
350 ms). The experiment was conducted on a single male
subject in the dorsal and volar left forearm and left palm. A
VAS score was registered to monitor the degree of pain.
The experiments showed an agreement between the model
and the IR recordings. The best agreement was found in
the longest solution durations. Common for all the tested
skin sites; the dispersion of the maximum temperatures was
greater in the IR recordings than the theoretical. Typically
the IR recordings maximum temperatures at each recording
site were spread across approx. 20 ◦ C whereas the theoretical only was spread across approx. 10 ◦ C. (IR: Min temp.:
42-51 ◦ C / Max temp.: 60-73 ◦ C - Theoretical: Min temp.:
48-52 ◦ C / Max temp.: 56-61 ◦ C) The spatial dispersion of
the temperatures were greater in the theoretical data than
the IR recordings.
The model showed that due to different thermal properties
of the skin (different thickness of the epidermis) in the forearm and palm, the heat transfer is different. This suggests
that the reason for the response latency of nociceptors in
glabrous skin partly can be found in the thermal properties
of the skin.
The following contents are freely available, but publication may only happen in agreement with the author.
.
Preface
The following pages are worksheets to accompany the paper Cutaneous Temperature Response after Infrared Radiation written by Ken Steffen Frahm, group 705E, seventh semester, fall 2007, at the Department of Health Science and
Technology, Aalborg University.
Aalborg University, December 2007
———————————–
Ken Steffen Frahm
.
Contents
I Worksheets
1
1
Cutaneous reception and processing
1.1 Anatomy and physiology of receptors in the skin . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Transport and processing of sensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The Bioheat transfer equation
2.1 Radiation . . . . . . . . . .
2.2 Conduction . . . . . . . . .
2.3 Convection . . . . . . . . .
2.4 The bioheat equation . . . .
2.5 Solving the bioheat equation
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Equipment and experiments
3.1 CO2 laser . . . . . . . . . . . . . .
3.2 Infrared camera . . . . . . . . . . .
3.3 High resolution ultra sound scanner
3.4 In vivo laser experiments . . . . . .
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Modelling in COMSOL multiphysics
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Worksheets
I
Cutaneous reception and
processing
1
This worksheet gives an overview of mechanisms involved in the perception and processing of skin pain. Of the different types of pain especially thermal pain will be in focus.
This worksheet is divided into two main sections, first the anatomy and physiology of the receptors in the skin including pain receptors, socalled nociceptors, will be described. Secondly the processing of the pain stimuli in the CNS and
brain will be described.
The generel senses in the human body is a term covering pain-sensation, thermal sensation, mechanical injury and
chemical disorders. Besides these senses the human body has the special senses; smell, taste, hearing, vision and
balance which will not be described in this paper.
When the nervous system registrers stimuli it is called sensation, when these sensations become present in our consciousness it is called perception. [Martini, 2004a]
1.1 Anatomy and physiology of receptors in the skin
The receptors in the skin can be divided into four groups 1) nociceptors (painreceptors) 2) thermoreceptors 3)
mechanoreceptors and 4) chemoreceptors. The nociceptors will be described more detailed later in this section. Basically the neurons which provide the sensation of any of these four groups are dorsal root ganglion neurons, the cell
body of these neurons are placed in the dorsal root of a spinal nerve. The neurons two branches reach the periphery and
the CNS, respectively. The type of stimuli which the neurons transduce to a nervous signal depends on the peripheral
terminal of the neuron. Neurons for transducing for example touch have speciel non-neural structures at their terminal
for transducing the signal, where neurons for transducing thermal and pain sensation are free nerve endings. Neurons
with specialised terminals are called epicritic and neurons with free nerve endings are called protopathic, generally
epicritic sensations are spatial specific, where as protopathic sensations are more crude. [Martini, 2004a] Crude in this
case means that the receptive fields are larger, so the stimuli at two different points has to be placed further apart from
each other than in areas which smaller receptive fields.
1.1.1 Anatomy - placement of the receptors in the skin
First it should be noted that these receptors, as mentioned above, can be found at several locations in the body not
just in the skin. For example the receptors can be found in internal organs, called visceral receptors. The receptors in
the skin are called somatic receptors. [Martini, 2004a] In this paper only the receptors in the skin will be described.
The receptors in the skin are usual placed in the dermis, a few in the epidermis. C type nociceptors are according
to Bromm and Treede [Bromm and Treede, 1983] terminating in the superficial skin layer (< 300 µm). Bromm and
Treede show that more than 90 % of the power from a CO2 (wavelength 10.6 µm) is absorbed in the superficial 50 µm
of the skin. However, this is sufficient for activating the C fibre nociceptors. One might conclude from this that some
of the C fibres are placed in the superficial part of the skin (< 50 µm). [Bromm and Treede, 1983] Tillman et al. 1995
found that the receptors of C fibres nociceptors, of macaca monkeys, were placed within depths ranging from 20 µm
to 570 µm, with a mean of 201 µm. Comparing their results with the anatomy of the skin, the receptors is placed in the
epidermis and the superficial layers of the dermis. [Tillman et al., 1995] It is unknown how directly their results from
the monkeys can be converting into the human anatomy.
1.1.2 Physiology - function of the receptors
As mentioned there exits different types of receptors in the skin. Three types of these receptors sense stimuli within
the normal range whereas the nociceptors sense extreme (and painfull) conditions. These three types are thermal,
mechano and chemo receptors.
Receptors are either tonic or phasic, a phasic receptor only fires if there is any change of the stimuli where a tonic
4
Cutaneous reception and processing
receptor always fires in the presence of a stimulus.
Some receptors can adapt to continues unpainful stimulaton. For example you do not notive the slight rumbling when
driving in a car. Besides adaption in the receptors, the CNS can also adapt to stimulations. Phasic receptors adapts to
a stimuli where tonic do not.
The signal being produced by these receptors are, like all nervous signals, frequency modulated. The nerve signal are
being produced in different types of specialized receptors. The six different types of receptors can be seen on figure
1.1. The letters in the following refer to the letters on figure 1.1
• Free nerve endings - a) usual found as nociceptors or thermoreceptors. These receptors are placed between the
epidermal cells. There is no structural difference between the cells which registrer pain or temperature.
• Root hair plexus - b) sense touch and displacement of hair. These receptors adapts very quickly which is the
reason why you do not notice the close touching your skin.
