Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms.

Abstract

A flexible and fast Monte Carlo-based model of diffuse reflectance has been developed for the extraction of the absorption and scattering properties of turbid media, such as human tissues. This method is valid for a wide range of optical properties and is easily adaptable to existing probe geometries, provided a single phantom calibration measurement is made. A condensed Monte Carlo method was used to speed up the forward simulations. This model was validated by use of two sets of liquid-tissue phantoms containing Nigrosin or hemoglobin as absorbers and polystyrene spheres as scatterers. The phantoms had a wide range of absorption (0-20 cm(-1)) and reduced scattering coefficients (7-33 cm(-1)). Mie theory and a spectrophotometer were used to determine the absorption and reduced scattering coefficients of the phantoms. The diffuse reflectance spectra of the phantoms were measured over a wavelength range of 350-850 nm. It was found that optical properties could be extracted from the experimentally measured diffuse reflectance spectra with an average error of 3% or less for phantoms containing hemoglobin and 12% or less for phantoms containing Nigrosin.

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Monte Carlo-based inverse model for calculating
tissue optical properties. Part I: Theory and
validation on synthetic phantoms
Gregory M. Palmer and Nirmala Ramanujam
A flexible and fast Monte Carlo-based model of diffuse reflectance has been developed for the extraction
of the absorption and scattering properties of turbid media, such as human tissues. This method is valid
for a wide range of optical properties and is easily adaptable to existing probe geometries, provided a
single phantom calibration measurement is made. A condensed Monte Carlo method was used to speed
up the forward simulations. This model was validated by use of two sets of liquid-tissue phantoms
containing Nigrosin or hemoglobin as absorbers and polystyrene spheres as scatterers. The phantoms
had a wide range of absorption ?0–20 cm?1? and reduced scattering coefficients ?7–33 cm?1?. Mie theory
and a spectrophotometer were used to determine the absorption and reduced scattering coefficients of the
phantoms. The diffuse reflectance spectra of the phantoms were measured over a wavelength range of
350–850 nm. It was found that optical properties could be extracted from the experimentally measured
diffuse reflectance spectra with an average error of 3% or less for phantoms containing hemoglobin and
12% or less for phantoms containing Nigrosin.© 2006 Optical Society of America
OCIS codes:
170.4580, 170.6510, 160.4760.
1.
Diffuse reflectance spectroscopy is sensitive to the
absorption and scattering properties of tissue. The
absorption coefficient is directly related to the con-
centration of the chromophores, and the scattering
coefficient reflects the size and density of scattering
structures in the tissue. However, the combined influ-
ence of the absorption and scattering events on dif-
fusely reflected light in tissue makes it difficult
to interpret the physically meaningful information
present in diffuse reflectance measurements, since a
change in measured diffuse reflectance could reflect a
changeinabsorption,scattering,orboth.Theabilityto
separately quantify the independent absorption and
scattering coefficients of tissue from a diffuse reflec-
tance spectrum will allow the extraction of the physi-
ological and structural properties of the sample that
influence the diffuse reflectance. Knowledge of these
Introduction
sources of optical contrast can provide insight into the
underlying mechanisms that permit diagnosis and
lead to an improvement in diagnostic algorithms.
Several groups have developed analytical and nu-
merical models to extract the absorption and scatter-
ing coefficients oftissue
measurements of diffuse reflectance spectra measured
in the ultraviolet–visible (UV–VIS) part of the elect-
romagnetic spectrum. In general, these methods can
becategorizedintothosethatuseananalyticalapprox-
imation of the transport equation such as the diffusion
equation,1–3a modified form of the Monte Carlo mod-
el,4and empirical methods (for example, neural net-
works).5,6The diffusion equation is valid for low to
moderate tissue absorption, i.e., for the case in which
tissuesarescatteringdominantandforrelativelylarge
photon travel paths, which is the case at red and near-
infrared(NIR)wavelengths.MonteCarlomodelingisa
numerical technique that is valid for a wide range of
absorption and scattering coefficients and photon
paths and thus can be used to model light transport
over the entire UV–VIS–NIR range. However, unlike
the diffusion equation, this technique is computation-
ally intensive. The previously developed models are
briefly discussed below.
