Journal of Mathematics and Statistics (3): 84-88, 005 SSN 549-3644 005 Science Pulications A Two Dimensional nfinite Element Model to Study Temperature Distriution in Human Dermal Regions due to Tumors K.R. Pardasani and Madhvi Shaya Department of Applied Mathematics, M.A.N..T., Bhopal, ndia Astract: n this study, a two dimensional infinite element model has een developed to study thermal effect in human dermal regions due to tumors. This model incorporates the effect of lood mass flow rate, metaolic heat generation and thermal conductivity of the tissues.the dermal region is divided into three natural layers, namely, epidermis, dermis and sudermal tissues. A uniformly perfused tumor is assumed to e present in the dermis. The domain is assumed to e finite along the depth and infinite along the readth. The whole dermis region involving tumor is modelled with the help of triangular finite elements to incorporate the geometry of the region. These elements are surrounded y infinite domain elements along the readth. Appropriate oundary conditions has een incorporated. A computer program has een developed to otain the numerical results. Key words: Rate of metaolic heat generation, thermal conductivity, lood mass flow rate, specific heat, tissue density NTRODUCTON A disturance-say, an increase in metaolic heat production due to some anormality-upsets the thermal alance. Heat is stored in the ody and core temperature rises. Core temperature continues to rise and these responses continue to increase until they are sufficient to dissipate heat as fast as it is eing produced, thus restoring heat alance and preventing further increases in ody temperatures. The rise in core temperature that elicits heat-dissipating responses is sufficient to reestalish thermal alance. When the disturance is too large, the whole ody is no longer ale to function. The study of relationships among various parameters, systems and organs of a human ody are of clinical importance for iomedical scientists doctors,in diagnosis and cure of certain diseases lie cancer etc and their treatment as well. Perl gave a mathematical model of heat and mass distriution in tissues. He comined the Fic's second law of diffusion and Fic's perfusion principle along with heat generation term to deduce the model. The partial differential equation derived y him is given y: ρ c T div ( KgradT ) + m c T T S t ( + ) () The effect of lood flow and effect of metaolic heat generation are given y the terms m c (T -T) and S, respectively. Where, tissue density, c specific heat of the tissue, T unnown temperature, t time, K thermal conductivity, m lood mass flow rate, c specific heat of lood, T lood temperature, S rate of metaolic heat generation. Perl [] solved a simple case of equation () for a spherical symmetric heat source emedded in an infinite, tissue medium. Cooper and Treze [] found an analytic solution of heat diffusion equation for rain tissue with negligile effect of lood flow and metaolic heat generation. Chao, Eisely and Yang [3] and Chao and Yang [4] applied steady state and unsteady state models with all the parameters as constant, to the prolem of heat flow in the sin and sudermal tissues. Treze and Cooper [5] otained a solution for a cylindrical symmetry considering all the parameters as constant and computed the thermal conductivity of the tissue. Saena [6,7] otained an analytical solution to one dimensional prolem taing position dependent values of lood mass flow and metaolic heat generation. Later on Saena and Arya [8] and Arya and Saena [9] initiated the use of finite element method for solving the prolem of temperature distriution in three layered and si layered sin and sucutaneous tissues. Saena and Bindra [0,] used quadratic shape functions in variational finite element method to solve a one dimensional steady state prolem. Later Saena and Pardasani [,3] etended the application of analytical and finite element approach to the prolems involving anormalities lie malignant tumors. Further Pardasani [,4] investigated the heat and water distriution prolem in sin and sucutaneous tissues. Jas [5] has also studied temperature distriution in cylindrical organs of human ody involving tumors using finite element method.no study has een carried out for infinite domains. We [6] studied thermal changes in infinite element domain of human peripheral regions due to tumors. Corresponding Author: K.R.Pardasani, Department of Mathematics & Computer Applications, M.A.N..T., Bhopal 84
