标准C中多层组织中光传输的蒙特卡洛建模——翻译版本

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MCML—Monte Carlo modeling of light transport in multi-layered tissues中文翻译及相关源码资料.zip （12个子文件）

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Monte Carlo modeling of light transport in multi-layered tissues in standard C.pdf 935KB

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1994原始代码

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ELSEVIER

Computer

Methods and Programs in Biomedicine 47 (1995) 131-146

computer methods

and programs

in biomedicine

CML - Monte Carlo modeling of light transport in

multi-la.yered. tissues

Lihong Wang* a, Steven L. Jacquesa, Liqiong Zhengb

‘Laser Biology Research Laboratory, Box 17, University

Texas d4.D. Anderson Cancer Center, 151.5 Holcombe Bouievard,

Houston, TX 77030, USA

bDepartment of Computer Science, University

Houston, Houston, TX 77204-3475, USA

Received 1 February 1995; revision received 11 April 1995; accepted 26 April 1995

Abstract

A Monte Carlo model of steady-state light transport in multi-layered tissues (MCML) has been coded in ANSI Stan-

dard C; therefore, the program can be used on various, computers. Dynamic data allocation is used for MCML, hence

the number of tissue layers and grid elements of the grid system can be varied by users at run time. The coordinates

of the simulated data for each grid element in the radial and angular directions are optimized. Some of the MCML

computational results have been verified with those of other thieories or other investigators. The program, including

the sou:rce code, has been in the public domain since 1992.

Keyvmxk:

Monte Carlo; Photon transport; Tissue optics; Standard C; Dynamic allocation

1. Intduction

Since Wilson

and

Adam [l] first introduced

Monte Carlo simulations into the field of laser-

tissue interactions, it has been widely used

to simu-

late light transport in tissues for various applica-

tions a.nd gone through several improvements.

12-71. Although multiple research groups have im-

plemented Monte Carlo simulations

in various

computer languages, our Standard C implementa-

tion of Monte Carlo modeling of photon transport

in multi-layered tissues (MCML), upon which a

* Corresponding author, Tel.: (713) 792-3664; Fax: (71.3)

792-3995, Email: lihong@laser.mda.uth.tmc.edu.

light beam is normally incident, is thle first one that

is portable to multiple computer platforms. We

also optimize the coordinates of the

scored physi-

cal quantities in each grid element, and allocate ar-

rays and matrices dynamically so that the number

of tissue layers and grid elements can be varied at

run time.

Monte Carlo simulation has been. used to solve

various physical problems besides laser-tissue in-

teractions. However, there is no distinct and well

established definition. We would

like to adopt the

definition by Lux et al. [S]: ‘In all applications of

t.he

Monte Carlo method, a stochastic model is

constructed in which the expected value of a cer-

tain random variable (or of a combination of

SSDI 0169-2607(95)01640-F

132

L. Wang et al. /Computer Methods and Programs in Biomedicine 47 (1995) 131-146

several variables) is equivalent to the value of a

physical quantity to be determined. This expected

,value is then estimated by the average of multiple

independent samples representing the random

,variable introduced above. For the construction of

the series of independent samples, random num-

bers following the distribution of the variable to be

estimated are used.’

Monte Carlo simulations offer a flexible, yet rig-,

orous approach to photon transport in turbid tis-

sues: which can score multiple physical quantities

simultaneously. The method describes local rules,

of photon propagation that are expressed, in the

simplest case, as probability distributions that de-

scribe the step size of photon movement between

sites of photon-tissue interaction, and the angles of

deflection in a photon’s trajectory when a scatter-

ing event occurs. However, the method is statis-

tical in nature and as such, relies on calculating the

propagation of a large number of photons (e.g.

100 000) by the computer. As a result, this method1

requires a large amount of computational time.

