Journal of Applied Mathematics and Computational Mechanics 03, (3), 39-46 DETERMINATION OF THE TEMPERATURE FIELD IN BURNED AND HEALTHY SKIN TISSUE USING THE BOUNDARY ELEMENT METHOD - PART I Katarzyna Freus, Sebastian Freus, Ewa Machrzak 3 Institute of Mathematics,Czestochowa University of Technology Institute of Computer and Information Science, Czestochowa University of Technology Częstochowa, Poland 3 Institute of Computational Mechanics and Engineering Gliwice, Poland katarzyna.freus@im.pcz.pl, sebastian.freus@icis.pcz.pl, 3 ewa.machrzak@polsl.pl Abstract. In the paper the burned and healthy layers of skin tissue are considered. The temperature distribution in these layers is described by the system of two Pennes equations. The governing equations are supplemented by the boundary conditions. On the external surface the Robin condition is known. On the surface between burned and healthy skin the ideal contact is considered, while on the internal surface limiting the system the body temperature is taken into account. The problem is solved by means of the boundary element method. Keywords: bioheat transfer, Pennes equation, boundary element method Introduction The boundary element method constitutes a very effective tool for numerical simulation of bioheat transfer processes. First of all, it assures the exact approximation of real shapes of the boundaries and also a very good exactness of boundary conditions approximation [, ]. These features of the BEM are very essential in the case of temperature determination in the burned and healthy tissue because small differences in temperature in these sub-domains require very accurate methods for predicting the temperature. In this paper two variants of the BEM are taken into account. For burned tissue, the classical boundary element algorithm for the Laplace equation is used, while for healthy tissue, the BEM algorithm for temperature-dependent source function is applied. These algorithms are coupled by the boundary condition on the contact surface between sub-domains. In numerical realization the linear boundary elements and the linear internal cells for burned tissue sub-domain are used. It should be pointed out that the burned sub-domain does not require the interior discretization. In the final part, the example of computations is presented and the conclusions are formulated.
40 K. Freus, S. Freus, E. Machrzak. Governing equations The burned and healthy sub-domains, as shown in Figure, are considered. Fig.. Domain considered The Laplace equation describes the steady temperature field in burned tissue x Ω : λ T = 0 () where x = (x, x ) are the spatial coordinates, λ is the thermal conductivity of burned tissue, T (x) denotes the temperature. The temperature field in healthy tissue is described by the Pennes equation [3] [ ] B B B met x Ω : λ T + G c T T + Q = 0 () where λ is the tissue thermal conductivity, T (x) is the tissue temperature, G B is the blood perfusion rate, c B is the specific heat of blood, T B is the arterial blood temperature, Q met is the metabolic heat source. On the external surface (cf. Fig. ) the Robin condition is assumed T x Γex _ : λ = α T Ta T x Γ Γ : λ = α T T [ ] [ ] ex _ ex _ 5 a (3) where T a is the ambient temperature, α is the heat transfer coefficient, T e / denotes the normal derivative (e =,) and n = [cosα, cosα ] is the normal outward vector. On the surface between sub-domains the continuity of heat flux and temperature field is taken into account
Determination of the temperature field in burned and healthy skin tissue using - part I 4 T T λ = λ x Γc : n n T = T (4) On the internal surface Γ in the Dirichlet condition is known x Γ : T = T (5) On the remaining boundaries the no-flux condition is accepted. in b. Boundary element method The problem has been solved by means of the boundary element method [, ]. The boundary integral equation corresponding to the equation () is the following (6) * * + ΓI ΓI ΓI ΓI B(ξ) T (ξ) q (ξ) T (ξ, x)d = T (ξ) q (ξ, x)d where Γ Ι = Γ ex_ Γ c, ξ is the observation point, the coefficient B(ξ) is dependent q x = λ T x / n is the heat flux. on the location of source point ξ, ( ) ( ) For the problem considered the fundamental solution T * (ξ, x) is the following * T ξ, = ln πλ r (7) where r is the distance between the points ξ and x. The heat flux resulting from the fundamental solution is defined as where * T (ξ, x) d (ξ, x) = λ = q πr d= ( x ξ )cosα + ( x ξ )cosα (9) (8) The integral equation corresponding to the equation () is the following (0) * * * + Γ = II Γ + II Ω ΓII Γ Ω II B(ξ) T (ξ) q T (ξ, x)d T q (ξ, x)d Q T (ξ, x)d where Γ ΙΙ = Γ c Γ ex_ Γ 3 Γ in Γ 4 Γ ex_5 (cf. Fig. ), and * * T T (ξ, x) Q = GBcBTB + Qmet, q = λ, q = λ ()
4 K. Freus, S. Freus, E. Machrzak The fundamental solution and heat flux resulting from the fundamental solution in the case discussed have a form GBc B T (ξ, x) = K 0 r πλ λ d GBc B GBc B q (ξ, x) = K r πr λ λ () where K 0 (. ), K (. ) are the modified Bessel functions of second kind, the zero order and the first order, respectively [, ]. 3. Numerical realization To solve the equations (6) and (0) the boundary Γ is divided into N elements Γ, =,,..., N and the interior Ω is divided into L internal cells as shown in Figure. Next, the integrals in the equations (6) and (0) are replaced by the sums of integrals over these elements, this means N N (3) = Γ = Γ B(ξ) T (ξ) + q T (ξ, x)d Γ = T q (ξ, x)dγ N * = N + Γ B(ξ) T (ξ) + q T (ξ, x)dγ = N L * * Γ + = N + Γ l= Ωl T q (ξ, x)d Q T (ξ, x)dω l (4) where N is the number of elements on the boundary limiting domain Ω. In the paper the linear boundary elements are applied. Finally one obtains the following systems of algebraic equations: for the burned region for the healthy tissue domain K K Gik qk= HikTk, i=,,..., K (5) k= k= K K Gik qk= HikTk + Pi, i= K+, K+,..., K (6) k= K+ k= K+ where K, K K are the number of boundary nodes located at the boundary limiting sub-domains Ω and Ω, respectively.
