Page 1
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211993
Properties of Photon Density Waves in Multiple
Scattering Media
Bruce J. Tromberg
University of California  Irvine
Lars O. Svaasand
University of Trondheim
TsongTseh Tsay
University of California  Irvine
Richard C. Haskell
Harvey Mudd College
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Recommended Citation
Tromberg, BJ, Svaasand, LO, Tsay, TT, Haskell, RC. Properties of photon density waves in multiplescattering media. Appl Opt.
1993;32(4): 607616.
Page 2
Properties of photon density waves in
multiplescattering media
Bruce J. Tromberg, Lars 0. Svaasand, TsongTseh Tsay, and Richard C. Haskell
Amplitudemodulated light launched into multiplescattering media, e.g., tissue, results in the propaga
tion of density waves of diffuse photons. Photon density wave characteristics in turn depend on
modulation frequency () and media optical properties. The damped spherical wave solutions to the
homogeneous form of the diffusion equation suggest two distinct regimes of behavior: (1) a high
frequency dispersion regime where density wave phase velocity V has a /; dependence and (2) a
lowfrequency domain where V, is frequency independent. Optical properties are determined for various
tissue phantoms by fitting the recorded phase () and modulation (m) response to simple relations for the
appropriate regime. Our results indicate that reliable estimates of tissuelike optical properties can be
obtained, particularly when multiple modulation frequencies are employed.
Key words: Photon density waves, frequencydomain photon migration, tissue optical properties,
multiplescattering media.
Introduction
Much of the driving force behind recent develop
ments in timedomain tissue optical spectroscopyl"2is
derived from studies of shortlightpulse propagation
in multiplescattering media.3'4
uous illumination techniques,5 6pulse propagation
methods can provide information about the distribu
tion of scatterers and absorbers in a single measure
ment.7'8
These optical properties may be used in a
variety of therapeutic and diagnostic techniques in
cluding imaging tissue structure,912monitoring phys
iology,'315and predicting optical dosimetry for laser
based procedures.'6
The conceptual basis for the timedomain approach
generally involves solutions to the radiative transfer
equation17'18using Monte Carlo simulation19'20and
diffusion theory approximations.4'2' Diffusionbased
methods provide relatively straightforward analytical
expressions that describe the shape of a diffusely
reflected or transmitted pulse in terms of the optical
properties of the medium.7
poral broadening of ultrashort pulses can be mathe
In contrast to contin
Thus the observed tem
B. J. Tromberg and TT. Tsay are with Beckman Laser Institute,
University of California, Irvine, Irvine, California 92715; L. 0.
Svaasand is with the University of Trondheim, 7000 Trondheim,
Norway; and R. C. Haskell is with Harvey Mudd College, Clare
mont, California 91711.
Received 4 February 1992.
00036935/93/04060710$05.00/0.
C 1993 Optical Society of America.
matically related to the large number of optical paths
available in multiplescattering media. Since the
introduction of losses (absorbers) reduces the average
path length, absorberdependent changes in pulse
propagation time can be used to calculate absorption
coefficients.7
Frequencydomain optical methods can be adapted
to diffusion theory models in a similar manner.
Fishkin et al.22first suggested that amplitude
modulated light propagates through homogeneous
multiplescattering media as diffuse waves with a
coherent front. These photon density waves can be
characterized by a phase velocity Vp and modulation
wavelength Xm that are primarily functions of media
optical properties. Diffuse wave properties bear no
relationship to corresponding electromagnetic wave
features, since, in turbid media, phase relationships
between optical waves vary in a rapid stochastic
manner.
The power and simplicity of frequencydomain
methods have been demonstrated by Lakowicz and
Bundt23who conducted tissue studies using frequen
cies to 3 GHz and Sevick et al.'3
measurements of hemoglobin saturation at a single
modulation frequency. More recently Patterson et
al.24derived frequencydomain analytic expressions
in semiinfinite media from the Fourier transform of
a timedomain relation.
main techniques are realtime recordings, which com
pared with timedomain methods, place less stringent
demands on the bandwidth of the light source and
who reported
In general frequencydo
1 February 1993 / Vol. 32, No. 4 / APPLIED OPTICS
607
Page 3
detector. When laser diodes and photomultiplier
tubes are employed, instrumentation costs can be
relatively modest. These useful analytical features
suggest the necessity for a thorough practical descrip
tion of frequencydomain measurements.