The Monte Carlo simulations are based on’

macroscopic optical properties that are assumed to

extend uniformly over small units of tissue

volume. Mean free paths between photon-tissue

interacnon sites typically range from 10-1000 pm,

and 100 pm is a typical value in the visible spec

trum [9]. For example, the simulations do not treat

the details of radiant energy distribution within

cells. The photons are treated as classical particles,

and the polarization and wave phenomenon are

neglected. The current version does not consider

anisotropic media, although the scattering can be

anisotropic.

The Monte Carlo simulations may be used for

both diagnostic and therapeutic applications of

lasers and other optical sources in medicine.

Folr

example, Monte Carlo-simulated diffuse reflec-

tance can be used to deduce optical properties of

tissues, which may be used to differentiate

cancerous tissue from normal tissue. Monte Carlo

simulated optical energy deposition inside tissue

may be used to compute light dosage for photody-

namic therapy of disease.

In the following sections, we will describe the

problem we are trying to solve (Section 2), describe

how to trace photons in tissues (Section 3) and

Photon Beam

Fig. 1. A schematic of the Cartesian coordinate system set up

on multi-layered tissues. The y-axis points outward.

score physical quantities (Section 4), and will pres-

ent a few MCML computational results (Section

5) followed by a section for conclusions (Section

6).

2. The problem and coordinate systems

The Monte Carlo simulation described in this

paper deals with the transport of an infinitely nar-

row photon beam, perpendicularly incident on a

multi-layered tissue (Fig. 1). The responses to the

infinitely narrow photon beam are called impulse

responses. Each layer is infinitely wide, and is

described by the following parameters: the

thickness, the refractive index, the absorption co-

efficient pa (cm-‘), the scattering coefficient p$

(cm-‘), and the anisotropy factor g. The refractive

indices of the ambient medium above the tissue

(e.g. air) and the ambient medium below the tissue

(if existing) need to be given as well. Although the

real tissue can never be infinitely widle, it can be so

treated on the condition that it is much wider than

the spatial extent of the photon distribution. The

tissue layers are parallel to each other.

The absorption coefficient cc, is defined as the

probability of photon absorption per unit in-

finitesimal pathlength, and the scattering coeftic-

ient is defined as the probability of photon

L. Wang et al. /Computer Methods and Programs in Biomedicine 47 (1995) 131-146 133

scattering per unit infinitesimal pathlength. For

the simplicity of notation, the total interaction co-

efficient pt, which is the sum of the absorption co-

efficient pa and the scattering coefficient ps, is

sometimes used. Consequently, the interaction co-

efficient means the probability of photon interac-

tion per unit infinitesimal pathlength. The

anisotropy g is the average of the cosine value of

the deflection angle [lo].

Three coordinate systems are used in the Monte

Carlo simulation at the same time. A Cartesian

coordinate system is used to trace photon move-

ments. The origin of the coordinate system is th.e

photon incident point on the tissue surface; the z-

axis is the normal of the surface pointing toward

the inside of the tissue; and the xy-plane ,is

therefore on the tissue surface (Fig. 1).

Since the infinitely narrow photon beam :is

perpendicular to the tissue surface of a multi-

layered tissue, the problem has cylindrical sym-

metry. Therefore, we set up a cylindrical coor-

dinate system to score internal photon absorption

as a function of r and z, where T and z are the radi-

al and z coordinates of the cylindrical coordinate

system, respectively. The cylindrical coordinate

system and Cartesian coordinate system share thle

origin and z-axis. The r coordinate of the cylindri-

cal coordinate system is also used for the diffuse

reflectance and diffuse transmittance as a function

of r and CY, where CY is the angle between the pho-

ton exiting direction and the normal to the tissue

surfaces (-z axis for reflectance and z axis for

transmittance).

A moving spherical coordinate system, whose z

axis is dynamically aligned with the photon pro-

pagation direction, is used for sampling of the pro-

pagation direction change of a photon packet. In

this spherical coordinate system, the deflection

angle 0 and the azimuthal angle J/ due to scattering

are first sampled. Then, the photon direction is up-

dated :in terms of the directional cosines in the

Cartesian coordinate system (see Section 3.7).

