Simulation of optical coherence tomography images by Monte Carlo modeling based on polarization vector approach

Simulation of optical coherence tomography images by Monte Carlo modeling based on

polarization vector approach

Mikhail Kirillin14

Igor Meglinski2 Vladimir Kuzmin

3 Ekaterina Sergeeva

1

and Risto Myllylauml4

1 Institute of Applied Physics RAS 603950 Nizhny Novgorod Russia 2 Jack Dodd Centre for Quantum Technology Department of Physics University of Otago

PO Box 56 Dunedin 9054 New Zealand 3 St-Petersburg Institute of Commerce and Economics 194021 St-Petersburg Russia

4 Optoelectronics and Measurement Techniques Laboratory University of Oulu PO Box 4500 90014 Oulu Finland

mkirillinyandexru

Abstract Monte Carlo method is applied for simulation of 2D optical coherence tomography (OCT) images of skin-like model Layer boundaries in skin model feature curved shape which agrees with physiological structure of human skin The effect of coherence properties of probing radiation on OCT image formation and speckles in the detected OCT signal is considered The developed model is employed for image simulation both for conventional and polarization dependent time-domain OCT modalities Simulation of polarized OCT signal is performed using vector approach developed previously for modeling of electromagnetic field transfer in turbid media

copy2010 Optical Society of America

OCIS codes (1104500) Optical coherence tomography (2905855) Scattering polarization (1705280) Photon migration (1703660) Light propagation in tissues

References and links

1 D Huang E A Swanson C P Lin J S Schuman W G Stinson W Chang M R Hee T Flotte K Gregory C A Puliafito and J D Fujimoto ldquoOptical coherence tomographyrdquo Science 254(5035) 1178ndash1181 (1991)

2 A F Fercher W Drexler C K Hitzenberger and T Lasser ldquoOptical coherence tomography ndash principles and applicationsrdquo Rep Prog Phys 66(2) 239ndash303 (2003)

3 J M Schmitt ldquoOptical coherence tomography (OCT) A reviewrdquo IEEE J Sel Top Quantum Electron 5(4) 1205ndash1215 (1999)

4 B E Bouma and G J Tearney Handbook of Optical Coherence Tomography (Marcel Dekker New York 2002)

5 V V Tuchin Handbook of Coherent Domain Optical Methods Biomedical Diagnostics Environment and Material Science (Kluwer Academic Boston 2004)

6 M J Yadlowsky J M Schmitt and R F Bonner ldquoMultiple-scattering in optical coherence microscopyrdquo Appl Opt 43(25) 5699ndash5707 (1995)

7 I V Meglinski ldquoModeling the reflectance spectra of the optical radiation for random inhomogeneous multi-layered highly scattering and absorbing media by the Monte Carlo techniquerdquo Quantum Electron 31 1101ndash1107 (2001)

8 M Yu Kirillin A V Priezzhev and R Myllylauml ldquoRole of multiple scattering in formation of OCT skin imagesrdquo Quantum Electron 38 486ndash490 (2008)

9 R R Meier J-S Lee and D E Anderson ldquoAtmospheric scattering of middle uv radiation from an internal sourcerdquo Appl Opt 17(20) 3216ndash3225 (1978)

10 C Lavigne A Roblin V Outters S Langlois T Girasole and C Roze ldquoComparison of iterative and monte carlo methods for calculation of the Aureole about a point source in the earthrsquos atmosphererdquo Appl Opt 38(30) 6237ndash6246 (1999)

11 E A Bucher ldquoComputer simulation of light pulse propagation for communication through thick cloudsrdquo Appl Opt 12(10) 2391ndash2400 (1973)

12 E Berrocal D L Sedarsky M E Paciaroni I V Meglinski and M A Linne ldquoLaser light scattering in turbid media Part I Experimental and simulated results for the spatial intensity distributionrdquo Opt Express 15(17) 10649ndash10665 (2007)

133285 - $1500 USD Received 12 Aug 2010 revised 8 Sep 2010 accepted 13 Sep 2010 published 29 Sep 2010(C) 2010 OSA 11 October 2010 Vol 18 No 21 OPTICS EXPRESS 21714

13 E Berrocal I V Meglinski D A Greenhalgh and M A Linne ldquoImage transfer through the complex scattering turbid mediardquo Laser Phys Lett 3(9) 464ndash468 (2006)

14 G Yao and L V Wang ldquoMonte Carlo simulation of an optical coherence tomography signal in homogeneous turbid mediardquo Phys Med Biol 44(9) 2307ndash2320 (1999)

15 M Yu Kirillin M V Shirmanova M A Sirotkina M L Bugrova B N Khlebtsov and E V Zagaynova ldquoContrasting properties of gold nanoshells and titanium dioxide nanoparticles for OCT imaging of skin Monte Carlo simulations and in vivo studyrdquo J Biomed Opt 14 021017 (2009)

16 B Karamata M Laubscher M Leutenegger S Bourquin T Lasser and P Lambelet ldquoMultiple scattering in optical coherence tomography I Investigation and modelingrdquo J Opt Soc Am A 22(7) 1369ndash1379 (2005)

17 B Karamata M Leutenegger M Laubscher S Bourquin T Lasser and P Lambelet ldquoMultiple scattering in optical coherence tomography II Experimental and theoretical investigation of cross talk in wide-field optical coherence tomographyrdquo J Opt Soc Am A 22(7) 1380ndash1388 (2005)

18 M Y Kirillin A V Priezzhev and I V Meglinski ldquoEffect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering mediardquo Quantum Electron 36(3) 247ndash252 (2006)