Previously we extended the diffusion theory model
to include photon density wave behavior over a range
of modulation frequencies and varying optical proper
ties.25 26
In this research, we solve analytically the
timedependent diffusion equation in infinite medium
conditions for sinusoidally amplitudemodulated
waves. Simplifications to our diffusion equation so
lutions are proposed and confirmed experimentally.
In this manner a general model is presented that
describes the unique characteristics of photon density
waves. Historically this approach has been applied
to a variety of physical phenomena including the
description of terrestial temperature oscillations.27
Although this discussion was confined to thermal
diffusion in lossless media, it provides an intuitive
picture of diffusely propagating density waves and
serves as a framework for photon diffusion in lossy
tissues.
Theory
Optical power in multiplescattering media, e.g., tis
sue, is typically characterized by the quantities (p, the
radiant energy fluence rate, and L, the radiance.
The radiant energy fluence rate,
f Ldfl,
Q=0
4,m
is defined as the optical energy flux incident on an
infinitesimally small sphere divided by the cross
sectional area of that sphere. Since the integration
is taken over all solid angles fQ, the fluence rate is a
measure of the total optical flux. The radiance L is
the optical energy flux in some direction per unit solid
angle per unit area orthogonal to that direction. In a
completely isotropic light field, L = qp/(4rr).
When the optical flux is viewed along the axis of a
solid angle element dfl, the irradiance E gives the flux
per unit area orthogonal to this axis and canbe
expressed as
r
(1)
(1)
E =J
L(l I n )dfl = rrL  4 '
=o4
where I and h are, respectively, the unit vectors along
the axis and the outward unit surface normal to the
solidangle element. As diffuse photons propagate
through tissue, the irradiance onto an element of
surface, e.g., a detector, will vary with respect to the
source location. When the surface normal points
toward the source, E is enhanced so that
j
E4+_2
Similarly, E is reduced when the surface normal
points directly away from the source and
4 2
The magnitudej of the transport vector j represents
the total deviation from a completely isotropic distri
bution. This deviation is a consequence of the diffu
sion process since the net transport of diffuse photons
in some direction must be expressed by a higher
radiance in that direction.
To satisfy the requirements for the directional
transport of diffuse photons, the radiance is ex
pressed as a series expansion of the form
(2)
L =
+ oj 1 + 
4,r
The constant a can be determined by combining Eq.
(1) with the fluxdependent expression for E:
Jn
E = ~ ~ + otj.
)(4 f)dfl= + 2a
2r [4
i ]+
T
h
(3)
thus yielding a = 3/4r.
results in the radiance series expansion21:
(p 3.^
L =
4,ir 4r
Substituting into Eq. (2)
+ 3j 1 +...
(4)
where the term p/41T corresponds to a completely
isotropic distribution and [3/(4Tr)]j I represents the
net transport of diffuse photons.
The migration of diffuse photons proceeds from
regions of high to low fluence rate and can be
expressed by Fick's law:
j = 
grad p,
(5)
where 4, the photon diffusion constant, is a composite
function of the scattering or and absorption, 13,
cients21:
coeffi
1
1
(6)
i
3[u(1 g) + 13]
= 3(ueff + )
and the effective scattering coefficient oreff is deter
mined by the average cosine of the scattering angle g.
To satisfy the diffusion approximation,
For diffuse photons migrating out of a unit volume,
the net flux is characterized by
eff >> 13.
divj = 
fi  1py + q.
at
(7)
Equations (5) and (7) can then be combined to yield
the timedependent diffusion equation7 28
XV2p
P
= qc,
(8)
608
APPLIED OPTICS / Vol. 32, No. 4 / 1 February 1993
Page 4
where c is the velocity of light in the medium (we
assume throughout that n
fat emulsions; therefore c
source term for the rate of photon generation per unit
volume, X = tc is the optical diffusivity, and 7 =
is the optical absorption relaxation time.