Zonios et al.1developed a forward model of light
transport based on the diffusion approximation,
which expressed the diffuse reflectance spectra as
fromfiber-optic-based
The authors are with the Department of Biomedical Engineer-
ing, 136 Hudson Hall, Box 90281, Duke University, Durham,
North Carolina 27708-0281. N. Ramanujam’s e-mail address is
nimmi@duke.edu.
Received 5 November 2004; revised 29 March 2005; accepted 8
May 2005.
0003-6935/06/051062-10$15.00/0
© 2006 Optical Society of America
1062 APPLIED OPTICS ? Vol. 45, No. 5 ? 10 February 2006

Page 2

functions of the wavelength-dependent reduced scat-
tering coefficient and absorption coefficient, an em-
pirically determined constant related to the probe
geometry, and an empirically determined adjustment
to the diffusion equation for use with high-absorption
coefficients. The empirical correction factors were de-
termined by use of tissue phantom studies. The
model was constrained to assume that hemoglobin is
the only absorber that contributes to the wavelength-
dependent absorption coefficient, and the reduced
scattering coefficient was constrained to be always
decreasing with increasing wavelength. A nonlinear
least-squares optimization routine was then em-
ployed to minimize the difference between the mea-
sured and the fitted diffuse reflectance spectrum. The
extracted reduced scattering coefficient was fit to Mie
theory to extract the scatterer size and scatterer den-
sity. This model was then used to extract the total
hemoglobin concentration, scatterer size, and scat-
terer density from diffuse reflectance spectra of
colon tissues measured over the wavelength range of
350–700 nm. Both total hemoglobin concentration
and scatterer size increased while scatterer density
decreased in adenomatous colon polyps compared
with normal tissues.
Finlay and Foster3improved on the standard dif-
fusion equation described above by using a more
complex model of light transport. They used a P3
approximation of the radiative transport equation,
which incorporates higher-order moments of anisot-
ropy than the standard diffusion approximation.
They found that this modification performed better
than the standard diffusion equation at relatively
short source–detector separations ??1 mm? and
high-absorption coefficients corresponding to those at
UV–VIS wavelengths. Furthermore, this model was
adaptable to complex fiber-optic probe geometries.
However, systematic deviations in the absorber
concentration were observed at source–detector
separations of less than 1 mm and for high absorber
concentrations.
Ghosh et al.2used a model based on the standard
diffusion approximation to extract the absorption and
scattering coefficients from spatially resolved dif-
fuse reflectance measurements. They measured the
spatially resolved diffuse reflectance at 12 source–
detectorseparations,rangingfrom1.2to12 mm,over
the wavelength range of 450–650 nm from human
breast tissue samples. The absorption and scattering
coefficients were determined from fits to the diffusion
approximation of the spatially resolved diffuse reflec-
tance at each individual wavelength. The results of
their study indicated that malignant tissues are more
absorbing and scattering than normal tissues are at
all wavelengths.
Thueler et al.4used a model based on Monte Carlo
simulations to extract absorption and scattering co-
efficients from spatially resolved diffuse reflectance
measurements. They used a scaling procedure to in-
crease the efficiency of their Monte Carlo model.
However, the scaling procedure was valid for only a
limited range of absorption7?albedo ?0.9?. They
measured spatially resolved diffuse reflectance mea-
surements of esophageal tissues, using ten source–
detector separations ranging from 0.3 to 1.35 mm,
over a wavelength range of 480–950 nm, and fit the
measurements at each wavelength to the Monte
Carlo model. Their results showed that there are sig-
nificant differences in the absorption and scattering
coefficients of the normal antrum and fundus of the
stomach.
