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Modified discrete particle model of
optical scattering in skin tissue
accounting for multiparticle scattering
Dirk H. P. Schneiderheinze, Timothy R. Hillman,
and David D. Sampson
Optical+Biomedical Engineering Laboratory
School of Electrical, Electronic and Computer Engineering
The University of Western Australia
35 Stirling Hwy, Crawley, Western Australia, 6009, Australia
[email protected]
http://obel.ee.uwa.edu.au
Abstract:
We rigorously account for the effects of multiparticle light
scattering from a fractal sphere aggregate in order to simulate the optical
properties of a soft biological tissue, human skin. Using a computational
method that extends Mie theory to the multisphere case, we show that multiparticle scattering significantly affects the computed optical properties,
resulting in a reduction in both scattering coefficient and anisotropy for the
wavelengths simulated, as well as a significantly enhanced forward peak
in the simulated phase function. The model is extended to incorporate the
contribution of Rayleigh scatterers, which we show is required to obtain
reasonable agreement with experimentally measured optical properties of
skin tissue.
© 2007 Optical Society of America
OCIS codes: (170.3660) Light propagation in tissues; (170.7050) Turbid media; (290.4020)
Mie theory; (290.4210) Multiple scattering; (290.5825) Scattering theory; (290.5850) Scattering, particles; (290.5870) Scattering, Rayleigh.
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1. Introduction
The interaction between electromagnetic waves and tissue is important for the development
of diagnostic and therapeutic applications of light in medicine. For example, scattering of
light within tissue simultaneously makes possible and limits high-resolution imaging techniques such as reflectance-mode confocal microscopy [1] and optical coherence tomography
(OCT) [2]. To better understand the interaction of light within tissue and, thus, the possibilities
of these and other imaging modalities, it is useful to have a model of tissue from which the
optical properties can be derived.
Currently, several models exist with which to simulate the optical properties of tissue. The
finite-difference time-domain (FDTD) method has been employed to simulate scattering from
single cells by direct solution of Maxwell’s equations [3,4]. For bulk tissue, Mie theory [5,6] has
been utilized by approximating the tissue with a fractal distribution of spheres [7, 8]. A fractalcontinuous distribution [9, 10] has been used to investigate light scattering in tissue based on
the sample refractive-index correlation function.
With the use of phase-contrast microscopy, it has been shown that the scattering centers
in biological tissue aggregate into complicated forms similar to those of fractal objects [11,
12]. A model based on a fractal size distribution of spherical scatterers is attractive in that it
provides an approximation to the complicated spatial refractive-index variations within bulk
tissue. Therefore, it provides the ability to simulate beam propagation through tissue and, thus,
to describe models of imaging modalities that rely on its scattering properties [13]. The optical
properties determined from the tissue model are also useful for Monte Carlo simulations of
light transport in tissue [14].
The discrete particle model of biological tissue [8] is an attractive and physically appealing
approach, however, it lacks a rigorous treatment of multiple and correlated scattering effects.
Multiple scattering refers to the situation in which the scattered wave from each particle contributes to the wave incident upon the other scatterers [15], whereas correlated scattering refers
to the interference effects between the scattered fields from all particles [16]. In this paper,
we define multiparticle scattering to include both multiple and correlated scattering effects.
The model’s use of a wavelength-independent packing factor [17, 18] is sufficient for describing correlated scattering among the small-sized particles of the fractal distribution that scatter
in the Rayleigh regime. However, the distribution contains important contributions from Mie
scatterers and their density makes the likelihood of correlated scattering, and especially multiple scattering, among these particles high. In addition, the effect of multiparticle scattering
upon the anisotropy of the discrete particle model has not been investigated to date. Using a
field-based computational method that extends Mie theory to the multisphere case, we simulated the optical properties of a fractal aggregate, further developing the discrete particle model
by incorporating the effects of multiparticle light scattering. We show that multiparticle scattering has important effects upon the optical parameters simulated, which include the scattering
coefficient, anisotropy, reduced scattering coefficient, and the phase function. In addition, we
demonstrate the importance of incorporating an enhanced contribution of Rayleigh scattering
to the model for reasonable agreement with measured data from skin tissue (dermis).