The homogeneous form of Eq. (8) (i.e., q = 0) can be
employed7and solved analytically for threedimen
sional harmonic waves of the form
1.33 for tissue phantom
2.26 x 108 m/s), q is the
/ Pc
exp(r/8)
exp(kr)
'p(r, t) = oP0 r
+ p
r )exp[i(kir
 t)]
(9)
where the first and second terms on the righthand
side of Eq. (9) are, respectively, the timeindependent
(dc) and timedependent (ac) components of a spa
tially r and temporally t varying source rate cp(r, t), )
is the angular modulation frequency, and kr and ki are
the real and imaginary components of the complex
angular wave number k(kr is attenuation and ki is the
phase per unit length). The dc penetration depth is
given by = 1/[31( + eff)]1l/
photon density wave phase velocity Vp and modula
tion wavelength Xm are by definition
2= (XT)1/2, and the
Xm = 2T/ki,
(10)
Vp = o/ki. (11)
These parameters describe the collective properties of
diffuse density waves, not individual photons. Spe
cifically Xm and Vp are, respectively, measures of the
minimum distance in space between regions with the
same phase of diffuse photon density and phasefront
propagation velocity.
The infinitemedium solutions to Eq. (8) can be
expressed in terms of the complex wave number29:
{[1 + (r)2]1/2 + 11/2,
(12)
(2X7)1/2{[1 + (0)2]1/2
11/2.
(13)
These relations are illustrated in Fig. 1. Although
specific tissuelike optical properties (X = 1.5 x 104
m2/s and r = 0.44 ns) were selected for the Fig. 1
simulation, the general appearance of w versus k
curves is independent of optical properties. Changes
in X and T simply alter the extent of the high and
lowfrequency regimes.
For example, in the highfrequency dispersion re
gime, w >> 1/T and kr = ki
conditions density wave properties are dominated by
scattering. They are heavily damped with a con
stant exp(27rr/Xm) attenuation; thus the amplitude
is reduced to 0.19% of its initial value over a distance
[/(2X)]'/
2.In these
120
800
k. 
/*
400
0
0
0.2 0.4 0.6
0.8
k (mm I
Fig. 1. Theoretical frequency (MHz) versus k (mm') response
derived from analytical solutions to Eq. (8) for the following optical
properties:
T =0.44 ns, creff = 50 cm', andX = 1500 m2/s.
of Xm. The corresponding phase velocity is given by
Vp = (o/ki = (2X6))112,
(14)
and Vp is proportional to 4io. As the modulation
frequency is reduced, light is collected from larger
regions and phase velocity decreases. This behavior
continues until Vp is independent of the modulation
frequency (o << 1/T), and wave properties are domi
nated by absorption (see below).
At higher frequencies there is greater attenuation
and light is collected from smaller regions. The time
required to achieve a steadystate photon distribution
is reduced and Vp increases with V;. Multiple scat
tering cannot occur, however, when the modulation
frequency is greater than the reciprocal of the average
time between scattering events, i.e., w << ceff = 1/TSC
Since the validity of diffusion theory is based on
multiple scattering, the upper frequency limit to
diffusion theory is imposed by the scattering relax
ation time 'r,. In tissues with Ueff
corresponds to 180 GHz.
In contrast to dispersionregime behavior, the prop
erties of lowfrequency waves are dominated by ab
sorption. Lowfrequency ( << 1/T) phase velocity
reaches a dispersionless lower limit independent of
modulation frequency. The solutions for k reduce to
50 cm', this
WT
r
(XT)/2
kr  (Xrl/ = 1/8,
(15)
(16)
and Vp = 2(X/T) /2.
behavior would occur at modulation frequencies that
are well below o = c (i.e., 360 MHz).
particularly small with respect to 1/T, the ac and dc
attenuation rates are equal. Figure 1 clearly illus
trates this frequencyindependent region where kr
approaches the reciprocal of the dc penetration depth
In Fig. 1 the onset of this
When 0 is
1 February 1993 / Vol. 32, No. 4 / APPLIED OPTICS 609
Page 5
B. As o increases, however, some frequency depen
dence can be observed. Provided that er < 0.7,
expanding kr in powers of UT and keeping the leading
term yield
1
[
L1 +
(XT)11
2
8
+
(17)
Thus the dispersionless attenuation ranges from a
frequencyindependent limit where kr
upper boundary where kr 1 + [(WT)
Frequencydomain measurements record phase
lag(+) and demodulation amplitude m with respect to
a source response. Since these parameters can be
defined in terms of the complex wave number,
m are simply functions of o, r, and the optical
properties
T and oreff:

therefore
1/8 to an
2/8].
and
Fig. 2. Frequencydomain instrument (see text for a detailed
description). BS, beam splitter; M, display monitor; r, distance.