Pfefer et al.5developed an empirical method for the
extraction of absorption and scattering coefficients
from spatially resolved diffuse reflectance measure-
ments. They employed a neural-network algorithm,
which was trained on phantoms, and then used this
neural-network model to extract optical properties
from another set of phantoms. The spatially resolved
diffuse reflectance was measured at six source–detector
separationsbetween0.23and2.46 mmatawavelength
of 675 mm from phantoms with absorption coeffi-
cients ranging from 1 to 25 cm?1and reduced scat-
tering coefficients ranging from 5 to 25 cm?1. The
neural-network model was able to extract the ab-
sorption and scattering coefficients of the phantoms
to within root-mean-square (rms) errors of ?2 and
?3 cm?1, respectively.
Amelink et al.6employed a method they referred to
as differential path-length spectroscopy to extract the
absorption and scattering coefficients from diffuse re-
flectance spectra of tissues measured with a specific
probe geometry. By subtracting the reflectance mea-
sured from two adjacent fibers at a given wavelength,
they found that the resulting output was sensitive to
a superficial portion of the tissue, and they were able
to express it as a simple function of the absorption
and scattering coefficients. They then constrained ab-
sorption and scattering coefficients in the model to be
functions of the hemoglobin absorption and Mie scat-
tering, respectively. They fit the diffuse reflectance
spectra measured over a wavelength range of 350–
1000 nm from normal and malignant bronchial mu-
cosa to this model. They found a statistically signifi-
cant decrease in blood oxygenation in malignant
tissue compared with normal tissues.
Table 1 summarizes the previously described
methods for extracting optical properties including
the light-transport model described in this paper, the
absorption range for which these models are valid,
the calibration requirement for the use of each
method, and the probe-geometry requirements. High,
moderate, and low absorptions correspond generally
to tissue absorption in the UV, VIS, and NIR wave-
length ranges, respectively. For multiple separation
probe geometries, the range of source–detector sep-
arations used in the study is provided. Table 1 shows
that the previously described methods are limited in
that they place one or more of the following con-
straints on their applicability: They are limited to
wavelengths corresponding to moderate to low ab-
sorption2,4(VIS and NIR wavelengths), they impose
constraints on the fiber-optic probe geometries used
or require multiple illumination–collection fiber
10 February 2006 ? Vol. 45, No. 5 ? APPLIED OPTICS1063

Page 3

separations,2–6or they require extensive phantom
studies for empirical calibration of the model.1,5
These limitations make these techniques difficult or
impossible to implement on the large body of existing
data that have been collected with a wide variety of
probe geometries over the UV–VIS wavelength
range. To address these concerns, this paper outlines
a method for extracting optical properties, which is
based on Monte Carlo modeling of light transport and
is valid for a wide range of optical properties, includ-
ing high absorption.8The method presented here is
validated for a specific probe geometry by use of
phantom studies; however, it imposes no inherent
constraints on the probe geometry used and can be
easily adapted to any arbitrary probe geometry pro-
vided a single phantom calibration measurement is
made.
2.Methods and Results
A.Forward and Inverse Models
Forward model. The forward model refers to the
model that relates the physiological and structural
properties of the tissue to its modeled diffuse reflec-
tance. Figure 1(a) shows the forward Monte Carlo
modelofdiffusereflectance.Themodelhastwosetsof
inputs, which are used to determine the absorption
and scattering coefficients. The concentration ?Ci? of
each chromophore (free parameter) and the corre-
sponding wavelength-dependent extinction coeffi-
cient ??i???? (fixed parameter) are used to determine
thewavelength-dependent
??a????, according to the relationship ?a??? ? ?
ln?10??i???Ci. Mie theory for spherical particles9is
absorptioncoefficient
Fig. 1.
Input boxes are gray, and output boxes are gray with a bold out-
line.
(a) Forward and (b) inverse models of diffuse reflectance.