19 V L Kuzmin and I V Meglinski ldquoMultiple scattering and intensity fluctuations in optical coherence tomography of randomly inhomogeneous mediardquo J Exp Theor Phys 105(2) 285ndash291 (2007)

20 R V Kuranov V V Sapozhnikova N M Shakhova V M Gelikonov E V Zagainova and S A Petrova ldquoCombined application of optical methods to increase the information content of optical coherent tomography in diagnostics of neoplastic processesrdquo Quantum Electron 32(11) 993ndash998 (2002)

21 M Yu Kirillin E Alarousu T Fabritius R Myllylauml and A V Priezzhev ldquoVisualization of paper structure by optical coherence tomography Monte Carlo simulations and experimental studyrdquo J Europ Opt Soc Rap Public 2 07031 (2007)

22 IM Sobolrsquo The Monte Carlo Method (The University of Chicago Press Chicago 1974) 23 A Ishimaru Wave Propagation and Scattering in Random Media (Academic New York 1978) 24 D Y Churmakov I V Meglinski and D A Greenhalgh ldquoInfluence of refractive index matching on the photon

diffuse reflectancerdquo Phys Med Biol 47(23) 4271ndash4285 (2002) 25 D Y Churmakov V L Kuzrsquomin and I V Meglinski ldquoApplication of the vector Monte-Carlo method in

polarisation optical coherence tomographyrdquo Quantum Electron 36(11) 1009ndash1015 (2006) 26 J W Goodman Statistical Optics (Wiley-Interscience 1985) 27 C Brosseau Fundamentals of Polarized Light a Statistical Optics Approach (New York John Wiley amp Sons

1998) 28 C F Bohren and D R Huffman Absorption and scattering of light by small particles (New York Wiley 1983) 29 X Wang and L V Wang ldquoPropagation of polarized light in birefringent turbid media a Monte Carlo studyrdquo J

Biomed Opt 7(3) 279ndash290 (2002) 30 S Bartel and A H Hielscher ldquoMonte Carlo simulations of the diffuse backscattering mueller matrix for highly

scattering mediardquo Appl Opt 39(10) 1580ndash1588 (2000) 31 M J Raković G W Kattawar M B Mehrubeoğlu B D Cameron L V Wang S Rastegar and G L Coteacute

ldquoLight backscattering polarization patterns from turbid media theory and experimentrdquo Appl Opt 38(15) 3399ndash3408 (1999)

32 D A Zimnyakov Y P Sinichkin P V Zakharov and D N Agafonov ldquoResidual polarization of non-coherently backscattered linearly polarized light the influence of the anisotropy parameter of the scattering mediumrdquo Waves Random Media 11(4) 395ndash412 (2001)

33 S V Gangnus S J Matcher and I V Meglinski ldquoMonte Carlo modeling of polarized light propagation in biological tissuesrdquo Laser Phys 14 886ndash891 (2004)

34 J M Schmitt A H Gandjbakhche and R F Bonner ldquoUse of polarized light to discriminate short-path photons in a multiply scattering mediumrdquo Appl Opt 31(30) 6535ndash6546 (1992)

35 E Akkermans P E Wolf R Maynard and G Maret ldquoTheoretical-Study of the Coherent Backscattering of Light by Disordered Mediardquo J Phys France 49(1) 77ndash98 (1988)

36 M J Stephen and G Cwilich ldquoRayleigh scattering and weak localization Effects of polarizationrdquo Phys Rev B Condens Matter 34(11) 7564ndash7572 (1986)

37 F C MacKintosh and S John ldquoDiffusing-wave spectroscopy and multiple scattering of light in correlated random mediardquo Phys Rev B Condens Matter 40(4) 2383ndash2406 (1989)

38 D A Zimnyakov and Y P Sinichkin ldquoA study of polarization decay as applied to improved imaging in scattering mediardquo J Opt A Pure Appl Opt 2(3) 200ndash208 (2000)

39 A Dogariu C Kutsche P Likamwa G Boreman and B Moudgil ldquoTime-domain depolarization of waves retroreflected from dense colloidal mediardquo Opt Lett 22(9) 585ndash587 (1997)

40 VV Tuchin Tissue optics light scattering methods and instruments for medical diagnosis (SPIE Press Bellingham 2000)

41 V L Kuzmin and I V Meglinski ldquoHelicity flip of backscattered circularly polarized lightrdquo Proc SPIE 7573 75730Z (2010)

42 P S Carney E Wolf and G S Agarwal ldquoStatistical generalizations of the optical cross-section theorem with application to inverse scatteringrdquo J Opt Soc Am A 14(12) 3366ndash3371 (1997)

43 V L Kuzmin and E V Aksenova ldquoA generalized Milne solution for the correlation effects of multiple light scattering with polarizationrdquo J Exp Theor Phys 96(5) 816ndash831 (2003)

44 V L Kuzrsquomin V P Romanov and L A Zubkov ldquoPropagation and scattering of light in fluctuating mediardquo Phys Rep 248(2-5) 71ndash368 (1994)


45 P K Milsom ldquoA ray-optic Monte Carlo description of a Gaussian beam waist ndash applied to reverse saturable absorptionrdquo Appl Phys B 70(4) 593ndash599 (2000)

46 V L Kuzrsquomin and I V Meglinski ldquoAnomalous polarization effects during light scattering in random mediardquo J Exp Theor Phys 110(5) 742ndash753 (2010)

1 Introduction

Nowadays optical coherent tomography (OCT) is well known as a reputable diagnostic technique for non-invasive imaging of complex turbid media such as biological tissues [1ndash5] Nevertheless a wide application of OCT in day-to-day clinical practice is significantly limited due to image blurring and in-depth signal decrease caused by multiple scattering of probing radiation within the biological tissues [6] To improve the OCT system design and image processing algorithms a systematic understanding of OCT image formation is required

