Modeling of light scattering in biotissue
Different statistical modeling techniques of radiation propagation in epithelial tissue are considered. The two main approaches are: the modified classical Monte Carlo method for light propagation in turbid medium and the coherent inverse ray tracing method. The classical Monte Carlo method was m...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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irk-123456789-1183712017-05-31T03:06:14Z Modeling of light scattering in biotissue Ivashko, P.V. Different statistical modeling techniques of radiation propagation in epithelial tissue are considered. The two main approaches are: the modified classical Monte Carlo method for light propagation in turbid medium and the coherent inverse ray tracing method. The classical Monte Carlo method was modified to take into account polarization of propagating radiation and birefringence, which can occur due to tissues commonly found under epithelium. As a supplementary method, used in the paper was a modified version of the classical ray tracing technique, which takes into account phase of radiation during its propagation in tissue. 2014 Article Modeling of light scattering in biotissue / P.V. Ivashko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 2. — С. 149-154. — Бібліогр.: 17 назв. — англ. 1560-8034 PACS 78.20.Fm, 87.10.Rt, 87.15.ak, 87.16.af http://dspace.nbuv.gov.ua/handle/123456789/118371 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Different statistical modeling techniques of radiation propagation in epithelial
tissue are considered. The two main approaches are: the modified classical Monte Carlo
method for light propagation in turbid medium and the coherent inverse ray tracing
method. The classical Monte Carlo method was modified to take into account
polarization of propagating radiation and birefringence, which can occur due to tissues
commonly found under epithelium. As a supplementary method, used in the paper was a
modified version of the classical ray tracing technique, which takes into account phase of
radiation during its propagation in tissue. |
| format |
Article |
| author |
Ivashko, P.V. |
| spellingShingle |
Ivashko, P.V. Modeling of light scattering in biotissue Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Ivashko, P.V. |
| author_sort |
Ivashko, P.V. |
| title |
Modeling of light scattering in biotissue |
| title_short |
Modeling of light scattering in biotissue |
| title_full |
Modeling of light scattering in biotissue |
| title_fullStr |
Modeling of light scattering in biotissue |
| title_full_unstemmed |
Modeling of light scattering in biotissue |
| title_sort |
modeling of light scattering in biotissue |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118371 |
| citation_txt |
Modeling of light scattering in biotissue / P.V. Ivashko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 2. — С. 149-154. — Бібліогр.: 17 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT ivashkopv modelingoflightscatteringinbiotissue |
| first_indexed |
2025-07-08T13:51:31Z |
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2025-07-08T13:51:31Z |
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1837087005150806016 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 149-154.
PACS 78.20.Fm, 87.10.Rt, 87.15.ak, 87.16.af
Modeling of light scattering in biotissue
P.V. Ivashko
Chernivtsi National University,
2, Kotsyubinsky str., 58012 Chernivtsi, Ukraine
E-mail: paul.ivashko@gmail.com
Abstract. Different statistical modeling techniques of radiation propagation in epithelial
tissue are considered. The two main approaches are: the modified classical Monte Carlo
method for light propagation in turbid medium and the coherent inverse ray tracing
method. The classical Monte Carlo method was modified to take into account
polarization of propagating radiation and birefringence, which can occur due to tissues
commonly found under epithelium. As a supplementary method, used in the paper was a
modified version of the classical ray tracing technique, which takes into account phase of
radiation during its propagation in tissue.
Keywords: Monte Carlo method, simulation, biological tissue, epithelial tissue,
polarization, birefringence.
Manuscript received 31.01.14; revised version received 24.04.14; accepted for
publication 12.06.14; published online 30.06.14.
1. Introduction
One of approaches for solving the problem of light
propagation in scattering media is the method of Monte
Carlo statistical simulation [1-5]. It is a set of techniques
that enables to find necessary solutions by repetitive
random sampling. Estimates of the unknown quantities
are performed with statistical means.
For the case of radiation transfer in scattering
medium, the Monte Carlo method implies repeated
calculations of the photon trajectory in medium based on
determining environmental parameters. Application of
the Monte Carlo method is based on the use of
macroscopic optical properties of the medium that is
considered to be homogeneous within small volumes of
tissue. The models based on this method can be divided
into two types: the models that take into account
polarization of radiation, and the ones that ignore it.
Simulation based on the previous models usually
discards the details of the radiation energy distribution
within a single scattering particle. This disadvantage can
be ruled out (in the case of scattering particles’ size
exceeding the wavelength) by using another method –
reverse ray tracing. This method, similar to that mentioned
above, is based on passing a large number of photons
through medium that is simulated. The difference is that
each scattering particle has a certain geometric topology,
and scattering is calculated using the Fresnel equations.