(a(
(a(
( = kir = rp[P + (1  g)]j}1/2(2
r kr
,0 2 1/2
1/2 x
,/dc)samlple=
lip
{ +
exp(krr)/ exp[r/(X)l/2]
r
, 
1
, (18)
Po
r

'/dc)source
((Pi /(PO)r=O
= expr e
r
ln(m) = r3[ + (1 
((I 1+ (,°)2]1/2 + 1/2 a)2_ * (20)
Materials and Methods
Instrumentation
The photon migration instrument is a modified multi
harmonic Fourier transform phase and modulation
fluorometer (SLM, Model 48000MHF, Champaign,
Ill.) illustrated in Fig. 2. Light (L) is provided either
by a watercooled argonion laser (Innova 905, Coher
ent, Palo Alto, Calif.) or an argonpumped dye laser
(Coherent Model 599) with DCM dye (Exciton, Inc.,
Dayton, Ohio). A Pockels cell (PC), driven either
directly by a frequency synthesizer or indirectly by
the amplified output of a harmonic comb generator
(HCG), is used to modulate light at single frequencies
or produce pulses with high harmonic content (multi
harmonic mode).
In multiharmonic operation, phase and modulation
data from 50 frequencies, up to 250 MHz, can be
acquired in a few seconds.30
comb generators (HCG1, HCG2, HCG3) receive in
puts from a frequency source FG at 5 and 5 MHz + 3
The three harmonic
Hz. The comb generator outputs are, in the time
domain, an impulse with a 5MHz repetition rate and,
in the frequency domain, a fundamental frequency (5
MHz) and its integer harmonics, 5, 10, 15 MHz, etc.
The impulse from HCG1 is amplified and applied to
the Pockels cell. This produces a pulse of light with
high harmonic content, which is focused onto a
600pmdiam fusedsilica fiberoptic probe (Fl). A
small portion of this light is diverted to a reference
photomultiplier tube (PMTr) (Hammatsu R928),
which allows phase and modulation locking of the
instrument. The optical fiber is used to direct the
bulk of the light into the sample, and a second fiber
(F2) collects the scattered light.
Scattered light is transmitted by F2 to the measure
ment photomultiplier tube (PMTm) (Hammatsu
R928). The gain of the photomultiplier tubes is
modulated by HCG2 and HCG3. Since the output of
these devices are the harmonic combs, 5 MHz + 3 Hz,
10 MHz + 6 Hz, 15 MHz + 9 Hz, etc., the sample's
phase and amplitude response at each harmonic is
contained within the crosscorrelation frequencies, 3,
6, 9, Hz etc. These 3Hz crosscorrelation notes are
sampled and digitized by a dualchannel analogto
digital converter (C). An array processor (AP) per
forms the transform that converts the digital data to
a frequency spectrum ranging from 3 to
Phase and modulation information from the high
frequency components of the harmonic comb func
tion (5 . . . 250 MHz) is contained within the
3... 150Hz spectrum. Phase and modulation val
ues are computed from the real and imaginary compo
nents of the transforms.
Singlefrequency readings are acquired simply by
eliminating the HCG circuit and tuning the fre
quency synthesizers to the region of interest.
Reference/sample measurements are interleaved, and,
as above, crosscorrelation detections is employed.
150 Hz.
Materials
All scattering measurements were conducted in a
30 x 30cm blackwalled cylindrical vessel filled with
10 L of an emulsified fat solution, Intralipid (KabiVit
rum, Inc., Clayton, N.C.). Fibers (flatcut faces,
610
APPLIED OPTICS / Vol. 32, No. 4 / 1 February 1993
I
g)U]1112 3 12
2
Page 6
600jim core diameter) were positioned in the center
of the liquid, parallel to each other. This central
location was selected carefully to simulate an infinite
medium. The distance between the source and col
lection fibers was systematically varied between 5 and
20 mm for each measurement series. Reference (i.e.,
source) measurements were recorded in air with
input and collection fibers facing each other.
A porphyrin compound, tetraphenyl porphine tetra
sulfonate (TPPS4) (Porphyrin Products, Logan, Ut.)
was added to the Intralipid so the effect of the
absorber could be quantitatively determined. A
514nm laserline filter (Corion Corporation, Hollis
ton, Mass.) was placed at the entrance to the PMT
housing to block TPPS4fluorescence and isolate the
scattered light.
