VARIATIONAL FINITE ELEMENT APPROACH TO A HEAT FLOW PROBLEM IN HUMAN LIMBS

An attempt has been made to study cross sectional temperature distribution in in-vivo tissues of a human limb employing variational finite element approach. The outermost surface of the limb isassumed tobe exposed to the atmosphere. The physiological andphysical parameters like rate ofmetabolic heat generation (rmhg), blood mass flow rate Cutoff) and thermal conductivity are assumed to vary in the subregions independently. Numerical results have been obtained for various cases of practical interest.


INTRODUCTION.
The temperature distribution in the peripheral region of limbs of a human body undergo frequent changes on account of changes in atmospheric temperature, the core being maintained at a nearly uniform temperature.The peripheral region comprises of non-uniform layers and has variable blood flow, metabolic heat generation and allied process.This intermediatory region play a very important role in maintaining a balance between the atmospheric variations and the interior consisting of intra-abdominal, intrathoracic and intracranial regions.Usually rectal and oesophageal temperatures are taken to represent core temperature.Oesophageal temperature taken at heart level is a good index of rapid changes of cardiac and aortic blood temperature.The temperature of the limbs and the surface layer of the trunk exhibits a wide variation of temperature.
This paper employs a variational finite element approach to study the temperature distribution in a normal cross-sectional region of a limb.Due to unsymmetric situations of large blood vessels passing through the core of the limb the inter-face has angular variation.The peripheral part of limb is directly exposed to atmosphere.
The peripheral part, assumed to be annular in geometry, has been approximated by the assembly of triangular elements of different sizes.Thus the circular boundaries are approximated by the polygonal one's.Different types of variations of parameters have been considered for different natural subregions such as stratum comeum, stratum germinitivum, dermis and underlying tissue (Montagana [1], Jarrett [2]  and Gray [3]).Finite element formulation provides necessary flexibility in taking care of different behavior of distinctly different subregions. 2.

MATHEMATICAL MODEL.
The rate of change of temperature u at a point in in vivo tissue at time is given by the following partial differential equation (Perl [4]) Ot where e,and K are respectively density, specific heat and thermal conductivity of the tissue; mb and Cb are blood mass flow rate and specific heat of the blood respectively; S is the rate of metabolic heat generation per unit volume and UA is the arterial blood temperature.Above equation has been modified and extensively used by Saxena [5], Saxena and Arya [6], Saxena and Bindra [7,8] in the thermal study of human skin and subcutaneous tissue.Here we employ the same for a human limb with circular symmetry.
The surface of the limb is assumed to be exposed to the atmosphere at temperature u The heat transfer coefficient between skin and the atmosphere may be due to convection, radiation and evaporation.Hence the boundary condition at skin surface can be put as -K Ou h(u u,,) + LE (2.2) where n is the direction of the normal, h is coefficient of convection, L is the latent heat of evaporation, E is the rate of sweat evaporation.At the inner boundary we put where d is a distance constant.The boundary condition (2.4) corresponds to the case of mirror symmetry in temperature distribution about x-axis (horizontal diameter).The variational form of equation (2.1) for a two dimensional steady state case along with the boundary condition (2.2) is given by (Myers [9]) all (0u) (0u) Here the problem region A with boundary if2 is a cylindrical limb with circular cross-sectional and symmetrical with respect to x-axis.The region of interest is semi-circular and is descretized into 150 triangular elements and 96 nodes as shown in Fig. 1.Here the angular points of each element are the nodal points.
The integral in equation (2.5) may be written as I I, +I,,, + t, +In Next we extremize I by differentiating it with respect to each nodal temperature ui and setting derivatives equal to zero.That is dl O, r-k,m,s, In view of the biology the region under study has been divided into one hundred and fifty triangular elements so that the integrals in equation (2.11) are computed as a sum over each of the elements Hence dl, o d--=,.---,r =k,m,s,. (2.13) Here (e) stands for the elements whose nodes are i, j and k.The expression on the right-hand side of equation (2.13) for r will be summed up for the elements on the boundary of the outer surface of the limb.Equations (2.13) will be commuted separately and then substituted into equation (2.11).For the (e)th element u,, uj and uk are the only temperatures to be taken into account.For this element I t'), I e) and Ie) will be function of these three comer temperatures only.Whereas Itu ) will be function of only two comer temperatures which lie on the outer boundary and element (e) adjoining this boundary.Consequently, the partial derivatives of I_ ") for r k,m,s, with respect to all other nodal temperatures will be zero. 3.

SOLUTION.
The following linear variation of temperature within each element is expressed as ut)-prC(') where pr=[1,x,y], C (')- Now u ') is equal to ui, uj and uk at the comers of the eth element whose nodal temperatures are ui, u and uk.Thus we have where With the help of equation (3.4), we can evaluate the integral L Further differentiating I with respect to each nodal temperatures and setting derivatives equal to zero we arrive at the following system of simultaneous equations WU -Z  (3.5)where W is a matrix of order (96 96) and Z is a matrix of order (96 1).Finally solving the system of simultaneous equations (3.5) we obtain the values of nodal temperatures (see Table 1). 4.
The constant 'd' can be assigned any value depending on the sample of the limb under study.Here we have taken d 0.625 cm The values of nodal temperatures have been obtained for two cases of atmospheric temperatures and are given in Table I.The temperatures at the nodes with numbers 1,7,13,19,25,31,37,43,49,55,67,73,79,85 and 91 (along the inner boundary) reflect the variation in the limb core temperature with respect to position.
The effect of this variation in limb core temperature on the other subregions is visible in Table I.On comparing the nodal temperature on outer surface and in each subregion, it is observed that these nodal temperatures vary considerably with the change in atmospheric temperatures and rate of sweat evaporation.The finite element method used here has made it possible to include more details of biology such as the wavy structure of the layers and different values of physical and physiological parameters in each subregion.A computer program was developed and executed on ICIM 6040 to perform these computations.

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

Table I .
Values of Nodal Temperatures for Different Cases

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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