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2. Theory
Following Schmitt and Kumar [8], we describe the scatterers within biological tissue in terms
of a distribution of spherical particles, of varying sizes but equal, uniform refractive indices
within a background medium (of constant refractive index). The scatterer size distribution for
dense fibrous tissue, such as the dermal layer of skin tissue, follows a fractal power law distribution [12]. If the sphere diameter is denoted d, the volume fraction density function in terms
of the sphere size is defined as:
K1 d 3−D f , dmin < d < dmax
(1)
η (d) =
0,
otherwise,
so that the fraction of the total volume occupied by spheres in the range [d, d + Δd] is equal to
η (d)Δd, D f is the fractal dimension, K1 is a scaling constant, d min is the smallest sphere size,
and dmax is the largest sphere size of the distribution. The total volume fraction of the aggregate
Fv is defined by the integral Fv ≡ 0∞ η (d) dd.
The variations in the refractive index of the model are determined using weighted averages
of typical refractive-index values for the constituents of biological tissue, to obtain background
and particle refractive indices, n̄ bkg and n̄p , respectively. For soft biological tissue, the background refractive index has been estimated to be approximately 1.35 [8], calculated by estimating the volume fractions of intracellular and interstitial fluids present in tissue. The particle
refractive index is determined by calculating a quantity known at the mean index variation,
Δn, based on the refractive indices and relative volume fractions of the collagen fiber, nuclei
and organelle constituents of dermis. This quantity has been estimated to be equal to 0.10 [8],
and the particle refractive index is given by n̄ p = n̄bkg + Δn = 1.45.
To be able to simulate the optical properties of the model, an aggregate was created by randomly distributing spheres throughout a spherical region of volume V . (A sphere is defined to
be part of the aggregate if its center is located within this region.) The volume V was chosen
to be suitably large so that the optical properties computed by Mie theory sufficiently converge
to those of the infinitely large aggregate. This also ensures that the statistical properties of bulk
tissue are approximated as well as possible, given the computational constraints of the simulation. The spheres corresponding to the smallest particle size of the distribution, that scatter
light within the Rayleigh regime, were excluded from the aggregate due to the computational
resource limitations of the simulation, however, their contribution to the total optical properties
was computed separately, as shown later. Similarly to Schmitt and Kumar [8], we chose a total
of 10 discrete sphere sizes. However, in contrast to their choice of diameters equally spaced on
a logarithmic scale, we sample sphere sizes linearly to best retain the fractal properties of the
distribution. We also chose particle sizes ranging from 50 nm to 20 μ m; the contribution of the
particle sizes outside this range to the total optical properties of the model is negligible. The
discrete particle volume fractions were weighted according to Eq. 1, so that their sum, the total
volume fraction Fv , was equal to 0.2. Examples of the aggregates created are shown in Fig. 1
and show the higher density of larger particles present in the aggregate with a lower fractal
dimension. The sizes of the two aggregates are different, since they have different particle densities but the number of spheres in each are approximately equal. For ease of comparison, they
are displayed in Fig. 1 with the same axis scale.
We employ a solution to the multisphere scattering problem provided by Xu [19–23], known
as the Generalized Multiparticle Mie solution (GMM) [24]. To model interaction (and, thus,
multiple scattering) effects between spheres, this method expands the scattered fields in terms
of vector spherical wave functions (VSWFs) that impinge on all other spheres within the aggregate. By also expanding the incident wave in terms of VSWFs, and making use of addition
theorems for these functions, multiple scattering effects can be solved analytically. By taking
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Fig. 1. Visual representation of example aggregates (excluding the smallest sphere size of
50 nm) for a fractal dimension of 3.5 and 3.7 shown in (a) and (b), respectively (scale in
micrometers).
account of the phase relation and direction of all scattered waves, correlated scattering in the farfield is solved by computing their interference. Inputs into the GMM FORTRAN program [23]
are the spatial locations, size, and relative refractive indices of all spheres, as well as the wavelength of the unpolarized incident plane wave. The output optical parameters include the total
scattering cross-section σ s,agg , anisotropy g agg , and the phase function Pagg averaged over all
azimuthal angles. The total scattering coefficient of the aggregate may then be calculated as:
μs,agg =
σsca
.
