Sensitivity of the skin tissue on the activity of external heat sources

Copyright c 23 Tech Science Press CMES, vol.4, no.3&4, pp.431-438, 23 Sensitivity of the skin tissue on the activity of external heat sources B. Mochnacki 1 E. Majchrzak 2 Abstract: In the paper the analysis of transient temperature field in the domain of biological tissue subjected to an external heat source is presented. Because of the geometrical features of the skin the heat exchange in domain considered is assumed to be one-dimensional. The thermophysical parameters of successive skin layers dermis, epidermis sub-cutaneous region are different, at the same time in sub-domains of dermis sub-cutaneous region the internal heat sources resulting from blood perfusion are taken into account. The degree of the skin burn results from the value of the so-called Henriques integrals. The first the second order sensitivity of these integrals with respect to the thermophysical parameters are analyzed. On the stage of numerical computations the boundary element method has been used. In the final part of the paper the results of computations are shown. keyword: bioheat transfer, burns prediction, sensitivity analysis, boundary element method 1 Introduction The susceptibility of human body to influence of external thermal interactions is very discriminated. It is caused, first of all, by the different values of thermphysical parameters of the skin - they are conditioned by the sex, age even the profession of concrete person. The same external heat flux the same its exposure time can give quite different effects e.g. the degree of burn according to the individual features. It seems that the methods of sensitivity analysis can be in such case very effective, because they give the information concerning the mutual connections between the ther- 1 Institute of Mathematics Computer Science Technology University of Czȩstochowa, 42-2 Czȩstochowa, Da browskiego 68, Pol, e-mail: moch@matinf.pcz.czest.pl 2 Dept. for Strength of Materials Computational Mechanics Silesian University of Technology 44-1 Gliwice, Konarskiego 18a, Pol, e-mail: maj@polsl.gliwice.pl mal processes proceeding in the domain analyzed the thermal effects in this case the damage of the skin for a large scale of individual thermophysical parameters - in particular if the higher order sensitivity is introduced into considerations. Thermal damage of skin begins when the temperature at the basal layer the interface between epidermis dermis rises above 44 o C 317K. Henriques 1947 found that the degree of skin damage could be predicted on the basis of the integrals τ I b I d τ P b T b exp P d T d exp RT b t RT d t dt 1 dt 2 /R K is the ratio of activation energy to universal gas constant, P b, P d 1/s are the pre-exponential factors, while T b, T d K are the temperature of basal layer the surface between epidermis dermis dermal base the surface between dermis subcutaneous region,, τ is the time interval considered. First degree burns are said to occur when the value of the burn integral 1 is from the interval.53 < I b 1, while the second degree burns when I b > 1, see Henriques 1947, Torvi Dale 1994. The third degree appears when the integral I d > 1. So, in order to determine the values of integrals I b I d the heating next the cooling curve for the basal surface dermal base must be known. 2 Mathematical model of thermal processes We consider the 1D heterogeneous domain in which one can distinguish the following sub-domains: epidermis of

