TIME DEPENDENT TEMPERATURE DISTRIBUTION MODEL IN LAYERED HUMAN DERMAL PART

Transcription

1 VOL. 8, No. II, DECEMBER, 0, pp TIME DEPENDENT TEMPERATURE DISTRIBUTION MODEL IN LAYERED HUMAN DERMAL PART Saraswati Acharya*, D. B. Gurung, V. P. Saxena Department of Natura Sciences (Mathematics), Schoo of Science, Kathmandu University Sagar Institute of Research and Technoogy, Bhopa, India *Corresponding address: Received 7 January, 0; Revised August, 0 ABSTRACT The paper deveoped appication of finite eement method with inear function in the study of temperature distribution in the ayers of derma partstratum corneum, stratum germinativum, papiary region, reticuar region and subcutaneous tissues as eements. The method is appied to obtain the numerica soution of governing differentia equation for one dimensiona unsteady state bio-heat transfer using suitabe vaues of parameters that effect the heat transfer in human body. The numerica resuts obtained are exhibited graphicay for various atmospheric temperatures for comparative study of temperature distribution profies. The oss of heat from the outer surface of the body to the environment is taken due to convection, radiation and sweat evaporation. Keywords: finite eement method, human derma part, bio-heat equation INTRODUCTION Human skin is a very compex tissue consisting of severa distinct ayers and components. Hence it exhibits compex materia behavior. It is the argest organ of the human body and it has severa functions. The most obvious function is protection of the body against externa infuences and it heps to reguate the body temperature. Skin temperature distribution of the human body is a compex interaction of physica heat exchange processes and the potentia for physioogica adjustment. Temperature infuences the functioning of bioogica systems. For humans, the core temperature varies within narrow bounds around 37 C. The human body acts ike an unsteady state system with a continuous input and not a reguar output. This continuous input provokes a mass storage in the body, eading for a reaction in order to burn it. This process is caed metaboism rate. The norma range of human body temperature varies due to metaboic rate of an individua. Higher metaboic rate causes higher norma body temperature and ower metaboic rate causes ower the norma body temperature. Other factors that might affect the body temperature of an individua may be the time of day or the part of body where the temperature is measured. The body temperature is ower in the morning due to the resting condition and higher at night after a day of muscuar activity and after food intake as we as the thermoreguatory processes of the body are carried on by different methods: convection, conduction, radiation and evaporation. Most of the heat of the body is ost due to convection because it is in contact with fuid such as air and water. Convective transfer by bood pays a key roe. Densities of bood vesses are very high and hence it is hard to compute a detaied temperature distribution in even a sma part of the body whie accounting for a of the bood vesses individuay. Fortunatey, the effects of the bood vesses can be described coectivey with some success by the hep of Pennes bio heat equation. So externa (cimate) and interna (metaboic) heat sources infuences body temperature. Heavy exercise, iness and not hot and humid but aso cod and windy environments ater body 66

2 VOL. 8, No. II, DECEMBER, 0, pp temperature outside the norma range. Ambient temperature, humidity, air movement and radiant heat from the sun as we as warm and cod surfaces contribute to cimate heat stress. Body temperature refects the carefu baance between heat production and heat oss. There is a continuous heat exchange between the body and the environment. Bio- directiona routes for heat exchange are: convection (Cv), conduction (Cd), and radiation (R). There are aso two unidirectiona routes: Metaboic heat (M) increases the therma oad, evaporation (E) decreases the oad. The net heat storage (S) formua is: S= M+/- Cv+/-Cd+/-R-E When the net heat storages (S) is positive, body temperature wi rise and when S is negative, it wi go down. The temperature mode on human body deas the study of temperature distributions on the ayers of human skin exposed to sources of temperature, e.g. surrounding temperature or any sources. So any irreguarities in the temperature distribution in derma ayers due to abnorma environment cause the disturbances in thermoreguation. Hence the study of temperature distribution has the cinica and theoretica importance. Temperature pays an important roe in the functioning of bioogica system. To predict tissue temperatures as we as the temperature of the skin ayers infuence of bood fow must be accounted. The temperature dependence of bioogica processes can be used to cinica effect. For exampe, hyperthermia treatment against cancer, cooing of the head to prevent hair oss as a side effect of chemotherapy and cooing of patients during major surgery to protect the brain are such exampes. Mathematica mode for temperature distributions have a roe both in treatment and diagnosis. They can aid in predicting the time in the course of treatment or giving information on the temperature where thermometry is acking. So numerica modeing and anaysis of the skin has numerous appications in medicine but aso in aesthetics, in computer, graphic, etc. MATHEMATICAL MODEL AND MODEL ASSUMPTIONS The mathematica mode used for bioheat transfer is based on the Pennes equation [8] which incorporates the effect of metaboism and bood perfusion in to the standard therma diffusion equation. This equation is written in simpified form as ρc T t = K. T + M(T a T) + S () where, ρ = Tissue density c = Tissue specific heat K = Tissue therma conductivity m b = Bood mass fow rate c b = Bood specific heat ω b = The bood perfusion rate T a = Arteria bood temperature M = ω b m b c b 67