Table 1.Summary of Methods for Extracting Optical Propertiesa
Author
Light-Transport
ModelAbsorption RangeCalibration Requirement Probe Geometry
Zonios et al. (Ref. 1)Modified diffusion
equation
P3approximation of
transport equation
Spatially resolved
diffusion equation
High to low
absorption
High to low
absorption
Moderate to low
absorption
Extensive phantom studies for
empirical calibration
Spectra normalized to a specific
wavelength
Spectra normalized to a specific
source–detector separation
Single source–detector
separation
Single source–detector
separation at ?1 mm
Multiple source–detector
separations: 1.2 to
12 mm
Multiple source–detector
separations: 0.3 to
1.35 mm
Multiple source–detector
separations: 0.23 to
2.46 mm
Specialized probe
geometry
Finlay and Foster
(Ref. 3)
Ghosh et al. (Ref. 2)
Thueler et al. (Ref. 4)Spatially resolved
Monte Carlo
approximation
Spatially resolved
empirical method
Moderate to low
absorption
Multiple phantoms required for
calibration
Pfefer et al. (Ref. 5)High to low
absorption
Extensive phantom studies for
empirical calibration
Amelink et al. (Ref. 6) Empirical method High to low
absorption
Internal calibration by means
of subtraction of signal from
two fibers
Calibration on a single
phantom
Palmer and Ramanujam
(this paper)
Monte CarloHigh to low
absorption
Requires only a single
source–detector
separation and
adaptable to any probe
geometry
aHigh, moderate, and low absorptions correspond generally to tissue absorption in the UV, VIS, and NIR wavelength ranges, respec-
tively. For multiple separation probes, the range of source–detector separations used in the study is provided.
1064APPLIED OPTICS ? Vol. 45, No. 5 ? 10 February 2006

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used to model scattering. The scatterer size and den-
sity are free parameters, and the refractive index is
fixed according to known values for phantoms. The
scattering coefficient ??s???? and the anisotropy factor
?g???? are then calculated at a given wavelength.
The optical properties ?a???, ?s???, and g??? of the
medium can then be input into a Monte Carlo model
of light transport to obtain the modeled diffuse re-
flectance for a given wavelength. The code used for
this purpose was adapted from that of Wang et al.10
Note that a Monte Carlo simulation would be re-
quired for each unique set of optical properties, thus
making this step of the forward model computation-
ally prohibitive. To increase the efficiency of this step,
a scaling approach previously described by Graaff et
al.11was incorporated such that a single Monte Carlo
simulation can be run for a particular set of optical
properties, the output of which can be scaled to any
set of optical properties. The method consists of run-
ning a single simulation for a given set of absorption
??a,sim? and scattering ??s,sim? coefficients and record-
ing the exit weight ?Wexit,sim?, net distance traveled
?rt,sim?, and total number of interactions for each pho-
ton (N) that exits the tissue surface. The scaling
method then uses these stored parameters to calcu-
late the new exit weight ?Wexit,new? [Eq. (1)] and net
distance traveled ?rt,new? [Eq. (2)] for a given photon
that had a different absorption ??a, new? and scattering
coefficient ??s,new? used in the same simulation. The
scaling relationships, taken from Graaff et al.11are
Wexit,new?Wexit,sim?
?s,new
?s,new??a,new
?s,sim??a,sim
?s,sim ?
N
,
(1)
rt,new?rt,sim?
?s,sim??a,sim
?s,new??a,new?.(2)
To further simplify the scaling process, it was as-
sumed that, for a given value of the reduced scatter-
ing coefficient, ?s? ? ?s?1 ? g?, the diffuse reflectance
would be the same for any values of ?sand g that
generate the same ?s?. This has been shown to be
valid over the range of g values present in human
tissue, i.e., for g values greater than 0.8.11,12By use of
this similarity relation and the scaling procedure out-
lined above, only a single Monte Carlo simulation
needed to be run to determine the output of a Monte
Carlo simulation for any set of optical properties. The
Henyey–Greenstein phase function was used in the
singleMonteCarlosimulation.Theparametersofthe
single Monte Carlo simulation were as follows: num-
ber of photons, 40 ? 106; ?s, 150 cm?1; ?a, 0 cm?1;
g, 0.8; model dimensions, 2 cm ?radius? ? 2cm
?depth?; refractive indices, 1.33 (medium for phan-
toms) and 1.452 (fiber-optic probe above medium).
Convolution was used to integrate over the illumi-
nation and collection fibers to determine the proba-
bility that a photon, traveling a fixed distance, would
be collected for a given probe geometry. This takes
advantage of the spatial invariance and rotational
symmetry present in a homogeneous medium. For a
pair of illumination and collection fibers, the proba-
bility of collection of a photon traveling a net distance
rtbetween the points of entering and exiting the me-
dium is given by
2?
max??ri, s?rt?rc?