V
(2)
For weakly-scattering subwavelength-diameter spheres, multiparticle scattering is dominated
by correlated scattering which may be described by the use of a packing factor [17, 18]. This
reduces the bulk optical cross-section of these scatterers so that their effective volume fraction
density function can be written as:
η (d) =
[1 − η (d)]4
[1 + 2η (d)]2
η (d).
(3)
By using the correction factor implicit in this equation, the contribution to the total scattering
cross-section from the smallest particle size initially excluded from the aggregate can be computed and included a posteriori. Scattering from skin tissue typically includes a strong Rayleigh
contribution [25–27], much greater than that predicted by the fractal distribution alone. As seen
in Eq. 3, the use of this correction factor does not change the wavelength dependence of the
Rayleigh scattering and, therefore, an added Rayleigh scattering component to the scattering
coefficient can be written without explicitly accounting for the scatterer sizes as:
μs,Rayleigh = K2 λ −4 ,
(4)
where K2 is a parameter that varies the magnitude of this additional component. We will assume that correlated scattering among Rayleigh scatterers does not significantly change the
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overall isotropic scattering behavior exhibited by these particles and, thus, the effect of their
contribution upon the total anisotropy can be derived as:
gtot =
gagg
.
K2 λ −4 /μs,agg + 1
(5)
The reduced scattering coefficient, defined as μ s = (1 − g)μs , is therefore given by the expression:
μs = (1 − gtot) μs,agg + K2 λ −4 .
(6)
The phase function is derived from the angular scattered irradiance (per unit incident irradiance)
of a medium. It is a probability distribution function for the scattering angle of a photon relative
to its incident direction. If multiparticle scattering effects are ignored, the volume-averaged
phase function for M particle sizes is given by:
P(θ ) =
∑M
i=1 μs (di )Pi (θ )
,
∑M
i=1 μs (di )
(7)
which is normalized so that 0π P(θ )2π sin θ dθ = 1. The phase function
for Rayleigh
scatterers
(given unpolarized incident light) may be written as PRay (θ ) = 163π 1 + cos2 θ , so that the
total phase function of the aggregate can be shown to be equal to:
K2 λ −4 163π 1 + cos2 θ + μs,aggPagg
.
(8)
Ptot (θ ) =
K2 λ −4 + μs,agg
The wavelength range simulated was between 400 and 1800 nm, incorporating both the “therapeutic window” and part of the near-infrared region of the electromagnetic spectrum. The wavelength increment of the simulation was chosen to be 200 nm, due to the substantial computation
time required by the program. (For a wavelength of 400 nm, a CPU time of approximately 160
hours was necessary to simulate an aggregate.) Larger incident wavelengths require less CPU
time due to the lower order of the incident and scattered field expansions necessary for convergence. A total of five random realizations of the aggregate corresponding to a particular fractal
dimension were created and the optical properties averaged for each data point. In computing
the optical properties without taking into account multiparticle scattering effects, we chose a
wavelength increment of 1 nm due to the substantially smaller computation time necessary.
3. Results and discussion
The wavelength dependence of the scattering coefficient, anisotropy, and reduced scattering
coefficient from the simulations of several fractal aggregates is shown in Fig. 2. The standard
error of the mean for each data point was not more than 0.35 mm −1 for the scattering coefficient
and 0.002 for the anisotropy. Based on spatial frequency analysis, the fractal dimension for a
particular soft tissue sample has been estimated to be around 3.6 [11, 12]. We chose three
fractal dimensions in the range 3.5 ≤ D f ≤ 3.7, which lead to optical properties consistent with
measured data for skin tissue.
