Thermal Relaxation Times in Biological Tissues Subjected to Pulsed Laser Irradiation

Size: px

Start display at page:

Download "Thermal Relaxation Times in Biological Tissues Subjected to Pulsed Laser Irradiation"

Caroline Hudson
6 years ago
Views:

1 9th AIAA/ASME Joint Thermophysics nd Het Trnsfer Conference 5-8 June 006, Sn Frncisco, Ciforni AIAA Therm Rextion Times in Bioogic Tissues Subjected to Pused Lser Irrdition Kyunghn Kim * nd Zhixiong Guo Rutgers, The Stte University of New Jerse Pisctw NJ, There re sever expressions for ccuting therm rextion time in tissues subjected to ser irrdition. Physicy this time constnt represents the time required for the temperture rise in heted region of tissue to drop to fctor of e -. In this stud we conduct combined rdition nd het conduction simutions to obtin this time constnt for sever different tissues, nd compre the resuts with other mthemtic expressions used for determining the therm rextion time. The scttering effect nd surfce convection re considered. A new therm rextion time expression is introduced in which its vue is two times of the therm rextion time used in the iterture. It is found tht the resuts predicted by this new expression mtch better with the predictions bsed on the pek temperture decying. C p = specific het cpcity c = speed of ight D = bem width G = incident rdition h = het trnsfer coefficient I = rdition intensity N = number of tot incident puses. OPD = optic penetrtion depth q r = het fux vector t = time T = temperture t = ser puse width p T = mbient temperture w = ngur weight x, y = coordintes α = therm diffusivity δ = chrcteristic ength ξ, η = direction cosines ω = scttering bedo Φ = phse function ρ = density σ = bsorption coefficient σ e ' s = extinction coefficient σ = reduced scttering coefficient τ = therm rextion time θ = non-dimension temperture Superscript Nomencture * Grdute Student, Deprtment of Mechnic nd Aerospce Engineering. Assistnt Professor, Deprtment of Mechnic nd Aerospce Engineering. Copyright 006 by the, Inc. A rights reserved.

2 Subscript b c = discrete ordinte = bckbody = coimted I. Introduction ince the dvent of sers in 960s, mny prctic nd potenti ppictions hve been investigted. Among S them, medic ser surgery nd/or tretment certiny beong to the most significnt dvnces of the pst sever decdes. Nowdys, cinic ser is widey used in ser microsurger nonbtive derm remodeing, tissue fusion, 3 nd photodynmic therp 4 etc. The gener go mong these ppictions is to provide proper het to trgeted region of tissue whie minimizing the coter therm dmge. There is n incresing demnd for effective nd sfe ser ppictions in medicine. The degree nd extent of tissue therm dmge depends on mny fctors. These fctors incude the mximum temperture reched nd the exposure time s we s the mechnic stress ppied on the tissue during heting. The mount nd rte of het deposition, couped with the properties of the tissue nd the surrounding medium govern the therm response t both the microscopic nd mcroscopic sces. The response is typicy modeed by the bio-het eqution. 5 For irrdition times ess thn bout 5 sec, the infuence of bood perfusion pys minor roe. 6 When bood perfusion is negected, the governing eqution is merey het conduction eqution, in which the rte of therm energy storge within voume must equ the oc voumetric bsorbed ser energy rte minus the oc voumetric osses of het fux by diffusion. A combined het rdition nd conduction modeing cn predict the temperture fied nd time histor nd the dt cn be used for evuting the ser prmeters. In reit nevertheess, simper modes re wys preferred for guiding the seection of ser input prmeters. The cssic theory of seective photothermoysis proposed by Anderson nd Prrish is genery dopted to determine the pproprite ser puse width. A common prctice is tht the irrdition durtion is ess thn the therm rextion time of the trget tissue to minimize therm dmge. 6,7 Time constnts in het conduction probems hve been intensivey studied. Furzikov 8 mthemticy expressed the therm rextion time s foows ( α ) τ = δ 4, () where δ is chrcteristic ength nd α is the therm diffusivity. δ is either the optic penetrtion depth of the incident rdition (for xi conduction) or the Gussin bem rdius t e (for rdi conduction). vn Gemert nd Wech 6 considered pre circuit of xi nd rdi het conduction with corresponding time constnts τ z, τ r, respectivey; nd suggested τ 0 = + () τ z τ r According to Eq. (), the over time constnt is smer thn the smest individu time constnt τ z orτ r. It is worth mentioning tht τ z ndτ r derived by vn Gemert nd Wech 6 re virtuy identic to Eq. () though the numeric fctors re sighty different. The optic penetrtion depth (OPD) is defined s the depth where the fuence rte drops to e of its origin rdince. 8 For purey bsorbing medi, OPD = / σ ccording to Beer s w. Becuse of the difficuty in the nysis of rdition het trnsfer in bsorbing nd scttering turbid medi ike bioogic tissues, most of the previous reserches 6-8 used Beer s w to represent the ser rdition trnsfer nd the scttering effect ws negected. For bsorbing nd scttering medi, corrected OPD cn be defined s OPD = /( σ + σ ' s ). Nowdys, gret progress hs been mde in the numeric soution of rdition het trnsfer prticipting medi nd the importnce of ccurte rdition nysis is being we recognized. The uthors considered the modeing of trnsient ser rdition trnsfer in tissues using the discrete ordintes method. 9- The OPD cn then be ccurtey predicted by the numeric method nd used to estimte time constnts. On the other hnd, the therm rextion time cn be physicy interpreted s the time required for the temperture rise ( T T bse ) in heted region of tissue to decrese to e - of its pek vue ( T pek T bse ). 7 Thus, one cn obtin therm rextion time bsed on the decy of the temperture t the ser spot. Certiny this time

