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Applied Thermal Engineering 59 (2013) 41e51

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

A two-temperature model for selective photothermolysis lasertreatment of port wine stainsq

D. Li a, G.X. Wang a,b,*, Y.L. He a,*, K.M. Kelly d, W.J. Wu a, Y.X. Wang c, Z.X. Ying c

a State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, ChinabDepartment of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, USAc Laser Treatment Center, Department of Dermatology, Medical School, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, ChinadBeckman Laser Institute, University of California, Irvine, CA 92612, USA

h i g h l i g h t s

� Local thermal non-equilibrium theory was adapted in field of laser dermatology.� Transient interfacial heat transfer coefficient between two phases is presented.� Less PWS blood vessel micro-structure information is required in present model.� Good agreement between present model and classical discrete-blood-vessel model.

a r t i c l e i n f o

Article history:Received 4 December 2012Accepted 6 May 2013Available online 15 May 2013

Keywords:Laser dermatologyPort wine stainsTwo-temperature modelPorous mediaLocal thermal non-equilibriumMonte-Carlo method

q This is an open-access article distributed undeCommons Attribution-NonCommercial-No Derivativemits non-commercial use, distribution, and reproductthe original author and source are credited.* Corresponding author. State Key Laboratory of M

gineering, Xi’an Jiaotong University, No. 28, West X710049, China. Tel.: þ86 29 8266 5930; fax: þ86 29

E-mail addresses: gwang@uakron.edu (G.X. Wan(Y.L. He).

1359-4311/$ e see front matter � 2013 The Authors.http://dx.doi.org/10.1016/j.applthermaleng.2013.05.00

a b s t r a c t

Selective photothermolysis is the basic principle for laser treatment of vascular malformations such asport wine stain birthmarks (PWS). During cutaneous laser surgery, blood inside blood vessels is heateddue to selective absorption of laser energy, while the surrounding normal tissue is spared. As a result, theblood and the surrounding tissue experience a local thermodynamic non-equilibrium condition. Tradi-tionally, the PWS laser treatment process was simulated by a discrete-blood-vessel model that simplifiesblood vessels into parallel cylinders buried in a multi-layer skin model. In this paper, PWS skin is treatedas a porous medium made of tissue matrix and blood in the dermis. A two-temperature model is con-structed following the local thermal non-equilibrium theory of porous media. Both transient and steadyheat conduction problems are solved in a unit cell for the interfacial heat transfer between blood vesselsand the surrounding tissue to close the present two-temperature model. The present two-temperaturemodel is validated by good agreement with those from the discrete-blood-vessel model. The charac-teristics of the present two-temperature model are further illustrated through a comparison with thepreviously-used homogenous model, in which a local thermodynamic equilibrium assumption betweenthe blood and the surrounding tissue is employed.

� 2013 The Authors. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Port wine stain (PWS) birthmarks are congenital vascular mal-formations which occur in approximately 0.3% of newborns. PWS

r the terms of the CreativeWorks License, which per-

ion in any medium, provided

ultiphase Flow in Power En-ianning Road, Xi’an, Shaanxi8266 5445.g), yalinghe@mail.xjtu.edu.cn

Published by Elsevier Ltd. All righ7

are composed of ectatic venular capillary blood vessels, with di-ameters ranging from 10 to 300 mm, buried within healthy dermaltissue. PWS can be treated by the pulsed dye laser (PDL) based onthe well-known theory of selective photothermolysis with hemo-globin serving as the target chromophore [1]. According to thetheory, the PWS blood vessels can be thermally damaged selec-tively due to their preferential absorption of laser energy, ascompared with normal skin tissues, which are minimally affected.

Numerical methods have been employed to simulate lasertreatment of PWS [2e4]. Most existing models simplify the com-plex skin anatomy into a multi-layer structure. The Multi-LayerMonte Carlo method (MLMC) [5] was used to simulate light prop-agation within tissues and thus quantify light energy deposition[6e8]. Energy deposition can then be input into the bio-heat

ts reserved.

Nomenclature

A array to score light energy absorptiona specific interfacial areac specific heatC correction factor for tissue optical properties in

homogeneous layersd diameterE incident laser fluenceF light fluence distribution within tissueg anisotropy indexh heat transfer coefficientk thermal conductivityKbd the ratio of the thermal conductivities of the blood and

the dermal tissuesN photon numberNu non-dimensional heat transfer coefficientQ volumetric heat generation due to laser energy

absorptionr radial coordinates free path length of the photonT temperatureT volume-averaged temperaturet timeV volumez axial coordinate

Greek symbolsa the thermal diffusivityε volumetric fraction of the chromophorema the absorption coefficientms the scattering coefficientmt the attenuation coefficients normalized timex random number

Subscripts and superscriptsa airb bloodbasal basal layer of the epidermisbasal/der interface between the basal layer and the dermal layerbd interface between the blood vessel and the dermisd dermisder/PWS interface between the dermal layer and the PWS layere epidermisepi/basal interface between epidermis without melanin and the

basal layerk bottom surface of the computation domainm melaninp pulsePWS the PWS layerREV representative elementary volume* normalized

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e5142

transfer equation as the energy source term. Solutions of the bio-heat equation yield temperature change of the skin tissues, fromwhich thermal damage from a selected thermal injury model canbe quantified [9].

Various skin models for laser treatment of PWS have beendeveloped and can be roughly divided into two categories, thehomogeneous model and the discrete blood vessel model. Thehomogeneous model refers to models that treat skin tissue withPWS as a homogeneous mixture of uniformly distributed blood andsurrounding dermal tissue with a given blood volumetric fraction[6]. The optical and thermal properties of the homogenous mix-tures are estimated based on the average of the correspondingproperties of the two constituents, weighted by their volumetricfractions. Detailed anatomic structure of the blood has not beentaken into account.