The wavelength dependence of the scattering coefficient, not taking into account multiparticle scattering effects, can be approximated by the power-law relation μ s ∝ λ 3−D f . This differs
from the relation reported by Schmitt and Kumar [8], which has an exponent of 2 − D f . The
discrepancy is the result of our sampling of the discrete particle sizes on a linear scale (as opposed to a logarithmic scale) to represent a fractal distribution. Our result is consistent with
the relation reported by Wang [28], in which the sphere sizes were also sampled linearly. This
simple relation does not hold, however, if multiparticle scattering is taken into account, and for
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(a)
(b)
0.98
Scattering coefficient (mm-1)
60
Anisotropy
0.96
0.94
Scattering
type
0.92
0.9
Fractal
dimension
0.88
0.86
{
{
400
600
800
Multiparticle
Non-multiparticle
Rayleigh & Multiparticle
Df = 3.5
Df = 3.6
Df = 3.7
50
40
30
20
10
1000 1200 1400 1600 1800
400
600
800
(c)
Reduced scattering coefficient (mm-1)
Wavelength (nm)
1000 1200 1400 1600 1800
Wavelength (nm)
Bashkatov et al. [27]
Prahl [32]
Simpson et al. [33]
Du et al. [34]
Troy et al. [35]
Chan et al. [36]
8
6
4
2
0
400
600
800
1000
1200
1400
1600
1800
2000
Wavelength (nm)
Fig. 2. Anisotropy (a) and scattering coefficient (b) for multiparticle (solid lines) and nonmultiparticle (dashed lines) scattering effects. The correspondence between fractal dimension and line color is indicated in the legend. The dotted lines in (a) shows the effect of the
Rayleigh contribution upon the anisotropy. The reduced scattering coefficient is shown in
(c) for multiparticle (solid lines) and an added contribution of Rayleigh (dotted lines) scattering. (No Rayleigh contribution is included for the aggregate of fractal dimension 3.7.)
Empirical data points (shown as markers) are from [27] and the references therein [32–36].
all fractal dimensions, it can be seen that the apparent exponent of the scattering coefficient
wavelength dependence is reduced. The reduction in magnitude of the scattering coefficient is
most significant at lower wavelengths, at which multiparticle scattering effects associated with
larger Mie scatterers dominate.
In general, the anisotropy of the aggregates is decreased due to multiparticle scattering; for
wavelengths above around 1000 nm, the effect is greatest for aggregates with lower fractal
dimensions. For wavelengths below this value, the contribution from small Rayleigh scatterers
becomes prominent due to the λ −4 dependence in Eq. 4, which for aggregates with larger
fractal dimensions, and hence larger relative volume fractions of small scatterers, reduces the
anisotropy. This strong reduction in anisotropy for shorter wavelengths, which is not present
in the discrete particle model of Ref. [8], is consistent with the empirical fit described by van
Gemert et al. [29] based on the measured data of Bruls and van der Leun [30], and that reported
by Beek et al. [31].
The reduction in the scattering coefficient is accompanied by a corresponding reduction in
the anisotropy; these effects partially cancel when calculating the reduced scattering coefficient.
Inspection of the experimentally measured data sets in [27, 32, 33] shows that the exponent of
the reduced scattering coefficient for shorter wavelengths approaches the Rayleigh limit of λ −4 ,
which is not accounted for in the fractal model since there is typically an underestimation of
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2
1
10
4
10
10
1
10
-2
0
10
5
10
15
Scattering angle (degrees)
-2
10-4
0
4
10
2
Phase function
Phase function
10
10
Phase function
4
Phase function
10
2
1
4
10
2
1
10-2
0
10
5
10
15
Scattering angle (degrees)
-2
10-4
20
40
60
80
100
120
140
160
180
0
20
40
60
80
100
120
140
Scattering angle (degrees)
Scattering angle (degrees)
(a)
(b)
160
180
Fig. 3. Phase functions of the aggregate for a wavelength of 633 nm and 1300 nm shown
in (a) and (b), respectively, with a fractal dimension of 3.6. The dashed lines (blue) and
solid lines (red) represent non-multiparticle and multiparticle scattering, respectively. The
dotted line (green) represents multiparticle scattering including a contribution of Rayleigh
scattering. The inset shows the phase function in the range from 0 to 15 degrees.