• Merkel cells and tactile discs - c) sense fine touch and pressure. These receptors are tonic and have very small
receptive field. The merkel cells are pressure sensitive and secrete chemicals which are noticed by the adjacent
tactile disks. The merkel cells are found in the superficial layers of the skin, at the papillary ridge between the
epidermis and dermis.
• Tactile corpuscle or Meissner’s corpuscle - d) like the merkel cells these sense touch and pressure and also lowfrequency vibration. They are most abundant in fingertips, eyelibs, lips and external genitalia. The corpuscles
are also placed superficially connected to papillary ridge. They rapidly adapt to stimuli making them phasic
receptors.
• Lamellated corpuscle or Pacinian corpuscle- e) sense deep pressure. The corpuscles are placed in the deep
dermis. The receptors are very fast adapting, these receptors adapt within a second to the new stimuli. This
means that they can sense minute vibration and high-frequency stimuli. Physiologically this corpuscle is very
similar to the Meissner’s corpuscle, however, not anatomically. The lamellated is constructed as a single dendrite
is placed in the center of severel collagen layers. This construction ensures that the dendrite only will be
stimulated by direct pressure, when the collagen layers are compressed.
• Ruffini corpuscle - f) are placed in the deep dermis. These receptors also registrer pressure and other distortion
of the skin. The receptors are tonic and do not adapt very much. The corpuscle surrounds collagen fibres which
are part of the dermis, so any distortion of these fibres will be registrered by the corpuscle.
[Kandel et al., 1991a] [Martini, 2004a]
The receptors described above most often produce nerve signals which reaches our consciousness and therefore these
cause perception of stimuli.
The superficial receptors; the Meissner’s corpuscle and the Merkel cells provide rather specific spatial sensing. Each
neuron is innervated from about 10-25 Meissner’s corpuscle or Merkel disks. These receptors have a reception area of
2 - 10 mm. The receptive fields for the Pacinian corpuscle and Ruffini corpuscle are much larger since the receptors
are placed deeper in the skin. The spatial sensitivity also depends on the density of receptors in the skin, the density is
1.1 Anatomy and physiology of receptors in the skin
highest at the fingertips < 5 mm and lowest on the back, thigh and calf 40 - 50 mm. [Kandel et al., 1991a]
The recognition of pressure and shape of object are computed from the input from several receptors.
Figure 1.1: The figure illustrates the six difference types of terminal structures on somatic sensory neurons. [Martini,
2004a]
As mentioned above three types of receptors registrer stimuli within the normal and unpainful range.
Thermoreceptors are free nerve endings which are in steady state tonic receptors, they continously fire discharges
when the temperature changes. Their activity increases but settles to steady state after some time giving them some
of the characteristics of phasic receptors. You usually do not notice the current room temperature, but if it suddenly
changes you will notice this. There exits receptors for registering both warm and cold, and there are no structural
difference between them. The cold receptors are three to four times as abundant as the heat receptors. The thermal
sensation is created from inputs from both the cold and warmth receptors. The receptors respond to temperatures
between 5 ◦ C and 45 ◦ C. Besides those found in the skin, thermoreceptors can be found in skeletal muscles, the liver
and the hypothalamus.
The mechanoreceptors can be subdivided into another three groups; tactile receptors, baroreceptors and proprioceptors. Tactile receptors which sense fine touch and pressure can be found as any of the six types of receptors seen on
figure 1.1. The density of the tactile receptors are greatest in the glabrous (hairless) skin, eg. in the palm and fingertips,
which provides these areas with high sensitivity. Most of the tactile sensors have specialised structures surrounding
the periphiral terminal of the neurons, figure 1.1 c) - f) only a few are free nerve endings, figure 1.1 a). [Kandel et al.,
1991a] Baroreceptors are free nerve endings placed in the epithel walls of an internal organ, eg. a blood vessel. The
sensation from baroceceptors are usually not perceived. Proprioceptors sense the position of joint, the tension in ligaments and muscular contraction. These receptors can for example be bare nerve endings placed in joint capsules.
Proprioceptors do not adapt to stimuli but continously send signals to the CNS.
5
6
Cutaneous reception and processing
Chemoreceptors respond to water soluble and lipid soluble substances in the surrounding fluid. These sensors are
typically placed in the larger blood vessel eg. in the carotid bodies and the aortic arch. These receptors do not send
signal to the primary sensory cortex so their messages do not reach our consciousness.
The sensation of pain does not occur by overstimulation of any of these receptors, for an example if the skin temperature exceeds 45 ◦ C the stimuli will not be sensed by the thermoreceptors but instead by the nociceptors, these will be
described in the following.
1.1.3 Nociceptors
Nociceptors are like thermal receptors free nerve endings, which have a large receptive fields which is the reason why
it can be difficult to pinpoint the exact source of a painful stimuli. Nociceptors are divided into the same three groups,
as the non-pain receptors are, being sensitive of 1) extreme temperatures, 2) mechanical destruction of tissue and 3)
nociceptors that recept the chemicals released by stressed cells or the presence of any alien chemicals. Intense stimuli
will often trigger all three types of receptors.
The nociceptors in the skin are like other receptors placed in superficial layers i.e. the dermis (some few in the
epidermis). Very simplifyed, nociceptors can be classified strictly as tonic receptors - meaning they continue producing
the nervous signals until the painful stimuli ceases. However, the CNS can adapt to a painful stimuli causing the
perception of pain to decline. [Martini, 2004a]
The information about the painful stimuli is sent to the CNS via two types of nervefibres. A fibres which are slightly
myelinated carry so-called first pain to the CNS. C fibres which are unmyelinated carry the slow pain, often described
as burning pain. The signal from the A fibres reach the CNS quickly and trigger somatic reflexes. [Kandel et al., 1991a]
Thermal nociceptors are A fibres with small diameter which conducts at a speed of 5-30 m/s. The fibres for mechano
nociceptors are also A similar to the fibres of the thermal nociceptors. Polymodal nociceptors which reacts to both
thermal, mechanical and chemical stimuli use C fibres which conducts at much lower speed, < 1 m/s. Very intense
stimuli triggers the polymodal receptors. [Kandel et al., 1991b] Nociceptive A fibres are categorised into δ and β fibres
depending on their conduction velocities, δ fibres have a lower conduction velocity ( 15 m/s) than β fibres ( 45 m/s).
