Indi an Journal of Biochemistry & Biophysics Vol. 36, October 1999, pp. 330-336

Monte Carlo simulation of photon scattering in biological tissue models

o Kumar, Susamma Chacko and Megha Singh*

Biomedical Engineering Division, Indian Institute of Technology, Madras 600036 , India

Monte Carlo simulation of photon scatteri ng, with and without abnormal tissue placed at variolls locatio ns in the rectangular, semi-circular and semi-elliptical tissue mQdels, has been carri ed out. The absorption coeffici ent of the ti ssue considered as abnormal is high and its scattering coefficient low compared to that of the control tissue. The placement of the abnormality at various locations within the models affects the trans miss ion and surface emission of photons at various locations. The scattered photons originating from deeper layc rs makc the maximum contribution at farther distances from the beam entry point. The contribution of various laye rs to photon scattering provides valuable data on variability of internal composition.Introduction

Introduction Optical techniques are attractive to probe internal

structures of tissues as these are non-invasive, utilise non-ionizing radiat ion, possess unique molecular contrast mechanisms, and are often technologically simple. A thorough understanding of both the therapeutic and diagnostic applications of laser light demands an accurate description of the transport of light in biological tissues. It has been demonstrated that Monte Carlo method can be used to provide a rea listic model of laser penetration in complex stmctures l

. This method is applied to the photon scattering in tissue-simulating models2 and it is observed that there is a distinct discontinuity in the surface emission profile of a two-layer modeI3.

4.

The normal and the malignant tissues in patients are clearly resolved by differences in optical and physiological properties). Optical path-length as measured in various models of the adult head by time­of-flight measurements agrees well with predictions from e ither Monte Carlo mode l or a finite-element method based on diffusion theory or a hybrid radiosity diffusion-theor/. Similarly by Monte Carlo simulation the photon migration in human breast tissues, bovine li ver and tissue models under ill vitro conditions have been analysed and the results were in good agreement with experimental data7

.

These observations show that the photon scattering process within the tissues and detection of backscattered photons at the surface provide the information on tissue structures theoretically" 8 as well as experi mentall y9. Recently, by our multi-layer

* AUl hor ror correspondence

imaging technique, we have further shown that the images of the objects embedded at deeper layers in models are reconstmcted from the photons received at a distance away from the laser beam entry point. By the same procedure. the images of var iolls tissue layers of human organs are also obtai ned 10. These backscattered photons may originate from a s ingl e layer or multi-layers of the ti ssue mode ls contributin g to the measured component at the surface. To au thors knowledge the details of this distribu tions are not known. Hence the objective of the present work is to apply Monte Carlo simulation to ana lyse the photon scattering process in tissue models of d ifferent size and shapes, with and without abnormalities embedded at various locations .

Materials and Methods

Tissue models For simulation purpose, rectangular, semi-c ircular

and semi-elliptical tissue models of he ight 1.0, 1.0 and 0.8 cm (referred to as control) , respective ly, were considered. The co-ordinates of the centres of the base of these model s were (0, 0) . Another ti ssue of hi gh

absorption coeffic ient of size 2mmx I mm (hereafter referred to as abnormality) was placed within the models at locations (0 .0, 0.9) (centred at y-ax is), (0. 1, 0.9) and (0.2, 0.8) for slab and circular models , whereas , for e lliptica l model the abnormality was located at (0 .0, 0 .7), (0 . 1, 0.7)' and (0.2, 0.6). These models , irrespective of their shape, with abnormality at three locations are referred to as mode ls I, 2 and 3, respective ly . The absorpti on and scatre ring coeffic ients of control (tissue I) and abnormality (tissue 2) were 3.358 and 9 .767 c m-I

, and 52.7 and

KUMAR e/ al.: MONTE CARLO SIMULATION OF PHOTON SCATTERING 331

41 cm- I, respectivel/ I. The normal incidepce of the

beam of 200, 000 photons considered at locations (0.0, 1.0), (0.0, 1.0) and (0.0, 0.8) for rectangular, semi-circular and semi-elliptical models, respectively. A schematic of the semi-elliptical geometry with grid used for simulation purpose is shown in Fig. I. Monte Carlo simulation technique

Monte Carlo simulation was performed by tracing individual photon histories l2

• A random number generator was used to sample discrete events from probability distributions derived from the interaction coefficients. The initial value of scattering angle e = 0, corresponded to M = cos e = I .