One critical shortcoming of the homogeneous model is that itfails to distinguish the difference in the temperature of the bloodand the surrounding dermal tissue. According to the theory of se-lective photothermolysis, blood should be heated up preferentiallyby laser irradiation due to stronger absorption of the chosenwavelength by blood, compared with surrounding dermal tissue. Itis therefore expected that the blood should have a temperature thatis significantly higher than that of the surrounding dermal tissues,leading to a local thermodynamic non-equilibrium condition. Thehomogeneous model, however, assumes one temperature for boththe targeted chromophores and surrounding tissues. In the case forPWS, such a local equilibrium assumptionmakes the homogeneousmodel undesirable.

Attention was then paid to direct simulation of laser heating ofthe blood vessels by applying so-called discrete blood vesselmodels [7,8,10]. PWS have complex anatomic structures, makingmodeling difficult. Early discrete models considered either a singleblood vessel [7] or an array of blood vessels, regularly arrangedwithin the dermis [8]. Multiple blood vessels randomly arrangedwithin the dermis have also been taken into consideration [10].The results concluded from these models have provided some

quantitative information regarding selective blood vessel heatingduring laser treatment of PWS, and have been used in developmentof PWS laser treatment protocols.

The discrete models mentioned above assume that all bloodvessels have a straight geometry and are parallel to the skin surface.Such an assumption significantly simplifies mathematical treat-ment of the problem. To make a more realistic PWS model, morecomplex structural models were attempted by Pfefer et al. [11] whoemployed information about complex geometric configurations ofPWS blood vessels obtained through biopsy. They also extended theMulti-Layer Monte-Carlo method to a Voxel-based three-dimen-sional method that could take into account the effect of the geo-metric complexity of the interface between dermal tissue andblood vessels [11]. Such a model could provide specific informationregarding selective heating of blood vessels during PWS lasertreatment. Unfortunately, we don’t currently have imaging mo-dalities that can map PWS blood vessels noninvasively, making thisthree-dimensional model impractical to use.

In this paper, we present a local thermodynamic non-equilibrium, two-temperature model for simulation of the ther-mal response of PWS to laser irradiation. We treat the PWS skin as aporous medium made of a tissue matrix buried with highly-absorbing chromophores (blood confined within the vessels).Two energy equations, one for the blood and the other for thedermal tissue, will be developed based on the local thermal non-equilibrium theory of porous media. We have shown that lightpropagation and energy absorption by skin tissue can still betreated by the well-developed Multi-Layer Monte-Carlo method,except that the energy deposition is modified to be separatelyscored in two phases respectively, which is consistent with thetwo-temperature model. We also show that the geometric config-uration of the chromophore can be represented, as a first approx-imation, by the volumetric fraction of the chromophores and alength scale, i.e., the average diameter of the blood vessels within aPWS. An approximate relation is also developed to estimate theinterfacial heat transfer between a blood vessel and surrounding

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e51 43

tissue by solving a simplified one-dimensional transient heat con-duction problem of a chromophore (blood vessel) within tissue.Finally, the present two-temperature model is compared with thediscrete-blood-vessel model and homogenous model to illustratethe applicability of the present local thermal non-equilibriummodel for laser dermatology.

2. Physical model and mathematical description

2.1. Problem description and basic assumptions

In laser treatment of PWS, the pulsed laser beamwith a chosenwavelength is fired from a handpiece and irradiates the skinsurface. As the light penetrates into the tissue, it continuouslyscatters and is absorbed along the path. Two major chromophoresthat can absorb the laser light at the chosen wavelength aremelanin in the basal layer of epidermis and hemoglobin (ordeoxyhemoglobin) in the red blood cells within the blood vessels.Other skin components are usually transparent to laser light at thechosen wavelength but do have a strong scattering effect. Thelaser energy absorbed by melanin and red blood cells in the bloodvessels can be converted to thermal energy, raising the temper-ature of the epidermis and the blood within the blood vessels. Ahigh blood temperature leads to the desired effectdthermaldamage to the walls of the blood vessels, while heating of theepidermis results in adverse effects. Due to the complexity of theanatomic structure of PWS and multi-physical and physiologicalreactions of the skin and blood under laser irradiation, simplifi-cations and assumptions have to be made in order to provide amathematical description of this physical process. Major as-sumptions in this work are given as follows:

1. The skin tissue containing PWS is assumed to be a multi-layered skin model composed of a normal multi-layer skinmatrix with the PWS blood vessels superposed on the matrix,see Fig. 1. The multi-layer skin matrix is simplified as a two-layered geometry, consisting of two parallel planar layers,which are the epidermal layer and the dermal layer withoutincluding any subcutaneous fat. The undulations of the skinsurface at the epidermaledermal interface are considered to beplanar. Additional anatomic structures of normal skin tissueincluding nerve endings, sweat glands, oil glands and hairfollicles are ignored. Melanin in the epidermis is an importantlight absorber and is included by inserting a melanin-filledbasal layer at the bottom of the epidermal layer. The melaninparticles are assumed to be homogeneously distributed in the

Fig. 1. Representative elementary volume (REV) for port wine stains in a four-layerskin model.

basal layer of the epidermis, simulating the situation of un-tanned skin [12]. The PWS blood vessels, on the other hand,are buried in the dermis below a small superficial portion ofuninvolved dermis and are assumed to be spread out andmixed with the dermal tissue, forming a PWS layer in the skinmodel.

2. The skin tissue is assumed to be a multi-layer turbid mediumwith light scattering and absorption occurring within thelayers. Reflection and refraction take place at the interfacesbetween layers. The well-known radiative transport equationcan be used to describe such light propagation. As a firstapproximation, all optical properties of the tissues are assumedto be constant and uniformwithin a given component and/or alayer. The radiative transport equations can be solved with thewell-developed MLMC method [5].

3. Blood perfusion and metabolic heat generation are notincluded in the energy equations because of the short pulse ofthe laser irradiation used. Early works demonstrated that theseeffects were small, compared with the heat due to laser energyabsorption [13]. As a first approximation, no phase change ofblood is considered when the blood temperature is higher thanthe boiling point of water.

4. The composite melanin-filled basal layer in the epidermis is ahomogeneous mixture of melanin particles and normalepidermal tissue. A local thermodynamic equilibrium condi-tion is assumed within the mixture during the process of lasertreatment of PWS so that both melanin and the epidermaltissues have the same temperature. Such an assumption isjustified since the thermal relaxation time of melanin particles(in nanoseconds) is much shorter than the pulse duration ofthe laser irradiation (in milliseconds) during treatment ofPWS.

5. The PWS layer is treated as a porous medium with blood ves-sels buried within the matrix of normal dermal tissues. Due toselective absorption of the laser energy by the blood, the layeris assumed to be at a local thermodynamic non-equilibriumcondition, i.e., the blood within the vessels and the surround-ing dermal tissue would have different temperatures. It isfurther assumed that the two energy equations of porous me-dia, derived based on the volume averaging technique, can beemployed to solve the macroscopic average temperatures ofthe two phases.

Of the five assumptions mentioned earlier, 1e4 are commonpractice and have beenwidely employed in the literature [7,14]. Thelocal thermodynamic non-equilibrium condition in the PWS layeris introduced here in laser PWS modeling. We have demonstratedthat such a condition is consistent with the principle of selectivephotothermolysis [1].

2.2. Governing equations and boundary conditions

With the above assumptions, the two normal epidermal anddermal layers are single-phase homogeneous media with constantand uniform thermal and optical properties. However, the basallayer with melanin and the PWS layer are two-phase mixtures. Themacroscopic energy equations derived based on the volume aver-aging technique are employed for the two mixtures, but one isunder local thermodynamic equilibrium condition and the other isunder local non-equilibrium. In the case of local equilibrium, one-energy equation is used with the properties of the mixture as theweighted averaging of the two components [14]. In the case of localnon-equilibrium, however, two energy equations, one for eachphase, were used [15,16]. The energy equations for each layer aregiven below:

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e5144

Epidermis without melanin (0 < z < zepi/basal)

recevTevt

¼ VðkeVTeÞ þ Qe

tp(1)

Basal layer with melanin (zepi/basal < z < zbasal/der)

�εmðrcÞm þ ð1� εmÞðrcÞe

� vTbasalvt

¼ V�ðεmkm þ ð1� εmÞkeÞVTbasal

� þ Qbasaltp

(2)

Dermis without blood (zbasal/der < z < zder/PWS)

rdcdvTdvt

¼ VðkdVTdÞ þ Qdtp

(3)

PWS layer (zder/PWS < z < zk)

εbrbcbvTbvt

¼ V�εbkbVTb

� � hbdabd�Tb � Td

� þ εbQbtp

(4)

ð1� εbÞrdcdvTdvt

¼ V�ð1� εbÞkdVTd

� þ hbdabd�Tb � Td

�þ ð1� εbÞQd

tp(5)

where the energy deposition Q is only true during the pulseduration, T represents the local temperature in the single phasemedium and T represents the intrinsic volume-averaged temper-ature of the given phase in the mixture:

T i ¼1Vi

ZVi

TidV (6)

where V is the volume and Vi is the volume of the phase i within thechosen representative element volume (see Fig. 1) and thesubscript i represents melanin (m), epidermal tissue (e), blood (b),or dermal tissue (d), respectively. For the basal layer in theepidermis, the two average temperatures, Te and Tm, are equal dueto the assumption of local thermodynamic equilibrium:

Tbasal ¼ Te ¼ Tm (7)

where Tbasal is the average temperature in the melanin-filled basallayer. Here the thermal and the optical properties of the mixturecan be obtained by the weighted average of each component.

The two volume fractions, εm and εb, in the two mixtures aredefined within a given representative elementary volume (REV),VREV (see Fig. 1), in the melanin-filled basal layer and PWS layerrespectively, by the following expressions:

εm ¼ Vm=VREV;basalεb ¼ Vb=VREV;PWS

(8)

where VREV is:

VREV;basal ¼ Vm þ Ve for the basal layerVREV;PWS ¼ Vb þ Vd for the PWS layer (9)

For the PWS layer, a local thermodynamic non-equilibriumcondition leads to a two-temperature model, i.e., an average tem-perature Tb for the blood and an average temperature Td for thenormal dermal tissue surrounding the blood vessels. Due to strongabsorption of laser light by the blood, the two temperatures, i.e., Tband Td, should be significantly different during laser irradiation.

Therefore, two energy equations should be written for each, as Eqs.(4) and (5).

It should be noticed that in the present skin model, dermaltissue is continuous within the entire dermal region. Therefore, thetemperature of the dermal layer (Td) without PWS and the tem-perature of dermal tissue (Td) within the PWS layer should becontinuous at the interface of the two layers. However, Td withinthe PWS represents a volume-averaged value, since the local tem-perature of the dermal tissues fluctuates greatly due to the exis-tence of the blood vessels.

In Eqs. (4) and (5), hbd and abd are the interfacial heat transfercoefficient and the interfacial area per unit volume, respectively.Both of them have to be determined separately from supplementalmodels, i.e., a closure problem to be discussed in detail in Section2.3 below. It is of interest to note that if one assumes Tb ¼ Td, i.e.,local thermodynamic equilibrium Eqs. (4) and (5) can be combinedtogether to form a one-temperature equation, i.e., the classicalhomogenous model widely used in the early stage of the field [6]:

�εbðrcÞb þ ð1� εbÞðrcÞb

� vTPWS

vt¼ V

�ðεbkb þ ð1� εbÞkdÞVTPWS�

þ εbQb þ ð1� εbÞQdtp

(10)

where TPWS is the average temperature of the PWS layer.The definitions of other variables in the above equations are

listed in the nomenclature, and will not be repeated here.On the skin surface (z ¼ 0), a convection heat transfer condition

is used to describe the cooling effect of the environment:

kevTevz

����z¼0

¼ ha½Teðr;0; tÞ � Ta� (11)

where Ta is the temperature of the environment, and ha, treated as aconstant, stands for the convective heat transfer coefficient be-tween the skin surface and air.

At the interface between the PWS layer and the dermal layerwithout blood (zder/PWS), a zero flux condition is used for Tb and acontinuum condition is used for Td:

8>>>><>>>>:

vTbvz

�����z¼zder=PWS

¼ 0

vTdvz

�����zþ¼zder=PWS

¼ vTdvz

����z�¼zder=PWS

(12)

Adiabatic conditions are adapted for other three boundaries.

2.3. Closure problem for the two-temperature model of the PWSlayer: estimation of the interfacial heat transfer (hbd and abd)

Eqs. (4) and (5) represent macroscopic energy transport of aporous medium based on the average temperature of each phase,with the effect of the detailedmicroscopic-level transport of energywithin the mixture lumped into two new variables, hbd and abd. hbdrepresent the rate of local energy exchange at the blood-vesselinterfaces, while abd is the specific interfacial area. Therefore, thevalues of these two new variables depend on the detailed size andanatomic structure of the blood vessels in the PWS. Mathemati-cally, there is a closure problem to provide the quantitative relationfor hbd and abd for any given PWS.

In the literature of porous media, significant efforts have beenmade to estimate hbd for various porous media [15e18], but the

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e51 45

majority of these studies targeted a liquid filled with porous solidmedia. Such work demonstrated that the interfacial heat transfercoefficient depended on the geometry, the size and the volumetricfraction of the solid phase. For fast transient heating due to laserabsorption, hbd, should really be a function of both the time andtemperature differences between the blood and surrounding tis-sues. To solve this transient problem, we have to go back to thespecific blood vessel configuration. However, PWSs are malforma-tions of venous capillaries with fairly complex shapes and sizes[19e22] and it is very difficult to predict the heat transfer betweenblood and dermis in a real PWS. As a first approximation in thisstudy, we simplify the irregularly distributed PWS blood vessels asparallel tubes, and regularly distributed in dermis with the samediameter db (which can be thought as an average diameter of bloodvessels). In this way, the dermis containing PWS blood vessels canbe recognized as repeating unit cells composed of normal tissueand a single blood vessel. Thermal analysis can then bemade in thisunit cell to provide quantitative information of interfacial heattransfer between the blood and surrounding dermal tissue. Bothtransient and quasi-steady conditions have been considered. Thedetails of the problem settings and solutions are given in theAppendix. The analysis provides non-dimensional heat transferrelationships that can be used to evaluate the interfacial heattransfer coefficient during and after laser irradiation, as summa-rized below for a case in which εb ¼ 6%:

Nubd ¼ hbddbkt

¼

8>>>>>>><>>>>>>>:

3:92141exp

� Dq0:05132

!þ 2:69397exp

� Dq0:00705

!þ 1:42507 t* � t*p

4

ln2Kbd

þ 1� ε1=2b

2ε1=2b

t* > t*p(13)

where db is the diameter of the blood vessels, Kbd¼ kb/kd is the ratioof the thermal conductivities of the blood and the dermal tissues.t* ¼ abt=d

2b and t*p ¼ abtp=d

2b are the non-dimensional time and

laser pulse duration, respectively, with tp the dimensional laserpulse duration and ab the thermal diffusivity of the blood, and Dq isthe non-dimensional temperature difference, defined as follows:

Dq ¼ Tb � TdQbd2b=4kb

(14)

Within such a uniform cell, the specific interfacial area abd canbe simply written as:

abd ¼ 4εb=db (15)

It is then evident that the information for both diameter (db) andthe volumetric fraction (εb) of the blood are needed to carry out anyquantitative analysis with the present two-temperature model.

2.4. Multi-layer Monte Carlo method for light propagation andenergy deposition

The radiative transport equation for light propagation in theskin model is solved by the Monte Carlo method. The Monte Carlomethod used here is the so-called MLMC, developed byWang et al.[5]. However, some modifications have to be conducted to make it

consistent with the present two-temperature model for laserheating of PWS.

In the MLMC method, the photon propagation through themulti-layer skin is calculated using the absorption and scatteringcoefficients, together with a random number x:

Ds ¼ �lnðxÞ=mt (16)

where Ds is the free path length, mt is the attenuation coefficient ofthe layer which is the sum of the absorption coefficient ma andscattering coefficient ms. Here we need to mention that, for themixed layers (basal layer of the epidermis and PWS), the averageabsorption coefficient ma and the scattering coefficient ms of themixture are weighted averages based on the local volumetricfraction of melanin or blood, respectively. Thus, in the melanin-filled basal layer, we have:

ma;basal ¼ εm$ma;m þ ð1� εmÞ$ma;e (17)

ms;basal ¼ εm$ms;m þ ð1� εmÞ$ms;e (18)

In addition, in the PWS layer, it is necessary to consider theoptical screening effect of the blood vessels [23]. Following Milanicet al. [24] a correction factor C, related to the absorption coefficientof the blood and the diameter of the blood vessels, can be intro-duced to account for such an effect. We have:

ma;PWS ¼ εb$C$ma;b þ ð1� εbÞ$ma;d (19)

ms;PWS ¼ εb$C$ms;b þ ð1� εbÞ$ms;d (20)

After each absorption event, the photon is scattered, and a newpropagation direction is then generated which can be predictedaccording to the probability of distributions of the deflection (po-lar) angle and azimuthal angle at each interaction site (based on theanisotropy index, g). In the mixed layers, the average anisotropyindex, g, of the mixture is again weighted based on the localvolumetric fraction:

For the basal layer with melanin:

gbasal ¼ εm$gm þ ð1� εmÞ$ge (21)

For the PWS layer with blood:

gPWS ¼ εb$C$ms;b$gb þ ð1� εbÞ$ms;d$gdεb$C$ms;b þ ð1� εbÞ$ms;d

(22)

here again the screening effect of large blood vessels is accountedfor by the same correction factor, C.

In the present MLMC simulation, the weight method is used inscoring the absorption of light energy at a given location [5]. Aftereach scattering event, a fraction of the photon weight would beabsorbed at the end location of each path. The initial photon energy

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e5146

is set to be a unit and reduced after each absorption event. Theabsorbed fraction (ma/mt) of the photon energy is scored in the ab-sorption array, A, for a given location. With sufficient photonpropagation, the light fluence, F, in each layer, within a represen-tative volume element can be calculated as follows:

F ¼ E$pr2pma;itp

$A

N$VREV; i ¼ e; basal; d and PWS (23)

where rp is the laser spot size (cm), tp the laser pulse duration (s),N the photon number, VREV the volume of the REV, and E the inci-dent laser fluence. The details of the MLMC method can be foundelsewhere [5].

The known light fluence distribution within tissue, F, can beconverted to the energy deposition by multiplying the light ab-sorption coefficient of a given layer:

Qi ¼ ma;iF i ¼ e; d; basal (24)

where the subscript i represents the epidermal layer and thedermal layer or the basal layer. In Eq. (24) a volume averaged ab-sorption coefficient has to be used for a mixed layer.

In the PWS layer, it is necessary to evaluate the volumetric heatgeneration for both phases separately, consistent with the presenttwo-temperature model (Eqs. (4) and (5)). A modification of theenergy scoring has also been applied to obtain the energy deposi-tion for each phase:

Qb ¼ ma;bFQd ¼ ma;dF

(25)

Fig. 2. Temperature distribution for the skin matrix (a) and the blood (b) immediatelyafter laser irradiation. Blood vessel diameter (80 mm), blood volume fraction (6%),melanin fraction in epidermis (25%), laser wavelength (585 nm), pulse duration(1.5 ms), spot diameter (6 mm), and laser fluence (3.1 J/cm2).

2.5. Numerical method

The above mathematical equations constitute a multi-region,two-dimensional axial-symmetric transient heat conductionproblem for pulsed heat generation. The problem is numericallysolved by a finite-volume method with a uniform grid. The solutionprocedure includes the following steps:

1. Solve photonweight deposition (A) and the laser fluence (F) foreach control element by using the MLMC method, with an in-house Monte-Carlo code.

2. For a given volume fraction of bloodwithin the PWS layer, solvethe transient heat conduction equation of a unit cell to evaluatethe interface heat transfer coefficient, hbd. For each time step,hbd at each location is estimated from Eq. (13) based on theknown Tb and Td.

3. Solve the nodal temperature equations with an implicit timescheme by the tri-diagonal matrix algorithm (TDMA) withblock correction.

The validity of the MLMC code and the temperature code hasbeen extensively tested based on simple cases found in the

Table 1Thermal and optical properties of skin components [10].

Physical properties Melanin Blood Dermis

Thermal conductivity, k (kW/(m K)) 0.34 0.55 0.41Specific heat, c (J/(kg K)) 3.2 3.6 3.5Density, r (kg/m3) 1120 1060 1090Absorption coefficient, ma (cm�1) 350 191 2.4Scattering coefficient, ms (cm�1) 470 467 129Anisotropy index, g 0.79 0.99 0.79

literature [25]. The dense grid is used to generate the grid-independent solutions.

3. Results

To illustrate the basic characteristics of the present two-temperature model, the above mathematical equations are solvedfor a case of clinical interest. We choose a laser wavelength of585 nm, typical for modern pulsed dye lasers [19]. The beamdiameter of a flat hat-beam profile is chosen as 6 mm with pulseduration of 1.5 ms [10]. The thickness of each layer is 50 mm for theepidermal layer without melanin, 10 mm for the melanin-filled

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e51 47

basal layer with amelanin volumetric fraction of 25%, 90 mmdermallayer without blood, and 850 mm PWS layer filled with 80 mmdiameter blood vessels with volumetric fraction of 6% [10]. Theinitial skin temperature is assumed to be at 33 �C. The ambienttemperature Ta is set to be 25 �C. The convective heat transfer co-efficient between the skin surface and air ha is assumed to be 10W/m2 K [10]. The optical properties of the epidermis, melanin, dermis,and blood at this wavelength are listed in Table 1.

Fig. 4. Comparison of temperature distributions immediately after laser irradiationbetween the present two-temperature model (lines) and the discrete model of Tunnellet al. [10]. Blood vessel diameter (80 mm), blood volume fraction (6%), melanin fractionin epidermis (25%), laser wavelength (585 nm), pulse duration (1.5 ms), spot diameter(6 mm), and laser fluence (3.1 J/cm2).

3.1. Thermal characteristics of the two-temperature model andcomparison with the discrete blood vessel model

Fig. 2 shows a spatial view of the calculated temperature of theskin matrix (a) and the blood (b) in the PWS layer at the end of the1.5 ms laser irradiation. Fig. 2a shows that there is a high temper-ature spike in the melanin-filled basal layer, much higher than thatof the melanin-free epidermis and dermis. At the same time, withinthe PWS layer, a high peak blood temperature followed by quickdecrease can be observed in Fig. 2b. Comparing Fig. 2b with Fig. 2a,as one would expect, the blood shows a significantly higher tem-perature than that of the surrounding dermal tissue.

The temperature distribution in the skin matrix (Fig. 2a) and theblood (Fig. 2b) can be plotted, as shown in Fig. 3 that plots thetemperature distribution of each layer along the tissue depth (z) atthe center of the beam spot. Fig. 3 shows a sharp temperature spike,Tbasal, caused by epidermal melanin heating, can be observed in themelanin-filled basal layer. In the PWS layer, however, significantdifferences in the calculated temperatures between the blood (Tb)and the tissue (Td) indicate that the present two-temperaturemodel can correctly reflect the selective heating of blood underlaser irradiation.

A quantitative comparison can be made between the presenttwo-temperature model and the discrete-blood-vessel model. Thelatter was proposed by Tunnell et al. [10] who employed a discrete-blood-vessel model that was made of randomly distributed parallelcylindrical blood vessels with diameter of 80 mm. The result of Fig. 4shows such a comparison immediately after 1.5 ms and 585 nmlaser irradiation. The colorful background in Fig. 4 is a two-dimensional view of a three-dimensional temperature presenta-tion taken from Tunnell et al. that shows significant overlapping oftemperature curves for neighboring blood vessels. As one can see,

Fig. 3. Temperature distribution for various tissue depths immediately after laserirradiation. Blood vessel diameter (80 mm), blood volume fraction (6%), melanin frac-tion in epidermis (25%), laser wavelength (585 nm), pulse duration (1.5 ms), spotdiameter (6 mm), and laser fluence (3.1 J/cm2).

Fig. 5. Comparison of temperature distributions 5 ms (a) and 10 ms (b) after laserirradiation between the present two-temperature model (lines) and the discrete modelof Tunnell et al. [10]. Blood vessel diameter (80 mm), blood volume fraction (6%),melanin fraction in epidermis (25%), laser wavelength (585 nm), pulse duration(1.5 ms), spot diameter (6 mm), and laser fluence (3.1 J/cm2).

Fig. 6. Comparison of temperature distributions along tissue depth (a) and dynamicvariation (b) between the present two-temperature model (TTM) and the homogenousmodel. Blood vessel diameter (80 mm), blood volume fraction (6%), melanin fraction inepidermis (25%), laser wavelength (585 nm), pulse duration (1.5 ms), spot diameter(6 mm), and laser fluence (3.1 J/cm2).

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e5148

in the epidermal layer, the calculated temperature by the presentmodel is almost the same as that calculated by Tunnell et al. For aPWS, the prediction of two-temperature model envelops the 2-Dtemperature field of blood vessels by Tunnell et al. The goodagreement shown in Fig. 4 for the two different models validatesthe present two-temperature model.

A further comparison between the present two-temperaturemodel and the discrete-blood-vessel model is made at two timepoints after laser irradiation in Fig. 5 that shows a comparison ofthe temperature distributions for various tissue depths at 5 ms (a)and 10ms (b) post-laser irradiation. Both Fig. 5a and b demonstratefairly good agreement between our two-temperature model andTunnell’s discrete model. Although the two-temperature modelshows a slower cooling rate than that predicted by the discretemodel (Fig. 5a), agreement between the two models becomesbetter at 10 ms and the subsequent period after laser irradiation(Fig. 5b). This discrepancy between the twomodels may result frominaccurate treatment of the interfacial heat transfer (abdhbd) be-tween the blood vessels and the surrounding tissues in the two-temperature model.

3.2. Comparison with the homogeneous PWS model

The present two-temperature model has many common fea-tures compared with the traditional homogeneous model used inearly modeling of PWS laser treatments [6]. Both models ignore thedetailed anatomic structure of the PWS blood vessels and treat theirregularly distributed blood vessels as homogenously distributedblood in the dermis. However the early homogeneous modelassumed a local thermodynamic equilibrium between the blood inthe blood vessel and the surrounding dermal tissue, and employedan average temperature for both blood and dermal tissue. As aresult, the homogeneous model was eventually replaced bydiscrete blood vessel models [7,14]. The present two-temperaturemodel removes the assumption of local thermodynamic equilib-rium and considers the temperature difference between the bloodand the dermal tissue. Such a model presents a more realistic viewof laser treatment of PWS.

The importance of considering local non-equilibrium in thePWS layer is demonstrated by comparing the results from thepresent two-temperature model with that from the traditionalhomogeneous model (Eq. (10)), as shown in Fig. 6 that shows thetemperature distribution for various tissue depths calculated bythe homogeneous model and the two-temperature model,respectively. As we can see from the figure, the temperaturepredictions by the present model are as exact as the homoge-neous model in the epidermis (0 < z < Zmel-epi/der) and dermiswithout blood (Zmel-epi/der < z < Zder/blood) as the thermodynamicequilibrium condition can be achieved in both layers. In dermiscontaining PWS (Zder/blood < z < Zk) and at the selected wave-length and pulse duration, the two-temperature model predictsthat the calculated average temperature of the blood (Tb) ismuch higher than that of the dermis (Td) due to the distinctselective heating of blood over the surrounding dermis. How-ever, with the homogenous model, only an average temperaturefor the blood and dermis (TPWS) can be calculated, as the localthermodynamic equilibrium condition was employed. Theaverage temperature of the PWS layer is lower than the bloodtemperature and higher than the dermal temperature calculatedby the two-temperature model. TPWS is related to the volumetricfraction of the blood, which in this study was selected to be 6%.As a result, the average temperature is closer to the dermaltemperature.

The dynamic temperature variations in the laser spot centerat dermal skin depth of 150 mm, as predicted by the two-

temperature and homogenous models, are shown in Fig. 6b. Aswe can see from the figure, in the two-temperature model, theblood and surrounding dermal tissue temperatures can be pre-dicted simultaneously. During laser heating, the blood tempera-ture (Tb) increases quickly due to selective energy absorption atthe selected wavelength. At the same time, dermal temperature(Td) increases little during laser heating. After laser heating, as aresult of heat conduction, energy absorbed by the blood wouldconduct to the surrounding dermis and, as a result, blood tem-perature gradually decreases and dermal temperature slowly in-creases simultaneously (volumetric fraction of the dermis is muchlarger than the blood). With a homogeneous model, however, onlyone average temperature (TPWS) can be predicted for the bloodand dermis, which is higher than the dermal temperature andlower than the blood temperature. During laser heating, theaverage temperature gradually increases as a result of energyabsorption by blood. After the laser irradiation, the average tem-perature gradually decreases. As time goes on (e.g. 20 ms after thelaser pulse), a local thermodynamic equilibrium condition isachieved between the blood and the surrounding dermal tissueand, as a result, the blood temperature predicted by the two-temperature model is almost equal to the dermal tissue temper-ature and the average temperature predicted by the homogeneousmodel.

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e51 49

4. Discussion

A discrete blood vessel model has been commonly used tocalculate the thermal process during laser treatment of PWS[7,10,14]. Such a model could provide an exact solution of detailedtemperature distributions within the blood vessels as well as sur-rounding tissues to laser heating. However, the results are onlyvalid if the blood vessels of PWS lesions are straight parallel cyl-inders, as assumed in the discrete blood vessel model. The localthermodynamic non-equilibrium heat transfer theory of porousmedia has already been used to model the flow and heat transfer inblood perfused biological tissue accounting for the thermal non-equilibrium induced by blood perfusion [26e28]. The porous me-dia heat transfer theory employs fewer assumptions compared toother bio-heat transfer models [27]. In this study, however, wepresent how the local thermodynamic non-equilibrium heattransfer theory can be implemented to model laser treatment ofPWS accounting for the thermal non-equilibrium between bloodand peripheral tissue induced by selective photothermolysis.

Without detailed anatomic structure, the complex geometricconfiguration of PWS blood vessels is quantified by a typical lengthscale such as the local average blood vessel diameter and localvolumetric fraction of blood vessels. Two volume-averaged tem-perature values, one for the blood temperature and the other forthe temperature of healthy dermal tissues surrounding the vessels,are defined in the REV. Two energy equations for blood and sur-rounding dermal tissue are constructed based on the local ther-modynamic non-equilibrium heat transfer theory. Both theselective light energy absorption of the blood and interfacial heattransfer between the blood and the surrounding tissue are includedin the present two-temperature model. The model provides anaverage behavior of selective laser heating of the blood as describedby the theory of selective photothermolysis. Mathematically, thetwo-temperature model is less complicated in terms of coding andcomputation and requires less PWS blood vessel micro-structureinformation, compared with the discrete-blood-vessel model. Theadvantage of the two-temperature model over the discrete modelbecomes even more significant as the geometric configuration of areal PWS becomes more complex.

To accurately estimate the thermal history within skin duringlaser treatment of PWS, however, quantitative information for theinterfacial heat transfer between the blood and surrounding tissueis essential in the present two-temperature model. In the literatureof porous media, significant efforts have been made to estimate hbdfor various porous media [15e18]. But the majority of these ana-lyses are for steady state flow problems. In the present two-temperature model, a simplified treatment is employed bysimplifying irregularly distributed PWS blood vessels as repeatingunit cells composed of normal tissue and single blood vessel par-allel tubes. A transient heat conduction problem is solved in thesingle cell for the interfacial heat transfer coefficient. Our as-sumptions deviate somewhat from clinical reality because PWSsare blood vessel malformations with fairly complex shapes andsizes [19e22]. However, we provide a reasonable approximation ofthe evaluated situations. Further work will be performed to solvethe closure problem for the two-temperature model in a morerealistic manner.

Our two-temperature model can also be applied to provide anaccurate description of other laser dermatologic surgery such as theshort-pulse laser heating process for removal of pigmented lesions,where a strong non-equilibrium exists between melanin and thesurrounding tissue, leading to a large temperature difference.However, due to the small size of melanin particles and the largenumber of these particles within the skin, traditional methods arenot accurate as they can only simulate the situation of a single

melanin particle [29]. We will apply this model to other dermato-logic surgery indications in the near future.