? cos?1?
1
2
?2ri
min?ri, s?rt?rc?
?s?x?cos?1?
s2??s?x?2?ri
2
2?s?x?s ?
2?s?x?rt ?dx,
rt
2??s?x?2?ri
2
(3)
where riis the radius of the illumination fiber, rcis
the radius of the collection fiber, s is the separation
between the centers of the illumination and the col-
lection fibers, and x is the spatial variable over which
the integral is taken (see appendix for derivation).
This equation was numerically integrated. To adapt
this to the fiber bundle used in this study (the geom-
etry of the fiber bundle is described below), the com-
mon end of the fiber bundle was imaged, and the
centers of each illumination and collection fiber in the
bundle were determined. Then the probe geometry
was integrated pairwise (for each illumination–
collection fiber pair) to determine the total probabil-
ity of collection. It was found that imaging the fiber
bundle to obtain the exact location of each illumina-
tion and collection fiber was necessary. Approximat-
ing the fiber bundle as solid rings of illumination and
collection fibers produced significant errors in the
model, likely because of imperfect physical placement
of the fibers within the bundle.
It was found that the scaling process and subse-
quent numerical integration for the probe geometry
of a large number of photons required approximately
1 s to complete, which, although much faster than
running an independent simulation, was still rather
slow for performing an inversion procedure. There-
fore the diffuse reflectance values for a range of op-
tical properties ??s, 5–500 cm?1; ?a, 0–200 cm?1;
g, 0.8) were determined ahead of time to form a
lookup table and cubic splines were used to interpo-
late between table values. The smallest increment
used in the lookup table was 0.1 cm?1for ?aand
2.5 cm?1for ?s.
Inversemodel. Figure1(b)showstheinverseMonte
Carlo model of diffuse reflectance. First, an initial set
of free parameters is input into the Monte Carlo-
based forward model. These include the absorber con-
centrations and the scatterer size and density. The
fixed parameters are the extinction coefficients of the
absorber and the refractive index of the scatterer and
the surrounding medium for a given wavelength.
These input parameters are used in the forward
model to generate the predicted reflectance as a func-
tion of wavelength. Next, the sum of squares error
between the predicted and measured reflectance is
computed. The free parameters are then iteratively
updated until the sum of squares error is minimized.
A Gauss–Newton nonlinear least-squares algorithm
10 February 2006 ? Vol. 45, No. 5 ? APPLIED OPTICS 1065
c

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provided in the MATLAB optimization toolbox
(Mathworks, Natick, Massachusetts) was used as the
optimization algorithm. To ensure convergence to a
global minimum, this procedure was repeated several
times for each sample, with a different, randomly
chosen set of starting parameters.
B.
The forward and inverse models were tested on ho-
mogeneous tissue phantoms with optical properties
representative of human tissues in the UV–VIS spec-
tral range.13Two sets of liquid homogeneous phan-
toms were created, and all measurements were made
the day the phantoms were made. The first set (phan-
tom set 1) consisted of five phantoms containing vari-
able concentrations of the absorber, hemoglobin, and
the scatterer, 1 ?m diameter polystyrene spheres
(07310-15, Polysciences, Inc., Warrington, Pennsyl-
vania).Thehemoglobinusediswatersolubleandwas
obtained by the supplier from human blood (H0267,
Sigma Co., St. Louis, Missouri). First, a solution with
a fixed volume density of polystyrene spheres sus-
pended in water was made and four titrations of a
fixed amount of hemoglobin were added, with a dif-
fuse reflectance measurement made after each titra-
tion. This produced a set of five phantoms (including
the phantom with no absorber) with similar scatter-
ing properties and a range of absorption properties.