432 Copyright c 23 Tech Science Press CMES, vol.4, no.3&4, pp.431-438, 23 thickness L 1 m, dermis of thickness L 2 L 1 subcutaneous region of thickness L L 2 - Fig. 1. The thermophysical parameters of these regions are equal to λ e W/mK thermal conductivity c e J/m 3 K specific heat per unit of volume, e1, 2, 3. coefficient, T is the ambient temperature. For conventionally assumed boundary Γ c limiting the system, the no-flux condition can be taken into account. Additionally, for t the initial temperature distribution is known: t: T e x,t p x. 3 The first-order sensitivity analysis q epidermis subcutaneous region L 1 L 2 L x dermis Figure 1 : Domain considered The transient bio-heat transfer in domain of skin is described by following system of equations, see Pennes 1948, Brinck Werner 1994, Shaw, Preston Bacon 1996, Liu Xu 2, Majchrzak 1998: x Ω e : c e T e t λ e 2 T e x 2 k e T b T e Q me 3 k e G e c B is the product of blood perfusion rate volumetric specific heat of blood, T B is the blood temperature Q me is the metabolic heat source. On the contact surfaces between the sub-domains the continuity conditions in the form The first order sensitivity analysis of bioheat transfer will be done with respect to the parameters λ e, c e, k e. These parameters we denote by p s, s 1,2,...,9. If the direct method of sensitivity analysis is used Dems 1986, Davies, Saidel Harasaki 1997, Majchrzak Jasiski 21, then we should consider nine additional boundary-initial problems resulting from the differentiation of diffusion equations boundary-initial conditions, this means x Ω e :c e U es t λ e 2 U es x 2 k e U es q Ve e 1,2,3 V 1s 1 λ 1 q t t x : λ 1 V 1s αu 1s 1 λ 1 q 1 t > t λ 1 U es U e1,s, e 1,2 x L e : 1 λ e q e V es 1 λ e1 q e1 V e1,s λ e λ e1 x L :V 3s t :U es 7 x Γ 1,2 : { q1 q 2 q b T 1 T 2 T b 4 x Γ 2,3 : { q2 q 3 q d T 2 T 3 T d 5 are given, q e λ e T e / x. On the skin surface the following condition is assumed U es T e, q Ve V es λ e U es x ce λ e c e Te λ e t 8 x Γ : { q1 q, t t q 1 αt 1 T t > t 6 1 λ e λ e k e T e k e T B Q me k e T B T e 9 q is the given boundary heat flux, t is the exposure time see Behnke 1984, α is the heat transfer Taking into account the form of functionals I b, I d the sensitivity of these integrals with respect to the parameters

Sensitivity of the skin tissue on the activity of external heat sources 433 p s should be calculated using the formulas I τ b I d τ P b P d RTb 2 exp U bs tdt 1 t RT b t RTd 2 exp U ds tdt 11 t RT d t U bs tu 1s L 1, tu 2s L 1, t 12 U ds tu 2s L 2, tu 3s L 2, t 13 The change of burn integrals connected with the change of parameter p s results from the Taylor formula limited to the first-order sensitivity, this means I b p s ± p s I b p s ±U bs p s 14 or I d p s ± p s I d p s ±U ds p s 15 The second-order sensitivity analysis The second order sensitivity analysis requires the differentiation of equations 7 with respect to the parameters p s. After the mathematical manipulations one obtains W es 2 W es x Ω e : c e λ e t x 2 k e W es Q Ve x : x L : Z 3s, t : W es Z 1s 1 λ 1 1 q V 1s λ 1 λ 1, t t Z 1s αw 1s 2 λ 1 λ 1 V 1s, t > t W es W e1,s 2 λ e x L e : V es Z es 2 λ e1 V e1,s Z e1,s λ e λ e1 16 W es U e, Q Ve 2 λ e λ e 1 λ e Z es λ e W es x U es c e k e U es t c e T e t k et e k e T B Q me 17 k e U es 18 The second order sensitivity of burn integrals with respect to the parameters p s is calculated using the formulas τ 2 I b p 2 s exp 2 I d p 2 s τ P b RTb 2t RTb 2t 2 T b t U bs tw bs t dt 19 RTb 2t P d exp RTd 2t RTd 2t T d t U ds tw ds t dt 2 RTd 2t W bs tw 1s L 1, tw 2s L 1, t 21 W ds tw 2s L 2, tw 3s L 2, t 22 If the second-order sensitivities are taken into account then the formulas 14, 15 can be developed to the more exact form: I b p s ± p s I b p s ±U bs p s 1 2 W bs p 2 s 23 I d p s ± p s I d p s ±U ds p s 1 2 W ds p 2 s 24