To score physical quantities in the Monte Carlo

simulation, we need to set up grid systems. For

scoring photon absorption, a two-dimensional

homogeneous grid system is set up in the

and z

directions. The grid separations are Ar and AZ in

the r ,and z directions, respectively. The total

numbers of grid elements in the r and z directions

are N, and N,, respectively. For scoring diffuse

reflectance and transmittance, a two-dimensional

homogeneous grid system is set up in the r and (Y

directions. This grid system can share the r direc-

tion with the grid system for photon absorption;

therefore, we need to set up one extra one-

dimensional grid system for the cliff&e reflectance

and transmittance in the (Y direction. The total

number of grid elements is N,. Since we always

chose the range of a! to be 0 I (1 5 I&!, the grid

separation is Aa = ?r/(2 N,).

Photon absorption, fluence, reflectance, and

transmittance are the physical quantities to be

simulated. The simulation propagates photons in

three dimensions, records photon deposition, A(r,

z), in the neighborhood of (r, z), and converts it

into probability of a photon being absorbed per

unit volume (cme3) at the end of tracing multiple

photons. The photon probability fluence (cmM2),

d)(r, z), which is the photon probability flow per

unit area, can be computed through. A(r, z). Dur-

ing the simulation, the absorption or fluence due

to the first photon-tissue interactions are singled

out and scored separately because they always

happen on the z-axis [ll]. The simulation also

records the escape of photons at the top and bot-

tom surface as reflectance, Rkr, a), and transmit-

tance, Ta(r, CX) (cmm2srW1), which is defined as the

probability of a photon escaping per unit area at

r per unit solid angle around CY. Similar to the ab-

sorption, the unscattered reflectancle or transmit-

tance are scored separately. Physical quantities of

lower dimensions can be computed through higher

dimensions. For example, photon absorption as a

function of z, A(z), can be obtained by integrating

A(r, z) over r.

In MCML, for consistency we use centimeter

(cm) as’the basic unit of length throughout the

s’imulation. For example, the thickness of each

layer and the grid separations in the r and z direc-

tions are in cm. The absorption coefficient and

s#cattering coefficient are in cm-‘.

In some of the discussions, the arrays will simply

be referenced by the location of the grid element,

e.g. (r, z) or

(r,

a), rather than by the indices of the

grid element, although the indices a.re used in the

program to reference array elements.

134 L. Wang et al. /Computer Methods

and Programs in Biomedicine 47 (1995) 131-146

3. Simulating photon propagation

This section presents the rules that define pho-

ton propagation in the Monte Carlo model as ap-

plied to multi-layered tissues. Some treatment is

based upon Ref. 2, which deals with a semi-infinite

tissue. ‘We consider multi-layered tissues, and our

approach is general enough to include clear media

(e.g. glass slides or water layers) as a special case.

Fig. 2 shows the basic flowchart for the photon

tracing part of the Monte Carlo calculation, which

has been implemented in ANSI Standard C. Many

boxes in the flowchart are direct implementations

of the following discussions.

3.1.

Sa,mpling

random variables

The Monte Carlo method relies on the random

sampling of variables from probability distribu-

tions. Several books [8, 12, 131 provide good refer-

ence for the principles of Monte Carlo modeling.

For random variable x, there is a probability

density function

p(x)

that defines the distribution

of x over the interval

(a, b).

This variable may be

the variable step size that a photon will take be-

tween photon-tissue interaction sites, or the ang1.e

of deflection that a scattered photon may experi-

ence due to a scattering event.

To simulate propagation, we want to be able to

choose a value for x repeatedly and randomly

based on a pseudo-random number generator pro-

vided by

computer. The computer generates a

random variable, [, which is uniformly distributed

over the interval (0, 1). The generally non-uniform

probability density function

p(x)

can be sampled

by solving the following equation for :y:

[8,7,12-141

(x)dx

= 4 for 5 C W)

(3.11)

In the following sections, Eq. 3.1 will be

repeatedly invoked for sampling propagation var-

iables.

3.2.