The A fibres conducting the signal from the nociceptors are mostly Aδ fibres. [Kandel et al., 1991a] [Treede et al.,
1998]
However, Treede et al. [Treede et al., 1998] [Treede et al., 1995] suggest that two very different types of A fibres
exist, which exhibit quite different characteristics. Treede et al. have done their research using macaques (Macaca
fascicularis) monkeys. The first type, type I has a higher heat threshold than type II. (>53 ◦ C vs. 46 ◦ C). The type
I nociceptors are found both in glabrous and hairy skin whereas type II is absent in glabrous skin. Type I fibres have
very high conduction velocity ( 25 m/s) in comparation type II conducts with a velocity of 14 m/s. But inspite of the
high conduction velocity of type I fibres, these fibres do not conduct first pain according to Treede et al. The reason
for this is the long response latency for type I fibres (∼5 sec.) in comparation it is only 0.22 sec. for type II. [Treede
et al., 1998] [Treede et al., 1995] Treede et al. also suggest that nociceptors are not stricly tonic receptors, some fibres
adapt. Type II A fibres and C fibres are very similar in many ways, they both adapt to a long duration stimuli and they
both have relatively short response latency. However, the most important difference between the two is the conduction
velocity, as mentioned C fibres have very low conduction velocity which means that only type II fibres are able to
carry first pain. [Treede et al., 1998] [Treede et al., 1995] Treede also found that no first pain could be registrered in
glabrous skin, which is consistent with the absense of type II fibres in these areas. [Treede et al., 1995]
Treede also suggest that polymodal receptors are not only C fibres but also Type II A fibers might be polymodal
receptors since these also are sensitive to capsaicin. [Treede et al., 1995] C fibres are found in both skin types and
exhibit adaption to stimuli. [Treede et al., 1995]
Type I fibres rather than type II fibres are capable to sense mechanical pain. Type II fibres have a very high mechanical
threshold, some research indicate that type II fibres are mechanically insensitive. Research have also shown that type
I fibres eventhough they do not carry thermal first pain, they are capable of carrying mechanical first pain. [Treede
et al., 1998]
As it can be noted from the above nociceptors and thermoreceptors are in many ways very similar. Both receptor types
are free nerve endings with no obvious structural differences. The axons which conduct the signal from the receptors
1.2 Transport and processing of sensation
to the CNS are only thinly myelinated, if myelinated at all. Not only are the neurons which conduct the signal
very similar, they also follow the same pathways from the peripheral terminal to the CNS. Since the mechanisms for
perception of temperature and pain are very similar thermal stimulation is often when investigating the human pain
perception.
The neurotransmitters being used by nociceptors are glutamate (an amino acid) and Substance P (a neuropeptide),
which are being released by the neurons to the CNS. [Martini, 2004a]
1.2 Transport and processing of sensation
The general senses of which pain sensation is one, is processed in the primary sensory cortex in the brain. Where the
special senses such as vision and taste are processed in other specialised regions of the cortex.
The signal from the peripheral receptors reach the primary sensory cortex through a number of pathways through the
peripheral nervous system (PNS), the CNS and in the brain. The sense of a peripheral stimuli is conducted through
three neurons before reaching the sensory cortex. The peripheral neuron which conducts the signal to CNS through
the dorsal root ganglion is called the first-order neuron. The second neuron which brings the signal from the CNS
to the thalamus in the brain, is called the second-order neuron. And finally the neuron which brings the signal from
the thalamus to the primary sensory cortex is called the third-order neuron. Somewhere along the pathway the signal
crosses from one side of the body to the other, eg. a stimuli sensed a the right arm in conducted through these neurons
and ends up on the left primary sensory cortex, and vice versa. [Martini, 2004a]
The exact pathway which the signal follows depends on the kind of signal. There exist three pathways for bringning
the signal from the periphery to the cortex. These are the posterier column pathway, the spinothalamic tracts, which
are subdivided into the anterior and lateral spinothalamic tracts and the spinocerebellar pathway. [Martini, 2004a]
Most mechano sensation such as fine touch, vibration and some proprioception are conducing through the posterier
column pathway. Sensation coming from the inferior half of the body is conducted in the most lateral part of the
pathway called the fasciculus gracilis where as the sensation from the superior half is conducted in the more lateral
part, called the fasciculus cuneatus. The axons of the second-order neurons cross over to the other side of the brain
stem just before reaching the thalamus, the pathway can be seen of figure 1.2 as a). [Martini, 2004a] [Kandel et al.,
1991a]
The spinothalamic tracts conduct the sensations of crude touch and pressure along with sensation of pain and temperature. The sensations of pain and temperature are conducted in the anterior tracts, where as the other sensations
are caried in the lateral tracts. The cross-over in these pathways happens immediately after the first-order neuron has
synapsed onto the second-order neurons which cross over before ascending in the spinal cord. This pathway can be
seen on figure 1.2 as b) and on figure 1.3 as c) [Martini, 2004a] [Kandel et al., 1991a]
The spinocerebellar pathway conducts sensation of proprioception from sensor placed inside muscle and order organs.
The pathway can also be subdivided into posterior and anterior spinocerebellar tracts, however, the sensation carried
in both are the same. This pathways seperate it self from the two other because the third-order neurons do not reach
the primary sensory cortex. Neither do any cross-over occur (actually some second-order axons cross but recross in
the cerebellum). The signal from the left side of the body is conducted to the cerebellum where second-order neurons
synapse to the left cerebellar cortex. No third-order neurons are present in this pathway. The sensations conducted
7
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Cutaneous reception and processing
through this pathway never reach our consciousness. This pathway can be seen on the right of figure 1.3. [Martini,
2004a]
Figure 1.2: The figure illustrates the posterior column pathway a) and the anterior spinothalamic tracts b). The firstorder neurons are illustrated as red, the second-order as white and third-order as black, see the legend.
[Martini, 2004a]
1.2 Transport and processing of sensation
Figure 1.3: The figure illustrates the lateral spinothalamic tracts c) and the spinocerebellar pathway, left. The firstorder neurons are illustrated as red, the second-order as white and third-order as black, see the legend.
[Martini, 2004a]
1.2.1 Processing
The signals transported to the brain are first processed in the thalamus and from there some of the information is sent
to primary sensory cortex. The thalamus filters some of the information and determines some of the characteristics of
the signal eg. the origin of the signal, the thalamus is responsible for sending the information to the correct part of the
sensory cortex. The primary sensory cortex is divided into regions each representing certain parts of the body. The
regions of the cortex vary in size accordingly to the importance of body part from which it receives stimuli, eg. the
regions covering the fingertips, and the lips and tongue are very large compare to for example the back, see figure 1.4.