The second set (phantom set 2) consisted of 25 phan-
toms containing variable concentrations of the same
scatterer and a different absorber, Nigrosin (N-4754,
Sigma Co., St. Louis, Missouri), to characterize the
model’s effectiveness by using an absorber with dif-
ferent absorption characteristics. For phantom set 2,
five base phantoms were created, each with a differ-
ent volume density of polystyrene spheres. For each
base phantom, four titrations of a fixed volume of a
Nigrosin solution were added, and a diffuse reflec-
tance measurement was made after each titration.
This produced a total of 25 phantoms (including the
phantoms with no absorber) with a range of scatter-
ing and absorption coefficients. The phantoms were
filled up to a height of at least 3 cm in a 1.6 cm
diameterplastic container.
dependent extinction coefficients for hemoglobin and
Nigrosin were measured with a spectrophotometer
(Cary 300, Varian, Palo Alto, California). It was as-
sumed that the oxygenation of hemoglobin was con-
stant throughout the course of the experiment. The
reduced scattering coefficient was determined from
Mie theory9by use of freely available software,14
given the known size ?1 ?m?, density, and refractive
index of the spheres (1.60) and the surrounding me-
dium, water (1.33). The refractive indices of polysty-
rene spheres have been reported to be constant to
within approximately 1% of this value over the wave-
length range used.15Tables 2 and 3 show the means
and ranges of the reduced scattering coefficient ??s??
and absorption coefficient ??a?, respectively, for each
of the phantoms in phantom sets 1 and 2 over the
wavelength range of 350–850 nm.
Tissue Phantom Models
Thewavelength-
C.
All diffuse reflectance spectroscopy measurements
were made with a spectrofluorometer (SkinSkan, JY
Horiba, Edison, New Jersey). This instrument
consists of a 150 W xenon lamp, double-grating exci-
tation, emission monochromators, and a photomulti-
plier tube (PMT). The illumination and collection of
light from the phantom were coupled through a fiber-
optic probe, consisting of a central collection core with
a diameter of 1.52 mm surrounded by an illumina-
tion ring with an outer diameter of 2.18 mm. The
instrument is designed for making fluorescence and
diffuse reflectance measurements through a fiber-
optic probe, and the standard probe and coupling
mechanism were used. Figure 2 shows a schematic of
the probe geometry used in this study, with the gray
central region containing the collection fibers and the
surrounding white region containing the illumina-
tion fibers. Both the illumination ring and collection
core are made up of 31 individual fibers, each with a
core–cladding diameter of 200 and 245 ?m. The nu-
merical apertures of the illumination and collection
fibersare0.125and0.12,respectively.Theadjustable
parameters of the system are the wavelength range
and increment and the signal-acquisition time. The
fixed parameters of the system are the excitation and
emission bandpasses, which are set at 5 nm, and the
Measurement System and Measurement Parameters
Table 2.
(?s=) for Each of the Phantoms in Phantom Sets 1 and 2 over the
Wavelength Range of 350–850 nma
Means and Ranges of the Reduced Scattering Coefficient
Phantom
Set
?s= Level
Fixed
Lowest
2
3
4
Highest
Mean ?s=
(cm?1)
?s= Range
(cm?1)
1
2
13.3
8.9
13.3
17.8
22.2
26.7
10.9–16.4
7.3–10.9
10.9–16.4
14.6–21.8
18.2–27.3
21.8–32.7
aNote that these are given for the phantoms for which no ab-
sorber had been added. The addition of absorber dilutes the scat-
terer and thus reduces the scattering coefficient, by as much as
16% and 20% for the hemoglobin and Nigrosin phantoms, respec-
tively.
Table 3.
of the Phantoms in Phantom Sets 1 and 2 over the Wavelength Range
of 350–850 nm
Means and Ranges of the Absorption Coefficient (?a) for Each
Phantom
Set
?aLevel
1
2
3
4
5
1
2
3
4
5
Mean ?a
(cm?1)
?aRange
(cm?1)
10
0.9
1.3
1.6
2.0
0
3.8
7.2
10.3
13.0
0–0
0.02–7.7
0.02–11.2
0.03–14.4
0.03–17.5
0–0
1.7–5.9
3.1–11.2
4.4–15.9
5.6–20.1
2
1066 APPLIED OPTICS ? Vol. 45, No. 5 ? 10 February 2006