MATHEMATCAL MODEL For a two dimensional steady state case, the partial differential equation () for heat flow in sin and sucutaneous tissues taes the form: ( K ) ( K ) + + m c ( T T ) + S 0 J. Math. & Stat., (3): 84-88, 005 () The tumor is characterized y uncontrolled rates of metaolic heat generation. The normal tissues are characterized y self-controlled rate of metaolic heat generation. n view of this the metaolic term S in eqn.() can e roen into two parts i.e. S S + W Where, S and W are self controlled and uncontrolled rate of metaolic heat generation respectively. We consider a rectangular composite region with oundaries a, and y c, y d with the following conditions. ε, at a, θ, at (3) T(,y) T, at y c, K h( T Ta ) + LE, at y d (4) The last condition incorporates the heat loss y radiation-convection and evaporation at the outer sin surface ecause the outer surface of the sin is eposed to the environment and heat loss taes place from this surface. where h is heat transfer coefficient, T a is the atmospheric temperature, L and E are respectively the latent heat and rate of sweat evaporation. Also the human ody maintains its core temperature at a uniform temperature (37 o C). T is the ody core temperature. The values of and are taen depending on the outward normal flow from the oundaries at a and. The region is divided into sufficiently large numer of two types of elements to match with geometry and to incorporate minute details of physiology.the triangular shaped finite elements and rectangular shaped infinite [7] elements. As the elements size is quite small, we tae linear variation of temperature within each element. The variational formulation of () for some general e th element along with the oundary conditions (3) and (4) is given y: [ ( ) + ( ( ) K K e ) + ( m c ) ( T T ) ( S + W ) T ] ddy + [ ( ) + ] + + h T Ta LET d ε T dy θ T dy τ τ τ3 (5) 85 Where, is the region contained in the e th element. Here the second integral of eq. (5) is valid for elements adoining the outer sin surface, eing the oundary of the e th element eposed to the environment and it is zero for all other elements. n the same way third and fourth integrals are valid only for the elements adoining the oundaries and respectively and taen equal to zero for the remaining elements. Following assumptions have een made for K and (m c ) : K K ( - y), (m c ) m ( - y ) For triangular finite elements: T c +c +c 3 y (6) Where T is equal to T i, T and T at the nodes of the e th element. Thus we have T i c + c i +c 3 y i, T c + c + c 3 y, T c + c + c 3 y n matri from this can e written as: T P C Where, c C c P c3 (7) i yi y y Where, C is otained from (7) and is given elow: C R T, Where R P (8) The epression (6) may also e written as: T P T C, Where P T [ y] (9) On sustituting the value of C from (8) in (9) we have T T P R T (0) The rate of metaolic heat generation is directly proportional to gradient of issue temperature. So when gradient of tissue temperature increases, rate of metaolic heat generation also increases and when the gradient of tissue temperature decreases, rate of metaolic heat generation also decreases. So the rate of self controlled metaolic heat generation for different triangular elements is prescried elow: (i) f y i y and y i > y S s (α - β y ) [ + q ( T i + T - T ) ]