9
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Cutaneous reception and processing
The size of the regions correspond to density of receptors found in the bodypart, eg. the fingertips have large density
of receptors and therefore a large region on the cortex. [Martini, 2004a]
Figure 1.4: The figure illustrates how the diffent regions of the primary sensory cortex is connected to certain parts of
the body. [M. Schmolesky, 2000]
Figure 1.4 illustrates how the socalled homonculus at the primary sensory cortex. It clearly shows that some parts of
the body fill more in our consciousness than other, and moreover the sensitivity in these regions are much higher than
other parts of the body.
The perception of stimuli which pain are much more present in our awareness than other types of stimuli. However,
sometimes the brain can filter the perception of pain from our consciousness eg. an athlete often do not feel an injury
until the game has finished.
Visceral receptors can cause stimuli to misinterpreted as originating from external stimuli. An example for this case
could be a myocardial infarction where you do not only feel pain from your chest, but the pain will also radiate into
the left arm, because the neurons carrying pain sensation from the left arm and myocardium converges on the same
neuron in the CNS.
1.2.2 Reflexes
Besides the processing which takes place in the primary sensory cortex a lot of processing happens in the inferior parts
of the CNS, i.e. the spinal cord. This processing also involves creation of a response, typically in the form of activiting
a motor unit. This form of processing is know as reflexes.
Monosynaptic reflexes typically is the stretSch of a muscle which is registrered by a muscle spindle (a form of mechano
receptor) which in the CNS synapses onto a motor neuron which causes the muscle to contract. The name monosynaptic comes from the fact that these reflexes only involve two neurons, the first-order neuron which brings the signal
1.2 Transport and processing of sensation
from the muscle spindle to the CNS and the second-order motor neuron which causes the muscle to contract. The
first-order neurons are large, highly myelinated type A fibres with high conduction velocity. [Martini, 2004b]
Figure 1.5: The figure illustrates how monosynaptic reflexes will make you contract a muscle if it is stretched. [Martini, 2004b]
Polysynaptic reflexes involves more neurons. An example could be the prick of a needle on your finger which will
make you remove the finger. Nociceptors sense the prick of the needle and send the signal through a first-order neuron
to the spinal cord, here a second-order neuron diverge the signal both sending the painful sensation to the brain and
also activating a motor neuron which cause muscles to contract which removes the hand from the needle. Some
polysynaptic reflexes will not only trigger the activation of a motor neuron. As seen on figure 1.6 the reflex will also
inhibit the motor neurons with the opposite function. As seen on figure 1.6 the reflex caused by stepping on a needle
will not only flex the leg but also inhibit the extensors in the thigh. On b) it can be seen how the reflex makes you
remove your foot from the needle but also makes you shift your weight to the other leg. Besides the activation of
11
12
Cutaneous reception and processing
the motor neurons, signal are sent to the brain for processing in the thalamus and sensory cortex. [Martini, 2004b]
Therefore we typically have already moved our foot from the needle before we become aware of the pain.
Figure 1.6: The figure illustrates how polysynaptic reflexes makes you remove your leg if you step on a needle.
[Martini, 2004b]
2
The Bioheat transfer equation
The bioheat transfer equation or simply the bioheat equation was first proposed 1960 based Pennes 1948 paper.
[Pennes, 1948] The equation describes the thermal exchange in and out of a certain biological compartment, and
if any, the storage of thermal energy. Using the first law of thermodynamics, the heat gained in a certain volume, Qgain
can be described as
Qgain = Qstorage + Qloss + W
(2.1)
[Welch and van Gemert, 1995a]
Where Qstorage and Qloss is the amount of heat either stored og lost from the volume. W describes the work performed
by the tissue in the volume. When considering situation involving laser irradiation of the skin or tissue, the work
performed by the tissue and the metabolic heat contribution can be neglected.
The thermal gain, Qgain , in the volume is equal to the integral of q(r,t) over the volumen V expressed as
Qgain =
Z
q(r,t)dV
(2.2)
V
Where q(r,t) is the heat gain per unit volume, in the spatial coordinate r at time t. Usually the heat gain from the
surroundings is negligible. [Welch and van Gemert, 1995a] But in problems considering laser irridation, the heat gain
from the surroundings will equal the energy gained from the laser. So q can be described as
q(r,t) = S(r,t)
(2.3)
[Welch and van Gemert, 1995a]
where S is the amount of energy contributed from the laser [W/m3 ]
The amount of heat energy can be expressed as the integral of temperature change per time unit over the volume,
multiplied by the intrinsic heat capacity, which is the product of the density, ρ and the specific heat, c, at constant
pressure.
Z
Qstorage =
V
ρc[∂T (r,t)/∂t]dV
(2.4)
[Welch and van Gemert, 1995a]
Accordingly to the second law of thermodynamics heat energy will flow from a volume with a high energy level to
a volume with a lower heat level. In the situation of irradiating tissue with a laser the temperature of the tissue will
increase causing heat to be distributed in the surrounding tissue and other surroundings. The exchange of heat in and
out of the volume can occur in at least three different ways; radiation, conduction and convection.
2.1 Radiation
Unlike the other types of heat exchange radiation differs, since it does not need a medium for energy transport.
Radiation af thermal energy is spread out over the entire electromagnetic spread at room temperature the waves of the
thermal radiation are mainly in the infrared region with wawelengths between 0.1 and 10 µm. [Serway and Jewett,
1995a] Truthfully radiation of heat energy occurs on the whole continuous spectrum.
The radiation of these electromagnetic waves is known as blackbody radiation. A blackbody is characterised by the
fact that it is a perfect emitter, it will emit the maximum amount of energy at a given temperature. However, in reality
most emitter will never have this perfect emission, these objects are called graybodies. Planck’s law states that the
monochromatic emissive intensity of a perfect blackbody, Wb , at wavelength λ [m] and temperature, T [K], equals
Wb (λ, T ) =
2πhc2
λ5 (ehc/λkB T ) − 1
[W /m3 ]
(2.5)
14
The Bioheat transfer equation
[Welch and van Gemert, 1995a] [Serway and Jewett, 1995a]
Where h is Planck’s constant = 6.626·10−34 [J·s], c is the speed of light = 3 ·108 [m/s] and kB is Boltzmanns’ constant
= 1.380 ·10−23 [J/K]. [Serway and Jewett, 1995a] The total emission power of the object is found by integrating the
intensity, Wb , in the whole spectrum, expressed as
Eb =
Z ∞
0
Wb (λ, T )dλ = σT 4 [W /m2 ]
(2.6)
[Welch and van Gemert, 1995a]
also known as the Stefan-Boltzmann law and the constant σ equals 5.67 · 10−8 [W /m2 · K 4 ].