3 constnt cn incorporte the infuence of reistic tissue surfce condition (free convection or forced cooing). Whieτ in Eq. () or () hs no such n dvntge. Nowdys, Cryogen spry cooing 3 hs become more nd more popur in ser surgery procedure to minimize therm dmge. The purpose of this study is twofod: () to investigte the infuence of rdition scttering on the prediction of OPD nd therm rextion time; nd () to compre time constnts predicted by different modes under vrious conditions. II. Mthemtic Formution Consider coimted ser puse incidence upon two-dimension (-D) bioogic tissue shown in Fig.. The oc temperture response of the tissue within short time period is formuted s T ( x, r ρ C p = q rd ( x,, (3) t r where q rd ( x, is the divergence of rdition het fux due to ser rdition bsorption nd cn be ccuted by where I b r q rd ( x, = σ (4π I G), (4) is the bck body emissive intensity of the tissue which is negigibe becuse the tissue cn be treted s cod medium s compred to the rge fux of ser bem. The incident rdition, G, is direction-integrted rdition intensity nd cn be obtined by the summtion of nge-discretized rdition intensity. To ccute the rdition intensity I in discrete ordinte, the time-dependent eqution of rdition trnsfer (ERT) in discrete-ordinte formt is introduced: b I c t I I + ξ + η + σ ei = σ es, =,,3,..., n (5) x y where ξ nd η re the direction cosines, scttering coefficients, nd S σ e is the extinction coefficient tht is the sum of the bsorption nd is the rditive source term from the ser rdition tht cn be expressed s S n ω i i i = ( ω ) Ib + w Φ I + Sc, =,,3,..., n (6) 4π i= Here, the scttering bedo is ω = σ e nd the scttering phse function is Φ ( sˆ sˆ ). A qudrture set of n discrete ordintes with the pproprite ngur weight w ( =,,.., n) is used. The ser source in Eq. (6) is the driving force of the trnsient rdition het trnsport nd cn be expressed s i S c ω = I c ( ξcξ + ηcη ), (7) 4π where the unit vector of ( ξc, ηc ) represents the coimted ser incident direction. The incident ser sheet with Gussin profie cn be expressed by I ( x, y) = I exp{ 4n [( t x / c) / t.5] } exp[ (y / D ) ] exp( σ x), c c0 p e 0 t 3t p (8) 3

