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Analytical model of light reflectance for extraction of the
optical properties in small volumes of turbid media
Roberto Reif,
1,
* Ousama A’Amar,
1
and Irving J. Bigio
1,2
1
Department of Biomedical Engineering, Boston University, 44 Cummington Street, 4th Floor, Boston,
Massachusetts 02215, USA
2
Department of Electrical and Computer Engineering, Boston University, 44 Cummington Street, 4th Floor, Boston,
Massachusetts 02215, USA
*Corresponding author: robreif@bu.edu
Received 9 April 2007; revised 4 August 2007; accepted 22 August 2007;
posted 23 August 2007 (Doc. ID 81960); published 9 October 2007
Monte Carlo simulations and experiments in tissue phantoms were used to empirically develop an
analytical model that characterizes the reflectance spectrum in a turbid medium. The model extracts the
optical properties (scattering and absorption coefficients) of the medium at small source-detector sepa-
rations, for which the diffusion approximation is not valid. The accuracy of the model and the inversion
algorithm were investigated and validated. Four fiber probe configurations were tested for which both the
source and the detector fibers were tilted at a predetermined angle, with the fibers parallel to each other.
This parallel-fiber geometry facilitates clinical endoscopic applications and ease of fabrication. Accurate
extraction of tissue optical properties from in vivo spectral measurements could have potential applica-
tions in detecting, noninvasively and in real time, epithelial (pre)cancers. © 2007 Optical Society of
America
OCIS codes: 170.3660, 170.3890, 170.4580, 170.6510, 170.7050.
1. Introduction
An important challenge in biomedical research is to
noninvasively characterize tissue conditions, for the
purposes of diagnosis of disease, such as (pre)cancer-
ous conditions, or for monitoring response to treat-
ment. In situ measurements of the optical properties
of tissue may reveal information concerning the mor-
phological and biochemical composition of the tissue.
Optical techniques are currently being used in vivo to
identify changes that occur in biological tissues [1–3]
associated with disease progression.
Tissues can be characterized by their optical prop-
erties, which are defined by the absorption coefficient
共
a
兲, the scattering coefficient 共
s
兲, the phase function
关p共兲兴, the anisotropy value 共g ⫽ 具cos 典兲 and the re-
duced scattering coefficient 关⬘
s
⫽
s
共1 ⫺ g兲兴. The
diffusion approximation to the Boltzman transport
equation is a method that has been used successfully
to determine the absorption coefficient and the re-
duced scattering coefficient in turbid media [4–6].
The validity of the diffusion approximation is limited
to media with higher scattering than absorption
共⬘
s
⬎⬎
a
兲, which is satisfied in biological tissues for
the wavelength region between 600–900 nm, and to
large separations between the source and the detec-
tor 共 ⬎⬎ 1兾⬘
s
兲. As a consequence, the collected pho-
tons travel through a large volume of tissue, and the
extracted optical properties represent average values
for the tissue volume probed. However, many clinical
settings require small fiber probes (e.g., endoscope
working channels typically have a diameter of
⬍3 mm). Different methods have been used to deter-
mine the optical properties in turbid media at small
source-detector separations [7–19], including some
based on the diffusion approximation [20,21]. Various
fiber optic probe designs have been developed for re-
flectance and fluorescence measurements [22–25]. In
many applications the tissues of interest are thin.
Most cancers arise in the epithelium, which is a su-
perficial tissue layer with a thickness, typically, of
100–500 m. Hence, sensitivity to the optical prop-
erties of the epithelial layer requires superficial mea-
0003-6935/07/297317-12$15.00/0
© 2007 Optical Society of America
10 October 2007 兾 Vol. 46, No. 29 兾 APPLIED OPTICS 7317
surement techniques. The penetration depth of the
collected photons depends on the fiber probe geo-
metry and the optical properties of the sample
[16,24,26,27], which may allow the interrogation
depth to extend beyond the epithelial layer.
Tilted fiber optic probes have been employed for
controlling the depth of penetration in tissue spec-
troscopy applications [26–30]. We put forward the
use of a new probe design for which the source and
the detector fibers are tilted with respect to the tissue
surface but, unlike previously published designs, the
fibers are kept parallel to each other as observed in
Fig. 1. This type of fiber probe geometry offers the
benefits of convenience for clinical applications with
endoscopes, and ease of fabrication. Further, we pro-
pose an innovative and simple analytical model that
can extract the absorption and scattering coefficients
of a turbid medium using a small source-detector
separation, based on a simple analysis of the reflec-
tance spectrum. The objective is to develop a diag-
nostic tool that would facilitate the determination of
biologically relevant parameters such as size and size
distribution of cellular organelles, tissue blood con-
tent, and hemoglobin oxygen saturation. The model is
developed and validated using Monte Carlo (MC)
simulations and experiments with tissue phantoms.
2. Materials and Methods
A. Monte Carlo Simulations
A MC code that simulates light transport within a
scattering and absorbing medium has been developed
based on previous codes [31,32], using a variance
reduction technique [33]. The MC code simulates a
probe that comprises two fibers (a source and a de-
tector), as depicted in Fig. 1. Each fiber has a core
diameter of 200 m and a numerical aperture (NA) of
0.22 in air. The fibers have a center-to-center sepa-
ration of 250 m. Both fibers can be tilted at an angle
() relative to the tissue surface normal, in the x–z
plane, such that the fibers are parallel to each other.
Four tilt angles have been investigated: 0°, 15°, 30°,
and 45°. The surfaces of the fibers were modeled to be
polished at an angle such that the faces of the fibers
were parallel to the surface of the medium. Thus, the
faces of the fibers had a circular or elliptical shape
depending on whether ⫽0or⬎0, respectively.
Photons were launched from points within the sur-
face of the source fiber into the medium, with an
angle within the NA and appropriate tilt angle of the
fiber (including index mismatch correction). The
launch point and angle of each photon were chosen by
using a random uniform distribution. The propaga-
tion of the photons within the medium was deter-
mined by the absorption coefficient, the scattering
coefficient, and the Henyey–Greenstein (HG) [34] or
modified Henyey–Greenstein (MHG) [9,35] phase
function. Unless otherwise indicated, throughout this
paper we will be referring to the HG phase function.