Due to a greater complication arisen when polarization and coherence properties of scattered laser radiation should be taken into account the radiation transfer cannot be described simply in terms of diffusion theory Moreover for a complex multi-layered media with non-planar layer boundaries encountered in practical applications it is rarely possible to find an analytical solution This situation is typical for many practical applications including OCT Numerical techniques such as Monte Carlo (MC) method are typically applied MC technique is well established in various biomedical [78] astronomical and meteorological applications [9ndash11] industrial sprays [11] and others A number of studies has been devoted in the past to analyze contribution of scattering orders into the image transfer [1213] and formation of OCT signal [814ndash19]

In current report we introduce MC-based technique applied for simulation of 2D OCT images of skin-like model with curved layer boundaries corresponding to the actual skin structure confirmed by OCT images of normal human skin obtained in vivo

2 Experimental OCT imaging of skin

The principles of OCT are widely described elsewhere [1ndash5] Experimental study was performed using the OCT system developed at the Institute of Applied Physics RAS (Nizhny Novgorod Russia) [2021] OCT setup operates at the central wavelength of 910 nm with spectrum width of 50 nm corresponding to axial and transversal spatial resolutions of 15 and 25 microm respectively The examples of typical OCT images of human skin obtained in vivo by conventional (Fig 1a) and polarization dependent (Fig 1b and Fig 1c) modalities exhibit complex multi-layered tissues structure

0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

1

2

3

4

0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

a b c

Fig 1 Experimental 2-D OCT images of human skin in vivo obtained for non- co- and cross- polarized modes (a) (b) and (c) respectively Upper Stratum Corneum (1) lower Stratum Corneum (2) epidermis (3) and dermis (4) are clearly distinguished in the OCT images

These images clearly show high scattering in upper Stratum Corneum layer manifested by bright area in non- and co-polarized modes However its brightness in the cross-polarization image is significantly smaller because of its small thickness which does not provide sufficient


depolarization Next layer lower Stratum Corneum seems relatively dark in all three images whereas underlying layer epidermis appears bright in the obtained OCT images in non- and co-polarized modes (see Fig 1a and 1b correspondingly) High scattering in this layer also results in significant depolarization degree which is evident by high intensity of this layer in the cross-polarization image (see Fig 1c)

3 Numerical simulation

31 Basic concept of Monte Carlo simulation of OCT signal

Principles of stochastic MC method for numerical calculation of radiation intensity scattered within a randomly inhomogeneous turbid medium are widely described in the literature [7ndash912ndash19] MC is based on the consequent simulation of a random photon trajectory within the medium between the point where the photon enters the medium and the point where it leaves the medium Simulation of the photon trajectories consists of the following key stages injection of the photon in the medium generation of the photon path-length generation of a scattering event definition of reflectionrefraction at the medium boundaries definition of detection and accounting for the absorption The photon free path s between the two successive elastic scattering events is determined by the Poisson probability density function [22]

( ) exp( )s s

f s smicro micro= minus (1)

where micros is the scattering coefficient Note that the parameter 1s

s micro= which is the average

scattering length depends on the size distribution of scatters their concentration and relative refractive index in respect to the surrounding medium

The probability that the photon free path exceeds s is defined as

( ) S

f s dsξinfin

prime prime= int (2)

Given the probability density function (1) it is easy to express the random magnitude s via the probability ξ

ln

s

sξmicro

= minus (3)

This is the key element of MC technique viz obtaining photon free path-length that consists of the computer generation of a random number ξ uniformly distributed in the interval [01]

Direction of the photon after single scattering is defined by the scattering phase function

( ) ( )

( )d

i si s

i s s

Gp

Gπ

minusminus =

minus Ωint4

n nn n

n n (4)

where

( ) ( )02

1(0) ( ) exp ( )

(4 )i s i s

G d ikε επ

minus = ∆ ∆ minus minusintn n r r n n r (5)

is the Fourier transform of the permittivity mutual correlation function ( )ε∆ r is the random

permittivity deviation at point r from the background value 0

2 k nπ λ= is a wave number

defined by central wavelength λ and average refractive index of the medium n i

n and s

n are

the unit vectors defining the direction of the photon prior and after the scattering event

respectively 2

2sini s

θminus =n n determine the direction transfer θ is the scattering angle

relative to the initial direction i

n The Henyey-Greenstein phase function which is the most


widely used model phase function for biotissues describes non-isotropic scattering depending

on the anisotropy parameter cosg θ= [23]

Described steps are repeated till the photon is detected arriving at the detector with the given area and acceptance angle or leaves the scattering medium or its total path exceeds the maximum allowed path The total number of the launched photons used in the simulation is typically ~10

7 The details of the reflection and refraction at the medium boundary and at the

interface between layers are given in [24] The OCT signal in terms of probing depth z can be presented as an interference term of

optical signals coming from the sample and reference arms [3425]

( ) 12

( ) Re ( ) r s cI z I I C z l= (6)

Here s

I and r

I are the mean intensities returning from the sample and reference arms of

the interferometer C(z lc) is the normalized coherence function and lc is the coherence length of probing radiation [26]

2

c

2ln2 l

λπ λ

=∆

(7)

where λ∆ is the full width at half maximum (FWHM) of the spectrum of source radiation

When performing the MC simulation described above the OCT signal detected at the definite transversal position of the probing beam (A-scan) is calculated for randomly polarized radiation as [18]