the number of particles that scatter in the Rayleigh regime. (Multiparticle scattering within the
aggregate reduces this exponent even further.) This demonstrates that, for human dermis, an
added Rayleigh scattering contribution to the discrete particle model will generally substantially improve its agreement with empirical data at low wavelengths. The value of K 2 , which
sets the magnitude of this contribution, was chosen by inspection so that the model gives reasonable results when compared to measured data for both the reduced scattering coefficient
and anisotropy. For the wavelength defined in micrometers, the value of K 2 was set to 0.07
and 0.03 for the aggregates with a fractal dimension of 3.5 and 3.6, respectively. The result of
this addition of Rayleigh scattering to these aggregates is shown in Fig. 2. There is reasonable
agreement between the reduced scattering coefficient of the model and that of experimental
data for human and porcine dermis, both of which are similar in structure [34, 37].
The shape and accuracy of the phase function can significantly impact the results of Monte
Carlo simulations of light transport in turbid media [38, 39]. The phase function for a specific
realization of the fractal aggregate is shown in Fig. 3 for wavelengths of 633 and 1300 nm. For
both wavelengths, there is a significant increase in the probability of direct forward scattering
(corresponding to a scattering angle θ = 0 ◦ ), due to the fact that the majority of the scattered
waves are in phase for this particular scattering direction [40,41] and, hence, is present for every
realization of the aggregate. Even though this forward peak increases the anisotropy slightly,
multiple scattering effects result in an overall reduction in anisotropy. The specific locations
of the local maxima and minima that form the high-frequency structure in the phase function
are dependent on the random positions of the spheres within the aggregate. These strong resonances, seen in Fig. 3, are not likely to be present in real tissues due to the averaging effects
of the non-spherical scatterers and random, continuous variations in refractive index. Averaging the phase function of many realizations of the aggregate, however, reduces the fluctuations
caused by these resonances and, thus, we expect our results to be applicable to simulating the
effects of real tissue. Additional Rayleigh scatterers increase the probability of direct backscattering (corresponding to θ = 180 ◦), which, as expected, is higher for lower wavelengths.
The close-packed nature of scatterers in soft biological tissue can sometimes be overlooked,
particularly when high-index contrast microspheres are used in experiments as tissue phantoms [42]. To achieve realistic optical properties, such phantoms have very small volume fractions and, thus, we expect multiparticle scattering effects to be negligible in such phantoms.
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The effects of multiparticle scattering are, broadly, to reduce the magnitude and exponent of the
wavelength dependence of the scattering coefficient, and to reduce the anisotropy. Comparison
with experimental data, however, demonstrates the importance of also adequately accounting
for an additional Rayleigh scattering contribution.
4. Conclusion
We have shown that, when taken into account, multiparticle scattering has a significant effect
upon the optical properties of a fractal soft tissue model. Combined with a Rayleigh scattering
contribution, the wavelength dependency of the resulting reduced scattering coefficient is in
reasonable agreement with experimentally determined values for skin tissue (dermis). A forward peak in the phase function due to correlated scattering is observed, however the effect of
multiple scattering reduces the total anisotropy over all wavelengths simulated. The model presented here, or variations of it to account for different tissue types, will provide a more accurate
basis for a range of tissue optics problems. Further to the results presented here, it is possible to
determine the Müller matrix, which relates the incident and scattered Stokes parameters, from
the amplitude-scattering matrix computed from the simulation. Thus, it would be possible to
model the depolarization of light propagating through the skin tissue model.
Acknowledgments
DHPS is supported by the William and Marlene Schrader Trust. The computational simulations
were run on an SGI Altix supercomputer provided by the ARRC computing facility at iVEC in
Western Australia.
#87674 - $15.00 USD
(C) 2007 OSA
Received 18 Sep 2007; revised 25 Oct 2007; accepted 25 Oct 2007; published 29 Oct 2007
12 November 2007 / Vol. 15, No. 23 / OPTICS EXPRESS 15010