The disadvantage of this method is that it can give reliable
results only if the particle size is much greater than the
wavelength (at least by an order of magnitude).
2. Simulation of polarized radiation propagation in
scattering medium by using the Monte Carlo method
The Monte Carlo method is used for implementing the
stochastic model, in which the expected value of the
random variable is equivalent to the value of the
physical quantity that must be determined. This expected
value is determined by averaging many independent
samples of the random variable. To build a series of
independent samples, random numbers are taken from
the probability distribution corresponding to the
distribution of the physical quantity.
Radiation is represented by a set of photons. Each
photon propagating inside medium has a certain amount
of energy, position, direction and polarization [6-9].
When modeling propagation of photons in environment,
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
149
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 149-154.
the following assumption is made: events of radiation
scattering by the particles are independent; distributed
environment, in which radiation appears, is one-axis
material with linear birefringence; dispersion in this
environment occurs on spherical particles that are
randomly distributed inside it.
During simulation, two coordinate systems are
used: the laboratory coordinate system that is bound to
the environment of propagation and the local coordinate
system that is bound to photon and changes after each
scattering event.
The geometry of the model is shown in Fig. 1.
Radiation in the form of a thin beam penetrates into
the medium towards the positive direction of the Z axis
(laboratory coordinate system). The slow linear
birefringence axis is directed along the axis X. In the
environment, photons are scattered by spherical
particles. This scattering is described by the Mie theory.
The detector is located in the XY plane of the laboratory
coordinate system and shown in the figure in grey color.
It captures only photons coming out from the
environment.
The whole algorithm is shown in Fig. 2.
The first step in modeling is to launch one of
several million photons in the medium. The reference
plane of polarization is chosen (vectors u and v of the
local coordinate system), and the Stokes vector
describing the polarization state of the incident beam is
selected [1].
Two single vectors are used to set the reference
plane: v = [0, 1, 0] and u = [0, 0, 1]. Vector u indicates
the initial direction of propagation of photon. For any
type of polarization, the incident beam is given by the
Stokes vector S = [IQUV], which plane of reference was
defined above.
Fig. 1. Geometry of the model.
Fig. 2. Block diagram of the algorithm.
Photon moves a distance ΔS that is calculated on
the basis of the random number ζ that lies within the
range (0, 1].
( )
t
S
μ
ς
−=Δ
ln . (1)
Here, sat μ+μ=μ is the interaction coefficient,
aμ – absorption coefficient, – scattering coefficient. sμ
The photon trajectory is characterized by directing
cosines [ ]zyx uuu ,, . The photon position is changed
from [ ]zyx ′′′ ,, to [ ]zyx ,, as follows:
.
,
,
Suzz
Suyy
Suxx
z
y
x
Δ+=′
Δ+=′
Δ+=′
. (2)
The birefringence effect is taken into account by
using a linear delay represented by the matrix:
( )
( )
( )
( ) ( ) ( ) ⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
ΔΔ−Δ
Δ⎟
⎠
⎞
⎜
⎝
⎛ Δ+⎟
⎠
⎞
⎜
⎝
⎛ Δ−⎟
⎠
⎞
⎜
⎝
⎛ Δ
Δ−⎟
⎠
⎞
⎜
⎝
⎛ Δ
⎟
⎠
⎞
⎜
⎝
⎛ Δ+⎟
⎠
⎞
⎜
⎝
⎛ Δ
=βΔ
cossinsin0
sin
2
cos
2
sin
2
sin0
sin
2
sin
2
cos
2
sin0
0001
,
22
2
22
4
2
4
2
2
4
22
4
CS
CCS
SSC
T , (3)
( )β= 2cos2C , , , ( )β= 2sin2S ( )β= 4sin4S
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
150
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 149-154.
where β is the azimuthal angle of the slow axis of
birefringence in the XY plane local coordinate system of
photon, Δ – delay that can be obtained as follows:
λ′
π
Δ=Δ
Sn 2 . (4)
Here Δn is the difference between the maximum
and minimum refractive indexes in the plane
perpendicular to the direction of photon, S – path that
went photon (react of scattering), – length of light
wave in the medium sample.
λ′
If you know the angle between the slow axis of
birefringence and direction of motion of photon, it can
be calculated by the formula:
( ) ( ) f
fs
fs n
nn
nn
n −
α+α
=Δ
22 sincos
, (5)
where ns and nf are the refractive indexes along the slow
and fast axes of birefringence, which satisfy the relation:
δ+= fs nn , (6)
where δ is the value of linear birefringence.