Assumptions made for the present simulation were: (1) the photons propagating through the tissue were neutral particles; (2) the tissue and abnormality were homogeneous; (3) monochromatic beam of photons was injected; (4) the scattering that takes place was purely isotropic.

In biological tissues (2) and (4) were not actually encountered. However, this approximation was acceptable as the significant contribution to measurement is from only a few mm around the source.

Steps involved in the simulation procedure were: (i) Photon-source generation: A stream of photons

of energy E and wavelength A were generated to represent a light beam which incidents normally on the tissue plane.

(ii) Pathway calculation: On the basis of the scattering and absorption characteristics of the medium and random distribution of scattering events, pathways were calculated.

y

+ G'

)(...- -- I I J I i J J I I

(iii) Following conditions were considered to calculate transmitted , absorbed and backscattered fractions : (a) If y < 0, the photon was transmitted. (b) If (y > I) for rectangular slab, and (y > rl" = ;j(a2

_Xl )

for semi-circle and (y > re = ;j(l_(x2/a1)b for semi­elliptical geometry, the photon was reflected, where a was radius for semi-circle and a and b were semi­major and semi-minor axes for semi-elliptical model, respectively. (c) If position (x, y) lies inside the model this leads to two conditions: Either the photon was absorbed if r2< I..I...I(/la+f..l.s) or was scattered. The new direction of the photon was to be determined by a third random number r3 gi ven by cos e = 2r3-1 where e was the scattering angle, /..Ia and /..Is were absorption and scattering coefficients of the tissue I .

After abnormality (t issue 2) was introduced, the path h!ngth within this was defined by its absorption

(/..I~ ) and scattering (/..I:f) coefficients. The following

conditions were considered for path length calculations: At the boundary due to change of refractive index either reflection or refraction took place. For this purpose another random number was generated which yielded the new position . Similar to the tissue model the photon underwent reflection at the surface lor transmission through the width of abnormality or undergoing absorption process after single or multiple interactions. The c~lculations were carried out similar to that of control tissues. Further details of implementation of this are shown in the flow chart ( Fig. 2). After transmission through tissue 2, the path length was calculated by the procedure as given in (i) to (iii).

r-C d . I"-- •

a b

,f ,,9

'\.h r\. \' [\ \ J j - +x

332 INDIAN J. BIOCHEM . BIOPHYS., VOL. 36, OCTOBER 1999

YES

Calculate

Start .. Input data No, ila, ils' h=\/O.8, R[n)=O,va\ue

Initialise X=0,Y=1I0.8,M= 1.0, n=O, \abe\[n]=O

I Ube~[n}: 1 1

Ilo:vru:.n=n+l\

J

Fig. 2-Flow chart of Monte Carl o simul ation of distribut ion of photons in ti ssue models o f di ffercnt ~eo l1letric, .

KUMAR el al.: MONTE CARLO SIMULATION OF PHOTON SCATTERI G

Determination of depth alld sur.face profiles

The transmitted fracti ons after their passage through successive locati ons along the depth (referred to as 'Iabel[n]') at 0.0, 0.4, 0 .8, 1.2 mm, etc., were determined . Similarly the backscattered (referred to as surface reflectance, R) and absorbed fractions with in the medium were determined.

The backscattered photons were emitted from various locations at the sUlface. As the depth increases, the transmitted fraction decreases . At each depth location the number of photons passing through this were determined and are presented as the transmitted fraction s. From the backscattered fraction s at various locations at the surface, originating from each depth region, the surface profiles for various models were constructed. The surface locations vary, depending on the geometry of the model. Due to curved surfaces of the semi-circular and semi-elliptical geometries there is a variation in the surface locations, as shown in Table I.

Monte Carlo simulation for each model was repeated ten times. The statistical variation of the data was calculated by student t-test.