2

0

1 c

2 2 ( ) cos (2 ) exp

phN

ii i

i

z LI z I W z L

l

πλ=

minus = minus minus sum (8)

where Nph is the number of photons launched I0 is a constant defined by instrumental properties of the OCT system Wi is the weight of i-th detected photon with optical pathlength Li and 2z is the optical pathlength in the reference arm If one neglects ldquospeckle structurerdquo of the OCT-signal defined by cosine item in (8) the result can be presented as a superposition of envelopes of partial detected photon contributions

2

0

1 c

2 ( ) exp

phN

ii

i

z LI z I W

l=

minus = minus

sum (9)

In order to simulate the 2D OCT image consequent A-scans are simulated with the definite transversal step in probing position The total number of simulated A-scans and the transversal step between them are predefined regarding the width (FWHM) of the probing beam diameter

32 Simulation of polarization dependent OCT signal

Polarization of an electromagnetic wave is typically described in framework of the Stokes-Mueller or Jones formalism [2728] Stokes-Mueller formalism was applied to study polarization in birefringent turbid media for potential application to polarization-sensitive optical imaging [29] The experimental and numerical studies were carried out to observe the backscattering polarization patterns presented in a form of the Mueller matrices [3031] The residual polarization degree of the backscattered light and its connection to the optical properties of the scattering medium was studied in [32] To explore the possibility of retrieving the birefringence properties of layered tissue with the depth-resolved polarization-sensitive OCT (PS-OCT) Jones formalism was implemented to reduce large calculation time typically required in the full Stokes-Mueller approach [33]


Schmitt et al proposed a method for discriminating short and long path photons that is based on the relationship between the polarization states of incident and forward scattered light [34] Discussing coherent backscattering and its dependence on the state of polarization of an incident linearly polarized light Akkermans et al assumed that for both states of polarization the corresponding intensity can be represented as a product of the intensity of a scalar wave which does not depend on the polarization state and a corresponding multiplicative factor (weighting function) describing polarization transfer [35] These factors are specific for a given polarization state and generally depend on the properties of scattering particles eg size shape [3637] The expression for the co- and cross-polarized intensities derived from the expression proposed in [37] has been successfully used in [38] whereas the experimental results [39] proved the adequacy of the Akkermansrsquo conjecture of time scales involved into depolarization process of the backscattered light

Consider a plane electromagnetic wave polarized in the x direction that enters the medium along positive direction of z axis normal to the interface By a co-polarized wave we understand a linearly polarized scattered wave with the same orientation of polarization as the incident wave and a cross-polarized wave is perpendicular to the incident wave direction of polarization [27] Thus waves scattered in xz and yz planes define co-polarized and cross-polarized components of the scattered electromagnetic wave respectively

To account for depolarization effect in PS-OCT images we adopted the polarization vector

formalism where the polarization is described in terms of a polarization vector P

undergoing a sequence of transformations after each scattering event [27] The trajectories of the polarized photons are weighted in accordance with their polarization state Within the far-field

or Fraunhofer approximation the polarization vector of the scattered wave 1iP minus

is transformed

upon the i-th scattering event into iP

as [2335]

1 1ˆ[ ] [ ] i i ii i i iP e e P I e e Pminus minus= minus times times = minus otimes

(10)

where ie

is the unit vector along the propagation direction between (i-1)th

and ith

scattering

events Note that although the expression (10) is rigorously introduced for the case of Rayleigh scattering it can also be applied as the first approximation in case of Rayleigh-Gans-Debye (RGD) scattering valid for soft scattering particles with the size comparable to or few times larger than the wavelength [23] Namely the size D of the particles should obey the relation (εr-1)Dλ ltlt 1 where (εr-1) is the relative fluctuation of dielectric permittivity between the scatterer (eg cell component such as nucleus or mitochondria) and the

surrounding medium (eg cytoplasm) Typically in biotissues the value of (εr minus1) is less than 01 [40] therefore RGD approximation is quite reasonable for the particles with the sizes of units of λ which are characterized by non-isotropic scattering phase function Recently it has been demonstrated that the expression (10) can be successfully applied with the use of Heyney-Greenstein phase function [41] At the same time implementation of Eq (10) for strongly scattering large inclusions may lead to some discrepancy in the final results however this approximation can be applied to qualitatively estimate the effects of

depolarization in OCT images Explicitly the tensor ˆ[ ]i iiI e e= minus otimesS

is presented as

2

2

2

1

1

1

iX iX iY iX iZ

i iX iY iY iY iZ

iX iZ iX iZ iZ

e e e e e

e e e e e

e e e e e

minus minus minus

= minus minus minus minus minus

S (11)

It guaranties the electromagnetic field remain transversal experiencing the ith

scattering event

The chain 1 1

( )n n

n minus=T S S S of projection operators i

S transforms the initial polarization 0P

upon a sequence of n scattering events to the final polarization nP


01 1n n n

P Pminus= S S S

(12)

Consequently propagation of co-polarized and cross-polarized components of electromagnetic field in the medium is described along the same trajectories obtained for the scalar field The vector nature of electromagnetic field can be taken into account by multiplying the statistical weight of each trajectory by the square of matrix element of tensor

T(n) xx

T ( )n for co-polarized component and yx

T ( )n for cross-polarized one

In order to link vector and scalar approaches in simulation of PS-OCT images it should be pointed out that in accordance with the optical theorem [42] the scalar approach yields [364344]

4

0

1( ) i s sk G d

sminus Ω =int k k (13)

whereas for the electromagnetic field [384243]

4

0 2

2 1( )

1 cosi s s

k G dsθ

minus Ω =+

int k k (14)

