Computation of Tissue Temperature in Human Periphery during Different Biological and Surrounding Conditions

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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)

588

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

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.

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

ρ ̅

= 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.

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Hence, the above equation (1) for three dimensional steady state case in cylindrical polar form with M= mb cb

can be put as,

+

II. MATHEMATICAL FORMULATION

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

+

+ (2)

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

K

h (u ) (3)

Where symbols are defined as under

h = Coefficient of heat transfer to environment, ua = environmental temperature.

L = Latent heat of evaporation.

E = rate of evaporation from the surface.

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)

u(a0, 𝜃) = function of 𝜃, (4)

Now, on non-dimensionalizing r and u and transforming the equations (2) and (3) in variational form with respect to radial coordinate we get

I = ∫ ̅ (

̅) ― ̅ (

) + ̅ + 2 ̅

] ̅ + a0[ h( a) 2

– α2 ]Ω (5)

Where τ is the region under study and Ω is the outer surface of the region,

̅ = , =( ) , (

), , =

LE/

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

= ∫_̅ ̅ ̅

(

̅ ) ― ̅ (

) +

̅ _{+ 2}_̅ _]_̅₊ ₍₆₎

̅ , are the values of K , M

and S in the th layer and

a0[ h( - a)2 – α2 ]Ω ,

III. 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

Layer No. K M S

I K1 m (1)

II K= ( ̅ - ̅) + _{( ̅ -}_̅ ₎ M=m ( ̅ - ̅ ) S=s _{( ̅ - ̅ )}

III K3 0 0

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Now the following variation for field variable is assigned in each layer

₌₊_̅ _,₍₇₎

in which

= ̅ ̅

= and is the value of in the th layer,

The integral I in equation (6 ) for each layer is evaluated and assembled to find

∑

] +

] + (9)

] +

] + , (10)

] +

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( )

̅

( – ̅ )

̅

( ̅ ̅ )

( ̅

̅) ,

( )

[ ]

Now I (equation 8) is extremized with respect to each to give the following system of partial differential equations,

∑ [ 𝜃 ]

(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 𝜃 𝜃 (13)

Where, if and are the temperatures at 𝜃

and 𝜃 respectively, then

(

) and (

)

The equations (12) are solved by using Fourier series for 𝜃 coordinate. Hence we take , for

∑

[ 𝜃

𝜃 ]

(14)

Equations (12 ) and (14 ) yield the following system of simultaneous equations

(15)

here, notations are defined as given below

[

]

̅ [

] ̅ [

]

̅

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̅ [ ]

[ ]

in which

, , ,

, = , =

=

= , ,

, = , , ,

, ,

The system of equation (15 ) is solved to obtain Fourier series constants which are given below;

̅ ]

̅ ,

,

, ,

Where

=

= ] _,_̅

, ̅ = , =

,

=

= ̅

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. NUMERICAL RESULTS AND CONCLUSION

The calculations have been performed for two different cases of atmospheric temperatures as given below:

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 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.

15 23

0.003 w1

0.0357 w2

0 w3

0.018 w1

0.018 w2

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REFERENCES

[1] Chatterji, C.C., Text book of Human Physiology, Medical Allied

Agency, ,2/1-5, 1985.

[2] Guyton A.C. and Hall J.E. , Text book of Medical physiology,

Elsevier,2008.

[3] Gary,H., Text Book of Gary`s Anatomy. Longmans Press,1973

[4] Pennes H.H., Analysis of tissue and arterial blood temperatures

in the resting human forearm. J Appl Physiol. 1998; 85: 5–34. 1948

[5] 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.

[6] Patterson A. M., “Measurement of temperature profiles in human skin,”S.Afr.J.Sc.72, 78-79, 1976.

[7] Saxena V. P., .and Bindra J. S., “Indian J. pure appl.Math.”, 18(9), 846-55, 1987.

[8] 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.

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[13] Pardasani K. R., and Shakya M., “Infinite element thermal model for human dermal regions with tumors." International Journal of Applied Sc. & computations, vol. 15 No., PP. 1-10, 1 2008.

[14] Gurung D.B. and Saxena V.P. Transient temperature distribution