Results

The photon scattering within the tissue medium depends on its optical parameters. The contributions of various locations in the control and models 1-3, on the tissue surface, vary. Due to curved nature of the semi-circular and semi-elliptical models the scattering within the tissues undergoes variation compared to that of rectangular slab model. Fig. 3 shows the backscattering pattern in semi-elliptical control and

Table I-Surface co-ordinates for measurement of backscattered photons for various models

Coordinates

X (em) Y (em)

Rectangular slab Semi-circular Semi-elliptical

0.0 1.0 1.0000 0.8000

0.1 1.0 0.9950 0.7960

0.2 1.0 0.9798 0.7838

0.3 1.0 0.9539 0.7632

0.4 1.0 0.9 165 0.7332

0.5 1.0 0.8660 0.7000

0.6 1.0 0.8000 0.6400

0.7 1.0 0.7141 0.5713

0.8 1.0 0.6000 0.4800

0.9 1.0 0.4359 0.3487

1.0 1.0 0.0000 0.0000

model s 1-3 . The variation in control tissue mode l is shown in Fig. 3a. At every region on the surface the total backscattered fraction is the resu lt of the contribution from various deeper layers which decreases with the increase in depth . For examp le at region I the contribution is from the top layer which decreases with the increase of depth. At layers farther away from the entry point of photons the upper layers do not make major contribution . For example at region 7 the max imum contributi on is from the depth 2.4 mm and nearby layers a lso contribute to the total backscattered fraction . With the increase in surface distance, deeper layers further contribute to the backscattered fraction s. For control model, the contribution is limited to a depth 3.2 mm. In model I due to placement of abnormality in the path of the beam this pattern is vari ed (Fig. 3b). There is a significant change (p < 0 .0005) due to abnormality in surface backscattering, within the region 2SxS4. Beyond this the effect of the abnormality on the surface backscattering is not observed. The contribution of the deeper regions below the abnormality to distant surface locations, is not affected.

The shifting of the abnormality to off-axis position further changes the backscattered pattern in model 2 (Fig. 3c) . This placement of the abnormality does not affect the beam which passes directly but alters the pattern of scattering of photons in the nearby regions. These photons have already been undergoing scattering originating from the entry point to a depth of 0 .07 cm. The contribution of region located at depths 7 .2SyS6.8 in presence of abnormality is significantly higher (p < 0.0(05) than that of control model, whereas, an opposite trend has been observed from the next layer at 6.8SyS6.4. But in terms of surface backscattering profile its effect at location beyond 3SxS4 is observed . By moving the abnormality to a deeper location (0 .2, 0.6 cm) the pattern further changes due to off-axis location of abnormality (Fig. 3d). The backscattered fraction is the maximum at region 5SxS6 and is higher than that of the control sample (p<0.0005). Thereafter the contribution of deeper layer is less than that of the controi which may be attributed to the increased absorption coefficient of the abnormality. The shift of abnormality to deeper layer shift the maximum of the surface backscattering away from the entry point.

The variation in surface backscattered frac tion at various locations from the entry point of the beam

334 INDI AN 1. BIOCHEM. BIOPHYS., VOL. 36, OCTOBER 1999

._---(a) (c)

10':

1 :

.1 dP

01 . 01 s::

.~ '.-1 . 001 ·

~

Q) \ .;.l .am .;.l (d ) .

cu 10 tJ

U)

1 ~

tJ

cu .1 ell

.01

.em

.CXXJ1 -0.0 1.0

4. 0

D e p t h mm )

Fig. 3-Backscattered photon distri bution at various locations on the ti ssue surface at nine locati ons (given in cm ) con tributed hy photons originating from different depths of a semi-elliptical ti ssue model in the absence (a) and presence of abnormalit y located at (O.O. 0.7) (b), (0.1,0.7) (c) and (0.2, 0.6) (d). The surface regions 1-9 covers the totol bac kscattered photons between 0-1 , 1- 2. 2-3. 3-4. 4-5. 5-6, 6-7 , 7-8 and 8-9 mm, respectively.

depends on the type of geometry used . Table 2 shows this variation at various distances (given as I, 3, etc. are in fact the regions between 0 to I, 2 to 3, etc.) along the surface for rectangular slab models. There is no significant variation between surface pro fi les a. observed for control and models I and 3. At some locations in model 2, a significant variation compared to the contro l i~ observed thus indicating that the presence of abnormality at a particu lar location affec ts the backscattering fraction.