434 Copyright c 23 Tech Science Press CMES, vol.4, no.3&4, pp.431-438, 23 4 The method of numerical solution The basic problem additional sensitivity problems have been solved using the I scheme of the BEM Banerjee 1994, Brebbia Dominguez 1992, Majchrzak 21. For mesh-free simulation of diffusion equations see also Golberg Chen 21, for heat conduction in anisotropic bodies Ochiai 21. The algorithm discussed concerns the following diffusion equation x Ω e : c e F e x, t t 2 F e x, t λ e x 2 S e x, t,e 1,2,3 25 F e denotes the temperature or functions resulting from the sensitivity analysis, while S e x,t is the source function. We introduce the time grid {t,t 1,...} with a constant step t t f t f 1 The I scheme of the BEM for equation 25 transition t f 1 t f leads to the formulas e 1,2,3 F e ξ,,t 1 c e 1 c e L e t f 1 c e F e t f 1 J e L e S e t f t f 1 ξ,,x, t f, t J e x, t dt ξ,,x, t f, t F e x, t dt ξ,,x, t f, t f 1 F e x, t f 1 dx x, t f 1 t f t f 1 are the fundamental solutions: xl e x xl e x ξ,,x, t f, t dtdx 26 ξ, x, t f, t 1 2 πa e t f t exp x ξ2 4a e t f t 27 ξ is the point in which the concentrated heat source is applied, a e λ e /c e, J e x,t λ e F e / x. The heat fluxes resulting from fundamental solutions are equal to J e ξ, x, t f, t ξ, x, t f, t λ e x λ e x ξ 4 πa e t f t 3 /2 exp x ξ2 4a e t f t 28 Assuming that for t t f 1,t f : F e x,tf e x,t J e x,t J e x,t one obtains the following form of equations 26 F e ξ, t g e ξ, L e J e Le, t g e ξ, J e Le 1, t h e ξ, L e F e Le, t h e ξ, F e Le 1, t u e ξw e ξ 29 t f h e ξ, x 1 Je dt c e t f 1 g e ξ, x 1 c e u e ξ t f x ξ er f c 2λ e L e t f 1 F e dt x ξ 2 a e t sgnx ξ x ξ erfc 2 2 a e t t exp πλe c e ξ, x, t f, t f 1 F e x, t f 1 dx x ξ2 4a e t 3 31 1 L e 2 x ξ2 exp F e x, t f 1 dx 32 πa e t 4a e t w e ξ L e S e x, t f 1 g e ξ,x dx 33

Sensitivity of the skin tissue on the activity of external heat sources 435 For ξ L e 1 ξ L e for each sub-domain considered one obtains the system of equations ge, g e,l e Je Le 1,t f g e L e, g e L e,l e J e Le,t he L e 1, 1 he L e 1,L e Le 1,t h e L e, h e L e,l e 1 F e Le,t ue L e 1 we L e 1 34 u e L e w e L e or g e 11 g e 12 g e 21 ge 22 ue u e L e Je Le 1,t J e Le,t h e f 11 h e 12 we w e L e h e 21 he 22 Le 1,t F e Le,t f 35 The continuity conditions for x L 1 x L 2 can be written in the form { F1 L1, t x L 1 : F 2 L1, t F b t J 1 L1, t J 2 L1, t A L 1, t 36 x L 2 : { F2 L2, t F 3 L2, t F d t J 2 L2, t J 3 L2, t B L 2, t 37 In the case of basic problem AL 1, t BL 2, t, for the sensitivity problems these functions result from the boundary conditions for basal layer dermal base see: equations 7 16. The final form of resolving system results from above continuity conditions the conditions given for x x L. So, for t t we have h 1 11 h1 12 g1 12 h 1 21 h1 22 g1 22 h 2 11 g2 11 h2 12 g2 12 h 2 21 g2 21 h2 22 g2 22 h 3 11 g3 11 h3 12 h 3 21 g3 21 h3 22 F 1,t F b t J 1 L1,t F d t J 2 L2,t F 3 L,t C,t f u 1 w 1 C,t u 1 L 1 w 1 L 1 g 2 11 A L 1,t f u 2 L 1 w 2 L 1 g 2 21 A L 1,t u 2 L 2 w 2 L 2 g 3 11 B L 2,t u 3 L 2 w 3 L 2 g 3 21 B L 2,t u 3 Lw 3 L 38 The value of C,t in the last equation is connected with the type of boundary condition given for x as in formula 6. The internal values of functions F e corresponding to time t f are calculated for the each layer of tissue, separately using the formula F e ξ, t g e ξ, J e Le 1, t g e ξ, L e J e Le, t h e ξ, L e F e Le, t h e ξ, F e Le 1, t u e ξw e ξ 39 5 Results of computations In numerical computations the following mean values of parameters have been assumed: λ 1.235 W/mK, λ 2.445 W/mK, λ 3.185 W/mK, c 1 4.368 1 6 J/m 3 K, c 2 3.96 1 6 J/m 3 K, c 3 2.674 1 6 J/m 3 K, c B 3.9962 1 6 J/m 3 K, G Be.125 m 3 blood/s/ m 3 tissue, e 2, 3, G B1, T B 37 C, Q me 245 W/m 3 for e 2, 3, while Q m1 Torvi Dale 1994. The thicknesses of successive skin layers:.1, 2 1 mm. Pre-exponential factors