Monte Carlo (MC) Model of Light Transport in Turbid Media

Transcription

1 Monte Carlo (MC) Model of Light Transport in Turbid Media M. Talib Department of Physics, University of AL Qadisiya Al Yahoo. Com Abstract: Monte Carlo method was implemented to simulation Random Photon Transport in turbid media. In this papaer provided formulas necessary to utilized Q Basic program to simulate transport laser in turbid media. The formulas necessary for implementation of the (MC) method in computer code are provided where this paper discusses internal reflection of photon at boundary, shows how the phase function may be used to generated new scattering angle. and calculated of photon (absorbed, scattered, reflected and transmited) to achieve the optimum operational circumstances for desirable change in the turbid medium. 1 Introduction Monte Carlo (MC) refers to technique first proposed by metropolis and Ulam to simulated physical processes using stochastic model (1). In irradiative a transport problem, The Monte Carlo method consist of recording photons histories as they are scattered and absorbed (2) This paper describes the (MC) method for modeling light transport in turbid media. The formulas necessary for implementation of the (MC) method in computer code are provided where this paper discusses internal reflection of photon at boundary, and shows how the phase function may be used to generated new scattering angle. 1

2 Q Basic program have been written according to (MC) method to simulate the interaction of laser with turbid medium; The out put results of the program have been compared with those published to insure the accuracy of the method. 2 Simulation Photon Propagation Once launched, The photon is moved distance S where it may be scattering, absorbed, propagation nudist, Internally reflected, or transmitted out of the media. The photon is repeatedly moved until it either escapes from or is absorbed by the turbid media. If the photon escapes from the media, The reflection or Transmission of the photon is recorded. If the photon is absorbed, The position and a new photon is lunched (3). This process is described in the flowing flowchart: 2

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4 3 Computer program Although multiple research groups have implemented (MC) simulations in various computer languages, Our standard Q Basic implementation of MC modeling of photon transport in turbid medium. 3 1 Steps of Computer program Step 1 Input parameters: The parameter of simulation shown in table 1. Step 2 The photon is injected orthogonally on to the medium at the origin. The photon position (x, y, z) is initialized to (0,0,0). Step 3 In this step the Randomized (RND) is ready. Step 4 In this step the direction cosines (µ x, µ y, µ z ) are set to (0,0,1) where the direction cosine are specified by taking the cosine of the angle that photons direction makes with each axis µ x = r. x^ (1 1a) µ y = r. y^ (1 1b) µ z = r. z^ (1 1c) where x, y, z are unite victor along axis (2) Step 5 If there is a refractive index mismatched interface between the medium and the ambient medium then some specular reflectance(r sp ) will occur. If the refractive indices of the outside medium and medium (n 1 ) and (n 2 ) respectively then the specular reflectance is specified: R sp = (n 1 n 2 ) 2 / (n 1 + n 2 ) 2 (1 2) 4

5 The photon weight, Initialized to (1), Is decreased by (R sp )for the photon packet to enter the medium : W = 1 R sp...(1 3) Step 6 The distance that photon will travel before interaction ( through absorption or scattering) with the turbid medium is generated as: S = S/ µ t..(1 4) S = Ln ζ Where S and ζ are the random distance and random number is uniformly disruption (Zero) and (One), µ t is total attenuation coefficient. µ t = (µa + µs) (1 5) S is the step size of photon (4). Step 7 The new coordinates of photon traveling distance S in the direction (µ x, µ y, µ z ) are given in this step by : x = x + µ x S..(1 6a) y = y + µ y S..(1 6b) z = z + µ z S..(1 6c) where (x, y, z) are the old photon position. Step 8 We decide whether the random (S) distance is greeted then db: db µ t S.(1 7) where S = Ln ζ (1 8) db is the distance between the current photon location (x, y, z) and the boundary of the current layer in the direction of the photon propagation is computed: (z 0 z) / µ z if µ z < 0 db = if µ z = 0 (1 9) (5) (z 1 z) / µ z if µ z > 0 5

6 Where (z o ),(z 1 ) and (z) are the coordinates of the upper, lower boundary and direction of photon propegation. Step 9 By Fresnel's formulas the internal reflectance (R(α i )) is calculated : (R(α i )) =½[sin 2 (α i α t )/sin 2 (α i + α t )+tan 2 (α i α t )/tan 2 (α i + α t ) ] (1 10) Where α i and α t are angle of incident and angle of transmission respectively. The value of α i is calculated: α i = cos 1 ( µ z ) (1 11) and the value of α t is given by Snell's low: n i sinα i = n t sinα t (1 12) The new photon direction (µ' x, µ' y, µ' z ) is: (µ' x, µ' y, µ' z ) = (µ x, µ y, µ z ) where µ x and µ y remain unchanged Step 10 We determine whether the photon is internally reflected or transmitted by generating random number(ζ) and comparing it with the internal reflectance, i.e. : If ζ R(α i ) then the photon is internally reflected If ζ > R(α i ) then the photon is transmited. Step 11 This step show that the photon is either absorbed or scattered, The probability that it is scattered is equal to the ratio of the scattering coefficient to the sum of the absorption and scattering coefficients ( albedo,a). If ζ is a random number uniformly distributed between (Zero) and (One), Then the photon is scattered if : ζ<µ a / µ a + µ s = a.(1 13) (2) Where a = µ a / µ a + µ s otherwise, the photon is absorbed. If the photon is scattered then a new photon direction is chosen based on the phase function, otherwise the photon is absorbed and the probability of photon absorption is: 6