The Stokes vector is multiplied by matrix before
each act of scattering. Absorption of radiation in the
environment is caused by the weight of photon W, which
is adjusted after each event of absorption in accordance
with the albedo of the medium.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
as
s
μ+μ
μ
=albedo . (7)
Albedo is partial probability of photon scattering.
The initial weight of photon is equal to 1. After the n-th
event of absorption, the weight is equal to ( )nalbedo .
When the photon weight decreases to critical levels, it is
completely absorbed. If photon comes from with a
certain weight W, the corresponding Stokes vector is
multiplied by W to account for photon attenuation.
WB
WU
WQ
WI
B
U
Q
I
⋅
⋅
⋅
⋅
= . (8)
Step 2. One of the most important steps in the
process of modeling is choosing the angles α (scattering
angle) and β (angle in the plane of scattering) (Fig. 3).
Choosing these angles is based on the phase
function. For polarized radiation , it
looks like:
[ ]00000 ,,, VUQIS =
( ) ( ) ( ) ( )
( ) ( ) .2sin
2cos,
02
0201
βα+
+βα+α=βαθ
Um
QmIm
(9)
Where and ( )α1m ( )α2m are the elements of the
scattering matrix:
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
αα−
αα
αα
αα
=α
34
43
12
21
00
00
00
00
mm
mm
mm
mm
M . (10)
The matrix determines the polarization properties
of the scattering element. In the case of spherical
scattering, the particles scattering matrix is symmetrical.
The elements ( )α1m , ( )α2m , and ( )α3m ( )α4m are
related to the scattering amplitudes:
( ) ( )2
1
2
21 2
1 SSm +=α ,
( ) ( )2
1
2
22 2
1 SSm −=α ,
( ) ( )∗∗ +=α 12123 2
1 SSSSm ,
( ) ( )∗∗ −=α 12214 2
1 SSSSm . (11)
We introduce some notations:
• particle size rkx = , where r – radius of the sphere,
λ
π
=
2k ;
• complex index of refraction ; km −ν=
( ) ( )( nnnn
n
ba
nn
nxmS τ+π
+
+
=θ ∑
∞
=1
1 1
12,, ) ,
( ) ( )( nnnn
n
ab
nn
nxmS τ+π
+
+
=θ ∑
∞
=1
2 1
12,, ) . (12)
Coefficients nπ and nτ are functions of θ=μ cos :
( ) ( ) ( )θπ
−
−θπ
−
−
θ=θπ −− 21 1
1
1
12cos nnn nn
n ,
( ) ( ) ( )[ ]
( ) ( ) ( ,sin12
cos
21
2
2
θτ+θπθ−−
−θπ−θπθ=θτ
−−
−
nn
nnn
n )
π≤θ≤0 , (13)
where
( ) 00 =θπ , ( ) 00 =θτ ,
( ) 11 =θπ , ( ) θ=θτ cos1 ,
( ) θ=θπ cos32 , ( ) θ=θτ 2cos32 . (14)
Fig. 3. Directions of photon scattering before and after
interaction.
151
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 149-154.
In the cases of and , these coefficients
are reduced to the following form:
0=θ π=θ
( ) ( ) ( )
2
100 +
=τ=π
nn
nn ,
( ) ( ) ( ) ( )
2
11 +
−=πτ=ππ−
nnn
nn . (15)
Coefficients an and bn are functions of m and x:
( )
( ) ( ){ } ({ }
( )
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
( ) ( )xwxw
x
n
m
yA
xwxw
x
n
m
yA
xma
nn
n
nn
n
n
1
1ReRe
,
−
−
−⎟
⎠
⎞
⎜
⎝
⎛ +
−⎟
⎠
⎞
⎜
⎝
⎛ +
= ,
( )
( ) ( ){ } ({ }
( )
)
( ) ( )xwxw
x
nyAm
xwxw
x
nyAm
xmb
nnn
nnn
n
1
1ReRe
,
−
−
−⎟
⎠
⎞
⎜
⎝
⎛ +
−⎟
⎠
⎞
⎜
⎝
⎛ +
= , (16)
where
( ) ( ) ( )xwxw
x
nxw nnx 21
12
−− −
−
= ,
( ) ( ) ( )xixxw cossin0 −= ,
( ) ( ) ( )xixxw sincos1 −=− . (17)
The expression for looks like: ( )yAn
( ) ( )
1
1
−
− ⎥
⎦
⎤
⎢
⎣
⎡
−+−= yA
y
n
y
nyA nn ,
( ) ( ) ( ) ( ) (
( ) ( )
)
qp
qqippyA 220 sinhsin
coshsinhcossin
+
+
= . (18)
The summation in expression (8) is reduced to
xn ≈ .