Therefore at every scattering event the multiplicative factor

2

2

1 cos θΓ =

+ (15)

should be included Finally for linearly polarized probing radiation the expression (9) can be re-written

separately for co- and cross-polarized detected OCT signal

2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n ico i xx i

i

z LI z I W T n

l=

minus = Γ minus

sum (16)

for co-polarized component and

2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n icross i yx i

i

z LI z I W T n

l=

minus = Γ minus

sum (17)

for cross-polarized one In (16-17) ni is the number of scattering events for i-th detected photon

4 Modeling of OCT images of skin

Basing on the principles of cross-polarization OCT signals formation described above we performed MC simulation of the OCT images of skin in non- co- and cross-polarized modes corresponding to those presented in Fig 1 Instrumental parameters used in simulation are taken in accordance with the characteristics of the reference experimental cross-polarization OCT setup described in section 2 The incidence of radiation at the surface of turbid medium is considered normal This situation corresponds to the case of collimated probing beam The case when imaging is performed with the focused Gaussian-shaped probing beam the waist of which is localized inside the medium is typically simulated using-ray-optics approach proposed by Milsom [45] For simulation of an A-scan 510

6 photons are employed 50 A-

scans are obtained to construct each 2D OCT image A sophisticated non-planar multilayered model of skin based on a number of experimental

OCT images of skin of a male thumb was used in simulations It is typically characterized by thick Stratum Corneum layer which can be separated into two parts upper thin layer consisting of chaotically oriented dry cells and lower thick layer consisting of ordered cells


(Fig 2) The underlying skin layers are represented by epidermis dermis with upper plexus dermis and dermis with lower plexus [8] The shape of the boundaries and the thickness of the layers are chosen basing on the observed experimental OCT images The optical properties of the layers are taken from [8] with a slight modification giving the larger weight to the values extracted from the experimental OCT images which mostly suits the simulation conditions The chosen optical properties represented in Table 1 cannot be considered as exact ones due to significant variations of reported values from subject to subject and depending on extraction method thus they can serve only as approximate values

00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm

Upper Stratum Corneum

Lower Stratum Corneum

Epidermis

Upper dermis with plexus

Dermis

Lower dermis with plexus

Fig 2 Layout of skin model used in the simulation

Table 1 Optical properties of thick skin layers (λ = 910 nm)

Skin layer Thickness

(mm) micro s (mmminus1) microa (mmminus1) g n

Upper Stratum Corneum 002 35 002 09 154

Lower Stratum Corneum 03 5 0015 095 134

Epidermis 015 12 002 085 14

Upper dermis with plexus 02 12 01 09 139

Dermis 02 7 007 087 14

Lower dermis with plexus 03 12 02 095 139

5 Results and discussion

Simulated OCT images of skin in non-polarized mode implemented using the expression (9) without account for ldquospeckle effectsrdquo are presented in Fig 3 for four various values of coherence lengths The coherence length of the probing radiation varies from lc = 5 up to 30 microm at the wavelength λ = 910 nm that is typical for super-luminescence diodes employed in OCT setups All figures are shown in the same color scale so the increase in coherence length results in increase of corresponding average signal intensity At the same time the axial resolution of the images decreases with the increase of coherence length thus yielding blurring of the image The obtained images qualitatively depict the essential features of the experimental images (Fig 1a) exhibiting bright epidermis layer below dark lower Stratum Corneum layer Such similarity indicates the adequacy of the chosen multilayer skin model The bright spots at the extrema of the sinusoidal boundary shape originate from strong back-reflection at the sections of layer boundaries normal to the incident probing beam direction


1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d

Fig 3 Simulated 2D OCT images of skin without account for ldquospeckle effectsrdquo for various coherence lengths of low-coherent light source 5 microm (a) 10 microm (b) 15 microm (c) 30 microm (d)

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d

Fig 4 Simulated 2D OCT images of skin with account for ldquospeckle effectsrdquo for various coherence lengths of low-coherent light source 5 microm (a) 10 microm (b) 15 microm (c) 30 microm (d)

Introduction of ldquospeckle effectsrdquo into Monte Carlo simulation process with accordance to expression (8) causes additional noise observed in the obtained OCT images for all considered coherence length values (Fig 4) These results are in qualitative agreement with the images simulated without account of the ldquospeckle noiserdquo which on the one hand confirms the adequacy of the proposed algorithm and on the other hand demonstrates its ability for


quantitative study of ldquospeckle effectsrdquo in OCT image formation In experimental setups ldquospeckle effectsrdquo are usually reduced by spatial averaging of the signal and additional numerical processing

At the next step we consider the implementation of expressions (1617) into MC simulation which allows to account for polarization state of photons The simulated A-scans (transversal position x = 0 microm) for the multilayer skin model are presented in Fig 5a At small probing depth the co-polarization signal is extremely close to the non-polarized one while their discrepancy grows with the depth which is the evidence of depolarization of the probing radiation

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized

Fig 5 A-scan of OCT image of skin obtained for non- co- and cross- polarized modes (a) 2D OCT images obtained for non- co- and cross- polarized modes (b) (c) and (d) respectively coherence length of low-coherent light source is 15 microm

Figures 5b 5c and 5d demonstrate the simulated OCT images obtained in non- co- and cross- polarization modes correspondingly the same as in Fig 1 which represents similar experimental results These images qualitatively agree with the experimental ones Note that both experimental and simulated images obtained in cross-polarization mode demonstrate the same features relatively low signal from the top boundary and bright epidermis layer However the simulated cross-polarization mode image exhibits lower absolute intensity compared to the experimental one which possibly originates from the assumption of soft


scatterers accounted in expression (10) In reality the rate of depolarization appears to be higher compared to the value accounted in the simulations