Photons were terminated from the simulation when
they had traveled farther than a specified distance
from the source fiber, had a path length larger than a
specified value, or had left the surface of the medium.
The specified distance and path-length values were
selected such that they had a negligible effect on the
results of the simulation. The simulation also ac-
counts for reflection at the surface of the medium due
to the mismatch of the index of refraction. Photons
were collected when they arrived at the surface of the
medium within the diameter, NA, and tilt angle of
the collection fiber. The index of refraction of the
fibers and the medium were set to 1.5 and 1.4, re-
spectively. Over 2 ⫻ 10
6
photons were tracked per
simulation.
The paths of photons traveling from a source to a
detector fiber depend on the optical properties of the
medium and on the fiber probe geometry. When the
fibers are tilted at an angle
f
, the light is bent inside
the medium to an angle
m
, which is described by
Snell’s law 共n
f
sin
f
⫽ n
m
sin
m
兲, where n
f
and n
m
are
the indices of refraction of the fiber and the medium,
respectively. Since the index of the fiber is typically
higher than the index of the medium, then
m
⬎
f
.
Figure 2 depicts the normalized voxel visitation his-
tory of the collected photons from a side view (as
indicated in Fig. 1) for the 0° and 45° fiber probe
configurations in a medium with
a
⫽ 0.1 cm
⫺1
, ⬘
s
Fig. 2. Normalized voxel visitation history of Monte Carlo simu-
lations for the (a) 0° and (b) 45° fiber probe configuration. The
dotted line depicts a depth of 300 m.
Fig. 1. Diagram of the fiber probe design used in both Monte
Carlo simulations and experiments in tissue phantoms.
7318 APPLIED OPTICS 兾 Vol. 46, No. 29 兾 10 October 2007
⫽ 10 cm
⫺1
, and g ⫽ 0.9. The dotted white line repre-
sents a depth of 300 m. It is observed that the 45°
fiber probe configuration is more sensitive to collect-
ing light in superficial volumes of tissue, compared to
the 0° fiber probe configuration. It is important to
note that the depth of penetration of the collected
photons is also dependent on the optical properties of
the tissue.
B. Instrumentation
The experimental setup for the reflectance mea-
surements consisted on a pulsed Xenon-arc lamp
(LS-1130-3, Perkin Elmer) as a broadband light
source, a spectrometer (S2000, Ocean Optics, Inc.)
and a fiber probe for the delivery and collection of the
light to and from the sample. Four fiber probe con-
figurations were fabricated with tilt angles of 0°, 15°,
30°, and 45°, as depicted in Fig. 1. The probes had two
multimode optical fibers (source and detector), which
are parallel to each other. The center-to-center sepa-
ration between the fibers was approximately 250 m.
Each optical fiber had a core diameter of 200 m and
a NA of 0.22 in air. The tips of the probes were pol-
ished parallel to the surface of the medium for each of
the angles.
The tilt angles of the fiber probes were verified by
inserting the angled tip in a dilute solution of tita-
nium dioxide and water, and measuring the output
angle of a HeNe 共632.8 nm兲 laser. Snell’s law was
used to determine that the probes were within 1.5° of
the expected tilt angle.
C. Liquid Tissue Phantoms
Liquid tissue phantoms were prepared using deion-
ized water, Intralipid-10% (Fresenius Kabi) as a
source of scattering and Indigo Blue dye (Daler-
Rowney) as an absorber. Intralipid-10% has been pre-
viously demonstrated to scatter light preferentially
in the forward direction [36]. The wavelength-
dependent extinction coefficient of the Indigo Blue
dye was measured using a spectrophotometer (Var-
ian, Cary-50), and the absorption spectrum, normal-
ized to the peak at 610 nm, is shown in Fig. 3. The
reduced scattering coefficients of the tissue phantoms
were determined using a method of spatially depen-
dent diffuse reflectance spectroscopy [37]. Known
amounts of dye were added to the phantom to obtain
the appropriate absorption coefficient. It is important
to note that the amount of dye solution added did not
exceed 0.5% of the volume of the phantom; therefore,
the alterations of the scattering properties were neg-
ligible. Unless otherwise indicated, the values of the
reduced scattering coefficient and absorption coeffi-
cient discussed in this paper are at a wavelength of
610 nm. Each phantom contained 200 ml of solution
and was placed inside a cylindrical container.
D. Calibration Phantom
A calibration liquid phantom was made by suspending
0.16 grams of titanium dioxide powder (J. T. Baker) in
200 ml of deionized water. Titanium dioxide has an
anisotropy value of ⬃0.5 in aqueous suspension [38].
No absorber was added to the calibration phantom.
The reduced scattering coefficient of the calibration
phantom was determined using the method of spa-
tially dependent diffuse reflectance spectroscopy [37].
E. Data Acquisition
Spectral measurements were obtained by subtracting
a dark measurement (lamp was not fired) from a light
measurement (lamp was fired). Averages of 15 mea-
surements were taken for each spectrum. Reflectance
values were calculated by dividing each spectrum by
the spectrum obtained with the calibration phantom.
The integration time of the spectrometer was the
same for the measurements taken by both the tissue
and the calibration phantoms, and it was varied be-
tween 8 and 60 ms such that the signal to noise ratio
was at least 25:1.
The measurements were taken by inserting the
fiber probe into the liquid phantoms. The diameter of
the fiber probe housing was large enough 共3mm兲
compared to the fiber separation 共250 m兲, such that
there was no difference between measurements
taken at the surface of the phantom or submerging
the probe inside the sample. The end of the fiber
probe was located more than 1 cm from the bottom
and walls of the container to avoid interference from
the boundaries.
3. Results
A. Reflectance as a Function of the Reduced Scattering
Coefficient
For the development of the model, the reflectance
from a medium that scatters light was analyzed
for conditions with very little absorption 共
a
ⱕ
0.01 cm
⫺1
兲. MC simulations were run for a medium
with an absorption coefficient of 0.01 cm
⫺1
and for
five values of reduced scattering coefficient (⬘
s
⫽ 5,
10, 15, 20, and 25 cm
⫺1
). The anisotropy value was set
to 0.9. Four different fiber tilt angles were modeled
Fig. 3. Absorption spectrum of Indigo Blue dye normalized to the
peak at 610 nm.
