Download Computation of Tissue Temperature in Human Periphery

International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
Computation of Tissue Temperature in Human Periphery
during Different Biological and Surrounding Conditions
M. P. Gupta
National Institute of Technical Teachers’ Training and Research, Bhopal
The study of heat transfer, under various conditions, in
the human body has been attempted by several
researchers [4-14]. Initially in 1948 Pennes devised his
bio-heat equation (known as heat sink equation) which
describes the blood to tissue heat transfer as if it all takes
place in the capillaries. Later in1962, Perl introduced a
mathematical model which elaborates heat and matter
distribution in body tissues. Since then these
mathematical models have been widely used to develop
theoretical and experimental concept for various
clinical/advanced research purpose. Earlier experimental
work was carried out by Patterson to determine
temperature profiles in the peripheral regions of human.
Mitchell et al. presented an analytical model to predict
temperature as a function of time in human legs.
Weinbaum et. Al. further developed the research to study
the heat flow in human limbs. Saxena and Pardasani
extended the research work to study the heat migration in
skin and dermal region of human body, Further, Saxena
and Bindra, Saxena and Gupta developed a Pseudo
Analytic method (PAM) for two dimensional case to
study the temperature distribution in human limbs. Here
the research work has been extended with more
complexity of the cross-section and biological parameters
to study the behaviour of tissue temperature during
cool(15℃)
and
moderate(23℃)
environmental
temperature for two dimensional case. The mathematical
model used here for computing tissue temperature in
Human periphery is based on the Perl’s model. This
mathematical model in equation form can be written as
Abstract-- A mathematical model is presented to
determine tissue temperature in the periphery of human
thigh during moderate(23℃) and cool(15℃) environment
temperature and the effect of sweat evaporation on heat loss
from human body is also measured. The region under study
is a circular cross-section of human thigh and further has
been partitioned into three concentric tissues layers on the
basis of biological properties. The outer surface of the
region is assumed to be exposed to moderate and cool
surrounding temperature and heat exchange takes place
with the environment through convection, radiation and
evaporation. The variation in biological parameters like
blood mass flow, thermal conductivity of tissues, specific
heat of the tissues are considered varying with respect to
position in different concentric tissues layers.
The
metabolic heat generation is assumed to vary with respect to
both position and temperature.
Keywords-- tissue temperature, periphery, tissues layers,
metabolic heat generation.
I.
INTRODUCTION
Temperature plays an important role in the functioning
of human biological system. The human body is
hypothetically divided into core and shell (Chatterji [1]).
The core consists of the brain and the internal organs in
the trunk and maintains its temperature of about 37℃.
Shell contains the remaining parts of the body ‘the
periphery’ surface tissues and limbs. The temperature of
this region is less constant. When environment conditions
vary, core temperature is maintained partly by behaviour
and partly by the body`s thermo- regulatory system. The
surface temperature of a man, in heat balance is always
lower than the core temperature (Guyton[2]). This means
that the arterial blood which flows to the outer shell
losses heat and returns as colder venous blood. As per
anatomy (Gary[3]) and Physiology of human body it is
found that temperature not only varies along the axial
direction but also varies along the angular direction in the
limbs. This is because the arteries carry blood from the
body core to the extremities of the limbs from where this
blood is carried back by veins to the core. The major
artery lies on one side of the limbs while the vein in the
other side of the limb. The blood coming in the vein is at
the lower temperature than the blood in the artery. Thus
one side of limb is at lower temperature than the other
side of the limb. Hence under varying environmental
conditions the temperature distribution in human
periphery is interesting for various practical purposes.
ρ̅
= div [ K grad u ] + mb cb (uA – u) + S
(1)
the symbols used in this equation has the meaning as
given below:
ρ= tissue density, ̅= tissue specific heat,
u= tissue temperature, t= time,
K= tissue thermal conductivity,
mb = blood mass flow rate,
cb = specific heat of blood,
uA= Arterial blood temperature ,
S = metabolic heat generation rate.
Geometrically, the human body organs are very close
to cylinder in shape and the projection of transverse cross
section is very close to circular in shape.
588
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
u(a0, 𝜃) = function of 𝜃,
Hence, the above equation (1) for three dimensional
steady state case in cylindrical polar form with M= mb cb
can be put as,
+
II.
Now, on non-dimensionalizing r and u and
transforming the equations (2) and (3) in variational form
with respect to radial coordinate we get
+
+
+
̅=
̅
̅
(
̅
) ―
+2 ̅
and S in the
a0[ h(
̅
+2 ̅
(5)
),
,
=
̅
(
]
) +
̅+
,
th
(6)
are the values of K , M
layer and
-
a)
2
– α2
III.
]Ω ,
SOLUTION
Since the region under study has been divided into
three concentric layers on the basis of their biological
properties, following assumptions in mathematical form
regarding M, S and K is taken
As per the physiology and anatomy of the human limb,
core temperature also vary along angular direction hence
at the inner boundary taking the following variation as
boundary condition, (0 ≤ 𝜃 ≤ 2π)
Layer No.
K
M
I
K1
m
III
̅ (
̅
( ̅ - ̅) +
( ̅ -̅ )
– α2 ]Ω
a)
),
=(
= ∫̅
h = Coefficient of heat transfer to environment,
ua = environmental temperature.
L = Latent heat of evaporation.
E = rate of evaporation from the surface.