7 P{absorption} ~ µ a S.(1 14) Similarly, the probability that the photon will scattering this distance is : P{scattering} ~ µ s S.(1 15) Step 12 Assuming three disjoint events in this step: (a) Absorption. (b) Scattering. (c) No interaction between the turbid medium and the photon. Where the sum of the three events equals unity. This implicitly assumes that the photon cannot be scattered and absorbed in the same propagation step. T0 determine if a photon is a scattered or absorbed random number ζ uniformly distributed between (Zero) and (One) is generated and compared with the probability of absorption if ζ < {absorption} Then the photon is absorbed and new photon is launched. The fraction the packet that is absorbed is fraction absorbed = µ a / (µ a + µ s ) = 1 (µ s / (µ a + µ s )) (1 16) P {absorption} ζ P {absorption} + P {scattering} = 1 a (2) then the photon is scattered and a new photon direction is chosen based on the phase function for the medium. (P(µ) = 1 g 2 / 2(1 + g 2 2gµ) 3/2 (1 17) Where µ = 1/2g [ 1 + g 2 [1 g 2 / 1 g + 2gζ] 2 ] (1 18a) for g 0 µ = 2ζ 1 (1 18b) for g = 0 where g and µ are unisotropic factor and the direction cosine If the photon is neither scattered nor absorbed then the photon has propagation the distance S without interaction. The new direction of photon scattered are : µ' x = Sinα i / [( 1 µ z ) 1/2 (µ x µ z CosΦ µ y SinΦ) + µ x Cosα i ].(1 19a) µ' y = Sinα i / [( 1 µ z ) 1/2 (µ y µ z CosΦ µ x SinΦ) + µ y Cosα i ].(1 19b) µ' z = Sinα i CosΦ ( 1 µ z ) 1/2 + µ x Cosα i.(1 19c) where Φ is the azimuthal angle (uniformly distributed within the interval [0,2π] thus we have : Φ = 2πζ..(1 20) (4) 7

8 Step 13 Russian roulette technique used to termination a photon in this step, when the weight falls below minimum (e.g.,0.001). The Russian roulette technique gives such a photon (with weight W) One chance in (m) (e.g., 10) of surviving with a weight (mw) or else its weight is reduced to Zero. Step 14 Up data results Step 15 Print results 4 result and discussion The optical Properties of the turbid medium which used in simulation are shown in table (1). Table (1):optical Properties of the turbid medium (6) Turbid medium λ (nm) µ a (cm 1 ) µ s (cm 1 ) g n t d (cm) Mussel rabbit A variable step size weighted MC model has been implemented and validated by comparison with published value, The result of simulation shown in table (2). 8

9 Table 2: Result of simulated photon in turbid media by Monte Carlo (MC) model. Photons simulated Published value (6) Relative percentage error Absorbed % Scattered % Reflected % transmitted % The output results of the program show that, there is relative percentage error of (5% as upper limit in the Reflected photons & 0.5% as lower limit in the Transmitted photons), Fifty thousand initial photons have been used in the simulation at initial weight equals One on the axis z and parallel to both on the x axis and y axis at original point (0, 0, 0). 9

10 References [1] S. Jacques, History of Monte Carlo Method.1990 [2] S. h. Prahl, Ph.D thesis, University of Texas at Austin, 1988, [3] S. h. Prahl, M. Keijzer, S. Jacques & A.J. Welch,,Proc.SPIE Is5(1989), [4] Sramachandran, Monte Carlo Techniques, (2003), 1, 2. [5] L. Wang & S. L. Jacques & L. Zheng, Meth.Prog.Biol, 1995, 47, [6] L. Oliveira & A. Lage, Monte Carlo Simulation for the Optical Transmittance in Biological Tissues during the Action of Osmotic- Agent

11 نموذج مونتي كارلو النتقال الفوتون في الوسط المضطرب مشتاق طالب الخالصة: استعملت طريقة مونتي كارلو لمحاكاة انتقال الفوتون عشوائيا في األوسااط المطابر ة ه اااا ال حاط يعباي التفاصيل األساسية ل ناء رنامج لمحاكاة تفاعل الليزر مع األوساط المطبر ةه حيط تم ناء رنامج حاسو ي لغة ال يسك المتقدم Basic) Q) لدراسة االنعكاس الداخلي عند حدود الوسط المتفاعل معه اإلشعاع وكاالك دراساة كيفية استخدام دالة البور لتوليد زوايا االستبارة كما تم حساب كام الفوتوناا) لالممات و المساتبار و المانعك و النافا( هدف التوصل إلى ظروف التشغيل المثلى إلحداث التأثير المرغوب في الوسط المطبربه 11