4 In which I c0 is the mpitude of the ser rdition intensity nd cn be djusted to et the tissue rech pressumed mximum temperture T t the ser spot center. T cn be expressed by ref ref T ref = T + i 3t p Nσ I c ( x = 0, y = 0 ρc p D /, dt (9) where Ti is the initi tissue bse temperture, nd N is the number of tot incident ser puses. The temperture is then nondimensionized s T Ti θ =. (0) T T ref i In the present stud we consider pused ser irrdition in short time period, sy ms. Within this time period, there re mny short puses. After tht, the ser is turned off nd the medium experiences merey het diffusion from the irrdition region to the surrounding region without extern het genertion. This process is governed by the het conduction eqution s foows: θ ( x, θ θ ρ C p = k +, () t x y where k is the therm conductivity. For the ccution of het conduction, the non-dimension temperture is initiy zero except in the pth of the coimted ser incidence. Eqution (), couped with the rdition het trnsfer modeing is empoyed for finding the initi temperture distribution ong the ser pth. The boundry conductions re specified beow. θ ( x, = 0, for y = 0, y = W, or x = L () T k = h ( T T ), for x = 0 (3) x For the nysis of rdition het trnsfer, refection nd refrction governed by Sne s w nd Fresne eqution, respective re considered t the tissue-ir interfce. At the rest boundries, diffuse refection is considered. For detis, pese refer to our recent pubictions. 9- The foowing ists sever ssumptions we dopted when we set up the present mode: () Therm rdition emission from the tissue is negected inside the tissue nd on tissue surfce. () The tissue optic nd therm properties re thermy stbe nd constnt during the het trnsfer processes. (3) Therm evportion nd phse chnge of tissue re not considered. (4) The heting of tissue by short-pused irrdition is treted by oc voumetric method ssuming therm equiibrium. The brupt eectron temperture rise during n utrshort time period is negected. In the present stud we consider -D tissue mode in Crtesin coordintes system. Thus, τ z nd τ r, in Eq. () re repced by τ nd τ, respectivey. τ is determined by the optic penetrtion depth; whie is dependent x y x on the bem width. Here we use two different therm rextion time definitions. The first one is n inversey verged time constnt defined s τ = + (4) τ τ x y The second one is the time required for the pek temperture rise θ pek (=) t the ser spot center to decrese to e -. We denote this therm rextion time s τ nd evute τ from time-dependent simution of 4

5 combined ser rdition nd het conduction. τ hs better physic interprettion thn τ. Further, τ is infuenced by the convective boundry condition t tissue-ir interfce. To sove the time-dependent eqution of the rdition trnsfer, the trnsient discrete ordintes method 9 (TDOM) bsed on S 0 scheme is empoyed. The tissue geometry is divided by uniform grid system of The soid nge is divided by qudrture set of n = 0 discrete ordintes. Detis of the numeric schemes hve been intensivey described nd evuted in our recent pubictions; 9- thus, they re not repeted here. The fuy expicit scheme is dopted to sove the het conduction eqution. The grid size is chosen the sme s the Pused ser 0 sheet W 0 L x T = 30 K rdition trnsfer probem nd the time step is seected s 0.00 sec. The ccution ws stbe. The simution ws exmined by nytic het conduction soutions nd the resuts were stisfctory. y T k = h( T T ) x III. Resuts nd Discussion For simpicit we consider squre tissue with L = W = 3.46 mm. The optic properties of the considered tissues re isted.0 in Tbe. The therm properties re ssumed s the tissue is composed with 70% wter 7, 4 : α = 0.4 mm 0.8 /s, 6 3 nd = J/ ( m K) ρc p ; if they re not otherwise specified. In the present study the reference temperture is = 65 C, T ref nd the bse temperture is T = = 37 bse T i T = 30 K Optic xis Tissue 0.5W T = 30 K Figure. Schemtic sketch of the irrdition tissue system. C. The mbient temperture is T = 3 C. 0. nd h is the het trnsfer coefficient. If not specified, the convective het trnsfer coefficient is seected s 5 h = 0 W/m K. 0.0 In Fig., the fuence vritions ong the optic xis re shown t ms time instnt for different optic properties of tissue. For Optic xis (mm) the fuy bsorbing medi ( σ = 0.mm or Figure. Fuence profies with vrious optic properties. σ =.0 mm ), σ ' s = 0 nd the numeric resuts re compred with the nytic soutions. An exceent greement ws found. Usuy scttering in tissue is very strong, prticury in the ner-infrred wveength. We consider the mode tissue with two different scttering coefficients (i.e., =.3 mm or =.6 mm ). It is seen tht the fuence drops fster with incresing scttering No rmized fuence (t m s ) coefficient. Since OPD is the ength for fuence drop to fctor of e, the OPD cn be mesured from Fig.. With the presence of scttering, OPD decreses. In prticur, the scttering effect is substnti for sm bsorption tissue. For exmpe, the OPD is 0 mm for the cse of σ = 0. mm when ony bsorption is considered; whie it is ess thn mm when the scttering effect is dded. Thus, both bsorption nd scttering effects must be incorported in the ccution of OPD. σ σ = mm = 0.mm 5 Anytic = 0. =.3mm =.6 mm