Step 3. We calculate the Stokes vector for the back
scattering at the angle ψ to its reference plane coinciding
with the plane of scattering. This is achieved by rotating
the matrix
( ) ( ) ( )
( ) ( )
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
ββ−
ββ
=β
1000
02cos2sin0
02sin2cos0
0001
R . (19)
Correction of the Stokes vector is as follows:
( ) ( )SRMS βα= . (20)
3. Simulation results for different values
of birefringence
In simulation (Figs 4-6), the following system
parameters [10-14] were used: the refractive index of the
scattering particle – 1.57, the refractive index of the
medium around the particles – 1.33, the diameter of the
scattering particles – 0.35 μm, the wavelength equal to
0.6328, absorption coefficient – 1 cm3, scattering
coefficient – 90 cm3, the thickness of the material along
the Z-axis – 0.2 cm, the number of photons – 1000000.
Fig. 4. The slow axis of birefringence of 1.331, fast axis 1.33.
Fig. 5. The slow axis of birefringence 10,331, fast axis 1.33.
Fig. 6. Distribution of the intensity of absorbed radiation in the
environment.
152
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 149-154.
4. Simulation of coherent radiation propagation
by using the reverse ray tracing method
The reverse ray tracing method is based on modeling the
passage of light beams (photon packets) from the
detector to the source (i.e. in the opposite direction)
through the scattering medium [15-17] that acts as a set
of polygonal objects. The goal of this simulation is to
obtain an interference pattern on the detector surface
(Figs 7 and 8). This pattern is formed by a set of points,
which are selected randomly on the surface of the
detector with normal probability distribution. At each
this point, the intensity of light is determined as a result
of interference of all rays that intersect in it and reach
the surface of the source. The direction of each ray is
randomly chosen on the surface of the unit sphere. Beam
moving in the environment interacts with the objects
placed in it. After collision with an object, we have two
types of new rays – the reflected ray and refracted one.
The number of necessary rays is largely determined by
the geometrical configuration of the system as well as
the absorption and scattering coefficients of environment
and objects in it. The redistribution of energy between
the reflected and refracted rays is calculated using the
Fresnel formulas:
( ) ( )
( ) ( )
2
21
21
coscos
coscos
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
θ+θ
θ−θ
=
ti
ti
s nn
nnR ,
( ) ( )
( ) ( )
2
21
21
coscos
coscos
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
θ+θ
θ−θ
=
it
it
p nn
nn
R ,
ss RT −=1 ,
pp RT −=1 . (21)
Here, R and T are the reflection and transmission
coefficients, respectively.
The energy changes associated with each photon
beam also occur because of environment absorption
around the objects and inside them. We discard the rays,
contribution of which to the interference pattern at the
detector surface is negligibly small and the rays that do
not reach the source. For each beam that has passed from
the detector to the source, the optical path is calculated.
This is needed to calculate the phase during interference
summation of all the rays.
The advantage of this method is the possibility of
modeling environments that contain objects of arbitrary
topology and the possibility of free positioning the plane
detector in space. Another feature of this design is that
the accuracy of the interference pattern is governed by
the number of rays that are traced from each point on the
detector and the density of registration dots.
Disadvantages include the fact that the size of the
objects should greatly exceed the wavelength (at least by
an order of magnitude) and the considerable time is
required for ray tracing.
Fig. 7. The interference pattern (radiation passing through a
series of cylindrical objects).
Fig. 8. The interference pattern (radiation passing through a
matrix of spherical objects).
The time required for simulation can be reduced by
distributing calculations on multiple data centers, which
can be done because the algorithm allows parallel
computations.
5. Conclusions
Thus, we have implemented two statistical modeling
techniques for radiation propagation in epithelial tissue.
The modeling results are in good agreement with
experimental data. But both models have some
limitations. Monte Carlo simulation is based on the Mie
theory for spherical particles and doesn’t handle
scattering particles of arbitrary geometry, which are
commonly found inside cells (even cell nuclei often have
elliptical form). The ray tracing method works for
scattering particles of arbitrary form, but their size must
be significantly greater than radiation wavelength, and,
since tracing is based on the Fresnel equations, this
model doesn’t take into account such phenomena as
dispersion at all.
Acknowledgements
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
This work was supported by grants F53/103-2013 from
the Fundamental Researches State Fund of Ukraine.
153
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 149-154.
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