3 VOL. 8, No. II, DECEMBER, 0, pp S = Metaboic heat generation rate The oss of heat from the skin surface due to convection, radiation and evaporation is considered. So the mixed boundary condition is K T x Skin surface = h (T T ) + LE () where, h = Combined heat transfer coefficient due to convection and radiation T L E = Surrounding temperature = Latent heat of evaporation = Rate of sweat evaporation The inner body core temperature T b is assumed to be 37 0 C. The thickness of stratum corneum, stratum germinativum, papiary region, reticuar region and subcutaneous tissue have been considered as,, 3, 3, 5 respectivey and T 0,, T, and T 5 = T b are the noda temperatures at a distances x = 0, x =, x =, x = 3, x = and x = 5. T (i), i =,, 3,, 5 be the temperature functions in the ayers stratum corneum, stratum germinativum, papiary region, reticuar region and subcutaneous tissue respectivey. T o 3 T T () T () Stratum Corneum Stratum Germinativum T (3) Papiary Region 5 T () Reticuar Region T 5 T (5) Subcutaneous Tissue Body core temperature = T b Figure : Schematic diagram of five ayers of derma part with noda points The anatomica structure of human derma part makes it reasonabe to consider M and S zero in stratum corneum. In the mode, the therma conductivity in the ayers of derma part is 68

4 VOL. 8, No. II, DECEMBER, 0, pp considered as constant. A the assumptions for parameters in the ayers of derma part can be summed up as: Quantity T T a K M S Outer Boundary Stratum Corneum (0 x ) Stratum Germinativum ( x ) Papiary Region ( x 3 ) Reticuar Region ( 3 x ) Subcutaneous Tissue ( x 5 ) T 0 T () = T 0 + T0 x T a () T = T + T x = 3T 3 + T 3 x = x = 0 K () () T a = 0 (3) T a = T b T a () = T b T 5 = 5 T 5 + T 5 (5) x T a 5 5 = T b M () = 0 S () = 0 K () M () = 0 S () K (3) M 3 = x K () M = x m m = x S 3 = x S = x K (5) M 5 = m S 5 = s s s s SOLUTION OF THE PROBLEM The variationa integra form of () in one dimensiona unsteady state case together with outer skin boundary condition () is given by I = 0 K dt dx + M(Ta T) ST + ρc T t dx + h T T + LET We write I separatey for the Five ayers; I for stratum corneum, I for stratum germinativum, I 3 for papiary region, I for reticuar region and I 5 for subcutaneous tissue, so () () () dt () () () () T I K M T T S T c dx dx A t 0 ht 0 Ta LET0 I () () dt () () () () T K M T A T S T c dx dx t () I 3 (3) (3) dt (3) (3) (3) (3) T K M T A T S T c dx dx t 3 (3) 69