6 The optic penetrtion depth for bsorbing nd scttering tissue my be pproximted s ( σ + ' ) OPD = / σ. (5) In Fig. 3, the OPDs estimted from this pproximtion re compred with the mesurements through compete numeric modeing. It is seen tht Eq. (5) is very good pproximtion when the 0 scttering coefficient is weker thn the Ccuted resuts bsorption coefficient. On the other side, when the bsorption coefficient is sm nd 8 Best fitted curves the bsorption is much weker thn the OPD = scttering, Eq. (5) shoud not be dopted. σ + σ s 6 In Fig. 4, the ctu fuence vrition in σ the y-xis on the tissue-ir interfce is s = 0 investigted. We compre the origin 4 incident bem vrince with those t ms = 0.6 mm with different scttering coefficients. The =.60 mm bsorption coefficient is fixed t σ = 0.mm. At ms, rdition equiibrium hs been chieved. It is seen tht 0 the scttering infuences the fuence profie in the y-xis s we. It sows down the fuence vrition in the y-direction nd resuts in n Absorption coefficient (mm - ) increse in the fuence width. However, this infuence is not s strong s tht in the OPD. Thus, one my sti use the origin bem rdius to represent the chrcteristic ength in Ccuted resuts the rdi direction. 6 Best fitted curves In the foowings we define the bem hf width t e - OPD = ( wl in Fig. 4) for estimting τ y σ + σ s nd use the numericy ccuted OPD for 4 σ = 0.mm nyzing τ x. Figure 5 shows the therm rextion times under vrious optic σ = mm properties. Cery τ x is infuenced by the σ = mm rdition het trnsfer, whie effect of rdition trnsfer on is ony obvious for Optic penetrion depth (mm) Optic penetrion depth (mm) very sm bsorption with strong scttering. With incresing bsorption, τ x decreses 0 exponentiy. With rge bsorption coefficient, the vrition of therm Reduced scttering coefficient (mm - ) rextion time ginst the scttering is sm. Figure 6 shows the tempor temperture Figure 3. Comprison of OPDs predicted numericy nd profies t sever octions for humn nyticy. dermis tissue. The corresponding optic properties 5 re σ = 0.7 mm nd = 3.55mm. It is seen tht the temperture drops rpidy t the center of ser spot (x = 0, y = 0.5W). The dropping speed decreses with incresing distnce. The soid bck mrks represent where the temperture in heted oction of the tissue decreses to e of its pek vue. The times in the mrks re then the therm rextion times t the seected octions bsed on the physic interprettion. It is seen tht therm rextion time t the ser spot center is the smest one; nd this time is seected s τ in the present study. The pek temperture t the ser spot center is so the mximum temperture of the tissue resuted from the irrdition. s 6