Similar vari ation along the surface of circular model (b) is also shown (Table 2) . The backscat terecJ fractions at various locations show similar pattern as that for rectangular slab bu t are higher upto 5 m Ill

(region between 4 and 5 mm ) followed by a decrease

with further increase of sUl{ace distance. A comparison between control and di ffere nt models show a significant variat ion between con trol and model 2.

A si mjlar vanatIOn in bac kscatterf'cJ fraction in elliptical models (c) is also observed (Table 2). Thi s is compclrabJe with that of ci rcular mode ls but hi gher than that o f rectangular model. For these models , thi s fract ion is comparable between control and models I and 3 but varies significan tly between control and model 2. From these observations it is e Vl cJ nt that irrespecti ve of the type of mode l, lhe surface backscattered fraction is inflllenc d by th t' location of the abnormality .

KUMAR e/ al.: MONTE CARLO SIMULATION OF PHOTON SCATTERING 335

Table 2-Surface backscattering(%) as measured at various distances from entry point for control and different models

Distance ( mm ) Control Model I Modell! Model III

a. Slab

I 40.603±O.120 40.654±0.094 40.626±O.120 40.634±0. I 10

3 2.083±O.O20 2.069±O.025 2.073±O.014 2.073tO.024

5 0.407±O.009 0.396±O.008 0.387±O.0 14h O.403±O.0(}<)

7 0.103±0.007 0. 10O±O.006 0.092±O.OO5< 0. I02±O.OO9

9 0.029±O.002 0.030±0.OO4 0.025±O.O02~ O.O29±O.O03

b. Semi-circular

I 42.926±0.076 42.9\\±O.064 42.9l9±O.064 42.92 1±0.O80

3 3.728±O.O68 3.792±O.068 4.082±O.O67" 3.727±O.O07

5 0.898±O.0 13 0.884±O.014 0.7 12±O.018h O.9 1I ±O.0 16

7 0.092tO.007 0.089±O.007 0.039±O.004" O.08JtO.OOT

9 0.00 I±O.OO I 0.00 I±O.OO I 0.000 0.00 I ±O.OO I

c. Semi-elliptical

42.445±O.OO8 42.455±O.O71 42.466±O.O95 42.437±O.O80

3 3.42I±O.065 3.435±O.061 3.502±O.049" 3.423±O.O63

5 0.903±0.021 0.888±O.O23 0.780±0.0 16c O.902±O.O2 1

7 0.164±0.008 0.158±O.0 10 0.090±0.005" 0. 157±O.OOX

9 0.006±0.00 I 0.006±O.00 1 0.002±O.OO I C 0.004±0.OO I

p-values: "< 0.01; h< 0.005; "< 0.0025;"< 0.00 1; c< 0.0005 ;

Discussion

Monte Carlo simulation has been used to relate the model depth and optical properties of tissues with the backscattering fraction on the surface . This study has in troduced the concept of photon nux paths to describe in an averaged statistical sense the escape of photons following their propagation through a medium. Thus the optically differentiable interna l structure can be assessed by photon nux measurements on the surface of the medium.

A va:'iati c n in differential path length factor (OPF) (relating a change in a measured attenuation in optical densi ty to a change in absorption ) with absorption and scattering is observed whi ch increases with increased scattering and decreases with increased absorption l J

. Our present simulation shows the higher transmitted fraction fo r hi gher scattering and lower absorption coefficients than the lower scattering and higher absorption coeffic ients.

The present observation shows that the deeply penetrated photons emit at farther regions from the beam entry point. To co llect photons scattered from deeper layers, the detectors are placed at farther distances from the beam entry point 14. 15 . The study of the relat ionshi p of surface reflectance measurements to optical properties of layereJ biological media

suggests that surface measurements are sensiti ve to optical properties of deeper tissues and the measurements at larger source detec tor separation tend to be independent of upper layer effects J().