J. Math. & Stat., (3): 84-88, 005 (ii) f y i y and y > y S s (α - β y ) [ + q ( T - T i + T ) ] (iii) f y i y and y > y S s (α - β y ) [ + q ( T i + T - T ) ] (iv) f y y and y > y i S s (α - β y ) [ + q ( T i - T i + T ) ] For rectangular infinite elements: For - ξ, - η M (ξ,η) +M (ξ,η) +M 3 (ξ,η) 3 +M 4 (ξ,η) 4 y M (ξ,η) y +M (ξ,η)y +M 3 (ξ,η)y 3 +M 4 (ξ,η) y 4 Where the mapping functions M, M, M 3 and M 4 are given y : M (ξ,η) ( - η)( - ξ) / ( - ξ), M (ξ,η) ( + η)( -ξ) / ( - ξ) M 3 (ξ,η) ( + η)( + ξ) / [( - ξ)], M 4 (ξ,η) ( - η)( + ξ) / [( - ξ)] T N (ξ,η) T i + N (ξ,η) T + N 3 (ξ,η) T + N 4 (ξ,η) T l sucutaneous tissue (SST) region of human ody has een divided into 0 layers. The epidermis consists of one layer. The dermis is divided into five layers and sudermal tissues are divided into four layers.the innermost layer is the core consisting of one, muscles, large lood vessels etc.the vertical cross section of sin and sudermal tissues with a solid tumor is shown in Fig. which is discretized into 55 triangular finite elements and 0 rectangular infinite elements. The epidermal layer is discretized into 5 triangular elements with infinite elements. The dermal layer is discretized into 9 elements with 5 infinite elements and also include 3 elements of tumor region. The sucutaneous tissues is discretized into 08 elements with 4 infinite elements Now using aove assumptions and epressions, the integrals are evaluated and assemled a given elow: 56 Σ (5) e Where the shape functions N, N, N 3 and N 4 are given y : N (ξ,η) ( - η)(ξ - ξ ) / 4, N (ξ,η) ( + η)(ξ - ξ ) / 4 N 3 (ξ,η) ( + η)( - ξ ) /, N 4 (ξ,η) ( - η)( - ξ ) / Let N e ( e ) () N total no. of elements. Now is minimized y differentiating it with respect to each of the nodal temperature and setting the derivatives equal to zero. / T 0 () Here, / T denotes the differentiation of with respect to each nodal temperature as given elow: ' (3) Also we define... i... [ T T T T T T ] ' T... i... (4) n Here n total numer of nodal points. NUMERCAL RESULTS AND DSCUSSON Here a uniformly perfused tumor is assumed to e situated in the dermal region of the ody. The sin and n 86 Fig. : Discretized section of sin The integral is etremised with respect to each nodal temperature T i (i ()3) to otain a following set of algeraic equations in terms of nodal temperature T i (i()3). X T Y (6) Here X, Y and T are respectively the matrices of order 3 3, 3 and 3. A computer program has een developed for the entire prolem in Microsoft Fortran Powerstation. Gaussian elimination method has een employed to solve the set of equations (6) to otain nodal temperatures which give temperature profiles in each suregion. The values of physical and physiological parameters have een taen from Cooper and Treze [], Saena and Bindra [0] and Pardasani and Saena [,3] as given elow:
K 0.030 Cal cm-min. C, K 0.0845 Cal cmmin. C, K 3 0.060 Cal cm-min. C M 0.003 Cal cm 3 -min. C, S 0.0357.Cal cm 3 - min, L 579.0, h 0.009 Cal cm -min. C The results have een computed for the following two cases of atmospheric temperatures and parameters m, S and E have een assigned the values as given in Tale. Tale : Values of m, S and E Atm. Temp M ma(m c ) ma S ma s cal cm 3 min Egm cm min T ( C) m cal cm 3 -min. C 5 C 0.003 0.0357 0.0 3 C 0.08 0.08 0.0,0.4X0-3 33 C 0.35 0.08 0.40-3, 0.480-3, 0.70-3 J. Math. & Stat., (3): 84-88, 005 The following values have een assigned to physical and physiological parameters in each suregion. Fig. 3: Graph etween temperature T (in C) and, y (in cm) at atmospheric temp. T a 3 C and E 0.0 gm cm for normal case i. Epidermis λ λ W β 0, K 0.03, m 0,q /(T a + T ), α, s S/6 ii. Dermis λ 3, λ.5, K 0.03, φ 4, φ 5, m m,w 0, q /(T a + T ), α 3.8, β 4.6, s S iii. Sudermal Tissues λ, λ W β 0, K 0.06, m m, q /T, α, s S iv. Tumor region λ, λ φ β s 0, K 0.036, φ, m m, α 5, W S, q 0 Fig. 4: Graph etween temperature T (in C) and, y (in cm) at atmospheric temp. T a 33 C and E 0.4 0 3 gm cm for normal case Fig. : Graph etween temperature T (in C) and, y (in cm) at atmospheric temperature T a 5 C and E 0.0 gm cm for normal case The various temperature profiles have een studied. Figure represents the temperature profiles for Fig. 5: Graph etween temperature T (in C) and, y (in T a 5 C and E 0.0 gm cm for a normal case cm) at atmospheric temp. T a 33 C and E (without tumor). Figure 3 shows temperature profiles 0.4 0 3 gm cm. Solid lines for normal for T a 3 C, E 0.0 gm cm for a normal case tissues and dashed lines for anormal tissues 87