10 October 2007 兾 Vol. 46, No. 29 兾 APPLIED OPTICS 7319
(⫽0°, 15°, 30°, and 45°). The absolute reflectance
共R
ABS
兲, is defined as the ratio of the collected light (I)
over the incident light 共I
0
兲, and is calculated using
Eq. (1):
R
ABS
⫽
I
I
0
⫽
兺
i⫽1
TPC
exp
共
⫺
a
l
i
兲
TPL
, (1)
where TPC is the total number of photons collected,
TPL is the total number of photons launched,
a
is
the absorption coefficient, and l
i
is the path length of
each collected photon. To analyze the reflectance ex-
perimentally, eight tissue phantoms were prepared
with different concentrations of Intralipid-10% in
deionized water. No dye was added; therefore the
absorption coefficient was assumed to be negligible
for the spectral range studied. The resulting reduced
scattering coefficients of the tissue phantoms varied
between 3 and 20 cm
⫺1
. Measurements were taken on
each phantom with each of the four fiber optic probe
configurations. It is difficult to determine experimen-
tally the absolute value of the incident light; however,
a relative value of the incident light can be obtained
by measuring the collected light from a calibration
phantom. The relative reflectance of a phantom
关R
P
REL
共兲兴 has been defined as the ratio of the absolute
reflectance of a phantom 关R
P
ABS
共兲兴 over the absolute
reflectance of a calibration phantom 关R
C
ABS
共
0
兲兴.
Throughout this paper, R
C
ABS
共
0
兲 represents the abso-
lute reflectance of the calibration phantom at
0
⫽ 610 nm, which will be a constant given that we
always use the same calibration phantom and wave-
length. If the incident light of the reflectance mea-
surements in the tissue and calibration phantom is
the same 关I
0
共兲 ⫽ I
0
共
0
兲兴, the expression for the rela-
tive reflectance will be independent of the incident
light, as described in Eq. (2).
R
P
REL
共
兲
⫽
R
P
ABS
共
兲
R
C
ABS
共
0
兲
⫽
I
P
共
兲
I
0
共
兲
I
0
共
0
兲
I
C
共
0
兲
⫽
I
P
共
兲
I
C
共
0
兲
, (2)
where I
P
共兲 is the collected light from the tissue phan-
tom and I
C
共
0
兲 is the collected light from the calibra-
tion phantom at 610 nm. Note that since R
C
ABS
共
0
兲 is a
constant, R
P
REL
共兲 is linearly proportional to R
P
ABS
共兲.
The relative reflectance of the tissue phantoms
were calculated, at the wavelength of 610 nm, as the
ratio of the tissue phantom measurement over the
measurement obtained with the calibration phan-
tom. The incident light on the tissue and calibration
phantom was the same.
The reflectance as a function of the reduced scatter-
ing coefficient, using the 45° fiber probe configuration,
obtained with MC simulations and experiments in tis-
sue phantoms is shown in Figs. 4(a) and 4(b), respec-
tively. It is observed that there is a linear relationship
between the reduced scattering coefficient and the re-
flectance. Equation (3) is a straight-line fit to the data
using linear least squares, where R共0兲 is the reflec-
tance when there is negligible absorption.
R
共
0
兲
⫽ a⬘
s
⫹ a
0
. (3)
Similar results were obtained with the 0°, 15°, and
30° fiber probe configurations (data not shown).
The determined values of the coefficients a and a
0
are presented in Table 1. The reason that the values
obtained with MC simulations and experiments in
tissue phantoms differ is because MC simulations
determine absolute reflectance values, and the exper-
iments determine relative reflectance values (i.e., we
did not attempt to measure the precise amount of
light emitted by the source fiber). The value of a
0
is a
function of the anisotropy value and phase function of
the scattering centers, as it will be discussed in Sub-
section 3.C.
B. Reflectance as a Function of the Reduced Scattering
and Absorption Coefficient
To further develop the model, the reflectance was an-
alyzed for conditions where the tissue phantom both
scatters and absorbs light. MC simulations were run
for a medium with 19 values of absorption coefficient
(
a
⫽ 0.1 to 10 cm
⫺1
) and for three values of reduced
scattering coefficient (⬘
s
⫽ 5, 10, and 20 cm
⫺1
). The
anisotropy value was set to 0.9. Four different tilt an-
gles were modeled (⫽0°, 15°, 30°, and 45°). The
absolute reflectance was calculated using Eq. (1).
For the experiments, liquid tissue phantoms were
prepared with three values of reduced scattering co-
efficient (⬘
s
⫽ 5.5, 10.6, and 20.7 cm
⫺1
) and 19 values
of absorption coefficient (
a
⫽ 0.12 to 8 cm
⫺1
). The
values of the optical properties, quoted at 610 nm,
were selected based on the range of values commonly
found in tissue [39]. Measurements were taken on
each phantom with each of the four fiber optic probe
Fig. 4. Reflectance as a function of the reduced scattering coeffi-
cient for the 45° fiber probe configuration in a nonabsorbing
medium obtained with (a) Monte Carlo simulations and (b) exper-
iments in tissue phantoms.
Table 1. Values for the Coefficients a and a
0
Obtained With Monte
Carlo Simulations and Experiments in Tissue Phantoms With a 0°, 15°,
30°, and 45° Fiber Probe Configuration
Fiber Tilt Angle
0° 15° 30° 45°
a
MC 1.2 ⫻ 10
⫺5
1.1 ⫻ 10
⫺5
1.1 ⫻ 10
⫺5
1.0 ⫻ 10
⫺5
Experiment 0.11 0.11 0.11 0.10
a
0
MC 0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
Experiment ⫺0.04 ⫺0.04 ⫺0.03 ⫺0.01
7320 APPLIED OPTICS 兾 Vol. 46, No. 29 兾 10 October 2007
configurations. The relative reflectance was calcu-
lated at the wavelength of 610 nm by dividing the
tissue phantom measurements by the measurements
obtained with the calibration phantom.
