Internat. J. Math. & Math. Sci.
VOL. 12 NO. 2 (1989) 403-408
403
PSEUDO-ANALYTIC FINITE PARTITION APPROACH TO TEMPERATURE
DISTRIBUTION PROBLEM IN HUMAN LIMBS
v.P. SAXENA
School of Mathematics and Allied Sciences
JiwaJi University
Gwa I i o r
and
J.S. BINDRA
Department of Applied Sciences
Mathematics Section
G.N. Engineering College
Ludhiana
(Received March I0, 1987 and in revised form September 8, 1988)
ABSTRACT.
The main object of this paper is to [ntroduce the Pseudo Analytic Finite
Partition method and to illustrate its use in solving physiological heat distribution
problems pertaining to limbs and slmllar body organs.
A two
d[mensional
circular
region resembling the cross section of a human or animal llmb is considered.
biological properties are assumed to vary along the radial direction.
The
The theoretical
model incorporates the effect of blood mass flow and metabolic heat generation.
The
region is divided into annular sub-regions and Ritz variational ffnlte element method
is applied along the radial direction, while for the angular direction, Fourier Series
has been used due to uniformity in each annular part.
KEY WORDS AND PHRASES.
Blood mass
flow rate,
rate
of
metabolic
heat
generation,
variational finite element method.
1980 AMS SUBJECT CLASSIFICATION CODES.
I. INTRODUCTION.
A human body maintains its body core temperature at a uniform temperature under
In order to maintain this core temperature,
atmospheric conditions.
parameters like rate of blood mass flow, rate of metabolic heat generation and,
thermal conductivity vary in response to changes in atmospheric conditions. However,
the
normal
in
extreme
parts
of
a
htan
body,
the
core
temperature
is
not
uniform
at
low
atmospheric temperatures, where the core temperature of limbs varies extensively as we
move away from the body core.
This may be because the arterial blood has cooled down
while travelling towards the extremities.
by W. Perl [I]
as given below_
The heat flow in fn-vivo tissues is given
404
V.P. SAXENA AND J.S. BINDRA
pc-
Div(K grad u) +
(I.I)
+ s
mbCb(Ub-U)
where ,O,c, K and S are respectively the density, specific heat, thermal conductivity
and, rate of metabolic heat generation in
m
tissues,
rate and specific heat of the blood respectively.
and c
b
b
are the mass blood flow
[I] derived and used this
W. Perl
Chao, Eisley and
equation to study simple problems of heat flow in tissue medium.
and Chao and Yang [3] also studied temperature distributions in infinite
Yang [2]
Cooper and Trezek [4] obtained a solution for a cylindrical symmetry
Saxena [5,6], Saxena and Arya [7,8], and Saxena
tissue mediums.
with all the parameters as constant.
and
[9,10] used this model
Bindra
subcutaneous
performed
tissues
for
a
using
one
to
analytical
dimensional
study temperature distributions in skin and
and
steady
numerlcal
state
techniques.
Later
case.
This
study
was
Arya [II], Arya and
Saxena [12] and Saxena and Bindra [13] investigated this problem for a twv dimensional
Here a cylindrical limb having
steady state case in skin and subcutaneous tissues.
circular
cross
section
with
layers
of
tissues
with
different
properties
is
The outer boundary is assumed to be exposed to the environment and heat
considered.
loss takes place due to convection, radiation and, evaporation.
The innermost solid
cross section is assumed to be at a known variable temperature.
This case may occur
when one side of the llmb contains major blood vessels and thus heated constantly by
the blood commlng out of main trunk.
2. MATHEMATICAL FORMULATION.
Equation (l.1) in polar steady state form can be written as:
u_r)
Krr
The
a
region
and a
is
+
K _ _.2Ur
2
+
divided
mbcb r(ub_u
into
N
+ rS
layers
(2.1)
0
with
inner
and
outer
radii
equal
to
respectively. (see Fig.l).
N
o
The boundary and Initial conditions imposed are
I-K
and
r
u(a ,8)
o
r-a
+ LE
h(U-Ua
(2.2)
N
f(8)
where f(0) is known.
u
(2.3)
o
Here h is the heat transfer coefficient and,
respectively the latent heat and rate of sweat evaporation.
L and E are
S is assigned temperature
dependent values given by
S-s
(Ub-U)/Ub,
where s is constant.
Thus equation (2.1) along with boundary conditions (2.2) and (2.3) is reduced to
descretlzed polar variational form as given below:
(2.4)
405
TEMPERATURE DISTRIBUTION PROBLEM IN HUMAN LIMBS
a
i
(I),
is the external radius of Ith annular region as shown in Fig.
ai/ao, ai(i--l(1)N)
Ki,
-
i
and v
a
z
(i)
are values of K, M and, v in the ith region,
+ s/u b)
(mbcb
0
r/a O’ v
(Ub-U)/Ub
v
a
--(Ub-Ua)/U b
The following linear shape functions have been taken for each region:
v( i )= aivi-I ai-lvi
a
a
i
v
i
vi_
a
a
+
i-I
i
(2.5)
r.
i-I
Evaluating integrals (2.4) and assembling these we get
N
r.
i-I
I
(2.6)
li
Now I is extremlzed with respect to v
N
i
l
Where
i
+
(Ej vj
i
and H
Ej Fj
i
i
and the following equations are obtained:
2v
i
--)
Fj
are
H
(2.7)
i
depending
constants
upon
physical
and
physiological
pa tame te r s.
