July 15, 2008 / Vol. 33, No. 14 / OPTICS LETTERS
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Simulation of polarization-sensitive optical coherence tomography images by a Monte Carlo method Igor Meglinski,1,* Mikhail Kirillin,2 Vladimir Kuzmin,3 and Risto Myllylä2 1
Cranfield Health, Cranfield University, Cranfield, MK43 0AL, UK Optoelectronics and Measurement Techniques Laboratory, University of Oulu, P.O. Box 4500, 90014, Oulu, Finland 3 St. Petersburg Institute of Commerce and Economics, 194021 St. Petersburg, Russia *Corresponding author:
[email protected]
2
Received April 28, 2008; accepted May 27, 2008; posted June 16, 2008 (Doc. ID 95558); published July 11, 2008 We introduce a new Monte Carlo (MC) method for simulating optical coherence tomography (OCT) images of complex multilayered turbid scattering media. We demonstrate, for the first time of our knowledge, the use of a MC technique to imitate two-dimensional polarization-sensitive OCT images with nonplanar boundaries of layers in the medium like a human skin. The simulation of polarized low-coherent optical radiation is based on the vector approach generalized from the iterative procedure of the solution of Bethe–Saltpeter equation. The performances of the developed method are demonstrated both for conventional and polarization-sensitive OCT modalities. © 2008 Optical Society of America OCIS codes: 110.4500, 290.0290, 290.5855.
Since Wilson and Adam [1] first introduced a Monte Carlo (MC) method into the field of laser-tissue interaction, this technique has been used extensively in a number of studies of photon migration for diverse optical diagnostic applications [2–5], including optical coherence tomography (OCT) [6–8]. Recently a MC technique has been widely used to simulate coherent phenomena of multiple scattering [9–11] and the changes of polarization of optical radiation scattered within the biological media [12–14]. Typically, within the MC algorithms the state of the polarization and its evolution during the propagation is described in framework of Stokes–Mueller or Jones formalism [12–14]. In this Letter, we introduce a new MC technique that is able to imitate two-dimensional (2D) polarization-sensitive OCT images of complex multilayered turbid scattering media avoiding implementation of Stokes–Mueller and/or Jones formalism. The concept of the approach is based on the development of a unified MC program as a natural extension for the most popular standard MC code [2]. Following this we consider propagation of photon packets along each possible trajectory within the medium. The free photon path between two successive scattering events is governed by Poisson distribution: f共li兲 = s exp共−sli兲, and defined as li = −−1 s ln , where is the probability that the mean free path l is no less than li, arbitrary value is chosen in the [0,1] interval using a random number generator, and s is the scattering coefficient: s = l−1. A new direction of the photon packet after each scattering event is determined by the Henyey–Greenstein scattering phase function [2], originally developed to approximate Mie scattering of light from particles with size comparable with the wavelength of the incident light. This expression is characterized by a so-called anisotropy factor g, which is equal to the mean cosine of the scattering angle g = cos (g 苸 关0 , 1兴, with g = 0 corre0146-9592/08/141581-3/$15.00
sponding to the isotropic scattering). A consideration of light scattering within an absorbing medium 共sl ⫽ 1兲 requires a proportional reduction of the statistical weight W of each photon packet according its traN jectory [15]: W = W0 exp共−兺i=1 ali兲, where W0 is the initial photon packet weight, a is the absorption coefficient of the medium, and N is the number of experienced scattering events. The photon packets fitting the given detection criteria [8], including size, numerical aperture, and detector position are taken into account. Completing the tracing of the photon trajectories for a large number of photons Nph (typically Nph ⬃ 107), the OCT signal is calculated as a convolution of distribution of the detected photons over their optical path lengths N li with the envelope of coherence function [16]: L = 兺i=1 Nph
冋 冉 冊册
I共z兲 = 2 兺 冑WrWs共Lj兲 exp − j=1
z − Lj lcoh
2
,
共1兲
where Wr and Ws are the total weights of the detected photon packets from the reference arm and scattering medium, respectively; lcoh is the coherence length of probing laser radiation; and z is the depth. Using a combination of the MC technique and the iteration procedure of the solution of Bethe–Salpeter equation [10], it has been shown that simulation of the optical path of a photon packet undergoing N scattering events directly corresponds to the Nth order ladder diagram contribution. In this correspondence we generalized the above mentioned MC technique for the direct simulation evolution of the polarization vector for each photon packet. The polarization is described in terms of the polarization vecជ undergoing a sequence of transformations after tor P each scattering event [17], and all the photon packets are weighted in accordance with the polarization state. In Rayleigh and Rayleigh–Gans approxima© 2008 Optical Society of America
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OPTICS LETTERS / Vol. 33, No. 14 / July 15, 2008
ជ tions the polarization vector of the scattered wave P i is transformed upon the ith scattering such that [17] ជ 兴 = 关Iˆ − eជ 丢 eជ 兴P ជ , ជ = − eជ 关eជ · P P i−1 i i i i−1 i i
共2兲
where eជ i is the unit vector aligned along the trajectory element of a photon packet after the ith scattering event. Thus, for simulating the electromagnetic field transfer we trace the transformation of the polarization vector of incident field P0 under the action of the chain of diadic operators: N
P = 共Iˆ − k−2ks 丢 ks兲 兿 共Iˆ − k−2kjj−1 丢 kjj−1兲P0 . 共3兲 i=2
The final stage of the parallel and perpendicular components of vector P determines the polarized and depolarized components of scattering radiation. In such a manner, we consider the propagation of polarized and depolarized components of the electromagnetic field along the trajectory defined for the intensity propagation (i.e., by scalar approach). However, and it should be pointed out here, based on the optical theorem [18] the scalar approach gives [19,20] k04
冕
˜ 共kជ ,kជ 兲d⍀ = G i s s
1 l
共4兲
,
whereas for the electromagnetic field in the limit of weak scattering [19,20] k04
冕
˜ 共kជ ,kជ 兲d⍀ = G i s s
1
2
l 1 + cos2
.