The reflectance as a function of the absorption co-
efficient for several reduced scattering coefficient val-
ues, using the 45° fiber probe configuration, obtained
with MC simulations and experiments in tissue
phantoms is presented in Figs. 5(a) and 5(b), respec-
tively. Similar results were obtained with the 0°, 15°,
and 30° fiber probe configurations (data not shown).
From the previous section, it was determined that
the reflectance is linearly proportional to the reduced
scattering coefficient when there is negligible absorp-
tion. Beer’s law states that the intensity should decay
exponentially as a function of the absorption coeffi-
cient; therefore, we expand the model using Eq. (4):
R
共
a
兲
⫽ R
共
0
兲
exp
共
⫺
a
具L典
兲
, (4)
where R共0兲 is given by Eq. (3), and 具L典 is the “mean
average path length” of the collected photons, as de-
fined in [40].
The mean average path length of the collected
photons can be described as being inversely propor-
tional to both the scattering and absorption prop-
erties of the medium, for small source-detector
separations. For that geometry, the collected photons
from a highly scattering medium reverse their direc-
tion close to the surface of the tissue and travel a
short path. Conversely the photons collected from a
low-scattering medium penetrate a longer distance
into the medium before scattering and reversing their
directions. The mean average path length of the col-
lected photons is also longer in a low-absorption me-
dium than in a high-absorbing medium because the
probability for a collected photon to travel a long path
is reduced for the latter case. The model for the mean
average path length is given by Eq. (5):
具L典 ⫽
b
共
a
⬘
s
兲
c
, (5)
where b and c are fitting coefficients. Equation (6) is
obtained by combining Eqs. (3), (4), and (5).
R
共
a
兲
⫽
共
a⬘
s
⫹ a
0
兲
exp
冉
⫺
a
b
共
a
⬘
s
兲
c
冊
. (6)
Equation (6) was used to provide the fit to the reflec-
tance measurements shown in Fig. 5, for which a, a
0
,
a
, and ⬘
s
were known, and the values of b and c were
determined (Tables 2 and 3). It is noted that for a
given tilt-angle fiber probe, the values of b and c agree
well for the three different reduced scattering coeffi-
cients, which helps to validate the model. The values
of b differ between the MC simulations and the ex-
periments, while the values of c agree well.
We repeated the MC simulations with the same
conditions described in Subsections 3.A and 3.B but
the indices of refraction of the fiber 共n
f
兲 and medium
共n
m
兲 were set to 1.46 and 1.33, respectively. The sim-
ulations were run only for the 0° and 45° fiber probe.
The value of the coefficients a and a
0
and the mean
value of the coefficients b and c are listed in Table 4
under column B. The values of the coefficients ob-
tained with MC simulations in Tables 1, 2, and 3 are
listed in Table 4 under column A. As a result, the
values of the coefficients do not depend on the indices
of refraction of the fiber and medium.
We also repeated the MC simulations with the same
conditions described in Subsections 3.A and 3.B but
with the center-to-center fiber separation set to
200 m. The simulations were only run for the 0° and
Fig. 5. Reflectance as a function of the absorption coefficient for
the 45° fiber probe configuration obtained with (a) Monte Carlo
simulations and (b) experiments in tissue phantoms.
Table 2. Values for the Coefficient b Obtained With Monte Carlo
Simulations and Experiments in Tissue Phantoms With a 0°, 15°, 30°,
and 45° Fiber Probe Configuration
Fiber Tilt Angle
0° 15° 30° 45°
MC
s⬘⫽5 0.33 0.32 0.32 0.32
s⬘⫽10 0.33 0.31 0.31 0.29
s⬘⫽20 0.30 0.30 0.30 0.28
Mean 0.32 0.31 0.31 0.30
Experiment
s⬘⫽5.5 0.20 0.26 0.25 0.26
s⬘⫽10.6 0.22 0.25 0.25 0.25
s⬘⫽20.7 0.23 0.25 0.23 0.24
Mean 0.22 0.25 0.24 0.25
Table 3. Values for the Coefficient c Obtained With Monte Carlo
Simulations and Experiments in Tissue Phantoms With a 0°, 15°, 30°,
and 45° Fiber Probe Configuration
Fiber Tilt Angle
0° 15° 30° 45°
MC
s⬘⫽5 0.21 0.21 0.21 0.20
s⬘⫽10 0.21 0.22 0.21 0.19
s⬘⫽20 0.20 0.20 0.20 0.19
Mean 0.21 0.21 0.21 0.19
Experiment
s⬘⫽5.5 0.20 0.20 0.20 0.20
s⬘⫽10.6 0.20 0.20 0.20 0.20
s⬘⫽20.7 0.20 0.20 0.20 0.20
Mean 0.20 0.20 0.20 0.20
10 October 2007 兾 Vol. 46, No. 29 兾 APPLIED OPTICS 7321
45° fiber probe. The values of the coefficients a and a
0
and the mean value of the coefficients b and c are
listed in Table 4 under column C. The values of a and
b depend on the center-to-center separation of the
fibers.
Finally, we reanalyzed the same experimental data
described in Subsections 3.A and 3.B, but we used a
calibration phantom prepared with deionized water
and Intralipid. The values of a obtained for the 0°, 15°,
30°, and 45° fibers were 0.19, 0.18, 0.18, and 0.18,
respectively, and the values for a
0
were ⫺0.06, ⫺0.06,
⫺0.05, and ⫺0.02, respectively. The values of the co-
efficients b and c were exactly the same as the values
listed in Tables 2 and 3, indicating that the calibration
phantom only affects the coefficients a and a
0
in a
linear manner. The fact that only the coefficients a
and a
0
are dependent on the calibration phantom is
reasonable, because a change in the calibration phan-
tom is reflected in the value of R
C
ABS
共
0
兲; therefore,
there is only a linear change in R
P
REL
共兲, as defined by
Eq. (2).