7 As fore-mentioned, the convective het trnsfer condition t the tissue-ir interfce wi ffect the vue of τ. To investigte this effect, sever vues of h re seected nd the tempor temperture profies t the ser spot center re compred in Fig. 7. Two types of tissue re chosen: dermis tissue nd hert tissue (endocrdium). We ssumed sme therm properties for the both tissues. Obvious the therm rextion time with higher h shows smer vue becuse prt of the het is reesed by the convection. Lrger convective het trnsfer rte provides more rpid cooing performnce. The effect of surfce convection on the hert tissue is more pronounced thn on the dermis tissue. The rdition coefficients of the dermis tissue re rger thn those of the hert tissue; thus, rdition effect predomintes in the dermis tissue. The therm rextion times in brin tissue (gry mtter) with vrying bem width re compred in Fig. 8. It is observed tht with incresing bem width the therm rextion time τ increses quick but the chnge in τ is y trivi. The predicted τ nd τ re between the vues of τ nd. Most importnt the x vues of τ nd τ mtch ech other very we when the bem width is ess thn 6 mm. Since τ 0 is hf of τ, the difference between τ 0 nd τ wi be rge. The use of τ 0 s the therm rextion time s previous reserchers 7 suggested is then not supported by the resuts of τ. Tbe ists the predicted therm rextion times bsed on the four definitions for sever bioogic tissues. x Normized f uence Therm rextion time (sec) Incident bem vrince (At ms) σ s = 0 = 0.6mm =.6 mm Bem 6 width8 0 Y (mm) Figure 4. Fuence vrition in the y-direction e - ine e - ine Absorption coefficient (mm - ) Figure 5. Therm rextion times in the x- nd y-directions. w L σ = 0.mm τ x = 0 = 0.6 mm = 0.65mm =.30 mm IV. Concusion The trnsient temperture response upon short-pused ser irrdition is numericy investigted. The emphsis ws pced on the comprisons of therm rextion times bsed on different definitions. The scttering effect shoud not be overooked in the determintion of the optic penetrtion depth bsed on the simution evutions. A new expression of τ is introduced nd found to cosey mtch with the prediction of τ which is bsed on the pek temperture decying with moderte het convection conditions. Thus, τ is recommended for use s the therm rextion time to be determined by the optic nd therm properties of tissue, couped with the ser input prmeters. τ hs better physic mening in representing the therm rextion time nd cn incorporte the reistic surfce cooing condition. However, the determintion of τ requires for compete modeing or experiment mesurement. 7

8 Figure 6. Trnsient temperture response t sever seected octions nd mesurement of τ. Figure 7. Infuence of surfce convection on trnsient temperture response nd τ. 8

9 30 Therm rextion time (sec) brin tissue-gry mtter ( σ = 0.63mm, σ = 0.7 mm s Bem width (mm) ) τ τ τ x 8 Figure 8. Comprison of therm rextion times predicted by different modes. Tbe. Optic nd therm properties of sever seected humn tissues. Wveength (nm) Absorption (mm ¹) Reduced Scttering (mm ¹) Diffusivity α (mm²/s) Humn dermis Cucsin 4, Humn ort 6, Humn hert Endocrdium 6, Humn iver 6, Humn brin gry mtter Humn uterus Humn brest Norm Tbe. The predicted therm rextion times for the sever humn tissues. Bem width D (mm) Therm Rextion τ x τ τ τ x τ τ τ x τ τ Time (sec) Dermis Aort Endocrdium Liver Gry mtter Uterus Brest