5 VOL. 8, No. II, DECEMBER, 0, pp I () () dt T K M TA T S T c dx dx t () () () () () 3 I 5 (5) (5) dt T K M TA T S T c dx dx t (5) (5) (5) (5) (5) 5 Evauating the integra I i, i =,,,5 with the hep of ayers wise assumptions, we get the foowing system of equations given beow I A BT 0 DT 0 ET FT 0T T0 T T0T t I A BT CT DT ET FTT T T TT t I3 A3 B3T C3T3 D3T E3T3 F3T T3 T T3 TT 3 t I A BT3 CT DT3 ET FT T 3 T T3T t I5 A5 B5T C5T5 D5T E5T5 F5T5 T T5 TT 5 t where A i, B i, D i, E i, F i, i 5 and C j, j 5 are a constants depending upon the vaue of physica and physioogica parameters of derma part, and A = ht ; B = LE ht ; D = K () A = 0; B = P ; C = P ; D = R ; ; E = R A 3 = T b m( 3 ) ( ) ; + h ; E = K() ; F = R ; F = K() B 3 = P ( 3 ) N ( ); C 3 = P ( 3 3 ) + N ( ); D 3 = R + Q ( ); E 3 = R + 8 Q ( 3 ) 3 ; F 3 = R + Q ( 3 ) 3 ; A = T b m ( ) ( ); B = P ( ) N ( ); C = P ( ) + N ( ); 70

6 VOL. 8, No. II, DECEMBER, 0, pp D = R 3 + Q ( ); E = R 3 + Q ( ); F = R 3 Q ( ); A 5 = mt b ( 5 ); B 5 = P 3 ( 5 ); C 5 = P 3 ( 5 ); D 5 = R + Q 3 ; E 5 = R + Q 3 ; F 5 = R + Q 3. Where, P = s( ) ; 6( ) R = Q = K () ; ( ) P = m ; ( 3 )( ) Q 3 = m( 5 ); 6 R = K(3) ; ( 3 ) N = T b m ; P 6( ) 3 = (T b m+s); Q = R 3 = K() ; ( 3 ) m ; ( 3 )( ) s. 6( ) R = K(5) ; ( 5 ) α = ρc ; β = ρc ( ) ; γ = ρc ( 3 ) ; μ = ρc ( 3 ) ; η = ρc ( 5 ) We differentiate I with regard to the noda temperatures T o,, T,, and T 5 and set di dt j = 0, j = 0,,, 3,. Since T 5 = T b (the body core temperature), we get the system of equations in matrix form: CT + PT = W (3) Where, C = 7

7 VOL. 8, No. II, DECEMBER, 0, pp T = T o T D F O O O F (E +D ) F O O, P = O F (E + D 3 ) F 3 O O O F 3 (E 3 + D ) F O O O F (E +D 5 ), T = T 0 T t T t t t NUMERICAL RESULTS AND DISCUSSION Parameter vaues used in the mode are as considered in Tabe beow. Tabe : Parameter vaues considered [3, ]. Parameter Vaue Unit K () w/m 0 C K () w/m 0 C K (3) w/m 0 C K () w/m 0 C K (5) w/m 0 C L.0 6 J/kg h 6.80 w/m 0 C E.90 - w/m ρ 050 kg/m C J/kg M w 0 C S 56.0 w/m 3 Tabe : Two sets of derma ayers Sets (m) (m) 3 (m) (m) 5 (m) I II And, for noda temperatures at t = 0, we are taking the equation T(x, t) = T(x, 0) + px considering initia temperature.87 0 C at skin surface because at norma atmospheric temperatures skin surface temperature is around.87 0 C. To sove the system of ordinary 7

8 Noda temperatures ( o C) Noda temperatures ( o C) KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY VOL. 8, No. II, DECEMBER, 0, pp differentia equation (3), we use Crank-Nicoson method. According to the method, the system of equation (3) can be written as C + t P T(i+) = C t P T(i) + tw () where t is the time interva. Using the above numerica vaues in equation () the profies for temperature distribution in the ayers of derma part so obtained is as shown in figures beow taking various atmospheric temperatures T T T 0 0 T Fig.: Noda Temperature for Set-I at T a = 5 0 C Fig. : Noda Temperature for Set-II at T a = 5 0 C The temperature distribution profies for noda temperatures T i (i=0,,, 3, ) for the two set of skin thicknesses at different atmospheric temperatures with time dependence are shown in figures-6. From these figures it has been observed that the curve for T i rise faster in set I and reach steady state case earier than set II. It is aso observed that steady state temperatures at these depths are higher for set I than to set II. These are due to ower thickness of set I than to set II. These figures revea that the temperatures Ti at high atmospheric temperatures are higher than ower atmospheric temperatures. This is due to the more effect on body temperature at higher atmospheric temperatures. So the thicknesses of derma ayers and different atmospheric temperatures have the significant effects in temperature profies. 73