C. Reflectance as a Function of the Anisotropy Value
and the Phase Function
At small source-detector separations, where the dif-
fusion approximation does not apply, the reflectance
of light is a function of the absorption coefficient, the
scattering coefficient, the phase function of the scat-
tering centers, the boundary conditions and the prop-
erties of the source and collection fibers. The phase
functions have typically been modeled using either
Mie theory [41], the HG approximation [34] or the
MHG approximation [9,35]. It has been previously
reported that the choice of phase function can have a
significant effect on the reflectance of light especially
for small source-detector separations [42,43]. When
using a monodisperse suspension of spherical parti-
cles, the Mie scattering presents very distinguishable
oscillations, but as the size distribution increases, the
Mie oscillations smooth out [44], and the signal is also
masked by a background of diffusely scattered light
from the underlying tissue [45]. To determine the
influence of the anisotropy value on the reflectance
measurements, we decided to use the HG approxima-
tion, because it has been commonly used to model the
phase function of scattering centers at small source-
detector separations [7,8,11,12,14,16,19,21,46,47].
MC simulations were run for a medium with an ab-
sorption coefficient of 0.01 cm
⫺1
and for a fiber probe
tilt angle of 0° and 45°. Fifteen simulations were
performed, for each tilt angle, using five different
anisotropy values (g ⫽ 0.2, 0.5, 0.75, 0.9, and 0.95),
and three different reduced scattering coefficient val-
ues (⬘
s
⫽ 5, 10, and 20 cm
⫺1
). The absolute reflec-
tance was determined using Eq. (1). For each reduced
scattering coefficient, the percentage variation (PV)
of the reflectance referenced to the reflectance with
g ⫽ 0.9 was calculated using Eq. (7), where R1isthe
reflectance at a given g value and R2 is the reflec-
tance at g ⫽ 0.9.
PV ⫽
共
R1 ⫺ R2
兲
R2
⫻ 100. (7)
The PV, for the 0° degree fiber tilt angle, as a function
of the anisotropy value for different reduced scatter-
ing coefficients is depicted in Fig. 6. Higher reduced
scattering coefficient values exhibit lower percentage
variation as a function of the g value, compared to
lower reduced scattering coefficient values. We note
that the percentage variation for all the values of ⬘
s
is less than 15% for typical biological tissue anisot-
ropy values 共g ⬎ 0.75兲. Similar results were obtained
with the 45° tilt angle.
We repeated the MC simulations described in Sub-
sections 3.A and 3.B to calculate the values of the
Table 4. Values for the Coefficients a, a
0
b, and c Obtained With Monte Carlo Simulations With the 0° and 45° Fiber Probe Configuration Under
Different Conditions
a
ABCDEFGH
Parameters
n
f
1.5 1.46 1.5 1.5 1.5 1.5 1.5 1.5
n
m
1.4 1.33 1.4 1.4 1.4 1.4 1.4 1.4
Separation (m) 250 250 200 250 250 250 250 250
Phase function HG HG HG MHG HG HG HG HG
g 0.9 0.9 0.9 0.9 0.2 0.5 0.75 0.95
Tilt angle
0°
a 1.2 ⫻ 10
⫺5
1.2 ⫻ 10
⫺5
1.4 ⫻ 10
⫺5
1.2 ⫻ 10
⫺5
1.1 ⫻ 10
⫺5
1.1 ⫻ 10
⫺5
1.2 ⫻ 10
⫺5
1.2 ⫻ 10
⫺5
a
0
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
1.5 ⫻ 10
⫺5
4.8 ⫻ 10
⫺5
1.7 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
b 0.32 0.30 0.26 0.31 0.36 0.32 0.31 0.31
c 0.21 0.21 0.21 0.21 0.20 0.20 0.20 0.19
45°
a 1.0 ⫻ 10
⫺5
1.1 ⫻ 10
⫺5
1.3 ⫻ 10
⫺5
1.0 ⫻ 10
⫺5
9.9 ⫻ 10
⫺5
1.0 ⫻ 10
⫺5
1.1 ⫻ 10
⫺5
1.0 ⫻ 10
⫺5
a
0
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
2.1 ⫻ 10
⫺5
4.0 ⫻ 10
⫺5
2.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
0.0 ⫻ 10
⫺5
b 0.30 0.30 0.25 0.30 0.35 0.32 0.30 0.30
c 0.19 0.20 0.20 0.20 0.20 0.21 0.21 0.21
a
Bold indicates the changes in the parameters referenced to the parameters described in column A. See text for details.
7322 APPLIED OPTICS 兾 Vol. 46, No. 29 兾 10 October 2007
coefficients a and a
0
and the mean value of the coef-
ficients b and c with the 0° and 45° fiber probes for the
anisotropy values of 0.2, 0.5, 0.75, and 0.95. The re-
sults for the anisotropy values of 0.2, 0.5, 0.75, and
0.95 are listed in Table 4 under columns E, F, G, and
H, respectively. The values of a
0
and b vary signifi-
cantly from the results in column A for low g values,
and do not depend on the anisotropy value for high g
values. For biological tissues we are only interested
in high anisotropy values.
To analyze the influence of the phase function on
our model, we altered our MC code to model a MHG
phase function [9,35]. The angular scatter distribu-
tion of the MHG 共p
MHG
兲 is described by Eq. (8).
p
MHG
共
, g
HG
,
兲
⫽p
HG
共
, g
HG
兲
⫹
共
1 ⫺
兲
3
4
cos
2
,
(8)
where g
HG
and p
HG
are the anisotropy value and the
angular scatter distribution of the HG phase func-
tion, respectively, and is a normalization factor. The
value of g
HG
was set to 0.9165 and was set to 0.9,
such that the anisotropy value of the MHG was 0.9.
Figure 7(a) plots the angular scatter distribution of
the HG and MHG phase functions where both anisot-
ropy values are equal to 0.9.
We performed MC simulations with the HG and
MHG phase function, for the 0° and 45° fiber probe,
with a ⬘
s
⫽ 5, 10, and 20 cm
⫺1
; g ⫽ 0.9; and
a
⫽ 0.01 cm
⫺1
. Figure 7(b) shows the PV between the
reflectance obtained with the HG and the MHG
phase functions, given by Eq. (7) where R1isthe
reflectance obtained with the MHG and R2isthe
reflectance obtained with the HG phase function at a
given ⬘
s
. We observe that higher ⬘
s
values present
smaller percentage variation between the two phase
functions.