J. Math. & Stat., (3): 84-88, 005 (without tumor). The fall in temperature profiles for a normal case (without tumor) is more for T a 5 C and E 0.0 gm cm as compared to that for T a 3 C, E 0.0 gm cm. This may e ecause the heat loss is more from the sin surface at low atmospheric temperatures due to the increase in temperature gradient at the sin surface. Figure 4 shows temperature profiles for T a 33 C, E 0.40-3 gm cm for a normal case (without tumor). The temperature profiles for T a 5 C and E 0.0 gm cm and T a 33 C, E 0.40-3 gm cm are close inspite of a difference of 8 C in atmospheric temperature and a small rate of sweat evaporation at higher temperature.this is due to large temperature gradient on the surface for low atmospheric temperature which causes more heat loss than that for smaller temperature gradients due to high atmospheric temperature. Figure 5 shows temperature profiles for T a 33 C, E 0.40-3 gm cm for a normal case (without tumor) and an anormal case (with tumor). The temperature profiles fall down as we move away from ody core to the sin surface. The numerical results for a normal case are comparale with those otained y Saena and Bindra [0]. An elevation in temperature profiles for sin and sudermal tissues with a tumor is oserved while comparing the profiles for the normal and anormal tissues.the maimum thermal disturances are seen in the region etween y 0.5 cm and y 0.7 cm. The points of change in the slopes of the curves are actually the unction of normal tissues and tumor region. This information is useful for ustifying the oundaries of tumor region and normal region. Such models can e developed to generate the thermal information which may e useful for iomedical scientists for diagnosis and treatment of cancer. REFERENCES. Perl, W., 963. An etension of the diffusion equation to include clearance y capillary lood flow. Ann. NY. Acad. Sci., 08: 9-05.. Cooper, T.E. and G.J. Treze, 97. A proe technique for determining the thermal conductivity of tissue. J. Heat Trans., ASME, 94: 33-40. 3. Chao, K.N., J.G. Eisley and W.J. Yang, 973. Heat and water migration in regional sin and sucutaneous tissues. Bio. Mech., ASME, pp: 69-7. 4. Chao, K.N. and W.J. Yang, 975. Response of sin and tissue temperature in Sauna and steam aths. Bio. Mech. Symp. ASME, pp: 69-7. 5. Treze, G.J. and T.E. Cooper, 968. Analytical determination of cylindrical source temperature fields and their relation to thermal diffusivity of rain tissues. Thermal Prolems in Biotechnol., ASME, NY., pp: -5. 6. Saena, V.P., 979. Effect of lood flow on temperature distriution in human sin and sudermal tissues. Proc. 9th Natl. Conf. Fluid Mechanics and Fluid Power, pp: 56-6. 7. Saena, V.P., 983. Temperature distriution in human sin and sudermal tissues. J. Theo. Biol., pp: 77-86. 8. Saena, V.P. and D. Arya, 98. Steady state heat distriution in epidermis, dermis and sudermal tissues. J. Theor. Biol., 89: 43-43. 9. Arya, D. and V.P. Saena, 98. Application of variational finite element method in the measurement of thermal conductivity of human sin and sudermal tissues. Proc. Natl. Acad. Sci., ndia, 5(A), pp: 447-450. 0. Saena, V.P. and J.S. Bindra, 984. Steady state temperature distriution in dermal lood flow, perspiration and self controlled metaolic heat generation. ndian. J. Pure and Appl. Math., 5: 3-4.. Saena, V.P. and J.S. Bindra, 989. Psuedo analytic finite partition approach to temperature distriution prolems in human lims. ntl. J. Math and Math. Sci. USA., : 403-408.. Pardasani, K.R. and V.P. Saena, 99. Effect of dermal tumor on temperature distriution in sin with variale lood flow. Bull. Math. Biol., USA., 53: 55-536. 3. Pardasani, K.R. and V.P. Saena, 989. Eact solutions to temperature distriution prolem in annular sin layers. Bull. Calcutta Math. Soc., 8: -8. 4. Pardasani, K.R., 988. Mathematical investigations on human physiological heat flow prolems with special relevance to cancerous tumors. Ph.D. Thesis, Jiwai University Gwalior, pp: 5-0. 5. Jas, P., 000. Finite element approach to the thermal study of malignancies in cylindrical human organs. Ph.D. Thesis, Barhatullah University, Bhopal. 6. Pardasani, K.R. and M. Shaya, 004. Two dimensional infinite element model of temperature distriution in human peripheral regions due to uniformly perfused tumors. Proc. Natl. Conf. Biomechanics, T, New Delhi, Nov. 9-. 7. Marques, J.M.M.C. and D.R.J. Owen. nfinite elements in quasi-static materially non-linear prolems. Computers & Structures, 8: 739-75. 88