Results and Discussion
Figure 3 illustrates the distance dependence for phase
(Figs. 3A and 3B) and modulation (Figs. 3B and 3D)
predicted by Eqs. (18) and (20). Data are presented
for various frequencies between 5 and 200 MHz in 2%
Intralipid ( = 650 nm). As can be seen in Figs. 3A
and 3B, the frequency response is predominantly
1 50
1 20
90
60
30
0
0.8
0.6
0.4
0.2
0
*
O
r=0.5cm
A
r = 1.0 cm
0
r = 1.5 cm vooo°
0o0 00000
000
301
0~~~~~
00000000
0 ~0
0
ee
000
 9
I I ! I I  i
0 45 90
Freq (MHz)
135 180 22!
0
l
I I
45 90 135 180
Freq (MHz)
nonlinear regardless of distance. This implies den
sity wave dispersion over most of the modulation
region. Furthermore small changes in fiber separa
tion substantially influence phase and modulation
values.
Redisplaying 4) and ln(m) as a function of distance
clearly demonstrates the predicted linear relationship
for all frequencies (Figs. 3C and 3D). Higher frequen
cies result in shorter modulation wavelengths or
more photon density fluctuations per unit distance.
Attenuation and phase lag increase as the modulation
wavelength is reduced; thus an increase in k, X, and
ln(m) are observed. As expectedthe distance linear
ity is maintained regardless of whether density waves
fall within the dispersion or nondispersion regimes.
In Fig. 4 phase and modulation are shown as a
function of frequency for fibers placed 1.0 cm apart in
0.4%, 2%, and 10% Intralipid ( = 650 nm). The
smooth curves through the data represent the best
nonlinear leastsquares fits to Eqs. (18) and (20).
Values for 13 and (reff can be calculated by using either
Eq. (18) or (20), provided that there is sufficient
nonlinearity (i.e., substantial dispersion behavior) for
reliable fits.
1 50 
120 
a
90 
60 
30 
0I
.I I
1
.I
0.4 0.7 1.3 1.
r (cm)
0.8 
0
0
so
0.6
0.4 
0.2 
0
225
0.4 0.7 1
r (cm)
1
1.3
6
1.6
Fig. 3. (A) Phase and (B) modulation versus frequency (5200 MHz) for various values of r. Linear fits to (C) phase and (D) modulation
versus r for selected frequencies (25, 100, 200 MHz). Measurements in 2% Intralipid; X = 650 nm; r = 0.5, 1.0, and 1.5 cm.
1 February 1993 / Vol. 32, No. 4 / APPLIED OPTICS 611
C
0
25 MHz
100 MHz
 200 MHz
p
p
p
I I I
>S@Oo@ g
B
0 000
004C1 00
0 00 0
00
0
00
* 0 g
0 00000
000
0000
000000
0 00000
00 000
0000
0 00
000 000
0 0 0 0 0
* r = 0.5 cm
r=1 cm
r = 1.5 cm
03
0
_ l l
D
0
25 MHz
0 100 MHz
200 Mhz
3
l
I I
I
.
I
I
l
Page 7
0 50 100
Freq (MHz)
Table 1. Calculated Absorption , and Effective Scattering r0 5
Coefficients for 0.4%. 2%, and 10% Intralipid Solutions
13 (cm')
Intralipid (%)
reff (cm' )
0.4
2
10
0.0078 ± 0.0019
0.0079 ± 0.0023
0.0078 ± 0.0013
5.8 ± 0.54
23 ± 5.3
120 ± 17
Note: Values are averages of six separate measurements, i.e.,
phase and modulation full fits at three sourcedetector distances.
illustrates that 13 is relatively insensitive to changes
in scattering solution concentration; however, five
fold Intralipid dilutions result in commensurate ceff
reductions.
The fullfit results obtained from Fig. 4 suggest
that the linearfrequency region (rr < 0.7) for 10%,
2%, and 0.4% Intralipid should be <20 MHz. In
these lowfrequency conditions the solutions for k,
given by Eqs. (15) and (17), yield the following
expressions for + and m:
. . . .
150 200
0
50 1 00 150
200
Frequency (MHz)
Fig. 4. (A) Phase and (B) modulation versus frequency for 10%,
2%, and 0.4% Intralipid; X = 650 nm, r = 1.0 cm.
through the data represent the best nonlinear leastsquares fits to
Eqs. (18) and (20).