10 References. Anderson, R.R., nd Prrish, J.A., Seective Photothermoysis: Precise Microsurgery by Seective Absorption of Pused Rdition, Science, Vo. 0 (4596), 983, pp. 54, 57.. Anderson, R.R., 003, Dermtoogic History of the Ruby Lser: The Long Story of Short Puses, Arch Dermto, Vo. 39 (), 003, pp. 70, Fock, S. T., nd Mrchitto, K.S., Progress Towrds Semess Tissue Fusion for Wound Cosure, Otoryngo Cin North Am., Vo. 38 (), 005, pp. 95, Aison, R.R., Bgnto, V.S., Cuenc, R., Downie, G.H., nd Sibt, C.H., The Future of Photodynmic Therpy in Oncoog Future Onco., Vo. (), 006, pp. 53, Weinbum, S., Xu, L.X., Zhu, L., nd Ekpene, A., A New Fundment Biohet Eqution for Musce Tissue-Prt I: Bood Perfusion Term, J. Biomech. Eng., Vo. 9 (3), 997, pp. 78, vn Gemert, M.J.C., nd Wech, A. J., Time Constnts in Therm Lser Medicine, Lsers Surg. Med., Vo. 9, 989, pp Choi, B., nd Wech, A. J., Anysis of Therm Rextion During Lser Irrdition of Tissue, Lsers Surg. Med., Vo. 9, 00, pp Furzikov, N. P., Different Lsers for Angiopsty: Thermo Optic Comprison, IEEE J. Quntum Eectron, Vo. 3, 987, pp Guo, Z., nd Kumr, S., Discrete Ordintes Soution of Short Puse Lser Trnsport in Two-Dimension Turbid Medi, Appied Optics, Vo. 40 (9), 00, pp Guo, Z., nd Kumr, S., Three Dimension Discrete Ordinte Method in Trnsient Rditive Trnsfer, J. Thermophys. Het Trnsfer, Vo. 6, 00, pp Guo, Z., nd Kim, K.H., Utrfst Lser Rdition Trnsfer in Heterogeneous Tissues with the Discrete Ordinte Method, App. Opt., Vo. 4, 003, pp Kim, K.H., nd Guo, Z., Utrfst Rdition Het Trnsfer in Lser Tissue Weding nd Sodering, Numeric Het trnsfer A., Vo. 46, 004, pp Frnco, W., Liu, J., Wng G. X., Neson, J. S., nd Aguir, G., Rdi nd Tempor Vritions in Surfce Het Trnsfer During Cryogen Spry Cooing, Phys Med Bio., Vo. 50 (), 005 Jn, pp. 387, Cohen, M. L., Mesurement of the Therm Properties of Humn Skin, A Review. J. Invest. Dermto., Vo. 69, 977, pp Diz, S.H., Aguir, G., Bsu, R., Lverni, E., nd Wong, B. J. F., Modeing the Therm Response of Porcine Crtige to Lser Irrdition, Proceedings of SPIE, Vo. 467, 00, pp Cheong, W.F., Prh, S.A., nd Wech, A. J., A Review of the Optic Properties of Bioogic Tissues, IEEE J. Quntum Eectron, Vo. 6, 990, pp Vvno, J.W., nd Chitsbesn, B., Therm Conductivity nd Diffusivity of Arteri W nd Atheroscerotic Pque, Lsers Life Sci., Vo., 987, pp Vvno, J.W., Cochrn, J.R., nd Dier, K.R., Therm Conductivity nd Diffusivity of Biomteris Mesured with Sef- Heted Thermistors, Int. J. Thermophys, Vo. 6, 985, pp Tromberg, B. J., Coquoz, O., Fishkin, J., Phm, T., Anderson, E. R., Buter, J., Chn, M., Gross, J. D., Venugopn, V., nd Phm, D., Non-invsive Mesurements of Brest Tissue Optic Properties Using Frequency-Domin Photon Migrtion, Phios. Trns. R. Soc. Lond. B: Bio. Sci., Vo., 35, 997, pp

ANALYZING EFFECT OF TEMPERATURE VARIATION ON SKIN SUB-LAYERS USING FINITE ELEMENT APPROACH

Origin Reserch Artice Aied sciences Interntion Journ of Phrm nd Bio Sciences ISSN 0975-699 ANALYZING EFFECT OF TEMPERATURE VARIATION ON SKIN SUB-LAYERS USING FINITE ELEMENT APPROACH MONA DWIVEDI 1 * AND