The MC simulations described in Subsections 3.A
and 3.B were repeated with the MHG phase function
for the 0° and 45° fiber probe. The values of the co-
efficients a and a
0
and the mean value of the coeffi-
cients b and c are listed in Table 4 under column D.
Only the value of a
0
appears to vary significantly from
the results presented in column A.
D. Extraction of the Optical Properties of a Tissue Model
from Monte Carlo Simulations
To validate the model described by Eq. (6), the optical
properties of a scattering and absorbing medium
were extracted from the reflectance obtained from
MC simulations (Subsection 3.D) and from experi-
ments in tissue phantoms (Subsection 3.E). The value
of a
0
is dependent on both the phase function and
anisotropy value, as described in Subsection 3.C.
The phase function and anisotropy value of the
scattering centers are wavelength dependent, and
are typically unknown in biological tissue; therefore,
to simplify the use of the model described in Eq. (6),
we set the value of a
0
to 0.
Two absolute reflectance spectra were generated
with MC simulations using the 45° fiber probe. Both
spectra have the same reduced scattering coefficient,
but different absorption coefficient values, which will
be referred to as Low
a
and High
a
spectra. The
wavelengths were selected for the range between 500
and 750 nm in 10 nm steps. The reduced scattering
coefficient was modeled using Eq. (9):
⬘
s
共
兲
⫽ d
⫺e
. (9)
The value of the exponent e was set to 1.1, based on
typical values found in biological tissue [48], and the
value of d was selected such that ⬘
s
共610 nm兲 ⫽
10 cm
⫺1
. The anisotropy value was held constant at
0.9 for all wavelengths. The absorption coefficient
was represented by Eq. (10):
a
共
兲
⫽ f
1
共
f
2
HbO
共
兲
⫹
共
1 ⫺ f
2
兲
Hb
共
兲兲
, (10)
where
HbO
共兲 and
Hb
共兲 are the extinction coeffi-
cients of oxyhemoglobin and deoxyhemoglobin, re-
spectively. The values of f
1
were selected such that
a
共610 nm兲 ⫽ 0.2 cm
⫺1
for the Low
a
spectrum, and
a
共610 nm兲 ⫽ 1cm
⫺1
for the High
a
spectrum. The
value of f
2
, which represents the oxygen saturation,
was set to 0.8.
Fig. 6. Percentage variation of the reflectance as a function of the
anisotropy value for the 0° fiber probe configuration obtained with
Monte Carlo simulations.
Fig. 7. (a) Angular scatter distribution of the HG and MHG phase
function. (b) Percentage variation of the reflectance between the
HG and MHG phase function as a function of the reduced scatter-
ing coefficient for the 0° and 45° fiber probe configuration obtained
with Monte Carlo simulations.
10 October 2007 兾 Vol. 46, No. 29 兾 APPLIED OPTICS 7323
To extract the optical properties from the MC sim-
ulation, Eq. (6) was fitted to the absolute reflectance
spectrum, for which the values of a, b, and c were
obtained from Tables 1, 2, and 3, respectively; d, e, f
1
,
and f
2
were the fitting parameters; and a
0
was set to
0. The parameters were bounded as 0 ⱕ d, e, f
1
and
0 ⱕ f
2
ⱕ 1.
Figure 8(a) shows the absolute reflectance spec-
trum and the least square fit to the model, for the
Low
a
and High
a
spectra. The actual and extracted
wavelength-dependent reduced scattering coeffi-
cients are shown in Fig. 8(b), and the actual and
extracted wavelength-dependent absorption coeffi-
cients are shown in Fig. 8(c). The values of hemoglo-
Fig. 8. (a) Absolute reflectance spectrum and model fit for a
Monte Carlo simulation with a 45° fiber probe configuration.
Actual and extracted (b) reduced scattering coefficient and (c) ab-
sorption coefficient.
Fig. 9. (a) Relative reflectance spectrum and model fit for exper-
iments in tissue phantoms with a 45° fiber probe configuration.
Actual and extracted (b) reduced scattering coefficient and (c) ab-
sorption coefficient.
7324 APPLIED OPTICS 兾 Vol. 46, No. 29 兾 10 October 2007
bin oxygen saturation 共f
2
兲 obtained were 0.826 and
0.836 for the Low
a
and High
a
, respectively.
E. Extraction of the Optical Properties of a Tissue
Phantom
Eight liquid tissue phantoms were prepared with dif-
ferent optical properties. Two reduced scattering co-
efficients were used, such that ⬘
s
共610 nm兲 was 9.2
and 17.5 cm
⫺1
, and four absorption coefficients were
used such that
a
共610 nm兲 was 0, 0.5, 2.8, and
8.5 cm
⫺1
. Measurements were taken with the four
fiber probe configurations.
Using Eq. (3), the values of a obtained in Table 1
and the determined values of the reduced scattering
coefficient of the calibration phantom ⬘
sC
, it was pos-
sible to determine theoretically the relative re-
flectance of the calibration phantom 关R
C
REL
共兲 ⫽
R
C
ABS
共兲兾R
C
ABS
共
0
兲 ⫽ a⬘
sC
共兲 ⫹ a
0
兴 when the absorption
coefficient is negligible. Because we do not know the
wavelength-dependent phase function and anisot-
ropy value of the calibration phantom, we simplify
the use of the model by setting the value of a
0
to 0.
The incident light at a given wavelength is not nec-
essarily equal to I
0
共
0
兲; therefore, the relative reflec-
tance of the tissue phantom as a function of
wavelength 关R
P
REL
共兲兴, can be calculated with Eq. (11).
R
P
REL
共
兲
⫽
R
P
ABS
共
兲
R
C
ABS
共
0
兲
⫽
R
P
ABS
共
兲
R
C
ABS
共
0
兲
R
C
ABS
共
0
兲
R
C
ABS
共
兲
R
C
ABS
共
兲
R
C
ABS
共
0
兲
⫽
R
P
ABS
共
兲
R
C
ABS
共
兲
R
C
REL
共
兲
⫽
I
P
共
兲
I
0
共
兲
I
0
共
兲
I
C
共
兲
a⬘
sC
共
兲
⫽
I
P
共
兲
I
C
共
兲
a⬘
sC
共
兲
, (11)
where I
P
共兲 is the collected light obtained from the
tissue phantom and I
C
共兲 is the collected light ob-
tained from the calibration phantom.