6 Conclusion

In this work original theory for scattering of polarized radiation is developed and numerical simulations of OCT images are performed with implementation of this theory Expressions for co- and cross-polarized radiation components contribution to OCT images of a biotissue sample are obtained To simulate PS-OCT images of skin a sophisticated model mimicking physiological skin structure is embedded into MC code Experimental OCT images of human skin are used as a reference The parameters for skin model are chosen basing on the values available from literature

We show that the developed improvement of MC code allows to simulate OCT images which are in good qualitative agreement with the experimental results Simulation can be performed both with and without account for ldquospeckle effectsrdquo which quantitatively demonstrates the role of these effects in OCT image formation for different objects The abilities of the developed code are illustrated by good correlation with the OCT images of human skin in vivo

The developed algorithm can be effectively utilized for simulation of polarization-sensitive OCT imaging of various scattering objects including biotissues and for evaluation of the role of various physical and instrumental factors such as speckles temporal andor spatial coherence properties of incident radiation etc Beside those the phenomena of optical anisotropy of tissues such as birefringence and optical activity that are of a high interest in framework of modern biomedical optical diagnostics and can be introduced in the proposed algorithm with easier eg similar to the approach presented in [46]

Acknowledgements

This project was supported in part by the Royal Society UK NATO (grant no PSTCLG979652) GETA graduate school and Tauno Toumlnning Foundation (Finland) E Sergeeva M Kirillin and V Kuzmin acknowledge Russian Foundation for Basic Research (grants 09-02- 97040 10-02-00744 10-02-00937) and grants of the President of Russian Federation МК-69820092 МК-112720102 The work is partly supported by FTP ldquoScientific and scientific-pedagogical personnel of innovative Russiardquo (projects 02740110839 02740110566 02740110437)


13 E Berrocal I V Meglinski D A Greenhalgh and M A Linne ldquoImage transfer through the complex scattering turbid mediardquo Laser Phys Lett 3(9) 464ndash468 (2006)

14 G Yao and L V Wang ldquoMonte Carlo simulation of an optical coherence tomography signal in homogeneous turbid mediardquo Phys Med Biol 44(9) 2307ndash2320 (1999)

15 M Yu Kirillin M V Shirmanova M A Sirotkina M L Bugrova B N Khlebtsov and E V Zagaynova ldquoContrasting properties of gold nanoshells and titanium dioxide nanoparticles for OCT imaging of skin Monte Carlo simulations and in vivo studyrdquo J Biomed Opt 14 021017 (2009)

16 B Karamata M Laubscher M Leutenegger S Bourquin T Lasser and P Lambelet ldquoMultiple scattering in optical coherence tomography I Investigation and modelingrdquo J Opt Soc Am A 22(7) 1369ndash1379 (2005)

17 B Karamata M Leutenegger M Laubscher S Bourquin T Lasser and P Lambelet ldquoMultiple scattering in optical coherence tomography II Experimental and theoretical investigation of cross talk in wide-field optical coherence tomographyrdquo J Opt Soc Am A 22(7) 1380ndash1388 (2005)

18 M Y Kirillin A V Priezzhev and I V Meglinski ldquoEffect of photons of different scattering orders on the formation of a signal in optical low-coherence tomography of highly scattering mediardquo Quantum Electron 36(3) 247ndash252 (2006)

19 V L Kuzmin and I V Meglinski ldquoMultiple scattering and intensity fluctuations in optical coherence tomography of randomly inhomogeneous mediardquo J Exp Theor Phys 105(2) 285ndash291 (2007)

20 R V Kuranov V V Sapozhnikova N M Shakhova V M Gelikonov E V Zagainova and S A Petrova ldquoCombined application of optical methods to increase the information content of optical coherent tomography in diagnostics of neoplastic processesrdquo Quantum Electron 32(11) 993ndash998 (2002)

21 M Yu Kirillin E Alarousu T Fabritius R Myllylauml and A V Priezzhev ldquoVisualization of paper structure by optical coherence tomography Monte Carlo simulations and experimental studyrdquo J Europ Opt Soc Rap Public 2 07031 (2007)

22 IM Sobolrsquo The Monte Carlo Method (The University of Chicago Press Chicago 1974) 23 A Ishimaru Wave Propagation and Scattering in Random Media (Academic New York 1978) 24 D Y Churmakov I V Meglinski and D A Greenhalgh ldquoInfluence of refractive index matching on the photon

diffuse reflectancerdquo Phys Med Biol 47(23) 4271ndash4285 (2002) 25 D Y Churmakov V L Kuzrsquomin and I V Meglinski ldquoApplication of the vector Monte-Carlo method in

polarisation optical coherence tomographyrdquo Quantum Electron 36(11) 1009ndash1015 (2006) 26 J W Goodman Statistical Optics (Wiley-Interscience 1985) 27 C Brosseau Fundamentals of Polarized Light a Statistical Optics Approach (New York John Wiley amp Sons

1998) 28 C F Bohren and D R Huffman Absorption and scattering of light by small particles (New York Wiley 1983) 29 X Wang and L V Wang ldquoPropagation of polarized light in birefringent turbid media a Monte Carlo studyrdquo J

Biomed Opt 7(3) 279ndash290 (2002) 30 S Bartel and A H Hielscher ldquoMonte Carlo simulations of the diffuse backscattering mueller matrix for highly

scattering mediardquo Appl Opt 39(10) 1580ndash1588 (2000) 31 M J Raković G W Kattawar M B Mehrubeoğlu B D Cameron L V Wang S Rastegar and G L Coteacute

ldquoLight backscattering polarization patterns from turbid media theory and experimentrdquo Appl Opt 38(15) 3399ndash3408 (1999)

32 D A Zimnyakov Y P Sinichkin P V Zakharov and D N Agafonov ldquoResidual polarization of non-coherently backscattered linearly polarized light the influence of the anisotropy parameter of the scattering mediumrdquo Waves Random Media 11(4) 395ndash412 (2001)