P b 1.43 1 72, P d 2.86A1 69 1/s for T b 317K Pb P d for T b < 317K. The initial temperature distribution has been assumed to be the parabolic one as in Figure 2. The skin subdomains have been divided into 1, 4 i 12 linear internal cells of dimensions 1 5,5 1 5, 8.333 1 5 m, respectively. The computations have been realized with time step t.5 s. The first example concerns the thermal processes in the tissue subjected to the boundary heat flux q 65 W/m 2 the exposure time t 18 s. For t t the Robin condition is assumed α 8 W/m 2 K, T 2 o C. The temperature field obtained for above input data is shown in Figure 2. On the basis of the knowledge of TL 1, t basal layer TL 2, t dermal base we found that the times to first second degree burn are equal 16.1 s 17.55 s, respectively. The third degree burn in this case does not appear.

436 Copyright c 23 Tech Science Press CMES, vol.4, no.3&4, pp.431-438, 23 Next, the sensitivity analysis of temperature field burn integral has been done with respect to the all thermophysical parameters. The part of the results obtained is shown in Figures 3, 4, 5, 6. In Figure 3 4 the distribution of sensitivity functions T/λ 1 T/c 2 are marked. The changes of the epidermis thermal conductivity are not very essential for the course of the burn integral I b. Figure 5 presents the course of I b for the values of λ 1 from the interval.21,.26 W/mK. One can see that the times of the burns apparitions are practically the same. The other situation takes place in the case of the parameter c 2. For c 2 12 J/m 3 K we observe the essential changes of integral I b - Figure 6. So, the influence of parameters on the course of thermal processes in the tissue domain is distinctly discriminated in numerical realization we assumed p s /p s const. The results above presented have been obtained using the first-order sensitivity analysis. The second example concerns the thermal processes in the tissue subjected to the boundary heat flux q 9 W/m 2 the exposure time t 4 s. The remaining input data are the same. Because of the big gradient of temperature heating rate, the second-order sensitivity has been considered. The computations previously discussed showed that the influence of the dermis conductivity λ 2 on the heat transfer process is also very essential the results of the second-order analysis concerning this parameter will be below presented. Figure 3 : Distribution of function T / λ 1 for χ,4 mm Figure 4 : Distribution of function T / c 2 Figure 2 : Temperature distribution Figure 5 : Course of burn integral its sensitivity with respect to λ 1

Sensitivity of the skin tissue on the activity of external heat sources 437 In Figure 7 the basic solution, this means the temperature field in the skin tissue is shown. The next Figure illustrates the distribution of the derivative 2 T / λ 2 2. The courses of the burn integral I d for the values of λ 2 from the scope.42,.47 W/mK are marked in Figure 9. One can see that the time of the third degree burn apparition is considerable differentiated. In order to verify the exactness of the algorithm proposed the extreme values of λ 2 have been directly introduced to the procedure solving the basic problem the differences between solutions were very small. Figure 8 : Distribution of function 2 T / λ 2 2 Figure 6 : Course of burn integral I b its sensitivity with respect to c 2 Figure 9 : The change of I d with respect to λ 2 6 Conclusions Figure 7 : Temperature field It seems that the application of the boundary element method to the solution of the problem discussed leads to the very effective numerical algorithm. The number of unknown parameters is not large only the boundary values. Next, the internal temperatures the internal values of functions U, W can be found at the optional set of points on the basic of the knowledge of boundary ones. The approximation of the real boundary conditions is exact it causes that the solution in the interior of domain is sufficiently exact, too. The results of computations presented in this paper show that the influence of different thermophysical parameters

