Pseudo-analytic Finite Partition Approach to Temperature Distribution Problem in Human Limbs

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. The biological properties are assumed to vary along the radial direction. 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.

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 the normal atmospheric conditions.
In order to maintain this core temperature, parameters like rate of blood mass flow, rate of metabolic heat generation and, thermal conductivity vary in response to changes in atmospheric conditions.However, 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.
The heat flow in fn-vivo tissues is given by W. Perl [I] as given below_ pc-Div(K grad u) + mbCb(Ub-U) + s (I.I) where ,O,c, K and S are respectively the density, specific heat, thermal conductivity and, rate of metabolic heat generation in tissues, m b and c b are the mass blood flow rate and specific heat of the blood respectively.W. Perl [I] derived and used this equation to study simple problems of heat flow in tissue medium.Chao, Eisley and   Yang [2] and Chao and Yang [3] also studied temperature distributions in infinite tissue mediums.
Cooper and Trezek [4] obtained a solution for a cylindrical symmetry with all the parameters as constant.Saxena [5,6], Saxena and Arya [7,8], and Saxena and Bindra [9,10] used this model to study temperature distributions in skin and subcutaneous tissues using analytical and numerlcal techniques.This study was performed for a one dimensional steady state case.
Later Arya [II], Arya and Saxena [12] and Saxena and Bindra [13] investigated this problem for a twv dimensional steady state case in skin and subcutaneous tissues.
Here a cylindrical limb having circular cross section with layers of tissues with different properties is considered.
The outer boundary is assumed to be exposed to the environment and heat 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.

MATHEMATICAL FORMULATION.
Equation (l.1) in polar steady state form can be written as: _r Krr The region is divided into N layers with inner and outer radii equal to a and a N respectively.(see Fig. l).
o The boundary and Initial conditions imposed are where f(0) is known.Here h is the heat transfer coefficient and, L and E are respectively the latent heat and rate of sweat evaporation.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) a i ai/ao, ai(i--l(1)N) is the external radius of Ith annular region as shown in Fig.
(I), Ki, i and v (i) are values of K, M and, v in the ith region, -a z (mbc b + s/u b) r/a v (Ub-U)/U b v a --(Ub-Ua)/U b 0 O' The following linear shape functions have been taken for each region: Evaluating integrals (2.4) and assembling these we get Where Ej Fj pa tame te r s.Now Fourier Series is applied to eliminate the 6 variable from the equation (2.7).
We take Here the coefficients Aoo Ano and Bno are known due to boundary condition (2.3).All the coefficients Aoi, Ani and Bni(i=l,2,...,N) are unknown.
Accordingly, the following system of linear equations is obtained: [Ej] (i--l(1) N, J I(I) N) are square matrices of order N.Here only a special case in which the annular cross-sectlon of the llmb has been divided into two layers i.e.N=2 is considered.These two layers have different biological properties.
The outermost layer is supposed to be made of mainly dead tissues.
Hence it does not have any metabolic heat generation or blood flow.These quantities have been prescribed in the inner annular part.The core of the llmb is assumed to have unsymmetrlc temperature.Parabolic variation of temperature along the circular boundary has been taken, so V F( and u are temperatures at where w 0 (u b-u)/u b w (u b u /u b 0=0 and O= respectively.Using F(0) in terms of Fourier series, the constants A Ano and B are determined and then substituted in the system of equations oo no (2 "I0).These equations are solved to find the values of Aol, Ani' and Bni which in turn are substituted in expressions (2.9) to obtain v I. Using (2.5) and (2.9), the temperature profiles are obtained for each sub-reglon.

NUMERICAL RESULTS
The following values of physical and physiological parameters and constants have been taken: KI--0.06Cal/cm-min.deg.C, K2=0.03Cal/cm-mln.deg.C., a -2.5 cm, el=5 cm, a2--7.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.These two curves are significantly different from the curve in Fig. 2.
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.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Fig. 2 .
Fig. 2. GRAPH SHOWING THE GEOMETRICAb REPRESENTATION OF THE BOUNDARY CONDITIONS.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation