International Journal of Heat and Mass Transfer 68 (2014) 278–294

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Analysis of radiative signals from normal and malignant human skinssubjected to a short-pulse laser

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.032

⇑ Corresponding author. Tel.: +1 321 674 7131; fax: +1 321 674 8813.E-mail address: kmitra@fit.edu (K. Mitra).

Arka Bhowmik a, Ramjee Repaka a, Subhash C. Mishra b, Kunal Mitra c,⇑a School of Mechanical, Materials and Energy Engineering, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, Indiab Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Indiac Department of Biomedical Engineering, Florida Institute of Technology, 150 West University Boulevard, Melbourne, FL 32901-6975, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 June 2013Received in revised form 16 September2013Accepted 16 September 2013Available online 15 October 2013

Keywords:Radiative heat transferShort-pulse laserTransmittanceReflectanceDiscrete ordinate methodMalignant human skin

Responses from a normal skin and a malignant human skin subjected to a low power short-pulse laser arestudied. Temporal variations of transmittance and reflectance that carry the signature of the skin condi-tions are analyzed. The normal skin is modeled based on the anatomical details available in the literature.The malignant skin represents commonly found skin lesions, viz., non-melanoma and cutaneous mela-noma. Optical properties of the skin such as the absorption and the scattering coefficients cover variouslesions, and they are within the therapeutic optical window of 600–1300 nm wavelength. Scattering ismodeled using Henyey–Greenstein phase function. The transient radiative heat transfer equation issolved using the modified discrete ordinate method.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In the recent past, consideration of thermal radiation in the char-acterization of optically participating media, such as tissues, hasbeen explored by many investigators [1–6]. Several facts that arenot revealed at normal spatial and temporal scales are understoodwell when these are investigated at lower scales. For example, withl as the characteristic length of the system and c = 3 � 108 m s�1 asthe speed of light, the temporal changes in transmittance and reflec-tance are not revealed if the radiative transport process is investi-gated at time scales O(l/c) = 10�6 s or higher [7]. This is truewhether the medium is under the influence of radiation for a shortor a long duration. However, if the observation time is lowered to10�9 s or lower, temporal changes in transmittance and reflectancebecome evident [7]. Further, if the incidence of the radiation sourceis also of the order of O(10�9 s) or lower, the temporal distributionsof the reflectance and transmittance are more distinct. This is for thefact that in the radiative transfer equation (RTE) the term (1/c)(oI/ot),where I is the intensity of radiation, can only be significant when thechanges are investigated at time scales O(10�9 s) or lower. This factbecomes a potential tool for characterizing an optically participatingmedium [1–7]. This attracted attention of many researchers towards

the possible application of thermal radiation in characterizing bio-logical tissues [1–6].

The properties for healthy and malignant tissues are different,since the cancer affected tissues exhibit a distinctive metabolicactivities, compositions and growth of cells as compared to thatof the normal tissue. Further, the properties and conditions ofthe tissues keep on changing with different growth phases ofmalignant cells. Thus, tissues having different optical propertiesyield different temporal variations of the transmittance and thereflectance when subjected to an external radiation source. Tem-poral variations in the transmittance and the reflectance carrythe signature of the tissue, and reveal its physiological state. Thus,for an optically participating human skin the optical window be-tween 600 and 1300 nm is of particular importance to opticaldetection of cancer. Since light in this wavelength regime scattersthrough several centimeters of tissues before being extinguished.One such observation by Anderson and Parrish [8] showed that1% of light can even penetrate the entire human chest at 605–850 nm wavelengths. The detailed information on human skinoptical properties can be obtained from [8–14].

In the literature, some works pertaining to the study of tempo-ral variations of transmittance and reflectance from optically par-ticipating medium are available [1–7]. Not specific to any tissuesand malignancies, these studies are of general nature. A goodamount of literature pertains to the applications of various

Nomenclature

c speed of lightg asymmetry factorG incident radiationI intensityIb blackbody intensityId diffuse intensityl lengthL order of approximation for Legendre polynomial PL

Mh number of discrete points over complete span of polarangle

n refractive indexq radiative heat fluxr positions geometric distance in the direction s of intensityS source termt timetp pulse widthtc cut-off periodTp time period of the pulse trainV volume of cellZ physical thickness

Greek symbolsd Dirac-delta functionl direction cosinek wavelengthja absorption coefficientrs scattering coefficientb extinction coefficients optical thicknessh polar angle

/ azimuthal angleX direction (h, /)DX solid angle (sinhdhd/)

Subscriptsc collimatedd diffusei incidentmax maximum valuemed mediumN northP cell centerS southref reflectancet totaltr transmittance

Superscriptsm index for discrete directions⁄ non-dimensional parameter

AbbreviationsCM cutaneous melanomaDOM discrete ordinate methodHG Henyey–GreensteinMDOM modified discrete ordinate methodNBCC nodular basal cell carcinomaRMC reverse Monte CarloRTE radiative transfer equationSCC squamous cell carcinomaSNR signal to noise ratio

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 279

numerical radiative transfer methods, such as the Monte Carlomethod [15,16], the discrete ordinate method (DOM) [7,17], thediscrete transfer method (DTM) [7,18], the finite element method(FEM) [19,20] and the finite volume method (FVM) [7,21] to thisclass of problems. Some experimental and numerical studies per-taining to the interaction of short pulse laser light with inhomoge-neous tissue phantoms [3,22], turbid medium [23], superficial skinmodels [24], normal human skins [25] and normal human breast[26] are available. The previous studies on skin melanoma and be-nign skin lesions are on active mode of thermal imaging [27] andon pulsed terahertz imaging [28,29], which has been proved tobe a potential non-invasive approach. Similarly, few importantinvestigations deal with the thermally induced skin injury [30]due to laser interaction, eye treatment using laser [31], thermallyinduced damage during retinal laser surgery [30,32], estimationof thermal load on eye using IR-thermogram [33] and on skin burn-ing due to external thermal load [34]. However, to the best of theknowledge of authors there is no study reported on the skin withand without malignant lesions, and for various types of malignan-cies therein when subjected to short pulse laser. The present work,therefore, aims at the study of temporal variations of transmittanceand reflectance of a biological tissue with and without malignancy,when subjected to short-pulse laser. In general, the radiative signa-tures due to laser-tissue interaction convey important informationabout in-vivo changes in tissue properties as well as tissue condi-tions, and can be quantified using a sensitivity study. Moreover,these output radiative signatures are different for different gradesand types of cancers. Thus, the variation in radiative signatures,due to the changes in properties and conditions (i.e., grades/growth of cancer) can be a means to understand any anomalies,and is, therefore, a potential dynamic technique for early detection

and staging of skin cancer. According to the guidelines given byAmerican Joint Committee for Cancer Staging, the increase in skincancer thickness/volume is regarded as the best predicator for can-cer staging [35]. Similarly, Clark et al. [36] and Breslow [37] de-scribed the prognostic relationship between the depth ofpenetration of skin cancer from epidermis into the dermis/subcutisand the survival expectancy. In view of the above details, presentstudy considered a model for the human skin cancer with differentgrowth stages (varying thickness) of lesion within the normal skin.

The motivation behind this work is to perform modeling basedfeasibility study to quantify the effectiveness of short-pulse laserfor early detection and staging of skin cancers. It is a well-known factthat, at present in most clinical settings, detection of subsurface can-cer at an early stage is a formidable challenge. It often involves mul-tiple biopsy trials to identify the types and grades. The detection andstaging of skin cancer using short-pulse laser based modality is a no-vel approach, since this will enable quick access of firsthand infor-mation without undergoing unnecessary biopsy trials. This articleperforms an extensive modeling and sensitivity study in order tomaterialize the argument described above. The present study is apart of the comprehensive research effort to device an approach todetect as well as stage skin cancer noninvasively.

In the present work, commonly found skin abnormalities, viz.,non-melanoma and melanoma skin cancers are considered. Theskin is modeled according to the available anatomical details,and the optical properties of the malignancies in the skin are con-sidered as reported in the literature. Temporal distributions oftransmittance and reflectance resulting from the incidence of lasersource having pulse-width of the O(1.0 � 10�9 s) and theO(1.0 � 10�12 s) are analyzed. Transient RTE is solved using themodified DOM (MDOM) [38].

280 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

2. Formulation

Schematic of the human skin consisting of epidermis, dermis,hypodermis and muscle layers is shown in Fig. 1a. Each layer iscomprised of sub-layers, and they differ in thicknesses and opticalproperties. Table 1 shows the thicknesses and optical properties ofthese layers [10,13,14,39]. The top layer of the inhomogeneousskin, i.e., the stratum corneum, is subjected to a short-pulse laser.The pulse-width of the laser can be O(1.0 � 10�9 s) or

(a)

(c)

(d)

Fig. 1. (a) Schematic of the multilayered skin, (b) schematic skin with different growth p(d) schematic of one dimensional control volume with representation of intensities use

O(1.0 � 10�12 s).The schematic of different growth phases of can-cerous lesion are shown in Fig. 1b. The temporal profile of the inci-dent laser pulse is the Gaussian one (as shown in Fig. 1c). Fig. 1ddepicts the one dimensional control volume of the physical domainwith the angular distribution of intensities and the heat fluxes.

While propagating through the radiatively participating skin,the absorption and the scattering of the laser light gives rise tothe temporal radiative signals, i.e., reflectance at the boundaryof incidence and transmittance at the opposite boundary.

(b)

hases of cancerous lesion, (c) temporal profile of the incident short-pulse laser andd in the MDOM.

Table 1Thickness and optical properties of the skin model at 785 nm [10,13,14,39].

Layers d (mm) n ja (mm�1) rs (mm�1) g

Stratum corneum 0.01 1.4 0.00089 18.95 0.8Living epidermis 0.08 1.4 0.19 18.95 0.8Papillary dermis 0.1 1.4 0.13 11.65 0.8Upper blood plexus 0.08 1.39 0.15875 15.485 0.818Reticular dermis 1.50 1.4 0.13 11.65 0.8Deep blood plexus 0.07 1.34 0.38875 46.165 0.962Dermis 0.16 1.4 0.13 11.65 0.8Hypodermis 3.0 1.44 0.008 11.44 0.9Muscle tissues 3.0 1.37 0.031 7.130 0.9

Table 3Optical properties of the skin lesion at 785 nm [45,46].

Skin carcinoma n ja (mm�1) rs (mm�1) g

Non-melanomaNodular basal cell carcinoma (NBCC) 1.4 0.035 8.140 0.8Squamous cell carcinoma (SCC) 1.4 0.062 6.680 0.8

MelanomaCutaneous melanoma (CM) 1.4 0.0075 9.185 0.8

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 281

Transmittance and reflectance are nothing but the radiative heatfluxes at the opposite boundaries, and they are obtained from thesolution of the transient RTE. With I as the intensity in any generaldirection s, ja as the absorption coefficient, rs as the scatteringcoefficient and b (=ja + rs) as the extinction coefficient, transientRTE is given by [7,40].

1cmed

@Iðr; s; tÞ@t

þ @Iðr; s; tÞ@s

¼ �bðrÞIðr; s; tÞ þ jaðrÞIbðr; tÞ þrsðrÞ4p

�Z

4pIðr; si; tÞUðs; siÞdXi ð1Þ

where Ib is the black body intensity, Uðs; siÞ is the scattering phasefunction, cmed is the speed of light in the medium (=c/n), c is thespeed of light in vacuum (=3.0 � 108 m/s) and n is the refractive in-dex. The first term on the right hand side of Eq. (1) accounts for theattenuation due to absorption and out-scattering. Augmentationowing to emission and in-scattering are represented by the secondand the third terms, respectively.

The transport of laser light (collimated radiation) Ic in the med-ium is governed by

1cmed

� �@Icðr; s; tÞ

@tþ @Icðr; s; tÞ

@s¼ �bðrÞIcðr; s; tÞ ð2Þ

It is to be noted that inside the medium, the laser light travels in thedirection s of its incidence, and it attenuates as per Eq. (2). Theattenuation of the laser light in the medium causes manifestationof the diffuse radiation which travels in all directions as shown inFig. 1d. Thus, in the medium, the intensity I in Eq. (1) is composedof the collimated intensity Ic and the diffuse intensity Id, i.e.,

Iðr; s; tÞ ¼ Icðr; s; tÞ þ Idðr; s; tÞ ð3Þ

Since the incident source of laser is short-lived, at a particular loca-tion in the medium, the laser light does not stay long. As a result,the manifested diffuse component at any location is also short-lived. The diffuse component of radiation that reaches the boundaryof incidence results in reflectance, and the attenuated component ofcollimated radiation and part of the diffuse radiation that reach theother boundary contribute to the transmittance (Fig. 1d).

Table 2Thickness and optical properties of the normal skin model at 520 and 840 nm [10,13,14,3

Layers d (mm) n k ¼ 520 nm

ja (mm�1) rs (

Stratum corneum 0.01 1.45 4.0 57.0Living epidermis 0.08 1.4 4.0 57.0Papillary dermis 0.1 1.4 0.5 50.0Upper blood plexus 0.08 1.39 2.45 50.0Reticular dermis 1.50 1.4 0.5 50.0Deep blood plexus 0.07 1.34 18.1 50.0Dermis 0.16 1.4 0.5 50.0Hypodermis 3.0 1.44 0.4778 33.7Muscle tissues 3.0 1.37 0.1366 8.8

Substitution of I = Ic + Id from Eq. (3) and left-hand-side terms ofEq. (2) in Eq. (1) results in

1cmed

� �@Idðr; s; tÞ

@tþ dIdðr; s; tÞ

ds¼ �bðrÞIdðr; s; tÞ þ Scðr; s; tÞ

þ Sdðr; s; tÞ¼ �bðrÞIdðr; s; tÞ þ Stðr; s; tÞ ð4Þ

where Sc and Sd are the source terms owing to the collimated anddiffuse radiations, respectively, and St (=Sc + Sd) is the total sourceterm. Source terms pertaining to collimated and diffuse radiationsare given as

Scðr; s; tÞ ¼rsðrÞ4p

Z4p

Icðr; si; tÞUðs; siÞdXi ð5Þ

Sdðr; s; tÞ ¼ jaðrÞIbðr; tÞ þrsðrÞ4p

Z4p

Idðr; si; tÞUðs; siÞdXi ð6Þ

In the problem under consideration, the short-pulse laser light ofmagnitude Ic,max incident on the top surface of the skin is the sourceof radiation. Owing to the intense scattering in biological medium,skin (Fig. 1a) can be approximated as an optically 1-D inhomoge-neous planar medium in which radiation is azimuthally symmetric.With incident angle of the laser source hc measured from the out-ward normal to the north boundary (Fig. 1a and d), in Eq. (5), Ic isgiven by [41]

Icðr; h; tÞ ¼ Ic;maxðh; tÞ expð�bðrÞscÞ

� exp �4t � sc

cmed� tc � ðN � 1ÞTp

tp

!2

ln 2

24

35� dðh

� hcÞ;0< t < 2tc ð7Þ

where sc = z/cos hc (Fig. 1d) is the geometric distance, N is the num-ber of pulses, Tp (=2tc) is the time period, tc = 3tp is cut-off timewhere tp is the pulse width and d is the Dirac-delta function.

With source term St (=Sc + Sd) known from Eqs. (5) and (6), thetemporal distribution of the diffuse intensity Id in the medium isobtained from the solution of Eq. (4). It is to be noted that evalua-tion of the source St (=Sc + Sd) requires knowledge of intensity

9,43].

k ¼ 840 nm

mm�1) g ja (mm�1) rs (mm�1) g

0.77 0.00091 17.6125 0.80.77 0.13 17.6125 0.80.77 0.105 10.625 0.80.79 0.15875 15.485 0.8180.77 0.105 10.625 0.80.96 0.4443 46.0625 0.9620.77 0.105 10.625 0.8

2 0.9 0.009 11.13 0.98 0.9054 0.029 6.71 0.9

0 5 10 15 2010-4

10-3

10-2

10-1

Ref

lect

ance

(q

ref*)

g = 0.0, t*

p = 0.3

τ1 = 0.5, τ

2 = 0.5

ω1 = 0.9, ω

2 = 0.1

Present (MDOM) RMC [47]

Dimesional time, t (ps)

(a)

0 5 10 15 2010-4

10-3

10-2

Ref

lect

ance

(q

ref*)

g = 0.0, t*

p = 0.3

τ1 = 0.5, τ

2 = 0.5

ω1 = 0.1, ω

2 = 0.9

Present (MDOM) RMC [47]

Dimensional time, t (ps)

(b)

Fig. 2. Comparison of temporal variation of reflectance q�ref from a two-layerinhomogeneous medium for scattering albedos (a) x1 = 0.9, x2 = 0.1 and (b)x1 = 0.1, x2 = 0.9 with [47] non-dimensional pulse width t�p ¼ 0:3.

(a)

(b)

0 2 4 6 8 10 12 14 16 18 20 22 24 260.0

5.0x10-3

1.0x10-2

1.5x10-2

2.0x10-2

2.5x10-2

tp = 1 ns single (N = 1)

train (N = 4)

Tra

nsm

ittan

ce (q

tr*)

Time, t (ns)

0 2 4 6 8 10 12 14 16 18 20 22 24 260.0

0.1

0.2

0.3

0.4

0.5

0.6

tp = 1 ns single (N = 1)

train (N = 4)

Time, t (ns)

Ref

lect

ance

(qre

f*)

Fig. 3. Temporal variations of (a) transmittance q�tr and (b) reflectance q�ref from thenormal skin subjected to single and train of 4 laser pulses having pulse width oftp = 1 ns.

282 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

distributions Ic and Id. With the variation of the collimatedintensity calculated from Eq. (7), the distribution of diffuse inten-sities Id are obtained from the solution of Eq. (4).

MDOM is an advanced version of the commonly used DOM [40].In the recent past, it has been successfully applied to varieties ofproblems in different geometry [7,38]. In the present work, thetransient RTE (Eq. (4)) is solved using the MDOM. The MDOM pro-cedure for calculating diffuse intensity Id is briefly described below.

Eq. (4) is first written for a discrete direction having indexm.Next, the first term @Id

@t in the left hand side of Eq. (4) is approxi-mated using the backward differencing scheme in time t. Withthese, in the simplified form Eq. (4) is written as

@Imd ðr; tÞ@s

þ AImd ðr; tÞ ¼ Sm

t ðr; tÞ þ BImd ðr; t � DtÞ ð8Þ

where B ¼ 1cmedDt and A = b(r) + B are constants.

For the physical domain (Fig. 1d) with direction cosinelm = cos hm, Eq. (8) can be written as

lm dImd ðr; tÞdz

þ AImd ðr; tÞ ¼ Sm

t ðr; tÞ þ BImd ðr; t � DtÞ ð9Þ

Integrating Eq. (9) over the discrete control volume DV = Dz � 1 � 1(Fig. 1d) results in

lm½Imd;SðtÞ � Im

d;NðtÞ� þ AImd;PðtÞDz ¼ Sm

t;PðtÞDzþ BImd;Pðt � DtÞDz ð10Þ

where for any direction having index m, Imd;N and Im

d;S are intensities atthe north and the south faces of the control volume having center P,(Fig. 1d) respectively, and Im

d;P and Smt;P are the volume averaged

intensity and the total source term, respectively. Owing to the azi-muthal symmetry, at any point in the medium, diffuse intensitieshave to be traced over the polar space 0 6 h 6 p: For 0 6 h < p=2,diffuse intensities are south bound (Fig. 1d), and the northbounddiffuse intensities are in the polar space p/2 < h < p. To facilitatesolution of Eq. (10), the number of unknown intensities are reducedby one. This is done by assuming cell center intensity to be the aver-age of the cell interface intensities, i.e.,

Imd;PðtÞ ¼

Imd;SðtÞ þ Im

d;NðtÞ2

ð11Þ

(b)(a)

(d)(c)

(f)(e)

0 1000 2000 3000 40000.0

2.0x10-3

4.0x10-3

6.0x10-3

8.0x10-3

1.0x10-2

1.2x10-2

Tra

nsm

ittan

ce (

q tr* )

Time, t (ps)

tp = 100 ps

N = 1 N = 4

0 500 1000 1500 2000 2500 3000 3500-0.040.000.040.080.120.160.200.240.280.320.360.400.440.48

Time, t (ps)

Ref

lect

ance

(qre

f* )

tp = 100 ps

N = 1 N = 4

0 200 400 600 800 10000.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

4.0x10-3

4.5x10-3

Time, t (ps)

Tra

nsm

ittan

ce (

q tr* )

tp = 10 ps N = 1 N = 4

0 50 100 150 200 250 300 350 400-0.04

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28tp = 10 ps N = 1 N = 4

Time, t (ps)

Ref

lect

ance

(qre

f* )

0 200 400 600 800 10000.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

tp = 1 ps

N = 1 N = 4

Time, t (ps)

Tra

nsm

ittan

ce (

q tr* )

0 20 40 60 80-0.010.000.010.020.030.040.050.060.070.080.090.100.110.12

tp = 1 ps N = 1 N = 4

Time, t (ps)

Ref

lect

ance

( qre

f* )

Fig. 4. Temporal variations of transmittance q�tr and reflectance q�ref from the normal skin subjected to single and a train of 4 laser pulses for (a) and (b) tp = 100 ps, (c) and (d)tp = 10 ps, and (e) and (f) tp = 1 ps.

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 283

In the solution, when lm > 0, marching starts from the north bound-ary and the diffuse intensity Im

d;N is comprised of the contributionfrom the collimated intensity and the reflected components ofnorthbound intensities coming from the sound boundary (Fig. 1d).The spatial marching in this case starts from the north boundary,and in the first control volume from that boundary, Im

d;N is knownfrom the radiative boundary condition given by [7]. The intensitiestravelling from north to south are called as southbound intensities.In this case, with Im

d;SðtÞ ¼ 2Imd;PðtÞ � Im

d;NðtÞ, Eq. (10) becomes

Imd;PðtÞ ¼

2lmImd;NðtÞ þ Sm

t;PðtÞdzþ BdzImd;Pðt � DtÞ

2lm þ Adz; lm > 0:0 ð12Þ

With Imd;P known for that control volume, Im

d;S is calculated which be-comes the known Im

d;N intensity for the next control volume. For all

discrete directions in 0 6 h < p=2; this process is repeated for all thecontrol volumes from the north boundary to the south boundary.Similarly, for the northbound diffuse intensities (lm < 0), for everydiscrete direction in the polar space p/2 < h < p, the spatial march-ing starts from the south boundary. The intensity at the southboundary are determined from the boundary condition given by[7]. In this case, for the first control volume from the south bound-ary, the known intensity is Im

d;S. Therefore, in this case withImd;NðtÞ ¼ 2Im

d;PðtÞ � Imd;SðtÞ, Eq. (10) becomes

Imd;PðtÞ ¼

2jlmjImd;SðtÞ þ Sm

t;PðtÞdzþ BdzImd;Pðt � DtÞ

2jlmj þ Adz; lm < 0:0 ð13Þ

Like calculation of southbound diffuse intensities, for all directionsin p=2 < h < p; and control volumes from the south boundary to

(a)

(c)(b)

0 5 10 15 201E-4

1E-3

0.01

0.1

tp = 1ps

Normal skinμm)μm)

CM (10NBCC (10SCC (10 μm)

Ref

lect

ance

(qre

f* )

Time, t (ps)

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.00.040

0.043

0.047

0.050

0.053

0.057

0.060

0.063

0.067

0.070

0.15 ps

0.00162

0.00

18

0.00

21

3.50 ps 3.65 ps

tp = 1ps

Normal skinCM (10 μm)NBCC(10 μm)SCC (10 μm)

Ref

lect

ance

(qre

f* )

Time, t (ps)

9 10 11 12 13 14 15 16 17 18 19 200.004

0.005

0.006

0.007

0.008

0.0090.01

0.02

tp = 1ps

Normal skin (m = -0.091)μm) (m = -0.0851 )μm) (m = -0.0764)

CM (10NBCC(10SCC (10 μm) (m = -0.0882)

Ref

lect

ance

( qre

f* )

Time, t (ps)

Fig. 5. Temporal variations of reflectance q�ref from the normal skin and malignancy (CM, NBCC and SCC) in the basal layer (thickness: 10 lm) of the skin; (a) full spectrum, (b)magnified view of the peak magnitudes of q�ref and (c) magnified view of the decaying signals.

284 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

the north boundary, Imd;S; I

md;P and Im

d;N are calculated. At any time level,computation of intensities Im

d;P using Eqs. (12) and (13) requireknowledge of the total source term Sm

t;P ¼ Smc;P þ Sm

d;P (Eqs. (5) and(6)) which are implicit function of intensities.

The scattering of radiation in the biological tissue is mostly inthe forward direction [12]. Thus, for the problem under consider-ation, i.e., in Eqs. (5) and (6) the scattering phase function Uðs; siÞis approximated by Henyey–Greenstein (HG) scattering phasefunction [40]. In the case of azimuthally symmetrical, this can bewritten in terms of Legendre polynomials PL as

Uðs; siÞ ¼ 1þXL max

L¼1

ð2Lþ 1ÞgLPLðcos hÞPLðcos hiÞ ð14Þ

where g is the asymmetry factor, L is the order of approximation. Inthe present work, the H–G scattering phase function has beenapproximated by considering the Legendre polynomial of 10th or-der, i.e., L = 10. It is fair enough to approximate the Legendre polyno-mial up to 10th order since the solutions have miniscule change withL > 10. The scattering function is used to evaluate the probableanisotropic distribution of incoming ray within the skin layers, andaccounts for in-scattering phenomenon to determine the sourcefunction in each discrete direction. It is worth-mentioning here that,the anisotropic factor g varies from layer to layer within the skin, andthe summation is considered up to the 10th order. In order to deter-mine the 10th order polynomial, the standard Legendre polynomial

expansion of polar angles has been considered. With this, the sourceterm (Eq. (5)) due to collimated radiation becomes,

Smc ðr; tÞ ¼

rsðrÞ4p

"Gcðr; tÞ þ 3gP1ðcos hmÞqcðr; tÞ

þX10

L¼2

ð2Lþ 1ÞgLPLðcos hmÞfIcðr; hc; tÞPLðcos hcÞdðhm � hcÞg#

ð15Þ

Similarly, the source term for diffuse radiation (Eq. (6)) takes theform,

Smd ðr;tÞ¼jaðrÞIbðr;tÞ

þrsðrÞ4p

Gdðr;tÞþ3gP1ðcoshmÞqdðr;tÞþX10

L¼2

fð2Lþ1ÞgLPLðcoshmÞ

�2pR hi¼p

hi¼0 Idðr;hi;tÞPLðcoshiÞsinhidhig

2664

3775ð16Þ

In Eq. (15), at any location in the medium, irradiation Gc and heatflux qc due to collimated radiation are computed from the following

Gcðr; tÞ ¼ Icðr; hc; tÞ ð17aÞ

qcðr; tÞ ¼ Icðr; hc; tÞ cos hcdðhm � hcÞ ð17bÞ

Similarly, the irradiation Gd and the heat flux qd owing to diffuseradiation are obtained from the following

(a)

(c)(b)

0 5 10 15 2010-4

10-3

10-2

10-1

tp = 1 ps

Normal skinμm)μm)

CM (160NBCC (160SCC (160 μm)

Ref

lect

ance

( q

ref* )

Time, t (ps)

2 3 4 5 6 7 80.01

0.02

0.03

0.04

0.05

0.060.07

3.65 ps

0.03

63

0.03

13

0.02

8

tp = 1 ps

Normal skinμm)μm)

CM (160NBCC(160SCC (160μm)

Ref

lect

ance

( q

ref* )

Time, t (ps)9 10 11 12 13 14 15 16 17 18 19 20

0.004

0.005

0.006

0.0070.0080.0090.01

0.02tp = 1 ps Slope

Normal skin (m = -0.09120)μm) (m = -0.08885)μm) (m = -0.09024)

CM (160NBCC(160SCC (160 μm) (m = -0.09017 )

Ref

lect

ance

( q

ref* )

Time, t (ps)

Fig. 6. Temporal variations of the reflectance q�ref from the normal skin and malignancy (CM, NBCC and SCC) spread to the living epidermis (thickness: 80 lm) and papillarydermis (thickness: 80 lm) of skin; (a) full spectrum, (b) magnified view of the peak magnitudes of q�ref and (c) magnified view of the decaying signals.

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 285

Gdðr; tÞ ¼ 2pZ hi¼p

hi¼0Idðr; hi; tÞ sin hidhi

� 4pXMh

m¼1

Imd ðr; tÞ sin hm sin

Dhm

2

� �ð18aÞ

qdðr; tÞ ¼ 2pZ hi¼p

hi¼0Idðr; hi; tÞ cos hi sin hidhi

� 2pXMh

m¼1

Imd ðr; tÞ sin hm cos hm sin Dhm ð18bÞ

where in Eq. (18), Mh is the number of discrete diffuse intensitiesconsidered over the polar space 0 6 h 6 p. Once the distributionsof collimated and diffuse intensities are known, normalized tempo-ral transmittance and reflectance signals are obtained from thefollowing

Transmittance : q�trð0; tÞ ¼qcð0; tÞ þ qdð0; tÞ

qc;maxðZ; tÞð19Þ

Reflectance : q�ref ðZ; tÞ ¼qdðZ; tÞ

qc;maxðZ; tÞð20Þ

It is to be noted here that, the transmittance, q�trð0; tÞ at the southboundary (i.e., at z = 0) is obtained from the knowledge of the

diffuse heat flux, qd(0, t) and the collimated heat flux, qc(0, t). Thediffuse heat flux, qd(0, t) is calculated from Eq. (18b) by consideringthe southbound ð0 6 h < p=2Þ diffuse intensities coming from thenorth boundary (Fig. 1d). Whereas, the reflectance at the northboundary (i.e., at z = Z) is obtained from the knowledge of the dif-fuse heat flux qd(Z, t). In this case, the diffuse heat flux, qd(Z, t) is cal-culated from Eq. (18b) by considering the northbound (p/2 < h < p)diffuse intensities coming from the south boundary (Fig. 1d). Sincethe residence time of radiation in the skin is very short O(10�11 s),the emission component of radiation owing to the temperature ofthe medium including its boundaries does not affect the transmit-tance and reflectance. Thus, in the calculation of transmittanceand reflectance at the south and the north boundaries, diffuse radi-ations are only from the medium.

The refractive indices of skin layers are all close to 1.4 (Table 1).Thus, inside the skin, radiation will travel in a straight path. How-ever, to avoid bending of incident laser light on the top boundary,and out coming intensities for transmittance and reflectance, theboundaries of the skin are covered with a gel having the samerefractive index, and the source and the detector are submergedwithin the gel covering skin. This is a common procedure amongmedical diagnostic techniques. It is to be further noted that radios-ities which consist of emission and reflection components of radi-ation from the two boundaries are also zero. The absence ofemission component as explained above, the nonexistence ofreflection component is owing to the fact that human skin acts like

(b)(a)

(d)(c)

0 100 200 300 400 500 6000.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

tp = 1.0 ps

λ = 520 nmλ = 785 nmλ = 840 nm

tp = 5.5 ps

Tra

nsm

ittan

ce (

qtr

* )

Time, t (ps)0 10 20 30 40 50

0.000.020.040.060.080.100.120.140.160.180.200.22

t p=

1 ps

t p =

5.5

ps

λ = 520 nmλ = 785 nmλ = 840 nm

Ref

lect

ance

( q

ref* )

Time, t (ps)

0 100 200 300 400 500 6000.0

1.0x10-3

2.0x10-3

3.0x10-3

4.0x10-3

5.0x10-3

tp = 25 ps

tp = 10 ps

λ = 520 nmλ = 785 nmλ = 840 nm

Tra

nsm

ittan

ce (

qtr

* )

Time, t (ps)0 20 40 60 80 100 120 140

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

t p =

25 pst p =

10 ps

λ = 520 nmλ = 785 nmλ = 840 nm

Ref

lect

ance

( q

ref* )

Time, t (ps)

Fig. 7. Temporal variations of the transmittance ðq�trÞ and the reflectance ðq�ref Þ from the normal skin at wavelengths, k ¼ 520 nm, 785 nm and 840 nm; (a) transmittance forsingle laser pulse of 1 ps and 5.5 ps, (b) reflectance for single laser pulse of 1 ps and 5.5 ps, (c) transmittance for single laser pulse of 10 ps and 25 ps, and (d) reflectance forsingle laser pulse of 10 ps and 25 ps.

286 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

a black surface [42]. Thus, the boundary intensities that are neededin the solution are zero.

3. Problem definition

In the present study, analysis has been done for normal andmalignant human skins. The properties of the normal and malig-nant skins are within the optical window of 600–1300 nm wave-length which is of particular importance for optical detection ofskin abnormalities [8]. The optical properties of skin presented inTable 1 [10,13,14,39] correspond to the wavelength of 785 nm. Ta-ble 2 [10,13,14,39,43] depicts the optical properties of the skin cor-responding to the wavelengths 520 nm and 840 nm.

Human skin primarily consists of three layers, viz., epidermis,dermis and hypodermis (or subcutaneous) (Fig. 1a). These layersare separated from various inner organs with the muscular layer.In the present work, thicknesses of the epidermis and the dermissame are the same as that of Wang et al. [39]. The thicknesses ofthe subcutaneous [14] and muscle layers are considered the same,and their refractive indices are taken from Bashkatov et al. [10].Thus, the total optical thickness of the normal tissue at 785 nmwavelength (Table 1) is s ¼

R z¼8:0mmz¼0 ðja þ rsÞdz is 83.00.

Cancer develops at the outermost layer of the skin, i.e., the epi-dermis, and it is named after the type of the cell. Nodular basal cellcarcinomas (NBCC) and squamous cell carcinoma (SCC) are the twotypes of non-melanomas. These non-melanomas occur most often,and they metastasize at a slower rate. On the other hand, the cuta-neous melanoma (CM) is the melanoma type of carcinoma which is

considered the most dangerous among all types of skin cancers[44]. The optical properties of non-melanoma and melanoma typeskin carcinomas are shown in Table 3 [45,46].

The malignancy in skin first starts in a thin layer of carcinomaslocated within the basal layer [44]. The basal layer is the lower partof the living epidermis (Fig. 1b). Later on, it spreads throughout theliving epidermis before it penetrates into the stratum corneumwhich comprises of dead cells and appears at the surface of theskin as a tumor [35]. The refractive index n and scattering asym-metry factor g of the malignant lesions are considered the sameas that of the living epidermis at early stages. However, theirabsorption and scattering coefficients are different.

4. Results and discussion

Numerical investigations have been done to analyze the tempo-ral radiative signals from the human skin with and without malig-nancies. Each skin layer has been divided into different number ofcontrol volumes resulting in equal cell size. For grid independentsolution, the grid size of Dz = 1.7 � 10�3 mm was found to be sat-isfactory, and a total of 40 angular divisions were found satisfac-tory for ray independent solution. The temporal stepDt ¼ Dz=cmed ¼ 8:0� 10�3 ps has been considered for time march-ing such that Dt� tp, and the incident angle of the laser lighthas been considered normal to the irradiated boundary, i.e.,hc = 00. At every time step, the convergence has been assumed tohave achieved when the change in the total source term St at allthe points for two consecutive iterations did not exceed the

)d()a(

)e()b(

)f()c(

0 5 10 15 200.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

tp = 1ps

dStratum corneum

= 0.01 mm

Variation in thicknessof Living epidermis 0.05 mm (37.5 % reduced) 0.08 mm (Original) 0.11 mm (37.5 % increased) 0.14 mm (75.0 % increased)

Ref

lect

ance

( q

ref* )

Time, t (ps)0 20 40 60 80 100 120

0.00

0.05

0.10

0.15

0.20

0.25

0.30

tp = 10 ps

dStratum corneum

= 0.01 mm

Variation in thicknessof Living epidermis 0.05 mm (37.5 % reduced) 0.08 mm (Original) 0.11 mm (37.5 % increased) 0.14 mm (75.0 % increased)

Ref

lect

ance

( q

ref* )

Time, t (ps)

2.0 3.0 4.0 5.0 6.0 7.0 8.00.020

0.030

0.040

0.050

0.060

0.070

dStratum corneum

= 0.01 mm

0.003 0.002

0.005

tp = 1ps

Variation in thicknessof Living epidermis

0.05 mm 0.08 mm 0.11 mm 0.14 mm

Ref

lect

ance

( q

ref* )

Time, t (ps)25 27 30 32 35 37 40 42 45

0.050.080.100.130.150.180.200.230.250.280.30

tp = 10 ps0.01 0.0055

0.005 dStratum corneum

= 0.01 mm

Variation in thicknessof Living epidermis

0.05 mm 0.08 mm 0.11 mm 0.14 mm

Ref

lect

ance

( q

ref* )

Time, t (ps)

9 10 11 12 13 14 15 16 17 18 19 200.000

0.005

0.010

0.015

0.020

tp = 1ps

dStratum corneum

= 0.01 mm

Variation in thicknessof Living epidermis

0.08 mm

Ref

lect

ance

( q

ref* )

Time, t (ps)

(m = -0.09025)(m = -0.091)(m = -0.09247)(m = -0.09755)

0.05 mm

0.11 mm

Slope

0.14 mm

50 60 70 80 90 100 110 1200.00

0.01

0.02

0.03

Slope(m = -0.0364)(m = -0.0363)(m = -0.0362)(m = -0.0361)

Variation in thicknessof Living epidermis

0.05 mm 0.08 mm 0.11 mm 0.14 mm

tp = 10 ps

dStratum corneum

= 0.01 mm

Ref

lect

ance

( q

ref* )

Time, t (ps)

Fig. 8. Comparative study of reflectance ðq�ref Þ from the normal skin at 785 nm wavelength due to the variation in the epidermal thickness, (a) and (d) full spectrums at 1psand 10ps pulse width, (b) and (e) magnified views of the peak magnitudes of q�ref at 1 ps and 10 ps pulse width, (c) and (f) magnified views of the decaying signals at 1ps and10 ps pulse width, respectively.

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 287

tolerance limit. The tolerance limit for nanosecond (ns) and pico-second (ps) pulse widths were 1 � 10�10 and 1 � 10�13, respec-tively. The space marching and the ray tracing procedures inMDOM are the same as that of the steady-state radiative transportphenomenon given in [38] and the time marching procedure fol-lows Mishra et al. [7].

Before we present and analyze results of the present study forvarious cases, we first validate our code against the results avail-able in the literature [47]. For a two-layer medium having differentoptical properties, i.e., optical thickness s and scattering albedo x,for asymmetry factor g = 0.0 and non-dimensional pulse widtht�p ¼ bctp ¼ 0:3, temporal variations of reflectance signal are com-pared in Fig. 2a and b. In Fig. 2a, results are compared for(s1, x1) = (0.5, 0.9) and (s2, x2) = (0.5, 0.1), while for comparisonmade in Fig. 2b, with the two layers swapped, the properties are

(s1, x1) = (0.5, 0.1) and (s2, x2) = (0.5, 0.9). In both the cases, re-sults of the present work obtained using the MDOM are in excel-lent agreement with that obtained using the reverse Monte Carlo(RMC) method [47].

4.1. Normal skin subjected to a single and a train of short-pulse laser

For skin properties as given in Table 1, the reflectance and thetransmittance from the normal skin have been analyzed forshort-pulse laser with pulse widths tp = 1 ns, 100 ps, 10 ps and1 ps for a single pulse (N = 1) and a train of 4 pulses (N = 4) in Figs. 3and 4. The temporal variations of the transmittance and the reflec-tance for pulse width tp = 1 ns are shown in Fig. 3(a) and (b),respectively. It can be observed from Fig. 3 that both the transmit-tance and reflectance replicate the incident short-pulse laser,

(d)(a)

(e)(b)

(f)(c)

0 5 10 15 2010-4

10-3

10-2

10-1

tp = 1ps

Variation in thickness of Dermis 1.910 mm (Original) 2.626 mm (37.5 % increased) 3.343 mm (75.0 % increased)

Ref

lect

ance

( q

ref* )

Time, t (ps)0 20 40 60 80 100 120

0.00

0.05

0.10

0.15

0.20

0.25

0.30Variation in thickness of Dermis

1.910 mm (Original) 2.626 mm (37.5 % increased) 3.343 mm (75.0 % increased)

tp = 10 ps

Ref

lect

ance

( q

ref* )

Time, t (ps)

2 3 4 5 6 7 80.0060.0070.0080.009

0.02

0.03

0.040.050.060.07

tp = 1ps

Variation in thickness of Dermis 1.910 mm (Original) 2.626 mm (37.5 % increased) 3.343 mm (75.0 % increased)

Ref

lect

ance

( q

ref* )

Time, t (ps)26 28 30 32 34 36 38 40

0.140.150.160.170.180.190.200.210.220.230.240.250.260.27

0.0051 0.0071

Variation in thickness of Dermis 1.910 mm (Original) 2.626 mm (37.5 % increased) 3.343 mm (75.0 % increased)

tp = 10 psR

efle

ctan

ce (

qre

f* )

Time, t (ps)

9 10 11 12 13 14 15 16 17 18 19 200.003

0.004

0.0050.0060.0070.0080.009

0.01

0.02tp = 1ps

Variation in slopethickness of Dermis

1.910 mm (m = -0.090) 2.626 mm (m = -0.091) 3.343 mm (m = -0.087)

Ref

lect

ance

( q

ref* )

Time, t (ps)50 60 70 80 90 100 110 120

0.00

0.01

0.02

0.03

Variation in thickness slopeof Dermis

1.910 mm (m = -0.0621) 2.626 mm (m = -0.0619) 3.343 mm (m = -0.0642)

tp = 10 ps

Ref

lect

ance

( q

ref* )

Time, t (ps)

Fig. 9. Comparative study of reflectance ðq�ref Þ from the normal skin at 785 nm wavelength due to the variation in the dermal thickness, (a) and (d) full spectrums at 1 ps and10 ps pulse width, (b) and (e) magnified views of the peak magnitudes of q�ref at 1 ps and 10 ps pulse width, (c) and (f) magnified views of the decaying signals at 1 ps and 10 pspulse width, respectively.

288 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

though their peak magnitudes are different. Transmittance ishighly attenuated with non-dimensional peak value �2.0 � 10�2

(Fig. 3a), while the same for the reflectance is �0.46 (Fig. 3b). Fora train of 4-pulses (N = 4), the peak magnitudes as well as temporalspreads corresponding to every pulse are the same as that for thesingle pulse (N = 1).The temporal spread owing to any pulse is al-most the same as that of the incident laser pulse. No cascading ef-fect is observed. This is for the reason that in travelling from thenorth boundary to the south boundary of the skin of thickness

8 mm, laser light takes just 1:4�8�10�3 m3�108 m s�1 ¼ 37:33� 10�12 s i.e.,

37.33 ps, and only after the effect of one pulse is over after a periodof about 6 ns, the effect of the second pulse starts. Lower value oftransmittance and a higher value of the reflectance is the resultof very high effective extinction coefficient b = ja + rs = 10375 m�1.

For single (N = 1) and train of four pulses (N = 4), temporalvariations of transmittance q�tr and reflectance q�ref signals from

the normal skin subjected to short pulse laser of pulse widthtp = 100 ps, 10 ps and 1 ps are shown in Fig. 4. Fig. 4a, c and e showq�tr variations for tp = 100 ps, 10 ps and 1 ps, respectively. For thecorresponding pulse widths, reflectance q�ref signals are shown inFig. 4b, d and f, respectively.

An observation of (Figs. 3a, 4a, c and e shows that for a single(N = 1) pulse, the peak magnitude of the transmittance q�tr signaldecreases with decrease in the pulse width tp. With tp = 1 ns andtp = 100 ps (=0.1 ns), the peak magnitude of q�tr is O(10�2), whilefor tp = 10 ps and 1 ps, the same are O(10�3) and O(10�4), respec-tively. Higher pulse width tp means more energy input to the med-ium, and hence is the observed trend of higher peak magnitudesfor higher tp. For N = 4, unlike tp = 1 ns (Fig. 3a), tp = 100 ps(Fig. 4a) and tp = 1 ps (Fig. 4e) cascading effect is visible fortp = 10 ps (Fig. 4c). This is for the reason that though in every casewith time period of the pulse-train Tp = 6tp, for tp = 1 ns and

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 289

tp = 100 ps, the effect of the second pulse manifests after the firstpulse has reached the south boundary at t = 37.33 ps. However,this is not so for tp = 10 ps and tp = 1 ps. In case of tp = 10 ps(Fig. 4c), the second pulse starts after Tp = 6tp = 60 ps and it is6 ps in case of tp = 1 ps (Fig. 4e). The cascading effect is visible fortp = 10 ps (Fig. 4c) since each pulse originates after 60 ps of the firstpulse whereas time taken to reach the south boundary is 37.33 ps,thus the effect of second pulse manifests at the south boundarybefore the effect of first pulse is over. In case of tp = 1 ps (Fig. 4e),each pulse originates after 6 ps of the first pulse but the time re-quired to reach the south boundary is more, i.e., 37.33 ps, due towhich the effect of second pulse manifests at the south boundaryafter the effect of first pulse is over. Thus, no cascading effect isobserved in case of tp = 1 ps. The cascading effect refers to the casewhen the effect of other pulse manifests at the boundary before theeffect of preceding pulse is over. The effect will appear as a stepincrement in the value from one pulse to another as can be seenfrom Fig. 4(c) and (f).

Peak magnitudes and temporal spreads of the reflectance q�ref

are too observed lower for lower value of the pulse width tp. Cas-cading effect too becomes visible only for tp = 1 ps (Fig. 4f). Thesuccessive increase in the peak magnitudes in the reflectance ismore prominent for tp = 1 ps (Fig. 4f). It is to be noted that unliketransmittance q�tr that in every case appear only after the laser lightpulse reaches the south boundary at 37.33 ps, the reflectance q�ref

manifests much before. As soon as the laser light enters the med-ium from the top boundary, it starts attenuating, and the mani-fested diffuse intensities give rise to reflectance q�ref signal at thetop boundary immediately. Though the temporal profile of theincident laser pulse is Gaussian is 6tp (Fig. 1c), significant energyinput to the medium takes place only for the time tp. This is thereason for manifestation of q�ref only after t � tp.

An observation of results in Figs. 3 and 4 has shown that thetransmittance signals are very weak. Further, in practical applica-tion, as far as any malignancy in the skin is considered, unlesssome portion of the affected skin is surgically removed and diag-nosed, owing to increased opacity of the tissues, bones and skinin the direction of travel of the laser light, no transmittance signalwill appear on the other side of the body part. However, the reflec-tance signal will remain present. In the following, we thus analyzethe temporal variations of reflectance for malignancies in the skin.In order to analyze the signals properly, we depict the results of asingle pulse having pulse width tp = 1 ps only. The signals from themalignant skin are easily distinguishable at shorter pulse width,and hence for comparison purpose, the results for 1 ps have beendepicted in the following sections. With malignant skin, the singlepulse provides better contrast than the multiple pulses; hence theresults are shown only for the single pulse.

4.2. Skin with and without malignancy subjected to a short-pulse laser

Cancer in the skin grows through several intermediate phasesbefore it manifests as a tumor on the skin surface. Initially, it ap-pears in the basal layer (thickness 10 lm), and then it equallyspreads to the remaining portions of adjacent tissues, i.e., the livingepidermis (thickness 70 lm) and papillary dermis (thickness80 lm). In its last phase, cancer penetrates through the stratumcorneum and appears at the surface of the skin as a tumor. In thefinal stage, when it appears on the skin, its nature can be ascer-tained through various available dermatological techniques. Thus,in the present work, consideration has been given to the diagnosisof early growth stages, i.e., the two intermediate stages of the can-cer in the skin. Initially, the results are provided for cancer only inthe basal layer. In this case, the remaining thickness of the skin isnormal. Next, the results are provided for the cancer spreadingequally on both sides of the basal layer, i.e., remaining living

epidermis (thickness: 70 lm) and the papillary dermis (thickness:80 lm) that makes the total thickness of 160 lm affected tissue.Fig. 1b depicts the schematic of different growth phases of cancer-ous lesion mentioned above. Results are shown in Figs. 5 and 6 fortwo intermediate stages.

In Fig. 5, temporal variations of reflectance are shown for cancerin the basal layer (thickness 10 lm). Results are provided for threetypes of malignancies, viz., CM, NBCC and SCC. These skin malig-nancies differ in their absorption and scattering coefficients (Ta-ble 3). To see the changes, the temporal variation of thereflectance for normal basal layer is also plotted. Fig. 5a showsthe entire spectrum, while the magnified views of the peak andthe decaying signals are shown in Fig. 5b and c, respectively.

An observation of Fig. 5b shows that, for normal skin, the peakmagnitude of reflectance q�ref appears at 3.65 ps. However, withmalignancies in the basal layer, the peak magnitudes manifestslightly ahead (0.05�0.15 ps) of the normal skin. These magnitudesare less than that for the normal skin. The differences in the peakmagnitudes of the reflectance q�ref of the normal skin and the skinwith CM, NBCC and SCC are 0.00162, 0.0018 and 0.0021, respec-tively. The trend is opposite in the decaying region (Fig. 5c).

With cancer spread equally on both sides of the basal layer upto total thickness of 160 lm, i.e., the remaining portion of the liv-ing epidermis (thickness: 70 lm) and papillary dermis (thickness:80 lm), temporal distributions of reflectance q�ref for skin with CM,NBCC and SCC are shown in Fig. 6a–c, respectively. In comparisonto the same malignancies in the basal layer (Fig. 5b), differences inthe peak magnitudes of q�ref between the normal and the skin withall three types of malignancies are more (Fig. 6b). For skin with CM,NBCC and SCC, these differences are 0.028, 0.0313 and 0.0363,respectively. In comparison to the malignancy only in the basallayer (Fig. 5b), the difference is approximately 17 times more. Like-wise, this difference is also significant in the decaying region(Fig. 6c). To have a better insight, in Figs. 5 and 6c, for every decay-ing signals, slope m obtained by a log-data fitting function has alsobeen provided. For all three malignancies, the –ve slope is margin-ally higher when the cancer has spread to the living epidermis aswell as papillary dermis (Fig. 6c). It is further observed fromFig. 6bthat the peak magnitudes of the reflectance q�ref with differ-ent malignancies appears almost the same time as for the normalskin, i.e., 3.65ps.Whereas, in the case of malignancy is only in thebasal layer of the skin (Fig. 5b) the peak magnitude of malignancyappears ahead of that of normal skin.

With the presence of malignancy in the skin, absorption andscattering coefficients go down (Tables 1 and 3). With emissionbeing absent within the tissue subjected to short-pulse laser, it isthe scattering that contributes to the reflectance. Therefore, as ob-served in Figs. 5 and 6, reduced scattering in the tissue results inlow magnitudes of the reflectance. Further, when the cancerspreads to the remaining part of living epidermis as well as papil-lary dermis, owing to the reduced scattering in more volume(160 lm) of the skin, reflectance reduces further. For cancer inthe basal layer as well as in case cancer spread to 160 lm of theskin adjacent to basal layer, peak magnitude is the maximum forskin with CM, and is the minimum for that with SCC. This trendis for the reason that with these malignancies, the scattering coef-ficients rs is the highest (9.185 mm�1) for skin with CM, and is thelowest (6.680 mm�1) for the skin with SCC. For all cases of themalignancies, and the normal skin, the peak magnitude appearsclose to the cut-off time of input laser light, i.e., 3 ps. This for thereason that the maximum intensity for the short-pulse laser lightwith Gaussian temporal profile is delivered at 3 ps. However,intensities with lower magnitudes in the first half of the pulsereaches any location in the living epidermis before the cut-off time,and the scattered radiation from these locations thus reach the topboundary of the skin at the start of time marching. Due to the

290 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

integrated effects, the maximum radiation reaches the top bound-ary close to the cut-off time of input laser light.

4.3. Sensitivity analysis

In this section, the sensitivity of the radiative signature due tothe variation in laser wavelength, variation in thickness of normalskin layers, change in different grades of cutaneous melanoma andthe effect of noise has been studied in detail.

4.3.1. Effect of laser light wavelengthIn general, the optical properties of the skin tissues vary with

wavelength as discussed in the literature [8–14]. Thus a sensitivitystudy has been carried out by considering the skin properties at520 nm, 785 nm and 840 nm, subjected to laser pulse having1 ps, 5.5 ps, 10 ps and 25 ps pulse width. Table 1 depicts the opticalproperties of the skin at 785 nm and Table 2 depicts the opticalproperties of the skin at 520 nm and 840 nm. A comparison oncharacteristic signals, viz., transmittance and reflectance from nor-mal skin subjected to laser of different wavelengths are shown inFig. 7. The temporal variation in transmitted and reflected signalssubjected to laser having single pulse of 1.0 ps and 5.5 ps at threedifferent wavelengths are depicted in Fig. 7(a) and (b), respectively.Similarly, the temporal variation in transmitted and reflected sig-nals subjected to laser having single pulse of 10 ps and 25 ps atthree different wavelengths are depicted in Fig. 7(c) and (d),respectively. Owing to the decrease in optical thickness of skin atk ¼ 840 nm as compared to that at k ¼ 785 nm, light diffuses to agreater depth at longer wavelength as compared to that at shorterwavelength, as can be seen from the variation in magnitude of

(

(b)

0 50.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Ref

lect

ance

( q

ref* )

Tim

2 3 4 5 6 7 80.01

0.02

0.03

0.04

0.05

0.06

0.07

0.02

8

0.00

7

0.02

0.002

3.65 ps

tp = 1ps

Normal skinμm)μm)μm)

CM (10CM (60CM(120CM(160μm)

Ref

lect

ance

( q

ref* )

Time, t (ps)

Fig. 10. Effect of affected volume on the temporal variations of reflectance q�ref from the sk(c) magnified view of the decaying signals.

transmitted signals (Fig. 7(a) and (c)). It is further observed thatat much shorter wavelength ðk ¼ 520 nmÞ no signals appear atthe south boundary. This is mainly due to the fact that: (a) theattenuation of the light at shorter wavelength is more as comparedto the longer wavelength as can be seen from Tables 1 and 2, and(b) skin is optically thick at the shorter wavelength ðk ¼ 520 nmÞcausing no/weak transmitted signals at the south boundary.Whereas, the reflected signals from skin at shorter wavelengthðk ¼ 520 nmÞ exhibit an increase in the peak magnitude as com-pared to the longer wavelength ðk ¼ 785 nm and 840 nm whensubjected to laser of 1 ps and 5.5 ps pulse widths as can be seenfrom Fig. 7(b). Since at shorter wavelength ðk ¼ 520 nmÞ laserpulse below 6 ps reflects the major amount of scattered light atthe surface of the skin from the stratum corneum. As a result,the peak magnitude of the reflected signals at shorter wavelength(k ¼ 520 nm) appears before those at longer wavelengths(k ¼ 785 nm and 840 nm) as can be observed from Fig. 7(b). Theobserved trend in the reflected signatures is opposite for laserpulse widths of 10 ps and 25 ps as can be seen from Fig. 7(d). Withthe increase in the laser pulse width above 10 ps, the peak magni-tude of the reflected signatures at longer wavelength (k ¼ 785 nmand 840 nm) increases from that of the shorter wavelength(k ¼ 520 nm). Because at pulse width above 10 ps the input energyand the duration for which the pulse persist at any location in-creases. As a result, the integrated effects of the maximum radia-tion reaching the top boundary from the skin layers at pulsewidth above 10 ps for laser of longer wavelength (k ¼ 785 nmand 840 nm) overshadows the effect of major radiation reachingthe top boundary from stratum corneum for laser of shorter wave-length (k ¼ 520 nm). Whereas the signals at shorter wavelength

a)

(c)

10 15 20

tp = 1ps

Normal skinμm)μm)μm)

CM (10CM (60CM (120CM (160 μm)

e, t (ps)

9 10 11 12 13 14 15 16 17 18 19 200.004

0.005

0.006

0.0070.0080.0090.01

0.02tp = 1ps Slope

Normal skin (m = -0.091)μm) (m = -0.0851)μm) (m = -0.0887)μm) (m = -0.0878)

CM (10CM (60CM(120CM(160μm) (m = -0.0888)

Ref

lect

ance

( q

ref* )

Time, t (ps)

in with CM: (a) full spectrum, (b) magnified view of the peak magnitudes of q�ref and

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 291

(k ¼ 520 nm) still appears ahead of that at longer wavelengths(k ¼ 785 nm and 840 nm) since majority of the light scatters fromthe stratum corneum.

4.3.2. Variation in epidermal and dermal thickness within the skin andits effect

Further, a sensitivity study has been performed with due con-sideration for variation in the thickness of normal skin layers,viz., variation in epidermal and dermal thicknesses, when sub-jected to short pulse laser of the order of picoseconds. The presentstudy has been carried out by considering laser of 785 nm only.Practically, the thickness of the skin layer varies throughout thebody depending on the type of the skin, thus a parametric studyon the interaction of the short-pulse laser light will be a usefulinformation for many future investigations. The human skin is gen-erally having a thickness of 2000–3000 lm [48]. The epidermalthickness within the skin varies between 50 and 150 lm, whereasthe dermal thickness varies between 1000 and 3000 lm. Thus, thepresent study has been carried out by varying the thickness of liv-ing epidermis and dermis by 37.5% and 75% of the thickness con-sidered by Wang et al. [39]. The transient response of thereflected signature due to the variation in the epidermal thicknesssubjected to laser pulse of 1 ps and 10 ps are shown in Fig. 8. Sim-ilarly, Fig. 9 depicts the transient response of reflected signaturedue to the variation in dermal thickness subjected to laser pulseof 1 ps and 10 ps. Fig. 8(a), (b) and (c) are the full spectrum, mag-nified view of peak magnitudes and magnified view of decayingsignals of the reflectance from the skin due to the variation in epi-dermal thickness for 1 ps laser pulse, respectively. Similarly,

(a

(b)

0 5 10.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

C

Ref

lect

ance

(qre

f* )

Time

2.0 3.0 4.0 5.0 6.0 7.00.040

0.043

0.045

0.048

0.050

0.052

0.055

0.057

0.060

0.062

0.065

0.067

0.070

Normal skin

Cutaneous Melanoma (CM)

κa=0.0075 mm-1

CM (σs=9.185 mm-1)

CM (σs=20.00 mm-1)

CM (σs=30.00 mm-1)

CM (σs=40.00 mm-1)

0.00420.0029

0.00

162

3.60 ps 3.65 ps

tp = 1 ps

Ref

lect

ance

(qre

f* )

Time, t (ps)

Fig. 11. Effect of the scattering coefficient of the malignancy (CM) in the basal layer ospectrum, (b) magnified view of the peak magnitudes of q�ref and (c) magnified view of t

Fig. 8(d), (e) and (f) are the full spectrum, magnified view of peakmagnitudes and magnified view of decaying signals of reflectedsignatures from skins with variation in epidermal thickness for10 ps laser pulse, respectively. It can be seen from Fig. 8(a), (b),(d) and (e) that, there is a considerable change in the characteristicsignals due to the associated change in the thickness of the livingepidermis for 1 ps and 10 ps laser pulse. There is a significant in-crease in the peak magnitude with the increase in the thicknessof the living epidermis. Similarly, there is a significant decreasein the peak magnitude due to the decrease in the thickness ofthe living epidermis as can be seen from Fig. 8(a), (b), (d) and (e).The distinguishing features are equally persisting in the decayingsignals at 1 ps and 10 ps laser pulse, as can be seen from Fig. 8(c)and (f), respectively. The full spectrum, the magnified view of peakmagnitudes and the magnified view of decaying characteristic sig-natures from the skin due to the variation in the dermal thicknessfor 1 ps laser pulse are shown in Fig. 9(a), (b) and (c), respectively.Whereas the full spectrum, the magnified view of peak magnitudesand the magnified view of decaying characteristic signatures fromthe skin due to variation in the dermal thickness for 10ps laserpulse are shown in Fig. 9(d), (e) and (f), respectively. It has been ob-served that the increase in dermal thickness exhibit nominal vari-ation in characteristic signatures, since dermal layers are located ata greater depth compared to epidermal layers. Considerablechanges in the peak magnitudes of the characteristics signatureshave been observed for 10 ps laser pulse (Fig. 9(d) and (e)) as com-pared to 1 ps laser pulse (Fig. 9(a) and (b)). Owing to the increase inthe dermal thickness, as can be seen from Fig. 9(e), there is smallincrease in the characteristic signatures for 10 ps laser pulse due

)

(c)

0 15 20

tp = 1 ps

Normal skin

utaneous Melanoma (CM)

κa=0.0075 mm-1

CM (σs=9.185 mm-1)

CM (σs=20.00 mm-1)

CM (σs=30.00 mm-1)

CM (σs=40.00 mm-1)

, t (ps)

9 10 11 12 13 14 15 16 17 18 19 200.004

0.005

0.006

0.007

0.008

0.0090.01

0.02

tp = 1 ps Normal skin (m = -0.091)

Cutaneous Melanoma (CM)

κa=0.0075 mm-1

CM (σs=9.185 mm-1) (m = -0.085)

CM (σs=20.0 mm-1) (m = -0.090)

CM (σs=30.0 mm-1) (m = -0.091)

CM (σs=40.0 mm-1) (m = -0.093)

Ref

lect

ance

( qre

f* )

Time, t (ps)

f the skin on the temporal variations of reflectance q�ref from skin with CM (a) fullhe decaying signals.

(a)

(b)

0 5 10 15 200.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2 4 6 80.01

0.02

0.03

0.04

0.05

0.06

tp = 1ps

Normal skin CM (10 μm) CM (60 μm) CM (120 μm) CM (160 μm)

Ref

lect

ance

( q

ref* )

Time, t (ps)

0 5 10 15 200.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2 4 6 80.01

0.02

0.03

0.04

0.05

0.06

tp = 1ps

Normal skin CM (10 μm) CM (60 μm) CM (120 μm) CM (160 μm)

Ref

lect

ance

( q

ref* )

Time, t (ps)

Fig. 12. Effect of noise on the characteristics signatures, i.e., the temporal variationsof reflectance q�ref from normal skin and different growth phases of CM (a) whiteGaussian noise of 55 dB with a standard deviation of 0.00175 and with mean nearlyequal to zero and (b) white Gaussian noise of 70 dB with a standard deviation of0.000321 and with mean nearly equal to zero (enlarged view of peak magnitude areshown in the insert).

292 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

to the increase in the energy of the input laser pulse and the dura-tion for which it persists at any location. Further, the distinguishingfeatures in the characteristic signatures are observed in the decay-ing signals as can be seen from Fig. 9(c) for 1 ps laser pulse andFig. 9(f) for 10 ps laser pulse.

4.3.3. Effect of different growth phases of cutaneous melanomaThe volume of the cancerous region increases with time. Start-

ing with a few cells, it continuously grows to other portions of thetissue before it has reached to the stage of metastasis when it startsspreading to other parts of the body. Further, thermophysical andoptical properties of a cancer vary according to its grade. For exam-ple, depending on the grade of the cancer, the scattering coefficientvaries over a wide range [49]. In the present work, for malignancyin the skin under consideration, in the following we present andanalyze results for varying (a) thicknesses of the malignant portionof the skin (Fig. 1(b)) and (b) scattering coefficient of the malig-nancy. Out of the three types of malignancies considered in thepresent work, consideration is given to the most harmful malig-nancy, i.e., the CM. Results are shown in Figs. 10 and 11.

With the optical properties given in Table 3, temporal variationsof reflectance q�ref for the skin with CM spread to four differentthicknesses, viz., 10 lm (i.e., in basal layer), 60 lm (i.e., 30 lm inliving epidermis and 30 lm in papillary dermis), 120 lm (i.e.,60 lm in living epidermis and 60 lm in papillary dermis) and160 lm (i.e., 80lm in living epidermis and 80 lm in papillary der-mis) are shown in Fig. 10(a)–(c). For the purpose of comparison, q�ref

variation of the normal skin is also shown. It is to be noted that thetotal thickness of the skin is the same as that in the previous cases.Over a period of 0 to 20 ps, q�ref variations are shown in Fig. 10(a),while Fig. 10(b) and (c) show the magnified view of same for 2–8 ps and 9–20 ps, respectively. An observation of Fig. 10(b) showsthat with increase in the infected volume of the skin, reflectancedecreases in the peak region. However, in the decaying region(Fig. 10c), opposite is the trend. The change in the temporal varia-tion of q�ref is thus indicative of the spread of cancer to the moredepths.

Results for the effect of the scattering coefficient rs on temporalvariation of q�ref from a skin with CM in basal layer are shown inFig. 11(a)–(c). Values of rs considered are 9.185, 20.00, 30.00 and40.00 mm�1. For the basal layer of a normal skin, rs = 18.95 mm�1

(Table 1). An observation of Fig. 11(a)–(c) reveals that with changein rs of the basal layer, there is a noticeable change in the peakmagnitude of q�ref . With an increase in rs, q�ref in the peak region alsoincreases. Though the trend is still opposite in the decaying region,effect of rs is marginal. When the rs of the CM is more than that forthe normal skin, q�ref is more. When they are comparable, q�ref pro-files are almost the same. When rs of the CM is lower than that forthe normal skin, q�ref becomes lower. As discussed before, reflec-tance is mainly due to the scattered diffuse radiation received atthe boundary of incidence. A higher value of the scattering coeffi-cient rs leads to increased scattering, and hence a higher value ofq�ref . The considerable increase in the scattering property of theharmful malignant lesion is well justified by the fact that thegrowth of the lesion is either because of increase in size of scatter-ers or due to the change in number density of scatterers in the af-fected tissues [49].

4.3.4. Effect of noise on cancer detectionIn the absence of the experimental data, the effect of noise is

incorporated in the model by adding white Gaussian noise, sayn(t) within the recorded signals. The effect of noise on the detec-tion is quantified by determining the signal to noise ratio (SNR)of the detected signals due to different growth phases of cutaneousmelanoma. In general, the SNR compares the magnitude of desiredsignals to that of the unwanted noise. The noise has been modelled

as white Gaussian noise of 55 dB and 70 dB with a standard devi-ation of 0.00175 and 0.00321, respectively. The mean is nearlyequal to zero for both types of noises. Fig. 12 compares the noisysignals from the normal skin to that of the skin with differentgrowth phases of CM. Fig. 12(a) presents the comparison for thecase of large noise in the signals, i.e., 55 dB. Fig. 12(b) presentsthe case with negligible noise within the signals, i.e., 70 dB. It hasbeen observed from Fig. 12(a) and (b) that, the changes in the can-cerous volume within the skin exhibit distinguishing features forthe case of signals with measurement error. The detection visibilitydue to different growth phases of cutaneous melanoma has beenquantified in terms of SNR based on the ratio of mean and standarddeviation of the desired signal, and is given by

SNR ¼ 10log10lab � ln

rn

� �in dB ð21Þ

Table 4Comparison of obtained SNR values for laser based detection technique with varyinggrowth phase of skin cancer.

Tissues Magnitude of SNR(in dB) for 55 dB noise

Magnitude of SNR(in dB) for 70 dB noise

CM (10 lm) �6.8357 �2.3094CM (60 lm) �33.9736 �6.1818CM (120 lm) �7.2505 �1.8374CM (160 lm) �5.2406 0.7128

A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294 293

where lab is the mean of the noisy signal from the abnormal skin,ln is the mean of noisy signal from the normal skin and rn is thestandard deviation of noisy signal from the normal skin. The com-puted SNR values are depicted in Table 4.

The practical implementation of pulse laser based modality forskin cancer detection can be achieved by the precise arrangementof various devices used in optics and by using the time gating tech-niques. The relevant details of these devices are given in the liter-ature [3,22]. The time gating technique consists of high speedstreak camera to capture the incoming instantaneous photons ofthe order of a picosecond or less from the tissue as discussed in[3,22]. In general, many challenges may prevail in real situationbut the major challenge might be in identifying the benchmark sig-nature emanating from the normal skin with due consideration ofintra- and inter-patient variability.

5. Conclusions

Radiative signatures from the human skin with and withoutmalignancy subjected to a short-pulse laser were analyzed. Theskin was modeled on the basis of anatomical details available inthe literature. Malignancies considered in the skins were CM, NBCCand SCC. In the early stage of the cancer, each of these malignan-cies was first considered in the basal layer (thickness: 10 lm),and then it was considered to have spread to the living epidermis(total thickness: 80 lm). Owing to a very high extinction coeffi-cient, the transmittance was very weak. However, because ofstrong scattering in the skin, reflectance was strong. Short-pulse la-ser with a ps (1.0 � 10�12 s) pulse-width was preferred over thathaving a pulse width of a ns (1.0 � 10�9 s). Temporal variationsof the reflectance from a 2-layer inhomogeneous planar mediumwere validated against the available results in the literature.Temporal variations of reflectance were found to be characteristicof the skin condition. With malignancies either in the basal layer orin the living epidermis, in the peak region, the reflectance was thehighest for skin with CM and the lowest with that for SCC. With liv-ing epidermis infected with any particular malignancy, reflectancewas more than its corresponding case of its infection of the basallayer. Further, reflectance was more when the infected volume in-creased, and for a given volume, it was higher when the scatteringcoefficient was more. The observed changes in the radiative signa-tures due to the associated changes in the optical properties andconditions can be a means to detect early traces of cancer in theskin.

References

[1] M.D. Feit, A.M. Rubenchik, B.M. Kim, L.B. da Silva, M.D. Perry, Physicalcharacterization of ultrashort laser pulse drilling of biological tissue, Appl. Surf.Sci. 127–129 (1998) 869–874.

[2] S. Kumar, K. Mitra, Microscale aspects of thermal radiation transport and laserapplications, Adv. Heat Transfer 33 (1999) 187–294.

[3] A. Trivedi, S. Basu, K. Mitra, Temporal analysis of reflected optical signals forshort pulse laser interaction with nonhomogeneous tissue phantoms, J. Quant.Spectrosc. Radiat. Transfer 93 (2005) 337–348.

[4] M. Jaunich, S. Raje, K. Kim, K. Mitra, Z. Guo, Bio-heat transfer analysis duringshort pulse laser irradiation of tissues, Int. J. Heat Mass Transfer 51 (2008)5511–5521.

[5] J. Boulanger, A.E. Akel, A. Charette, F. Liu, Direct imaging of turbid media usinglong-time back-scattered photons, a numerical study, Int. J. Therm. Sci. 45(2006) 537–552.

[6] A.Y. Sajjadi, K. Mitra, M. Grace, Ablation of subsurface tumors using an ultra-short pulse laser, Opt. Lasers Eng. 49 (2011) 451–456.

[7] S.C. Mishra, P. Chugh, P. Kumar, K. Mitra, Development and comparison of theDTM, the DOM and the FVM formulations for the short-pulse laser transportthrough a participating medium, Int. J. Heat Mass Transfer 49 (2006) 1820–1832.

[8] R.R. Anderson, J.A. Parrish, The optics of human skin, J Invest. Dermatol. 77(1981) 13–19.

[9] A.N. Bashkatov, E.A. Genina, V.I. Kochubey, V.V. Tuchin, Optical properties ofhuman skin, subcutaneous and mucous tissues in the wavelength range from400 to 2000 nm, J. Phys. D: Appl. Phys. 38 (2005) (2000) 2543–2555.

[10] A.N. Bashkatov, E.A. Genina, V.V. Tuchin, Optical properties of skin,subcutaneous, and muscle tissues: a review, J. Innov. Opt. Health Sci. 4(2011) 9–38.

[11] W.F. Cheong, S.A. Prahl, A.J. Welch, A review on optical properties of biologicaltissues, IEEE J. Quantum Electron. 26 (1990) 2166–2184.

[12] M.J.C. Van-Gemert, S.L. Jacques, H.J.C.M. Sterenborg, W.M. Star, Skin optics,IEEE Trans. Biomed. Eng. 36 (1989) 1146–1153.

[13] R.C. Simpson, M. Kohl, E. Essenpris, M. Cope, Near-infrared optical propertieson ex vivo human skin and subcutaneous tissues measured using Monte Carloinversion technique, Phys. Med. Biol. 43 (1998) 2465–2478.

[14] A.N. Bashkatov, V.V. Tuchin, E.A. Genina, M.M. Stolnitz, D.M. Zestkov, G.B.Altshuler, I.V. Yaroslavsky, Monte Carlo study of skin optical clearing toenhance light penetration in the tissue, in: V.V. Tuchin (Ed.), ComplexDynamics and Fluctuations in Biomedical Photonics IV, Proceedings of SPIE,vol. 6436, 2007, pp. 64360Z1�64360Z12.

[15] P-f. Hsu, Effects of multiple scattering and reflective boundary on the transientradiative transfer process, Int. J. Therm. Sci. 40 (2001) 539–549.

[16] Z. Guo, J. Aber, B.A. Garetz, S. Kumar, Monte Carlo simulation and experimentsof pulsed radiative transfer, J. Quant. Spectrosc. Radiat. Transfer 73 (2002)159–168.

[17] J.M. Wang, C.Y. Wu, Transient radiative transfer in a scattering slab withvariable refractive index and diffuse substrate, Int. J. Heat Mass Transfer 53(2010) 3799–3806.

[18] P. Rath, S.C. Mishra, P. Mahanta, U.K. Saha, K. Mitra, Discrete transfer methodapplied to transient radiative transfer problems in participating medium,Numer. Heat Transfer A 44 (2003) 183–197.

[19] T. Okutucu, Y. Yener, A.A. Busnaina, Transient radiative transfer inparticipating media with pulse-laser irradiation – an approximate Galerkinsolution, J. Quant. Spectrosc. Radiat. Transfer 103 (2007) 118–130.

[20] T. Okutucu, Y. Yener, Participating media exposed to collimated short-pulseirradiation – a Laguerre–Galerkin solution, Int. J. Heat Mass Transfer 50 (2007)4352–4359.

[21] J.C. Chai, P.-F. Hsu, Y.C. Lam, Three-dimensional transient radiative transfermodeling using the finite-volume method, J. Quant. Spectrosc. Radiat. Transfer86 (2004) 299–313.

[22] C. Das, A. Trivedi, K. Mitra, T. Vo-Dinh, Experimental and numerical analysis ofshort-pulse laser interaction with tissue phantoms containinginhomogeneties, Appl. Opt. 42 (2003) 5173–5180.

[23] M.Q. Brewster, Y. Yamada, Optical properties of thick, turbid media frompicosecond time-resolved light scattering measurements, Int. J. Heat MassTransfer 38 (1995) 2569–2581.

[24] K.M. Katika, L. Pilon, Steady-state directional diffuse reflectance andfluorescence of human skin, Appl. Opt. 45 (2006) 4174–4183.

[25] G. Pal, A. Dutta, K. Mitra, M.S. Grace, A. Amat, T.B. Romanczyk, X. Wu, K.Chakrabarti, J. Anders, E. Gorman, R.W. Waynant, D.B. Tata, Effect of lowintensity laser interaction with human skin fibroblast cells using fiber-opticnano-probes, J. Photochem. Photobiol. B 86 (2007) 252–261.

[26] A. Bhowmik, R. Singh, R. Repaka, S.C. Mishra, Conventional and newlydeveloped bioheat transport models in vascularized tissues: a review, J.Therm. Biol. 38 (2013) 107–125.

[27] M. Pirtini Çetingül, C. Herman, Quantification of the thermal signature of amelanoma lesion, Int. J. Therm. Sci. 50 (2011) 421–431.

[28] R.M. Woodward, B.E. Cole, V.P. Wallace, R.J. Pye, D.D. Arnone, E.H. Linfield, M.Pepper, Terahertz pulse imaging in reflection geometry of human skin cancerand skin tissue, Phys. Med. Biol. 47 (2002) 3853–3863.

[29] E. Pickwell, B.E. Cole, A.J. Fitzgerald, M. Pepper, V.P. Wallace, In vivo study ofhuman skin using pulsed terahertz radiation, Phys. Med. Biol. 49 (2004) 1595–1607.

[30] A.J. Welch, The thermal response of laser irradiated tissue, IEEE J. QuantumElectron. 20 (1984) 1471–1481.

[31] E.H. Ooi, W.T. Ang, E.Y.K. Ng, A boundary element model of the human eyeundergoing laser thermokeratoplasty, Comput. Biol. Med. 38 (2008) 727–737.

[32] A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, H.Lubatschowaski, Nonlinear side effects of fs pulses inside corneal tissue duringphotodisruption, Appl. Phys. B 74 (2002) 419–425.

[33] J.H. Tan, E.Y.K. Ng, U.R. Acharya, C. Chee, Study of normal ocular thermogramusing textural parameters, Infrared Phys. Technol. 53 (2010) 120–126.

[34] E.Y.K. Ng, H.M. Tan, E.H. Ooi, Prediction and parametric analysis of thermalprofiles within heated human skin using the boundary element method, Phil.Trans. R. Soc. A 368 (2010) 655–678.

[35] F.L. Greene, C.C. Compton, A.G. Fritz, J.P. Shah, D.P. Winchester, AmericanCancer Staging Atlas, Springer, New York, 2006. pp. 195–234.

294 A. Bhowmik et al. / International Journal of Heat and Mass Transfer 68 (2014) 278–294

[36] W.H. Clark Jr., L. From, E.A. Bernardino, M.C. Mihm, The histogenesis andbiologic behavior of primary human malignant melanomas of the skin, CancerRes. 29 (1969) 705–727.

[37] A. Breslow, Thickness, cross-sectional areas and depth of invasion in theprognosis of cutaneous melanoma, Ann. Surg. 172 (1970) 902–908.

[38] S.C. Mishra, H.K. Roy, N. Misra, Discrete ordinate method with a new and asimple quadrature scheme, J. Quant. Spectrosc. Radiat. Transfer 101 (2006)249–262.

[39] S. Wang, J. Zhao, H. Lui, Q. He, H. Zeng, Monte Carlo simulation of near infraredautofluorescence measurements of in vivo skin, J. Photochem. Photobiol. B 105(2011) 183–189.

[40] M.F. Modest, Radiative Heat Transfer, second ed., New York, 2003.[41] R. Muthukumaran, S.C. Mishra, Analysis of the transport of a train of short-

pulse radiation of Gaussian temporal profile through a 2-D participatingmedium, Heat Transfer Eng. 30 (2009) 1197–1207.

[42] J. Steketee, Spectral emissivity of skin and pericardium, Phys. Med. Biol. 18(1973) 686–694.

[43] H. Zeng, C. MacAulay, D.I. McLean, B. Palcic, Reconstruction of in vivo skinautofluorescence spectrum from microscopic properties by Monte Carlosimulation, J. Photochem. Photobiol. B 105 (1997) 234–240.

[44] A.C. Border, Squamous-cell epithelioma of the skin-A study of 256 cases, Ann.Surg. 73 (1921) 141–160.

[45] E. Salomatina, B. Jiang, J. Novak, A.N. Yaroslavsky, Optical properties of normaland cancerous human skin in visible and near-infrared spectral range, J.Biomed. Opt. 11 (6) (2006) 0640261–0640269.

[46] A. Garcia-Uribe, E.B. Smith, J. Zou, M. Duvic, V. Prieto, L.V. Wang, In-vivocharacterization of optical properties of pigmented skin lesions includingmelanoma using oblique incidence diffuse reflectance spectrometry, J. Biomed.Opt. 16 (2011) 0205011–0205013.

[47] X. Lu, P.F. Hsu, Reverse Monte Carlo simulations of light pulse propagation innon-homogeneous media, J. Quant. Spectrosc. Radiat. Transfer 93 (2005) 349–367.

[48] D.D. Sampson, T.R. Hillman, Optical coherence tomography, in: G. Palumbo, R.Pratesi (Eds.), Lasers and Current Optical Techniques in Biology, Royal Societyof Chemistry, Cambridge, 2004, pp. 549–554.

[49] J.R. Mourant, A.H. Hielscher, A.A. Eick, T.M. Johnson, J.P. Freyer, Evidence ofintrinsic differences in the light scattering properties of tumorigenic andnontumorigenic cells, Cancer 84 (1998) 366–374.