Smooth curves
Optical properties derived from this method are
displayed in Fig. 5 and summarized in Table 1. Each
13
and creff value is an average of phase and modulation
estimates at three fiber distances. Figure 5 clearly
0.015
150
GEL f'°1
 F A tt
20
0.01
g0
E
2
0~ ~ ~ ~~~~~~~~~~~6
0.005
30
0
3 6 9
12
% Intralipid
Fig. 5. Summary of optical property measurements in 10%, 2%,
and 0.4% Intralipid estimated from nonlinear phase and modula
tion fits. eff is linearly proportional to and 13 is independent of %
Intralipid. All measurements were made at X = 650 nm; r = 0.5,
1.0, and 1.5 cm.
3
2 (Jeff
e(2p)1/2
1/2
( = ir
r,
ln(m) = kr  (XT)1/2
3
2 Cf
1/2
c(2p)1i/2
ro2
40c
X
.
Linear fits to + versus X and n(m) versus W2furnish
constant slopes, mi, and Miln(m), respectively, which
can be used to calculate 13
(independent of r):
(23)
4c X mi(m)
Linearfit 13 values obtained from 5, 10, 15, and
20MHz data at six different sourcedetector dis
tances are summarized in Table 2. These results are
in reasonable agreement with fullfit 1 values; how
ever, since only a few frequencies were available
to satisfy oAr < 0.7, fullfit methods should be more
accurate. For example, assuming that the reported
0.4% Intralipid13 is accurate, rigorous fulfillment
of the XT < 0.7 criterion would require an upper
frequency limit of only 13 MHz. The superior linear
fit precision is also a consequence of the limited
number of modulation frequencies. Extremely low
absorption coefficients are calculated from small val
ues of mi, and mln(m), which in turn are determined
Table 2. Absorption Coefficients and Frequencyindependent
Velocities Determined from LowFrequency
Phase
Data Where T < 0.7
Intralipid (%)
f3 (cml)a
Vp (cm/s)
0.4
2
10
0.0040 + 0.00018
0.0050 ± 0.00022
0.0074 ± 0.00051
7.61 x 108
3.74 x 108
1.73 x 108
aValues of , are averages of six sourcedetector distances.
612 APPLIED OPTICS / Vol. 32, No. 4 / 1 February 1993
150
120
0)
R0
90
60
30
0
0.6
0.6
0.4
=0
0
0.2
0
(21)
(22)
Page 8
from close to the minimum number of points required
for a straight line. Thus m<, and MiIn(m) appear to be
precise, when in fact they are determined from too
small a data set.
Figure 6A illustrates the lowfrequency phase re
sponse for each Intralipid dilution (r = 1 cm). When
this behavior is linear, the phase velocity can be
obtained from the distance dependence of mi, dis
played in Fig. 6B. The linear appearance of ne,
versus r for each Intralipid concentration suggests
medium homogeneity. Since ki = 4)/r [Eq. (18)] and
V = w/ki [Eq. (11)],
mig = +/w = r/Vp.
(24)
Thus the reciprocal of the mi, versus r slope (Fig. 6B)
yields Vp. This is the lower limit for phase velocity.
VP remains independent of o up to modulation fre
quencies where XT < 0.7. In the region where WT =
1 (i.e., 0.7 < (or < 1.4), ki is on the edge of the Fig. 1
parabola and Vp begins to display some frequency
dependence. For
dependence described by Eq. (14). Frequencyindepen
T >> 1, Vp assumes the full ()1/2
0.6
0.4
0.2 +
0
2.0 107
I
I
1
5.0 107 8.0 107
a) (rad/s)
1.1
dent Vp values are summarized in Table 2. Density
wave phase velocities range from as little as 0.0076c
in 10% Intralipid to a high of only 0.034c in 0.4%
solution.
Figure 7 illustrates that, for constant absorption,
the phase velocity is linearly proportional to 1/VaLeff.
Accordingly a fivefold increase in percent Intralipid
results in a C5 phasevelocity decrease. Vp reduction
in turn leads to shorter modulation wavelengths (Xm),
greater phase lag (), and enhanced ln(m).
practical terms lowering Vp can lead to improved
absorber detectability, as illustrated by the concentra
tiondependent sensitivity improvements of Fig. 6A.
However, these exceptionally low phase velocities are
rarely observed in real tissues. Despite the scatter
ing similarities between Intralipid and some tissue,
absorption differences, typically of an order of magni
tude, lead to substantial Vp enhancement in vivo.