33 S V Gangnus S J Matcher and I V Meglinski ldquoMonte Carlo modeling of polarized light propagation in biological tissuesrdquo Laser Phys 14 886ndash891 (2004)

34 J M Schmitt A H Gandjbakhche and R F Bonner ldquoUse of polarized light to discriminate short-path photons in a multiply scattering mediumrdquo Appl Opt 31(30) 6535ndash6546 (1992)

35 E Akkermans P E Wolf R Maynard and G Maret ldquoTheoretical-Study of the Coherent Backscattering of Light by Disordered Mediardquo J Phys France 49(1) 77ndash98 (1988)

36 M J Stephen and G Cwilich ldquoRayleigh scattering and weak localization Effects of polarizationrdquo Phys Rev B Condens Matter 34(11) 7564ndash7572 (1986)

37 F C MacKintosh and S John ldquoDiffusing-wave spectroscopy and multiple scattering of light in correlated random mediardquo Phys Rev B Condens Matter 40(4) 2383ndash2406 (1989)

38 D A Zimnyakov and Y P Sinichkin ldquoA study of polarization decay as applied to improved imaging in scattering mediardquo J Opt A Pure Appl Opt 2(3) 200ndash208 (2000)

39 A Dogariu C Kutsche P Likamwa G Boreman and B Moudgil ldquoTime-domain depolarization of waves retroreflected from dense colloidal mediardquo Opt Lett 22(9) 585ndash587 (1997)

40 VV Tuchin Tissue optics light scattering methods and instruments for medical diagnosis (SPIE Press Bellingham 2000)

41 V L Kuzmin and I V Meglinski ldquoHelicity flip of backscattered circularly polarized lightrdquo Proc SPIE 7573 75730Z (2010)

42 P S Carney E Wolf and G S Agarwal ldquoStatistical generalizations of the optical cross-section theorem with application to inverse scatteringrdquo J Opt Soc Am A 14(12) 3366ndash3371 (1997)

43 V L Kuzmin and E V Aksenova ldquoA generalized Milne solution for the correlation effects of multiple light scattering with polarizationrdquo J Exp Theor Phys 96(5) 816ndash831 (2003)

44 V L Kuzrsquomin V P Romanov and L A Zubkov ldquoPropagation and scattering of light in fluctuating mediardquo Phys Rep 248(2-5) 71ndash368 (1994)




1 Introduction






0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

1

2

3

4

0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

a b c








( ) exp( )s s






( ) S

f s dsξinfin



ln

s

sξmicro

= minus (3)



( ) ( )

( )d

i si s

i s s

Gp

Gπ

minusminus =

minus Ωint4

n nn n

n n (4)

where

( ) ( )02

1(0) ( ) exp ( )

(4 )i s i s

G d ikε επ






n and s

n are


respectively 2

2sini s











( ) 12

( ) Re ( ) r s cI z I I C z l= (6)

Here s

I and r



2

c

2ln2 l

λπ λ

=∆

(7)



2

0

1 c

2 2 ( ) cos (2 ) exp

phN

ii i

i

z LI z I W z L

l

πλ=



2

0

1 c

2 ( ) exp

phN

ii

i

z LI z I W

l=

minus = minus

sum (9)











is transformed


as [2335]


(10)

where ie


and ith

scattering




is presented as

2

2

2

1

1

1

iX iX iY iX iZ

i iX iY iY iY iZ

iX iZ iX iZ iZ

e e e e e

e e e e e

e e e e e

minus minus minus


S (11)


scattering event

The chain 1 1

( )n n





01 1n n n

P Pminus= S S S

(12)


T(n) xx




4

0

1( ) i s sk G d



4

0 2

2 1( )

1 cosi s s

k G dsθ

minus Ω =+

int k k (14)


2

2

1 cos θΓ =

+ (15)



2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n ico i xx i

i

z LI z I W T n

l=

minus = Γ minus

sum (16)


2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n icross i yx i

i

z LI z I W T n

l=

minus = Γ minus

sum (17)









00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm



Epidermis


Dermis








Epidermis 015 12 002 085 14


Dermis 02 7 007 087 14





1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements





1 Introduction






0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

1

2

3

4

0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

0 200 400 600 800 1000

500

1000

1500

0

10

20

30

40

50

60

a b c








( ) exp( )s s






( ) S

f s dsξinfin



ln

s

sξmicro

= minus (3)



( ) ( )

( )d

i si s

i s s

Gp

Gπ

minusminus =

minus Ωint4

n nn n

n n (4)

where

( ) ( )02

1(0) ( ) exp ( )

(4 )i s i s

G d ikε επ






n and s

n are


respectively 2

2sini s











( ) 12

( ) Re ( ) r s cI z I I C z l= (6)

Here s

I and r



2

c

2ln2 l

λπ λ

=∆

(7)



2

0

1 c

2 2 ( ) cos (2 ) exp

phN

ii i

i

z LI z I W z L

l

πλ=



2

0

1 c

2 ( ) exp

phN

ii

i

z LI z I W

l=

minus = minus

sum (9)











is transformed


as [2335]


(10)

where ie


and ith

scattering




is presented as

2

2

2

1

1

1

iX iX iY iX iZ

i iX iY iY iY iZ

iX iZ iX iZ iZ

e e e e e

e e e e e

e e e e e

minus minus minus


S (11)


scattering event

The chain 1 1

( )n n





01 1n n n

P Pminus= S S S

(12)


T(n) xx




4

0

1( ) i s sk G d



4

0 2

2 1( )

1 cosi s s

k G dsθ

minus Ω =+

int k k (14)