5. Conclusion

In this paper, a two-temperature model for laser surgery of PWSis created by treating PWS as a porous medium made of a tissuematrix buried with the blood in the dermis and following the localthermal non-equilibrium theory of porous media. The thermalmodel includes two energy equations, one for the blood and theother for non-absorbing tissues. A relation between non-dimensional heat transfer coefficient and non-dimensional timedifference have been developed for interfacial heat transfer be-tween the blood and the tissue by performing a numerical analysisin a unit cell with simplified blood vessel geometry. The techniquehas been also devised in applications of the MLMC method forlaser propagation and energy deposition within a two-phase PWSlayer. The calculated results of the present two-temperature modelhave demonstrated clearly that the local thermal non-equilibriumoccurring during laser treatment of PWS is consistent with thetheory of selective photothermolysis. The present model mayprovide a new way to numerically simulate laser treatmentof PWS.

Acknowledgements

The work is supported by Joint Research Fund for OverseasChinese Scholars and Scholars in Hong Kong and Macao(51228602), the Specialized Research Fund for the Doctoral Pro-gram of Higher Education (20120201130006) and the FundamentalResearch Funds for the Central University (2011jdhz35, xjj2012115).KM Kelly is supported by National Institutes of Health (HD065536).

Appendix. Calculation of normalized interfacial heat transfercoefficient Nubd between the blood and normal tissue

Here we construct a microscopic heat transfer unit cell tocalculate the interfacial heat transfer coefficient, hbd, between theblood and the surrounding dermal tissue for transient heatingduring and after laser irradiation. It is further assumed that the sizeof the blood vessels is small, and as a result, the assumption ofuniform heating within the blood with a rate of heat generation, Qb,can be employed. Cylindrical geometry for PWS blood vessels isassumed as one-dimensional heat transfer.

In each unit cell, the energy equations can be written in terms ofthe blood (b) and the surrounding dermal tissue (d), respectively:

rbcbvTbvt

¼ kbr

v

vr

�rvTbvr

�þ Qb 0 � r � db

2(A.1)

rdcdvTdvt

¼ kdr

v

vr

�rvTdvr

�þ Qd

dd2

< r � dk2: (A.2)

where db is the diameter of the blood vessels and dk is the equiv-

alent diameter of the microscopic unit cell, with dk ¼ db=ε1=2b . Eqs.

(A.1) and (A.2) can be non-dimensionalized with the length (db/2),the character time (s ¼ d2b=ab), and the characteristic temperature

difference (DT ¼ Qbd2b=ð4kbÞ):

vqbvt*

¼ 1�r*� v

vr*

��r*� vqbvr

�þ 1 0 � r � 1 (A.3)

D. Li et al. / Applied Thermal Engineering 59 (2013) 41e5150

vqd*¼ 1�

*� v

*

��r*� vqd� þ Qd 1 < r � 1

1=2 (A.4)

Fig. A1. Calculated Nubd vs Dq for cylindrical blood vessels (εb ¼ 0.06) during transientheating due to an internal heat source within the blood.

rk

o

1 2

r1

r2

dk/2

db/2

Fig. A2. A thermal circuit for one-dimensional heat conduction between the blood andthe surrounding dermal tissue.

vt abd r vr vr Qb ε

where ab is the thermal diffusivities of the blood and abd is the ratioof the thermal diffusivities of the blood and the dermal tissues.

The corresponding boundary conditions become:

r* ¼ 0;vqbvr*

¼ 0 (A.5)

r* ¼ 1;

8><>:

qb�1; t*

� ¼ qd�1; t*

�Kbd

vqbvr*

����r*¼1

¼ vqdvr*

����r*¼1

(A.6)

r* ¼ 1ε1=2;

vqdvr*

¼ 0 (A.7)

It is assumed that the entire micro-volume is initially at a uni-form temperature so the initial condition becomes:

t* ¼ 0; qb�r*;0

� ¼ qd�r*;0

� ¼ 0 (A.8)

The interfacial heat transfer coefficient, hbd, between the bloodand the surrounding tissue is defined based on the heat transferbetween the blood and the surrounding tissue and the meantemperatures of the chromophores and the tissue. A non-dimensional form is given as follows:

Nubd ¼ hbddbkd

¼�2

vqdvr*

����r*¼1

qb � qd(A.9)

where the dimensionless mean temperatures of the blood (qb) andthe dermal tissue (qd) are defined as follows:

qb ¼ 12

Z10

r*qbdr* (A.10)

qd ¼ 12

1

ε1=2b

� 1

! Z1=ε1=2b

1

r*qddr* (A.11)

For a given chromophore and its volumetric fraction (εb), theabove problem can be solved numerically. The results provide arelationship between Nubd and dimensionless temperature differ-enceDq ¼ qb � qd. Fig. A1 shows the typical Nubd for cylindricalblood vessels. The curve can be numerically fitted to a mathemat-ical relationship that can be used to estimate hbd in Eqs. (4) and (5)for the two-temperature model simulation.

A further simplified form for Nubd can be derived for the casewithout internal heating, by assuming quasi-steady state one-dimensional heat conduction between the blood and the sur-rounding tissue. In this case, a simplified thermal circuit can beconstructed as shown in Fig. A2 with the average temperatures ofthe blood and the surrounding dermal tissue located at the middleof region, respectively. Then one has:

For a cylindrical chromophore:

Nubd ¼ hbddbkb

¼ 4

ln2Kbd

þ 1� ε1=2b

ε1=2b

(A.12)

Finally, we can combine the two cases mentioned earlier for the

interfacial heat transfer during (t < tp) and after laser irradiation(t> tp), where tp is the laser pulse duration, as given in Eq. (13) for acylindrical blood vessel.

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