For graybodies both the monochromoatic emission power and the total emission power are less than those for blackbodies. Which for the monochromatic emission power can be expressed as
W (λ, T ) = ελWb (λ, T ) [W /m2 ]
(2.7)
[Welch and van Gemert, 1995a]
Where ελ is thermal emissity at a given wavelength. ελ is also less or equal to 1, for a perfect blackbody ελ = 1.
Similar the total emission power at a given temperature can be calculated by multiplying the right expression in Eq.
2.6 with the gray body emissivity ε being less than 1.
E(T ) = εσT 4 [W /m2 ]
(2.8)
[Welch and van Gemert, 1995a]
The gray body emissivity is averaged over the thermal radiation spectrum, weighted by the monochromatic emissive
power, Wb . For the human body, the gray body emissivity, ε is approximately at 0.97.
In aspects considering laser-skin interaction the contribution from radiation is neglible, which means it can be left out
of the bioheat equation.
2.2 Conduction
Conduction of heat through a medium is due to internal temperature gradient inside the medium. The conduction
of heat requires directly contact between the parts of the medium in which the conduction occurs. The transfer of
heat happens at the molecular and atomic level, no mass of the medium is moved in conduction. The heat energy is
conducted via the vibration and movement of the molecules.
The conduction of heat through a medium, Q can expressed using the Fourier equation which states that the amount
of heat energy being conducted is directly proportional to A the cross-sectional area perpendicular to the direction of
conduction; T2 − T1 the temperature difference in the medium; ∆t the time in which the conduction occurs; ∆L the
length of path over which the heat is being conducted. When taking into acount that different media have different
thermal properties, a proportionality constant must be added. By adding this constant k, the thermal conductivity of
the medium [W /(m · K)] the Fourier law can be expressed as
Q = −kA(T2 − T1 )∆t/∆L [J]
(2.9)
[Welch and van Gemert, 1995a]
The negative sign can be explained by remembering the second law of thermodynamics, which states that heat energy
always will flow from a medium with high level of energy to a medium with a lower level of energy.
The heat flow [W /m2 ] can be expressed as the conduction of heat, per cross-sectional area and time unit
fc =
Q
= −k(T2 − T1 )/∆L [W /m2 ]
A∆t
(2.10)
[Welch and van Gemert, 1995a]
When limiting ∆L to a infinitesimally small length, the expression (T2 − T1 ) kan be expressed as the temperature
gradient, ▽T . Which gives
fc = −k▽T [W /m2 ]
(2.11)
2.3 Convection
15
[Welch and van Gemert, 1995a]
Since k for biological tissue depends on the nonhomogenity and anisotropic properties of the tissue, k is expressed as
second-order tensor, however, since the variation of k often is small and thus it can be described as a scalar, which
simplies the problem significantly since tensor calculations can be very difficult. Integrating Eq. 2.11 over the crosssection area will provide us with the power of heat being conducted through the medium.
Qc = −
Z
A
k▽T (r,t) · n̂dA [W ]
(2.12)
[Welch and van Gemert, 1995a]
n̂ is the normal vector to the surface, dA.
When dealing with the transient heat due to laser irradiation, heat conduction will be used for calculating how the heat
spreads through out the tissue from the point of laser radiation.
2.3 Convection
Convection is the transfer of heat due to bulk movement of a fluid (liquid or gas). In biological problems usually due
to blood flow, but in principle heat gain or loss due to convection could also be caused by adjacent air flow. Convection
occurs when a fluid is heated by an adjacent solid (or another fluid) and the fluid moves, transporting the heat energy to
another location where other solids (or fluids) can be heated by the transported energy, stored in the fluid. Convection
can be explained as the following four processes
1. Conduction of heat from a solid into an immidiately adjacent fluid.
2. Absorptions and storage of the conducted energy in the fluid, by the elevating the fluid particles internal energy.
3. Movement of the energy-rich particles to another location with lower energy
4. Transport of the energy by by bulk movement of the fluid.
[Welch and van Gemert, 1995a]
The heat flow from a solid into a fluid is expressed using Newton’s law for forced and free convection, which states
that the heat flow, f equals
f = h(T2 − T1 )
(2.13)
[Welch and van Gemert, 1995a]
Where T2 and T1 is the temperature of the solids surface and the temperature of the fluid, respectively. The constant,
h, is the convective heat transfer coefficient, which depends on the type of convection, either free or forced. Free
convection occurs due to density differences in the fluid, these differences originates from different temperatures
within the fluid. Forced convection is caused by mechanical forces, eg. the bloodflow is caused by the pressure
differences caused by the mechanical work of the heart. The values for h during forced convection is in the area of 50
- 20,000 W /(m2 · K) for liquid, i.e. blood.
Convection plays an important role for sustaining the bodys temperature at a acceptable level. E.g. the temperature of
muscles being exercised without being perfused can rise as high as 45 ◦ C, without the perfusion to cool the muscles.
The interaction between the tissue and the blood is important for regulating the tissue temperature. To express the
amount of heat being contributed to a tissue volume, qb from a liquid one can use Fick’s principle, saying that the
amount of substance taken by the volumen, is equal to the difference between the arterial level of substance and the
venous level, times the bloodflow, wb it can all be combined into
qb = ρb cb wb (Tart − Tven ) [W ]
(2.14)
[Welch and van Gemert, 1995a]
The blood flow is important for regulating the bodys temperature because the blood leave a certain tissue volume will
have the same temperature as the tissue itself (providing that diameter of the vessels are of a suffucient size). The
16
The Bioheat transfer equation
temperature of the venous blood, Tven will therefore equal the tissue temperature, T , which means that Eq. 2.14 can
be ridden as
qb = ρb cb wb (Tart − T ) [W ]
(2.15)
Summed over the volume of the tissue compartment being investigated the result is
Qb =
Z
V
ρb cb wb [Tart (r,t) − T (r,t)]dV [W ]
(2.16)
This equation is a good approximation of the thermal energy being transported by the bloodflow, in the smaller vessels. In the larger other things might have to be taken into acount. The presence of such large vessel which change the
problem considerably.