Determination of the temperature field in burned and healthy skin tissue using - part I 43 The system of equations (5) and (6) can be written in the matrix convention and Gq= HT (7) Gq= HT+ P (8) Details of the calculation of matrix elements appearing in equations (7) and (8) are described in [3]. Fig.. Discretization of boundaries and interior Ω The equations (7) and (8) can be written in the following form for the burned region ex_ ex_ ex_ q ex_ T G Gc = H Hc qc Tc for the healthy tissue domain (9) qc Tc ex _ ex _ q T 3 3 q _ 3 4 _ 5 T ex in ex ex _ 3 in 4 ex _ 5 c = G G G G G G in Hc H H H H H + P in q T 4 4 q T ex _ 5 ex _ 5 q T where: ex_ ex_ T, q are the vectors of functions T and q at the boundary Γ ex_ of domain Ω, Tc, Tc, qc, q c are the vectors of functions T and q on the contact surface Γ c between sub-domains Ω and Ω, (0)
44 K. Freus, S. Freus, E. Machrzak ex _ 3 in 4 ex _ 5 ex _ 3 in 4 ex _ 5 T, T, T, T, T, q, q, q, q, q are the vectors of functions T and q at the boundary Γ ex_ Γ 3 Γ in Γ 4 Γ ex_5 of domain Ω. The condition (4) written in the form qc = qc = q Tc = Tc = T () should be introduced to equations (9), (0). Next, coupling these systems of equations and taking into account the remaining boundary conditions, one has ex _ T T q ex _ ex _ ex _ α G H Hc Gc 0 0 0 0 0 T ex _ ex _ 3 in 4 ex _5 ex _5 3 = 0 Hc Gc αg H H G H αg H T in q 4 T ex _5 T ex _ αg T a = ex _ ex _5 in α G Ta + αg Ta + HTb + P Finally, the system of equations () can be written in the form () AX= B (3) where A is the main matrix, X is the unknown vector and B is the free terms vector. The system of equations (3) enables the determination of the missing boundary values. Knowledge of nodal boundary temperatures and heat fluxes allows to calculate the internal temperatures at the optional points selected from the burned and healthy tissue sub-domains [, ]. 4. Results of computations The rectangular domain of dimensions L L (L = 0.0 m) shown in Figure has been considered. The shape of internal surface Γ c is defined by the parabola with vertex (0.0, 0.06). The following input data have been assumed: thermal conductivity of burned tissue λ = 0. W/(mK), thermal conductivity of healthy tissue λ = 0. W/(mK) [4], blood perfusion rate G B = 0.5 kg/(m 3 s), specific heat of blood c B = 400 J/(kgK), arterial blood temperature T B = 37 C, metabolic heat
Determination of the temperature field in burned and healthy skin tissue using - part I 45 source Q met = 00 W/m 3, heat transfer coefficient α = 0 W/(m K), ambient temperature T a = 0 C, boundary temperature T b = 37 C (cf. condition (5)). In Figure 3 the temperature distribution in the domain considered is presented, while Figure 4 illustrates the course of temperature on the external surface of skin tissue. Fig. 3. Temperature distribution Fig. 4. Temperature distribution on the external surface Conclusions The heterogeneous domain being the composition of burned and healthy layers of skin tissue has been considered. The temperature distribution has been described by the system of two Pennes equations with different thermophysical parameters. The problem has been solved by means of the boundary element method. The algorithm proposed allows one to determine the temperature field in burned and
46 K. Freus, S. Freus, E. Machrzak healthy tissue and can be used as part of a computer system that enables one to predict the shape of the burn wound [5]. Acknowledgement The article and research are financed within the proect N R3 04 0 sponsored by the Polish National Centre for Research and Development. References [] Brebbia C.A., Dominguez J., Boundary Elements. An Introductory Course, CMP, McGraw-Hill Book Company, London 99. [] Machrzak E., Boundary Element Method in Heat Transfer, Publ. of the Techn. Univ. of Czest., Czestochowa 00 (in Polish). [3] Pennes H.H., Analysis of tissue and arterial blood temperatures in the resting human forearm, Journal of Applied Physiology 948,, 93-. [4] Romero Mendez R., Jimenez-Lozano J.N., Sen M., Gonzalez F.J., Analytical solution of a Pennes equation for burn-depth determination from infrared thermographs, Mathematical Medicine and Biology 00, 7, -38. [5] Machrzak E., Dziewoński M., Nowak M., Kawecki M., Bachorz M.,.Kowalski P., The design of a system for assisting burn and chronic wound diagnosis, [in:] E. Piętka, J. Kawa (eds.), ITIB 0, LNCS 7339, Springer-Verlag, Berlin-Heidelberg 0, 0-7.