Equation (6) was then fitted to R
P
REL
共兲 for the
wavelengths of 500–750 nm. The reduced scattering
coefficient was modeled by Eq. (9) and the absorption
coefficient was modeled with Eq. (12):
a
共
兲
⫽ f
1
dye
共
兲
, (12)
where
dye
共兲 is the extinction coefficient of the Indigo
Blue dye. The fitting coefficients were bounded by 0
ⱕ d, e, f
1
; the values of a, b, and c were obtained from
Tables 1, 2, and 3, respectively; and a
0
was set to 0.
Figure 9(a) shows the relative reflectance spectrum
and the least square fit to the model, using the 45°
fiber optic probe configuration, for ⬘
s
共610 nm兲 ⫽
17.5 cm
⫺1
and
a
共610 nm兲 ⫽ 0.5 and 8.5 cm
⫺1
. The
actual and extracted wavelength-dependent reduced
scattering coefficients are shown in Fig. 9(b), and the
actual and extracted wavelength-dependent absorp-
tion coefficients are shown in Fig. 9(c).
Tables 5 and 6 present the mean and standard
deviation values of the absorption and reduced scat-
tering coefficients over 15 measurements determined
with all the fiber probe configurations in the eight
phantoms at 610 nm, respectively. The errors in es-
timating the absorption and scattering coefficients
between the mean
a
and actual
a
was less than
Table 5. Mean and Standard Deviation Over 15 Measurements of the Absorption Coefficient Extracted From Experiments in Tissue Phantoms with
a 0°, 15°, 30°, and 45° Fiber Probe Configuration at 610 nm
a
Actual
a
Actual
s
⬘
Fiber Tilt Angle
0° 15° 30° 45°
0 9.2 0.00 ⫾ 0.00 0.00 ⫾ 0.00 0.00 ⫾ 0.00 0.00 ⫾ 0.00
17.5 0.00 ⫾ 0.00 0.00 ⫾ 0.00 0.00 ⫾ 0.00 0.00 ⫾ 0.00
0.5 9.2 0.58 ⫾ 0.03 0.58 ⫾ 0.01 0.49 ⫾ 0.00 0.58 ⫾ 0.02
17.5 0.58 ⫾ 0.02 0.58 ⫾ 0.02 0.49 ⫾ 0.02 0.53 ⫾ 0.02
2.8 9.2 2.71 ⫾ 0.04 3.18 ⫾ 0.03 2.62 ⫾ 0.02 2.89 ⫾ 0.04
17.5 2.70 ⫾ 0.04 3.10 ⫾ 0.02 2.45 ⫾ 0.14 2.82 ⫾ 0.03
8.5 9.2 7.98 ⫾ 0.14 8.75 ⫾ 0.12 8.01 ⫾ 0.16 8.53 ⫾ 0.19
17.5 8.39 ⫾ 0.06 8.56 ⫾ 0.52 8.23 ⫾ 0.05 8.57 ⫾ 0.08
a
Units are in cm
⫺1
.
Table 6. Mean and Standard Deviation Over 15 Measurements of the
Reduced Scattering Coefficient Extracted From Experiments in Tissue
Phantoms With a 0°, 15°, 30°, and 45° Fiber Probe Configuration at
610 nm
a
Actual
s
⬘
Actual
a
Fiber Tilt Angle
0° 15° 30° 45°
9.2 0 8.7 ⫾ 0.3 8.8 ⫾ 0.1 9.1 ⫾ 0.1 9.6 ⫾ 0.1
0.5 9.0 ⫾ 0.3 9.6 ⫾ 0.1 9.0 ⫾ 0.1 9.1 ⫾ 0.2
2.8 8.5 ⫾ 0.3 9.5 ⫾ 0.2 9.7 ⫾ 0.0 9.2 ⫾ 0.3
8.5 9.3 ⫾ 0.3 9.6 ⫾ 0.2 9.3 ⫾ 0.1 8.8 ⫾ 0.3
17.5 0 16.9 ⫾ 1.1 17.4 ⫾ 0.5 17.4 ⫾ 0.2 18.3 ⫾ 0.6
0.5 17.4 ⫾ 0.9 19.0 ⫾ 1.0 17.3 ⫾ 0.2 17.3 ⫾ 0.9
2.8 16.4 ⫾ 0.7 18.4 ⫾ 0.4 17.6 ⫾ 1.1 18.3 ⫾ 0.4
8.5 18.5 ⫾ 0.4 17.1 ⫾ 2.0 17.2 ⫾ 0.2 16.4 ⫾ 0.4
a
Units are in cm
⫺1
.
10 October 2007 兾 Vol. 46, No. 29 兾 APPLIED OPTICS 7325
20% and between the mean ⬘
s
and actual
s
was less
than 10%.
The analysis was also performed by using a cali-
bration phantom prepared with Intralipid and deion-
ized water, which has different optical properties
than that of a titanium dioxide suspension. The er-
rors in estimating
a
and ⬘
s
were slightly reduced;
however, the disadvantage of using Intralipid as a
calibration phantom is that its optical properties can
change with time as the solution degrades.
4. Discussion and Conclusions
A simple analytical model for describing the reflec-
tance of a scattering and absorbing medium has been
presented. The model is successful in describing the
reflectance spectrum at small source-detector sepa-
rations and is not limited to a parameter range with
scattering much larger than absorption, a limitation
that generally plagues models based on diffusion
theory.
MC simulations indicate that tilting both the
source and the detector optical fibers, such that the
fibers are parallel to each other, enhances the sensi-
tivity to superficial volumes of tissue, while main-
taining a fiber-probe geometry that is convenient for
clinical applications and is easy to fabricate. Further
studies will assess the depth of penetration of the
photons as a function of the tissue optical properties
and fiber tilt angle. The model has been shown to
work well for fiber probe tilt angles between 0° and
45°.