When considering other problems than laser-tissue interaction, convection is the dominant factor in biological heat
transfer.
2.4 The bioheat equation
Combining Eq. 2.3, 2.12, and 2.16 and replacing them in Eq. 2.1 will give the following result
Z
V
S(r,t)dV =
Z
V
ρc[δT (r,t)/δt]dV −
Z
A
k▽T (r,t) · n̂dA −
Z
V
ρb cb wb [Tart (r,t) − T (r,t)]dV
(2.17)
[Welch and van Gemert, 1995a]
By applying the divergence theorem and observing that Eq. 2.17 must be true for any volume element, the equation
can be ridden as
ρc[δT (r,t)/δt] = ▽[k▽T (r,t)] + ρb cb wb [Tart (r,t) − T (r,t] + S(r,t)
(2.18)
This equation is known as the bioheat equation and solving this for varies volumes can be used as a way of modelling
the heat transfer through out biological tissue. The bioheat equation should be solved using an initial condition, which
specifies the temperature in the tissue at a given time. Besides the initial condition a boundary condition, of all the
boundaries of the tissue. The boundaries can either be a prescribed temperature or a prescribed heat exchange. Using
a prescribed temperature the surface temperature at the boundary is either constant, or described as a function of the
location and/or time. The prescribed heat exchange across a boundary is either constant or described as a function of
location and/or time. Other and more complex boundary conditions occur e.g. when either the two solid volumes at
the boundary are not in perfect contact or the boundary at a moving surface between a solid and fluid, caused by the
change of phase of the medium e.g. the evaporation of water in an ablation process. [Welch and van Gemert, 1995a]
2.5 Solving the bioheat equation
There exits several approaches for solving the bioheat equation. First the bioheat equation seen in Eq. 2.18 can be
reduced if some assumptions are made. When considered laser skin interaction, the laser pulse is often very brief and
both radiation and convection can be considered neglible. This reduces the equation into the following
ρc
∂T (r,t)
= k▽2 T (r,t) + S(r,t)
∂t
(2.19)
[Welch and van Gemert, 1995b]
The different expressions in the equation is explained above.
2.5.1 Green’s function
The Green’s function is used in many of the analytical models developed for solving the bioheat equation. Most
solutions involve finite element modelling however in some cases it is possible to solve the equation analytically. In a
2.5 Solving the bioheat equation
17
infinite medium the temperature rise can be described as
∂T
= α▽2 T (r,t)
∂t
(2.20)
[Welch and van Gemert, 1995b]
k
Where α is the thermal diffusivity, defined as α = ρc
The initial temperature through out the medium are described
via the function f (r), where r is the three dimensional ccordinate (x,y,z). It can be seen that the equation above is the
same as Eq. 2.19 where the input from the laser is zero. This simply describes the total temperature change in the
medium. [Welch and van Gemert, 1995b]
Some solution only calculates the temperature change perpendicular to the surface, abriviated the z-direction. Then
Eq. 2.20 can be expressed as
δ2 T
∂T
(2.21)
=α 2
∂t
δz
[Welch and van Gemert, 1995b]
The boundary conditions of Eq. 2.20 can be expressed as seen below
T (r, 0) =
f (r)
(2.22)
lim T (r,t) =
0
(2.23)
lim T (r,t) =
0
(2.24)
t→∞
|r|→∞
In other words the initial temperature at time 0 is described using the f (r) mentioned above. For time going toward
infinity the temperature will become zero and for the coordinate r going toward infinity the temperature will go to
zero. It can be understood like the initial temperature rise will have no effect whenever either time or the coordinate
goes toward infinity.
Eq. 2.20 can be expressed as
T (r,t) =
Z ∞Z ∞Z ∞
−∞ −∞ −∞
g(r,t, r′ ,t ′ ) = 0) f (r)dx′ dy′ dz′
(2.25)
[Welch and van Gemert, 1995b]
by using Green’s theorem. The function g(r,t, r′ ,t ′ ), called the Green’s function, has the following properties
lim g(r,t, r′ ,t ′ ) = 0, for all r except r’
(2.26)
lim g(r,t, r′ ,t ′ ) = 0
(2.27)
|r − r′ |2
1
exp(−
)
4α(t − t ′ )
8(πα(t − t ′)3/2 )
(2.28)
|r|→∞
and
t→t ′
Green’s function can be expressed as
g(r,t, r′ ,t ′ ) =
[Welch and van Gemert, 1995b]
The temperature rise at location r at time t can be expressed as
T (r,t) =
1
Q
|r|2
)
exp(−
ρc 8(παt 3/2 )
4αt
(2.29)
Where Q is the amount of energy delivered [J] at coordinate r′ = 0 and at time t ′ = 0. This is the reason why both r′
and t ′ are eliminated from the equation. [Welch and van Gemert, 1995b]
18
The Bioheat transfer equation
2.5.2 Other analytical methods
Other solutions than Green’s function can be applied. Brugmans et al. discuss two different ways of solve a heat
transfer problem. [Brugmans et al., 1991] The first method which they refer to as The Surface Heat Flux Model. They
assume that the radiation from a CO2 laser is completely absorbed at the tissue surface and no radiation penetrates the
surface. The heat flux at the surface abbreviated f [W/m2 ] at the air-tissue surface, at the coordinate, z = 0 and in the
time interval from t = 0 to t = tL . The temperature at time and coordinate can then be expressed as
√
2 f αt
z
(2.30)
∆T (t, z) =
ier f c( √ ), 0 ≤ t ≤ tL
k
2 αt
[Brugmans et al., 1991]
where ier f c(x), called the inverse complimentary error function, is defined as
2
1
2
ier f c(x) = √ e−x − x(1 − √
π
π
[Brugmans et al., 1991]
The flux at the surface, f is described as
f=
Z x
−∞
2
e−y dy)
q
tL
(2.31)
(2.32)
Where q is the surface energy flux [J/m2 ], at time tL . Eq. 2.30 can, for times much larger than tL meaning a long time
after the laser irridiation has stopped, be rewritten as
b
∆T (t, z = 0) ∼
= √ ,t >> tL
t
(2.33)
q
b= p
πkρc
(2.34)
where b is
[Brugmans et al., 1991]
The other solution discusses by Brugmans et al. the socalled Time Constant Model assumes that the temperature
increase during the laser radiation and the temperature decrease after the radiation have the same exponential time
constant. For a CO2 laser scheme (beam size » penetration) the model predicts the following temperature
∆T (t, z) =
Where Φ(z) is the fluence rate at depth z
τµa Φ(z)
(1 − e−t/τ ), 0 ≤ t ≤ tL
ρc
(2.35)
Φ(z) = Φ(0)e−µa z
(2.36)
and τ is the axial time constant defined as
τ=
ρc 4 2
)
(
k πµa
(2.37)
[Brugmans et al., 1991]
Neither of the models proposed by Brugmans et al. are easily applicable. Instead the bioheat equation is often solved
using the finite element method.