K=
) +
2
Further, on the basis of biological properties, circular
cross-section is partitioned into three concentric layers
with radii a0 (core of the limb), a1, a2 and a3 (outer surface
of the region)
in r direction. Hence the descretized
variational integral(5) for each sub-region is represented
by
Where symbols are defined as under
= ̅
,
(3)
II
(
LE/
(2)
)
̅
Where τ is the region under study and Ω is the outer
surface of the region,
Heat flow inside human body occurs when the
temperature of the body surface is lower than that of the
body interior. The body supply to the skin is the chief
determinate of heat transport to the skin. Heat loss occurs
by the processes of radiation, Conduction Convection
and evaporation. Hence at outer surface the boundary
condition is given by
h (u
̅ + a0[ h(
]
Here assuming that the physiological parameters are
symmetrical with respect to Z-direction but varies along
angular and radial direction. The region under study is a
transverse cross section of human thigh which is
geometrically circular in shape. Therefore for study state
two dimensional case, Perl's heat equation, in polar
cylindrical coordinates reduces to the following form
K
̅ ( ̅) ―
I= ∫
MATHEMATICAL FORMULATION
+
(4)
M=m
K3
(̅ - ̅ )
0
̅
589
S
(1)
S=s
(̅ - ̅ )
0
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
Now the following variation for field variable
assigned in each layer
=
̅ ,
+
is
(
̅
(7)
(̅
in which
̅
=
̅
̅
̅
)
),
(
=
layer,
and
is the value of
in the
)
th
[
The integral I in equation (6 ) for each layer is
evaluated and assembled to find
]
Now I (equation 8) is extremized with respect to each
to give the following system of partial differential
equations,
∑
∑[
]+
(9)
]+
,
(12)
Where
and
are constants depending
on various constants and parameters involved in above
equations. At the inner boundary (core of the thigh), the
following parabolic variation is assigned to
along
angular direction;
]+
]+
]
𝜃
(10)
𝜃
]+
𝜃
𝜃
(13)
Where, if
and
are the temperatures at 𝜃
and 𝜃
respectively, then
(11)
(
(
)
)
(
and
)
The equations (12) are solved by using Fourier series
for 𝜃 coordinate. Hence we take , for
̅
̅
̅
∑ [
(
– ̅
𝜃
)
𝜃 ]
̅
Equations (12 ) and (14 ) yield the following system
of simultaneous equations
̅
̅
̅
̅
̅
(15)
̅
here, notations are defined as given below
[
(
(
(14)
̅
̅
̅
̅
]
)
̅
)
590
[
]
̅
[
]
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
̅
[
Using the values of above Fourier coefficients in
equation (14 ) we get the value of . Now the tissue
temperature can be obtained by the relation
]
[
,
]
IV.
in which
,
,
,
NUMERICAL RESULTS AND C ONCLUSION
The calculations have been performed for two
different cases of atmospheric temperatures
as
given below:
,
,
.
15
,
0.003 w1
,
=
,
0.0357 w2
=
0 w3
=
=
,
23
0.018 w1
0.018 w2
w3
,
Where,
=
,
,
,
=
,
,
,
,
The numerical results have been obtained by using
the following clinical and experimental values of
physical and physiological constants/parameters for
human body :
,
.The system of equation (15 ) is solved to obtain
Fourier series constants which are given below;
̅
⁄
,
]
The graph has been plotted for different values of
angle (for 𝜃
. The drop in tissue
temperature is shown in Figure 1. On analyzing the
graph, we find that the tissues temperature drop sharply
when outer surface of the thigh is exposed to
environment with the condition
at
23 as compared to
no evaporation at
15 . The temperature
gradient also increases as we move from inner boundary
to outer surface. This indicates that as evaporation
increases at the outer surface of the region understudy
the heat loss from human body also increases and we
conclude that evaporation play significant /major role in
the process of heat loss from the human body. The
variation in tissues temperature in angular direction, in
each layer reflects the effect of assumed boundary
condition at the core of thigh and this effect decreases as
we move towards outer surface from the core.
]
̅
]
̅
̅
,
,
,
,
Where
=
=
,̅ =
,
] ,
=
̅
,
,
=
=
̅
591
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
Saxena V. P., and Pardasani K. R., “Effect of dermal tumors on
temperature distribution in skin with variable blood flow,”
Mathematical Biology, USA. Vol. 53, No.4, 525-536,1991.
[9] Mitchell J. W., Galvez T. L., Hengle J., Myers G. E., and
Siebecker K. L., “Thermal response of human legs during cooling
“J.Appl.Physiology,U.S.A., 29 (6), 859-865, 1970.
[10] Weinbaum S, Jiji LM. A new simplified bioheat equation for the
effect of blood flow on local average tissue temperature. J
Biomech Eng.;107:131–139.1985
[11] Saxena V. P, and Bindra J. S., “Pseudo-analytic finite partition
approach to temperature distribution problem in human limbs,”
Int. J. Math.Sciences. Vol. 12, 403- 408, 1989.
[12] Saxena V. P.., and Gupta M. P., “Steady state heat migration in
radial and angular direction of human limbs,” Ind. J. Pure. Appl.
Math. 22(8),657- 668, 1991.
[8]
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Chatterji, C.C., Text book of Human Physiology, Medical Allied
Agency, ,2/1-5, 1985.
Guyton A.C. and Hall J.E. , Text book of Medical physiology,
Elsevier,2008.
Gary,H., Text Book of Gary`s Anatomy. Longmans Press,1973
Pennes H.H., Analysis of tissue and arterial blood temperatures
in the resting human forearm. J Appl Physiol. 1998; 85: 5–34.
1948
Perl W., “Heat and matter distribution in body tissues and
determination of tissue blood flow by local clearance methods,”
J.Theo. Biol. 2, 201- 235, 1962.
Patterson A. M., “Measurement of temperature profiles in human
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592
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)
[13] Pardasani K. R., and Shakya M., “Infinite element thermal model
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