438 Copyright c 23 Tech Science Press CMES, vol.4, no.3&4, pp.431-438, 23 on the course of the process is discriminated. The quantitative valuation of these problems can be done first of all on the basis of sensitivity analysis. The quite good results can be, as a rule, obtained using the first-order approach, while the second-order analysis is useful in the case of parameters especially responsible to the course of thermal processes. The next stage of the investigations in this scope will concern the sensitivity analysis with respect to geometrical parameters. Especially the epidermis thickness can have the essential importance on the thermal damage of the skin. Acknowledgement: The paper was sponsored by KBN Grant No 8 T11F 4 19. References Banerjee, P.K. 1994: Boundary element methods in engineering, McGraw-Hill Company, London. Behnke, W.P. 1984: Predicting flash fire protection of clothing from laboratory tests using second-degree burn to rate performance, Fire Materials, 8, pp.57-63. Brebbia, C.A.; Dominguez, J. 1992: Boundary elements. An introductory course, Computational Mechanics Publications, McGraw-Hill Book Company, London. Brinck, H; Werner, J. 1994: Estimation of the thermal effect of blood flow in a branching countercurrent network using a three-dimensional vascular model, Journal of Biomechanical Engineering, 116, pp. 324-33. Davies, C.R.; Saidel, G.M.; Harasaki H. 1997: Sensitivity analysis of one-dimensional heat transfer in tissue with temperature-dependent perfusion, Journal of Biomechanical Engineering, Transactions of the ASME, 119, pp. 77-8. Dems, K. 1986: Sensitivity analysis in thermal problems-i: variation of material parameters within fixed domain, Journal of Thermal Stresses, 9, pp. 33-324. Golberg, M.A.; Chen, C.S. 21: An efficient meshfree method for nonlinear reaction-diffusion equations, CMES: Computer Modeling in Engineering & Sciences, Vol. 2, No 1, pp. 87-96. Henriques, F.C. 1947: Studies of thermal injuries V. The predictability the significance of thermally induced rate processes leading to irreversible epidermal injury, Archives of Pathology, 43, pp. 489-52. Liu, J.; Xu, L.X. 2: Boundary information based diagnostics on the thermal states of biological bodies, International Journal of Heat Mass Transfer, 43, pp. 2827-2839. Majchrzak, E. 1998: Numerical modelling of bio-heat transfer using the boundary element method, Journal of Theoretical Applied Mechanics, 2, 36, pp. 437-455. Majchrzak, E. 21: Boundary element method in heat transfer, Technological University of Czestochowa Publishers, Czestochowa in Polish. Majchrzak, E.; Jasiski, M. 21: Sensitivity analysis of bioheat transfer in 2D tissue domain subjected to an external heat source, Acta of Bioengineering Biomechanics, 3, 2, pp. 329-336. Ochiai, Y. 21: Steady heat conduction analysis in orthotropic bodies by triple reciprocity BEM, Computer Modeling in Engineering & Sciences, Vol. 2, No 4, pp. 435-446. Pennes, H.H. 1948: Analysis of tissue arterial blood temperatures in resting human forearm, Journal of Applied Physiology, 1, pp. 93-122. Shaw, A.; Preston, R.C.; Bacon, D.R. 1996: Perfusion corrections for ultrasonic heating in nonperfused media, Ultrasound in Medicine Biology, 22, 2, pp. 23-216. Torvi, D.A.; Dale, J.D.: 1994 A finite element model of skin subjected to a flash fire, Journal of Biomechanical Engineering, 116, pp. 25-255.