2.5.3 Finite element modelling
As the name indicates the solution uses a finite number of elements to solve the problem. The finite element modelling or method (FEM) technique is not only used for solving the bioheat equation but also in many other engineering
problem. FEM offers a approximate solution for all the elements in the model. [Welch and van Gemert, 1995b]
The advantage of FEM are many it can be to object which have varying properties e.g. varying both temporally and
2.5 Solving the bioheat equation
spatially perfusion rates. Another advantage of the FEM is that it can be used for modelling complex structures e.g.
those found in the different skin layers in the body. [Welch and van Gemert, 1995b]
The disadvantage of FEM is the complexity of the models. This calls for the need of computers to do the calculations,
several program exists for this purpose, and one in particular will be used in this project, the COMSOL Multiphysics
3.4 (previously known as FEMLab) a program for solving FEM problems in engineering schemes.
19
.
Equipment and experiments
3
3.1 CO2 laser
The Synrad 57-1 100 W laser used for the experiments is a carbon dioxide (CO2 ) gas laser producing infrared radiation
with a wavelength of 10.6 µm. The word laser is an abbreviation for Light Amplification by Stimulated Emission of
Radiation. The stimulated emission occurs when a metastable exited atom is hit by a photon and the atom returns to
an unexcited state, after this two photons with equal energy, direction, and phase exist, see figure 3.1.
Figure 3.1: The figure illustrates the principle in a gas laser. The gas is being excited by an external energy input
and both stimulated and spontaneous emission occurs. The two mirros confines the photons and there is a
build up of photons and the amount of stimulated emission. [Serway and Jewett, 1995b]
The photon hitting the atom has to have an energy equal to difference between the exited and normal state of the
atom, if not nothing happens. Instead of the stimulated emission an stimulated absorption can occur, then energy of
the photons energy is absorped by the atom, the chance of either stimulated emission or absorption is the same. After
stimulated emission the two photons can excite other atoms and cause further stimulated emission (or absorption) and
a chain reaction is under way. [Serway and Jewett, 1995b]
In order for the laser to produce output there must be a build up of photons in the gas. To ensure this three conditions
must be satisfied
• More atoms must be excited than in ground state - this ensures that the number of photosn emitted is greater
than number of photons absorbed. [Serway and Jewett, 1995b]
• The excited state of the gas must be a metastable state - which means that it is probable stimulated emission
occurs before spontaneous emission. [Serway and Jewett, 1995b]
22
Equipment and experiments
• Further more the photons emitted must be confined long enough so they induce stimulated emission with other
atoms, this is done by having the gas in a tube with a mirror at one end at semi mirror at the other, the semi
mirror has a small hole, the photons escaping through this hole form the laser beams, see figure 3.2. [Serway
and Jewett, 1995b]
Figure 3.2: The figure illustrates the principle in a gas laser. The gas is being excited by an external energy input and
both stimulated and spontaneous emission occurs. The two mirros confine the photons and there is a build
up of photons and the amount of stimulated emission. [Serway and Jewett, 1995b]
The laser used had a tickle pulse which excites the gas to level just below the creation of a laser beam. This should
ensure that when the energy input increases above the threshold for emission there is no rise time to speak of. [Synrad,
1998] The laser beam has a 90 % Gaussian profil (the top 90 %).
To calculate the energy of each photon from the laser, one can use Plancks law. To do this the frequency f of each
photon must be known
f=
c
3E8 m/s
=
= 2.83E13 Hz
λ
10.6 µm
(3.1)
where c is the speed of light, 3E8 m/s nad λ is the wavelength of the photons 10.6 µm.
Plancks law states that the energy of photon with frequency f has the energy E = h f , h is Plancks constant. This
means that each photon from the CO2 laser has the energy
E = h f = 6.626E − 34 J · s · 2.83E13 Hz = 1.87E − 20 J = 0.12 eV
(3.2)
3.1.1 Problems using the CO2 laser
Two issues came to attention when conducting experiments with the laser. First it was discovered that the tickle pulse
did not ensure that no rise time was present. The exact increase of risetime was not found since no power metres with
sufficient fast response was available in the lab. This especially had an effect on short shoot durations.
Secondly the laser was equipped with a scanner head to control the direction of the beam. However, there was found
a large crack in the lens of the scanner head (Ø ∼ 1 cm) this crack made the beam uncollimated and made the exact
3.2 Infrared camera
shape (no longer Gaussian) of the beam difficult to model, if not impossible. The crack in the lens can be seen on
figure 3.3
Figure 3.3: The photograph displays the crack (black arrow) in lens of the scanner head. The crack made the laser
beam uncollimated and difficult to model.
3.2 Infrared camera
The Agema thermovision 900 infrared camera, used for the thermographic recordings, records blackbody radiation
in the spectrum 2 - 5.4 µm. [FLIR systems, 2005a] The blackbody radiation is temperature dependent and generally
increases with temperature see Eq. 2.6 on page 14. The infrared camera translates the blackbody radiation into temperature.
The cameras sensor is liquid nitrogen cooled, otherwise the blackbody radiation from the sensors themself would
blind the sensors. When starting the camera it takes some time before the cooling is sufficient. [Wikipedia foundation,
2007b] The cameras sensors are made of Indium antimonide (InSb). [FLIR systems, 2005b] [Wikipedia foundation,
2007a]
The camera can record with a framerate of up to 30 Hz. The reolution of each frame is 272 x 136. [FLIR systems,
2005b]
3.3 High resolution ultra sound scanner
To measurement the thickness and profile of the epidermis and dermis was made using a 50 Mhz Dermascan (Cortex
technology) ultrasound scanner was used. The scanners transducer emits high frequency sound waves and measures
their reflections in the tissue, the longer it takes a reflection to reach the receiver, the deeper the reflection occured
(assuming homogene sound velocity through out the tissue). Since ultrasound is very purely conducted through gasses
23
24
Equipment and experiments
(air) a scan gel was used to ensure good contact between the skin and the probe.