The model has been defined by Eq. (6), and the
coefficients a, a
0
, b, and c have been listed in Tables
1, 2, and 3. The value of the coefficient a differs be-
tween the MC simulations and the experiments in
tissue phantoms because the former measures abso-
lute reflectance while the latter measures relative
reflectance. The values of a and a
0
depend on the
optical properties of the calibration phantom. If a
calibration phantom with different optical properties
is used, new values for a and a
0
should be determined
experimentally as described in Subsection 3.A. A
change in the optical properties of a calibration phan-
tom is reflected in the value R
C
ABS
共
0
兲, which trans-
lates to a linear change of R
P
REL
共兲, as defined by Eq.
(2). Therefore, only the values of a and a
0
are affected
by the details of the calibration phantom, while the
values of b and c remain the same, as it was demon-
strated by comparing the results of using a calibra-
tion phantom made with Intralipid instead of
titanium dioxide. Although any wavelength of the
calibration phantom could be used as a reference, in
this study the value of 610 nm was chosen, because it
matches the absorption peak of the Indigo Blue dye.
Typically, reflectance measurements are calibrated
by dividing the tissue spectral measurement by a
spectrum measured with a spectrally-flat diffuse-
reflector reference material instead of a calibration
phantom measurement as described in this paper.
The advantages of using an immersion-type liquid
calibration phantom as a reference, as described in
this paper, are:
(1) When using a diffuse reflector, the distance
between the fiber probe and the reflector surface
must be fixed and repeatable if absolute values for
optical properties are sought, as opposed to submerg-
ing the fiber probe in the liquid calibration phantom,
which minimizes the possibility of error and is easily
repeatable.
(2) A fiber probe with tips faceted at 45° exhibits
total internal reflection when the fiber tip is in air;
however, most of the light emerges when the fiber tip
is in contact with water (or tissue).
The model for the mean average pathlength ex-
pressed by Eq. (5) was determined intuitively and
verified empirically. Although an analytical deriva-
tion for the mean average path length is not provided,
it has been shown in this paper that Eq. (5) enables
successful modeling of the reflectance spectrum of a
turbid medium with remarkable accuracy, when it is
incorporated in the model defined by Eq. (6).
For a given tilt-angle fiber probe, the values of the
coefficients b and c agree well for different values of
a
and ⬘
s
, which validates the model, as observed in
Tables 2 and 3. The value of c is determined to be
between 0.19 and 0.21 by both the MC simulations
and the experimental results; however, there is dis-
agreement for the value of b. Although we have not
uniquely determined the source for that discrepancy,
we note that it is consistent with the fact that the
center-to-center fiber separation of the fiber probes
fabricated were approximately 238 m, instead of
250 m as used in the simulations. Table 4 indicates
that the value of b depends on the fiber separation, as
observed when comparing b values obtained with
probes that have a separation of 200 and 250 m. The
value of b increases for smaller anisotropy values but
the experimental tissue phantoms are highly forward
scattering; therefore, we do not attribute the varia-
tion of b to small anisotropy values.
Based on MC simulations and the results pre-
sented in Table 4, we can determine that the coeffi-
cients of the model do not depend on the indices of
refraction of the fiber and the medium, and if the
same phase function is used, the coefficients are not
affected for different values of highly forward scat-
tering anisotropies. However, the coefficient a
0
is af-
fected by the phase function and by low anisotropy
values.
At small source-detector separations, the reflec-
tance spectrum is dependent on the details of the
phase function and the anisotropy value of the scat-
tering centers. The reflectance was calculated for
different anisotropy values using the HG approxima-
tion and was also calculated using a MHG phase
functions with an anisotropy value of 0.9. It was de-
termined that the reflectance is sensitive to the an-
isotropy value and phase function for low-scattering
coefficients, but relatively insensitive for high scat-
tering coefficients, where the photons undergo more
7326 APPLIED OPTICS 兾 Vol. 46, No. 29 兾 10 October 2007
scattering events before being collected, which allows
the photons to lose track of their original directions.
This effect can be observed in Figs. 6 and 7(b). It was
also determined that for anisotropy values typically
found in tissue 共g ⬎ 0.75兲, the reflectance varies less
than 15% for all values of ⬘
s
.
The model was tested by reconstructing the optical
properties of a spectral reflectance obtained from MC
simulations. The extracted values for the reduced
scattering and absorption coefficients agree well with
the actual values [Figs. 8(b) and 8(c)]. The values of
the hemoglobin oxygen saturation extracted were
also close to the actual values. This validates the
inversion algorithm with an approach that can be
used in vivo in biological tissue with typical values of
optical properties.
The model was tested in tissue phantoms, and the
values of the extracted optical properties agree well
with the actual values (Tables 5 and 6). The errors for
the absorption coefficient were less than 20%, while
the errors for the reduced scattering coefficient were
less than 10%. Equation (11) depends on R
C
REL
共兲,
which is calculated theoretically by setting a
0
⫽ 0.
The data was reanalyzed by using a calibration phan-
tom made from Intralipid, for which the results were
slightly improved, indicating that the real values of
a
0
in Intralipid, for all wavelengths, are closer to 0
than is the case for a suspension of titanium dioxide.
The disadvantage of using Intralipid as a calibration
phantom is that its optical properties are less stable
over time.
The phase function and anisotropy value of real
tissue is not well-defined and is wavelength depen-
dent; therefore, the specific value of a
0
would be un-
known. By setting a
0
to 0 we are assuming that the
effect of the phase function of the scattering centers is
small throughout the model; hence, the model is a
simplification of light transport in a turbid medium.
Further studies will asses the validity of the model
for a wide range of phase functions and anisotropy
values typically found in tissues. Nevertheless, we
were able to reconstruct the reduced scattering and
absorption coefficients with good accuracy.
The model is applicable for fiber probe configura-
tions with various tilt angles, for the geometry de-
picted in this paper. Fiber probes with this geometry
are compatible with endoscope channels, which will
allow measurements in tissues such as the colon and
esophagus. The model has been empirically derived
and validated using both MC simulations and exper-
iments in tissue phantoms.
The authors acknowledge support from the National
Institutes of Health (NIH) fellowship F31CA119916,
NIH grant U54 CA104677, and the Boston University
Photonics Center.
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