Representation of a photon packet

Data structures are an important part of the

program. Logically related parameters are

organized by structures in C such that the program

Launch photon

(dimensionless s = 01 -4--

pT$$

Find distance to boundary db

(

End

Fig. 2. Flowchart for tracing photons in multi-layered tissues

with the Monte Carlo simulation.

is easier to write, read, maintain and modify. The

parameters for a photon packet are grouped into

a. single structure defined by:

L. Wang et al. /Computer Methods and Programs in Biomedicine 47 (1995) 131-146

135

The location of a photon packet described by the

Cartesian coordinates (x, y, z) is represented by

the structure members x, y, Z. The traveling direc-

tion of a photon packet described by the direc-

tional cosines (cc,, pY, cc,) is represented by the

structure members UX, uy, UZ.

A simple variance reduction technique, implicit

photon capture [15], is used to improve the effi-

ciency of the Monte Carlo simulation. This tech-

nique allows one to equivalently propagate many

photons as a packet along a particular pathway

simultaneously. Each photon packet is initially

assigned a weight,

equal to unity. The current

weight of the photon packet is denoted by the

structure member w.

The member dead, initialized to be 0 when the

photon packet is launched, represents the status of

a photon packet. If the photon packet has exited

the tissue or has not survived a Russian roulette

(to be discussed) when its weight is below the

threshold weight, the member dead is set to 1. It is

used to signal the program to stop tracing the cur-

rent photon packet. The type Boolean is not an in-

ternal data type in ANSI Standard C, and is

defined to be type char in the header file of

MCMI,.

The member s is the step size in dimensionless

unit, which is also called ‘optical distance’ antd

defined as the integration of the interaction coef-

ficient CL, over the photon pathway. [16] In a

homogeneous medium, the optical distance is

simply the photon pathlength multiplied by the in-

teraction coefficient. The member scatters is used

to store the number of scatterings experienced by

a photon packet. When photon weight is scored,

the member scatters is used to identify the unscat-

tered reflectance or transmittance, or the first in.-

teractians inside the tissue.

The member layer is the index to the layer where

the photon packet resides. It is defined for compu-

tation e:fficiency, although the layer can always be

identified according to the z coordinate of the pho

ton packet and the geometric structure of the

media. ‘The member layer is updated only when the

photon packet crosses layer interfaces.

3.3. Photon launching

The photon is injected orthogonally onto the tis,-

sue at the origin, which corresponds to a col-

limated infinitely narrow beam of photons. The

photon position (x, y, z) is initialized to (0, 0, 0),

and the directional cosines (CL,, pYcLy, pZ) are set to

(0, 0, 1). The weight is initialized to 1, and several

other structure members including dead, scatters,

and layer in the structure PhotonStruct are also in-

itialized.

When the photon is launched, if there is a

refractive-index-mismatched interface between the

tissue and the ambient medium, then some

specular reflectance will occur. If the refractive in-

dlices of the outside medium and tissue are nl and

n2, respectively, then the specular reflectance,

Rsp,

is specified: [17, 181

= h - n212

(4 + n212

(3.2)

If the first layer is clear medium, which is on top

olf a layer of medium whose refractive index is n3,

multiple reflections and transmissions on the two

bjoundaries of the clear layer are considered. The

specular reflectance is then computed by:

j;,, = rl + (1 -

rJ2r2

1 - rir2

(3.3)

where rl and r2 are the Fresnel reflectances on the

two boundaries of the clear layer:

r, = (4 - n2j2

h + n212

r? = b3 - n212

. (n3 + fi212

(3.4)

(3.5)

The photon weight, initialized to 1, is decreased

R,,

for the photon packet to enter the medium:

W= 1 - R,,

(3.6)

If Eq. 3.2 is used, the structure member layer is

set to the first layer. Otherwise, Eq. 3.3 is used,

and the structure member layer is set to the second

layer. If, for any reason, there are multiple con-

secutive layers of clear media and they cannot be

combined into one layer due to mismatch of

refractive indices, the photon packet will be laun-

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