Now Fourier Series is applied to eliminate the
6 variable from the equation (2.7).
We take
v
o
A
v
Aoi
+ I
oo
i
+ I
n=l
Here the coefficients
the
(A
n=l
and
coefficients
no
CosnO + B
no
(AniCosnO
+ B
ni
and
(2.9)
Sinn0)
Aoo Ano and Bno are
Aoi, Ani
(2.8)
Sinn0)
known due to boundary condition (2.3).
Bni(i=l,2,...,N)
are
unknown.
All
Accordingly,
following system of linear equations is obtained:
o
o
E A
o
--G
EA =C
n
2
EB
2
n
(2.10)
=G
v
where
[Ej]
(i--l(1) N,
J
I(I) N) are square matrices of order N.
the
V.P. SAXENA AND J.S. BINDRA
40
v
0
An--
[Aoll
Ej,_ Aol Ani Bnl
earlier.
Bn-- [Bnll
[Ani]
and G
i
[Gi
G
and
i=l(1)N
where
vectors.
are constants and they depend on the parameters mentioned
Here only a special case in which the annular cross-sectlon of the llmb has
These two layers have different
been divided into two layers i.e. N=2 is considered.
The outermost layer is supposed to be made of mainly dead
biological properties.
tissues.
are
Hence it does not have any metabolic heat generation or blood flow.
The core of the llmb is
quantities have been prescribed in the inner annular part.
assumed to have unsymmetrlc temperature.
These
Parabolic variation of temperature along the
circular boundary has been taken, so
V
where w
o
(u
0
F(
2
b- u)/u b
(u
w
0=0 and O= respectively.
A
Ano
oo
(2 "I0).
and B
are
no
w
2
b
o
u
Using
-I-
(- 2
B) /u b
.2)
w
and u
-I-
2
o
are temperatures at
F(0) in terms of Fourier series,
determined and
then substituted
(2.9) to obtain v I.
the
constants
the system of equations
in
These equations are solved to find the values of
turn are substituted in expressions
w
2
Aol, Ani’
and
Bni
which in
Using (2.5) and (2.9), the
temperature profiles are obtained for each sub-reglon.
3. NUMERICAL RESULTS
The following values of physical and physiological parameters and constants have
been taken:
KI--0.06
Cal/cm-min. deg. C,
a -2.5 cm,
o
el=5
cm,
a2--7.5
K2=0.03
Cal/cm-mln. deg. C.,
cm, L-579 Cal/gm,
/cm2_mi n. deg. C, Ub=37 oC,
Ml=(mb Cb)l--0.003 Cal/cm3-min. deg. C, M2-(mb Cb)2--0
(S)i=si=0.0357 Cal/cm3-min, (S)2=s2--0,
o
o
o
u=30 C, u--34 C, Ua=15 E--0,
h--O.009 Cal
Graphs have been plotted between u and 8 and the temperature profiles have been
shown in fig. 2, 3 and, 4.
Fig. 2 is a geometrical representation of the boundary
conditions and figures 3 and 4 give temperature variation in direction around the two
annular partitions.
Fig. 2.
These two curves are significantly different from the curve in
There is a slow rise in the temperature at 0=0.
This rise becomes sharper
later on and obviously the temperature takes on its maximum value at
--.
The linear
variation of temperature with respect to r is easily seen by comparing the figures 3
and 4.
The number of computations involved in this method are less as compared to those
in variational finite element method for two dimensional case.
TEMPERATURE DISTRIBUTION PROBLEM IN HUMAN LIMBS
407
34’
32
\
/
\
/
28-
Fig. I.
ANNULAR CROSS-SECTION OF A HUMAN
LIMB DIVIDED INTO N LAYERS. SHADED
PORTION IS SOLID CIRCULAR SECTION.
27"
6
2
6
6
2
6
Fig. 2. GRAPH SHOWING THE
GEOMETRICAb REPRESENTATION OF
THE BOUNDARY CONDITIONS.
28-
27
26
GRAPH BETWEEN U. AND (R), SHOWING THE
TEMPERATURE
AROUND THE
INNER ANNULAR PARTITION.
VARIATIONS
,
GRAPH BETWEEN UgAND
SHOWING THE TEMPERATURE -VARIATIONS
AROUND THE OUTER ANNULAR PARTITION.
Fig. 4.
Fig. 3.
V.P. SAXENA AND J.S. BINDRA
408
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I.
2.
3.
4.
5.
6.
7.
8.
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9.