9 Noda temperatures ( o C) Noda temperatures ( o C) Noda temperatures ( o C) Noda temperatures ( o C) KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY VOL. 8, No. II, DECEMBER, 0, pp T 0 8 T T 0 T Fig. 3: Noda Temperature for Set-I at T a = 37 0 C Fig. : Noda Temperature for Set-II at T a = 37 0 C T T 0 T 0 5 T Fig. 5: Noda Temperature for Set-I at T a = 5 0 C Fig. 6: Noda Temperature for Set-II at T a = 5 0 C 7

10 VOL. 8, No. II, DECEMBER, 0, pp CONCLUSION The resuts are comparabe with those obtained in [] who considered ony three ayers of skin epidermis, dermis and subcutaneous tissue. The differences in the resuts whatsoever may be, due to the extension up to five ayers of derma part Stratum Corneum, Stratum Germinativum, Papiary Region, Reticuar Region and Subcutaneous Tissue. Our mode gives better profies for temperature distribution in derma ayers. This is because the mode has incorporated more feasibe ayers and has taken significant biophysica parameters. The epidermis ayer contains five ayers and dermis contains two ayers. So bioogicay derma region has eight ayers. We may further extended the mode considering the eight ayers as eight eements in one dimensiona case. It may be extended for two dimensions and three dimensions. The study may be carried out for predication of therma response for humans exposed to hot or cod environments. The mode may be extended by considering the fied variabe as quadratic or of higher order poynomia in FEM technique to get better temperature profies. The different temperature profie can be viewed by considering this mode at cinica situation, indoor cimate and risk under stressfu condition. It is hepfu for the medica scientist to suggest and evauate for new treatment strategies. It is aso hepfu to maintain indoor temperature. The presented mode can be extended to expore severa probems mentioned above. REFERENCES [] Cooper T E & Trezek G J, A probe technique for determining the therma conductivity of tissue, J. Heat transfer, ASME, 9(97)33. [] Guyton C & Ha E, Text book of medica physioogy, Esevier, India, 009. [3] Gurung D B, Mathematica study of abnorma thermo-reguation in human derma parts, Ph. D. Thesis, Kathmandu University, 007. [] Gurung D B, Saxena V P & Adhikary P R, FEM approach to one dimensiona unsteady state temperature distribution in the derma parts with quadratic shape function, J. App. Math and Informatics, 7(-)(009). [5] Gurung D B, Thermoreguation through skin at ow atmospheric temperatures, KUSET, 5() (009). [6] Gurung D B & Saxena V P, Periphera temperature distribution in human subjects exposed to wind fow, Internationa ejournas of Mathematics and Engineering 73(00) 78. [7] Myers G E, Anaytic methods in conduction heat transfer, MC. Graw Hi Book Co., New York, 97. [8] Pennes H H, Anaysis of tissue and arteria bood temperature in the resting human forearm, J. App. Physio., (98)93. 75

11 VOL. 8, No. II, DECEMBER, 0, pp [9] Crank J, The mathematics of diffusion, Oxford Science Pubications, second edition, 975. [0] Rao S S, The finite eement method in engineering, Esevier, a division of Reed Esevier India Private Limited, (009). [] Per W, Heat and matter distribution in body tissues and the determination of tissue bood fow by oca cearance method, J. Theor. Bio, (96)0. [] Saxena V P, Adhikary P R & Gurung D B, Mathematica estimation of unsteady state burn damage due to hot temperature, The Nepai Math Sc. Report, 7(-)(007). [3] Rao Y V C, Theory and probems of thermodynamics, New Age Internationa (P) Limited, second edition, (000)3. 76