the exact solutions for pointlike Rayleigh scattering particles. For normal incidence and exact backscattering detection the generalized Milne solution for the electromagnetic field gives the ratio of polarized and depolarized intensities I储 / I⬜ ⬇ 1.92 [19]. Numerical stimulation suggests that I储 / I⬜ ⬇ 1.93 with all significant digits. The generalized Milne solution gives the depolarization ratio DP= I储 − I⬜ / I储 + I⬜ = 0.31 [19], while the results of our simulation suggest DP⬇ 0.326. The close value DP= 0.33 was obtained in [9]. The performances of this extended computational approach for simulating 2D conventional OCT and polarization-sensitive OCT images are shown in Fig. 1 for the skin model with six nonplanar layers [4,16] (see Fig. 1a). The optical properties of the layers are s: 35, 5, 10, 10, 7, 12 共mm−1兲; a: 0.02, 0.015, 0.02, 0.1, 0.7, 0.2 共mm−1兲; g: 0.9, 0.95, 0.85, 0.9, 0.87, 0.95; and refractive index n: 1.54, 1.34, 1.4, 1.39, 1.4, 1.39, respectively. The source-detector parameters are chosen in accordance with the parameters of the OCT system described in [21]: wavelength = 900 nm, lcoh = 15 m, and numerical aperture 0.2. To simulate OCT images the sequential A-scans are simulated with the defined 20 m transversal scanning step. The results of simulation of 2D OCT and polarization-sensitive OCT images for both copolar-
共5兲
˜ 共kជ , kជ 兲 is the scattering phase function; kជ and Here, G i s i ជks are the wave vectors for incident and scattered of complex-conjugated fields, respectively; and k0 = 2 / , where is the wavelength. Therefore, the extra multiplicative factor ⌫=
2
共6兲
1 + cos2
should be taken into account at every scattering event. Finaly, the expressions for the intensities of the copolarized 共I储兲 and cross-polarized 共I⬜兲 components valid for Henyey–Greenstein phase function are Nph
I储共兲 =
W j⌫ N P 2 , 兺 j=1 j
储
j
Nph
I ⬜共 兲 =
2 W j⌫ N P ⬜ , 兺 j=1 j
j
共7兲
where N is the number of scattering acts for the jth photon packet and denotes the dependence on any variable of interest such as depth, time, number of scattering events, etc. This MC algorithm has been validated against an accepted analytic solution for a semi-infinite medium [11], and cross validated for a slab geometry with an absorbing inclusion. The results are compared with
Fig. 1. (Color online) Schematic presentation of a skin model used in the simulation. a: 1, upper stratum corneum (average thickness 0.02 mm); 2, lower stratum corneum 共0.18 mm兲; 3, epidermis 共0.2 mm兲; 4, upper blood net dermis 共0.2 mm兲; 5, reticular dermis 共0.8 mm兲; 6, deep blood net dermis 共0.6 mm兲. 2D simulated images for b, convention OCT and c, copolarized and d, cross-polarized OCT modalities, respectively.
July 15, 2008 / Vol. 33, No. 14 / OPTICS LETTERS
ized and cross-polarized modes are presented in Figs. 1b–1d. The OCT image obtained in copolarized mode (see Fig. 1c) is similar to the OCT image obtained for the convention OCT (Fig. 1b), whereas the intensity of the image obtained in cross-polarized mode (see Fig. 1d) is much lower compared with the copolarized mode (Fig. 1c). This is due to the origin of the OCT signal formed mostly by so-called “snake” or low-scattering-order photons [22]. Therefore, contribution of the photon packets depolarized owing to multiple scattering in the formation of the final OCT image is relatively small in upper layers and significantly rises for deep layers. Thus, based on the generalization of iterative procedure of the solution of Bethe–Saltpeter equation, a new vector-based MC approach has been developed. For the first time to our knowledge using a MC technique we have imitated 2D OCT and polarizationsensitive OCT images of a skin model with nonplanar layers’ boundaries. The performances of the developed method are shown both for conventional and polarization-sensitive OCT modalities. The current approach allows considerably simplified simulation of polarized low-coherent optical radiation propagation within turbid scattering media avoiding implementation of Stokes–Mueller and/or Jones formalism, with the substantial reducing of the computational resources. In a similar manner this approach can be applied for circular polarization allowing one to avoid a cumbersome calculation of the Mueller matrix. The technique can provide detailed information on polarization propagation within/outside the medium, scoring various physical quantities simultaneously. The technique will find a number of straightforward applications related to the noninvasive diagnostics of various disperse random media, including biological tissues, polymers, liquid crystals, and others. The authors acknowledge the support of UK Biotechnology and Biological Sciences Research Council (project BBS/B/04242) and Royal Society, NATO (project PST.CLG.979652), GETA graduate school, and Tauno Tönning Foundation, Finland.
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