The 50 Mhz scanner had high resolution but lacks penetration depth compared to lower frequencies. But the penetration depth of 3 mm were sufficient to measure the thickness and profile of the epidermis and dermis. The scanner had
a resolution of 25 x 60 µm. [Cortex technology, 2007]
In the scans of the glabrous skin (the palm) it was difficult to identify the dermis due to its high concentration of
collagen, which only provide a small reflection of the ultrasound.
3.4 In vivo laser experiments
The experimental setup can be in figures 3.4 and 3.5
Figure 3.4: The photograph displays experimental setup used. The scanner head directs the beam onto the object
(here water, for in vivo experiments the skin).
3.4 In vivo laser experiments
Figure 3.5: The photograph displays experimental setup used. The infrared camera is placed as close to to perpendicular to the surface (normally skin, here the water in the small tub).
25
26
Equipment and experiments
3.4.1 Skin injuries
Due to the crack in the lens of the scanner head the beam was quite intense some places. This leads to some skin juries
on the test subject. Immidiately after the experiments the skin was red and irritated, see figures 3.6 and 3.7
Figure 3.6: The photograph displays the skin at site 1 and 3 immidiately after the experiments, the skin is very red
and a few blisters are beginning to form.
3.4 In vivo laser experiments
Figure 3.7: The photograph displays the skin at site 4 immidiately after the experiments, the skin is red and irritated
in a large area (larger than the beam size).
27
28
Equipment and experiments
Ten days after the experiments wounds was formed on the skin surface. On a few sites blisters had formed in the days
following the experiments.
Figure 3.8: The photograph displays the skin at site 1 and 3, ten days after the experiments, wounds and scars have
formed on the skin surface.
3.4 In vivo laser experiments
Figure 3.9: The photograph displays the skin at site 4 ten days after the experiments, blisters have formed and bursted
and some wounds and scars can be seen. The red square seen on the skin has nothing to do with the laser
stimuli.
On both figures it can be seen how some of the wounds are shaped like a halfcircle which is the same shape seen when
tested the laser on thermograhic paper.
29
.
Modelling in COMSOL
multiphysics
4
In COMSOL multiphysics the finite element modelling of the heat transfer was made using the heat tranfer conduction
mode. COMSOL multiphysics offer to model both conduction and convection. It is assmued that any radiation of heat
is negligible and hence only heat transfer in the most superficial skin layers were being investigated where blood flow
is either absent or scarce, therefore convection is negligible, hence the heat transfer is due to conduction.
The model was created using a 2D axial setup. A 2D axial model will minimize the computational burden compared
to a 3D model.
To model the epidermis and dermis at each site two connected rectangles was drawn, forming two seperate compartments, each representing epidermis and dermis. The shape of the rectangles was based on high resolution ultrasound
scans. An image from one ultrasound scan can be seen below
Figure 4.1: The figure illustrates the image from a high resolution ultrasound scan. The white line is the epidermis
and the structure below that is the dermis. The black area above the epidermis is ultra sound gel.
32
Modelling in COMSOL multiphysics
After the two comparments had been created, using the draw mode, and a mesh was created, using the mesh mode.
The mesh divides the compartments into many smaller compartments or elements, hence the name finite element
modelling. The two compartments are displayed on figure 4.2
Figure 4.2: The figure illustrates the two compartments representing epidermis and dermis, the thin layer at the top is
epidermis. The mesh seen divides the compartments into a finite number of elements.
The finer the mesh the more accurate model and the larger computational burden. For example when calculating the
maximum temperature in an area the exact temperature will depend on the quality of the mesh.
When having created the meshed compartments the physical properties of each compartment is defined using the
subdomain mode. In each compartment the following properties was set, the density ρ, the specific heat c, the thermal
conductivity k, the initial temperature T and the heat term Q. The values used for the two compartments can be seen
in table 4.1
33
Property
ρ [kg/m3]
J
]
c [ kg·K
k [W/(m·K)]
Epidermis
1200
3600
0.21
Dermis
1200
3800
0.58
Table 4.1: Properties for the epidermis and dermis used for modelling
Depending on the type of skin modelled, the initial temperature varied. In the foream it was 34 ◦ C and in the palm
it was 31 ◦ C, the initial temperature was the same in both epidermis and dermis. The heat term Q describes the heat
energy contributed to the skin from the laser. The laser beam had a Gaussian profile and a part of the Q expression
describes the 3D Gaussian shape of the laser beam (actually the laser is only 90 percent Gaussian but for the modelling
it was assumed it had perfect Gaussian profile). The expression has been normalized to unity, making it easy to model
different output powers from the laser, simply by changing Pin, the laser power setting. Even though the model are
in 2D axial symmetry the expression is in 3D because the modelling answer is in 3D due to the rotations symmetry.
Thus, the normalization part of the expression is squared, and can be expressed as
1
−r2
( √ )2 · exp(
)
2 · σ2
σ 2π
(4.1)
By using Beers law and neglicting any surface reflection or photon scattering in the tissue, the intensity decays exponentially from the surface and can be expressed as
µa · exp(µa · z)
(4.2)
[Welch and van Gemert, 1995c]
Where µa is the absorbtions coefficient. The neglection of photon scattering is acceptable due to the CO2 low penetrations depth. Since z is zero at the surface and decreases with depth, Eq. 4.2 expresses a decay not increase.
The two expressions seen above is combined into the following expression
1
−r2
) [W /m3 ]
Q = Pin · µa · exp(µa · z) · ( √ )2 · exp(
2 · σ2
σ 2π
(4.3)
This expression is entered as a scalar expression Q and the heat term in each compartment is set to Q.
The constants used in Q can be seen in label 4.2.
Constant
µa
σ
Value
50000
0.0035
Unit
1/m
m
Table 4.2: Constant used for modelling
After having defined the heat term and tissue properties the solve parametres is set. The model is solved for one
second, with at time step of 0.001 second. The absolut tolerance is set to 0.0001.
Afterwards the model is solved and in the post processing mode it is possible to extract several different plots e.g.
surface plots which display the temperature vs. time.
.
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