2

2

1 cos θΓ =

+ (15)



2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n ico i xx i

i

z LI z I W T n

l=

minus = Γ minus

sum (16)


2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n icross i yx i

i

z LI z I W T n

l=

minus = Γ minus

sum (17)









00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm



Epidermis


Dermis








Epidermis 015 12 002 085 14


Dermis 02 7 007 087 14





1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements







( ) exp( )s s






( ) S

f s dsξinfin



ln

s

sξmicro

= minus (3)



( ) ( )

( )d

i si s

i s s

Gp

Gπ

minusminus =

minus Ωint4

n nn n

n n (4)

where

( ) ( )02

1(0) ( ) exp ( )

(4 )i s i s

G d ikε επ






n and s

n are


respectively 2

2sini s











( ) 12

( ) Re ( ) r s cI z I I C z l= (6)

Here s

I and r



2

c

2ln2 l

λπ λ

=∆

(7)



2

0

1 c

2 2 ( ) cos (2 ) exp

phN

ii i

i

z LI z I W z L

l

πλ=



2

0

1 c

2 ( ) exp

phN

ii

i

z LI z I W

l=

minus = minus

sum (9)











is transformed


as [2335]


(10)

where ie


and ith

scattering




is presented as

2

2

2

1

1

1

iX iX iY iX iZ

i iX iY iY iY iZ

iX iZ iX iZ iZ

e e e e e

e e e e e

e e e e e

minus minus minus


S (11)


scattering event

The chain 1 1

( )n n





01 1n n n

P Pminus= S S S

(12)


T(n) xx




4

0

1( ) i s sk G d



4

0 2

2 1( )

1 cosi s s

k G dsθ

minus Ω =+

int k k (14)


2

2

1 cos θΓ =

+ (15)



2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n ico i xx i

i

z LI z I W T n

l=

minus = Γ minus

sum (16)


2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n icross i yx i

i

z LI z I W T n

l=

minus = Γ minus

sum (17)









00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm



Epidermis


Dermis








Epidermis 015 12 002 085 14


Dermis 02 7 007 087 14





1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements









( ) 12

( ) Re ( ) r s cI z I I C z l= (6)

Here s

I and r



2

c

2ln2 l

λπ λ

=∆

(7)



2

0

1 c

2 2 ( ) cos (2 ) exp

phN

ii i

i

z LI z I W z L

l

πλ=



2

0

1 c

2 ( ) exp

phN

ii

i

z LI z I W

l=

minus = minus

sum (9)











is transformed


as [2335]


(10)

where ie


and ith

scattering




is presented as

2

2

2

1

1

1

iX iX iY iX iZ

i iX iY iY iY iZ

iX iZ iX iZ iZ

e e e e e

e e e e e

e e e e e

minus minus minus


S (11)


scattering event

The chain 1 1

( )n n





01 1n n n

P Pminus= S S S

(12)


T(n) xx




4

0

1( ) i s sk G d



4

0 2

2 1( )

1 cosi s s

k G dsθ

minus Ω =+

int k k (14)


2

2

1 cos θΓ =

+ (15)



2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n ico i xx i

i

z LI z I W T n

l=

minus = Γ minus

sum (16)


2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n icross i yx i

i

z LI z I W T n

l=

minus = Γ minus

sum (17)









00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm



Epidermis


Dermis








Epidermis 015 12 002 085 14


Dermis 02 7 007 087 14





1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements









is transformed


as [2335]


(10)

where ie


and ith

scattering




is presented as

2

2

2

1

1

1

iX iX iY iX iZ

i iX iY iY iY iZ

iX iZ iX iZ iZ

e e e e e

e e e e e

e e e e e

minus minus minus


S (11)


scattering event

The chain 1 1

( )n n





01 1n n n

P Pminus= S S S

(12)


T(n) xx




4

0

1( ) i s sk G d



4

0 2

2 1( )

1 cosi s s

k G dsθ

minus Ω =+

int k k (14)


2

2

1 cos θΓ =

+ (15)



2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n ico i xx i

i

z LI z I W T n

l=

minus = Γ minus

sum (16)


2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n icross i yx i

i

z LI z I W T n

l=

minus = Γ minus

sum (17)









00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm



Epidermis


Dermis








Epidermis 015 12 002 085 14


Dermis 02 7 007 087 14





1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements



01 1n n n

P Pminus= S S S

(12)


T(n) xx




4

0

1( ) i s sk G d



4

0 2

2 1( )

1 cosi s s

k G dsθ

minus Ω =+

int k k (14)


2

2

1 cos θΓ =

+ (15)



2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n ico i xx i

i

z LI z I W T n

l=

minus = Γ minus

sum (16)


2

2

0

1 c

2 ( ) ( ) exp

ph

i

N

n icross i yx i

i

z LI z I W T n

l=

minus = Γ minus

sum (17)









00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm



Epidermis


Dermis








Epidermis 015 12 002 085 14


Dermis 02 7 007 087 14





1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements




00 02 04 06 08 10

12

10

08

06

04

02

00

Z m

m

Xmm



Epidermis


Dermis








Epidermis 015 12 002 085 14


Dermis 02 7 007 087 14





1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements



1

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d


200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

a b

c d






200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements





200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

200 400 600 800 1000

0

500

1000

0

10

20

30

40

50

60

b c d

a

00 02 04 06 08 10 12

0

10

20

30

40

50

60

70

OC

Tsig

na

ldB

Z mm

Non-polarized

Co-polarized

Cross-polarized





6 Conclusion




Acknowledgements




6 Conclusion




Acknowledgements



Documents

Simulation of optical coherence tomography images by Monte Carlo modeling based on polarization vector approach