This study also shows tha t the insertion of abnormality leads to changes in the surfJce emiss ion profile at farther regions as we ll as in the fracti on fo r deeper penetration . One of the parameters of the measured intensity profile, the full wiJ th half

maxi mum (FWHM) reached a maximum for IJ.s =

15 cm·1 and decreased s light ly for highe r Il ~. In fact .

it followed fundamentally from the form of the Green's function of the di ffus ion equati on whi ch

implied that the penetrati on depth increases when IJ., is increased. An opposite effect takes place wi th an insertion of ti ssue of low scat tering coeffi c ient as observed in our present observat ions. Thes,~ findings are also in agreement with that as obtained by Monte Carlo simu lation in a turbid rnedium J<,.

The surface backscattered fracti on depends on the geometry of the mode l and dec:reases with the increase of surface distance from the beam entry poi nt. These observations ,1re in agreement with that as observed by Bon ner (' I ([/ 17. For all these mode ls the photons re-emitted cl ose \0 the ir entry point are

336 INDIAN J. BIOCHEM . BIOPHYS. , VOL. 36, OCTOBER I<)<)l)

mostly from the upper layer, whereas, those re­emitted far from the entry point are from the deeper ti ssue beneath the top layer, depending on the penetration depth . Distinguishing features appear only in the regions of the emission profiles, depending on the location of the abnormality at a particular depth 10.

The overall effects of the absorption by the bottom layers on the emission profile appear at larger distances from the entry point, and that of the top layers close to the photon injection point' . Our present observation supports these findings and are further experimentally. verified by photon scattering studies by locating the abnormalities at various depths in a model lo

.

References Maarek J M, Gilbert J, De Cosnac B B, Lansiart A & Hung B M (1984) Ann. Biomed. Eng. 12, 281-303

2 Wilson B C & Adam G (1983) Med. Phys. 10,824 - 830 3 Nossal R, Kieter 1, Weiss G H, Bonner R, Taitelbaum H &

Havlin S (1988) Appl. Optics 27, 3382 - 3390 4 Taitelbaum H, Havlin S & Weiss G H (1989) Appl. Oplics 28,

2245-2249

5 Fishkin J B, Coquoz 0 , Anucrson E R. BrcnlH.: r M & Tromberg B J ( 1997) Appl O/Jlics 36, 10-20

6 Okada E, Firbank M, Schweige r M. Arridf!c S R, Corc M & Delpy D T ( 1997) ApI'1 Optics 36, 2 1-31

7 Bevilacqua F, Marquet P, Coquoz 0 & Dcpeursin ge C ( 19Sl7) Appl. Optics 36 , 44- 51

8 Wilson B C & Jacques S L ( 1990 ) IEEE J. Quont. Electronics 26,2186-2199.

9 Shanthi S & Singh M ( 1997) Met!. Bioi. Eng. CO/llPll t. 35 , 253-258

10 Chacko S & Singh M ( 1999) Med. Bioi. Eng. COInpIII. 37 , 278-284

II Singh M & Chacko S ( 1997 ) CIIlTent ScienCl' 73, 101 5 - 10 19

12 Sobol 1 M (1 974) Th e Monte Carlo Melhod pp. 45-51 Uni v. of Chicago Press., Chicago

13 Delpy D T, Cope M, Pvan del' Zee, Arridge S, Wray S & Wyatt J (1988) Phys. Med. Bioi. 33, 1433-1 442

14 Schmitt J M, Zhou G X, Walker E C & Wall R T ( 1990 ) 1. 01'1. Soc. Am. A 7, 2141- 2153

15 Cui W & Ostrander L E (1992) IEEE Trans . Bioll1ed. Eng 39 , 194-201

16 GraafR, Koelink M H, De Mul F F M, Zijl stra W G & Dassel A C M & Aarnoudse J G (1993) AWl. Oplics 32, 426-434

17 Bonner R F, Nossal R, Havlin S & Weiss G 1-1 (1987 ) .1.01'1. Soc.Am. A 4, 423-433