Figure 8 illustrates the effect of the TPPS4ab
sorber on phase (Fig. 8A) and modulation (Fig. 8B) in
10% Intralipid (X = 514 nm). Modulation frequen
cies to 165 MHz are shown at a sourcedetector
separation of 7.5 mm. There is distinct nonlinearity
to the 4) and m versus frequency curves in the absence
of TPPS4. However, as the absorber is added, the
frequency response flattens out. More specifically,
phase decreases and attenuation increases [i.e., larger
(m) values] with increasing absorber concentration.
This occurs because, with the addition of the ab
sorber, measured photons follow shorter paths to the
detector. Thus fewer scattering events are recorded,
and a decrease in phase delay and demodulation is
observed. When the scattering coefficient is held
constant, the increasing 13 results in higher phase
velocities (Vp oc ) and longer modulation wave
lengths (m).
In high absorption conditions (i.e., when AT << 1),
m [Eq. (19)] may be close to (or equal to) unity since kr
approaches 1/8. If there is sufficient demodulation
to satisfy Eq. (22), , can be determined from Eq. (23).
Redisplaying the phase (Fig. 9A) and ln(m) (Fig. 9B)
In
9
6 1 09
9
0.25 0.75 1.25
r (cm)
1.75
8.0 108
0.
2.25
Fig. 6. A, lowfrequency phase (rad) versus o (rad/s) response for
r = 1.0 cm. The slopes of the linear fits are m, (see text). B,
phase slope (mi,) from A for several fiber separations; the reciprocal
of each B slope is the lowfrequency phase velocity (see text).
Measurements in 10%, 2%, and 0.4% Intralipid; X = 650 nm.
5.7 1 8
3.3 103
1.0 108

'
' I
I I '
I
'
d
0.05 0.15 0.25
( 1 /ff)1
/2
0.350.45
Fig. 7. Phase velocity (determined from Fig. 6B) versus (/aeff)
for 10%, 2%, and 0.4% Intralipid; X = 650 nm.
shown is predicted by diffusion equation solutions.
1/2
The linear relation
1 February 1993 / Vol. 32, No. 4 / APPLIED OPTICS 613
0.4 0.%
%0 2%
10%
A
0
B
0 10%~~~~~~


2 %
0.4%
I I I I I
.
I
1 8
toC
Page 9
; A
000
0
0
0
0~~~~
C3 0~~~~
C3 0 ~ ~
~ ~ ~
0
; 3
_
_0Af
1 zLg/r0L
o Z * *
o~~1 no added absorber
*
120
Freq (MHz)
 C3
0
0
60
120
0
Freq (MHz)
0
3

°000~~
0
*O~~~~°
0 0~~~~~~~~~~~r
0 _g/mL
60
1 80
l Cg/m
o
no added absorber
a
l_
60
12
18
1 00
a
aC,
75
50
25
0
1.1
0.90~
0
0.7
0.5
0.3
0
0.45 
xm
CD
0)
CO
0.
0.3 
0.15  o
1
0 __
1.0 108
80
0.08 
0.06 
E
C0.04 
0.02
Fig. 8. A, Phase and B, modulation versus frequency response for
10% Intralipid with and without 1 pug/mL of TPPS4absorber; r =
0.75 cm, X = 514 nm.
as functions of w and W2, respectively, illustrates the
utility of this linearfit approach.
Linearfit absorption coefficients were determined
for 1, 2, and 4g/mL
Intralipid. Nonlinear fits were used to calculate the
absorption coefficient of pure 10% Intralipid. Aver
age f3 values, obtained from at least five separate
distance measurements, are displayed as a function of
[TPPS4] in Fig. 10. The molar concentration of
TPPS4in Intralipid can be calculated from the rela
tionship between the linear absorption coefficient 3
and the molar extinction coefficient , i.e., 1 = 2.3EC,
where (TPPS4) = 2.5 X 104 M1cm' at 514 nm.
These results, summarized in Table 3, show reason
able agreement between calculated and actual values
of concentration.
The linear appearance of Fig. 10 indicates that
relatively low 13 values (0.02 absorbance units, 8 x
107 M) can be reliably determined despite the pres
ence of substantial scattering. In fact scattering
enhances detectability by slowing Vp and reducing Xm
to dimensions that approach 1/13. Assuming diffu
sion behavior, the accuracy of highly determinations
is generally limited by signal quality. Most tissue
measurements (in the red and nearIR spectral re
TPPS4 solutions in 10%
0
0
I
,
1
.I
2.2 108
3.5 108
co (rad/s)
4.7 108
6.0 1 08
B
01
0
.
+
B1 2 8 g/mL
1 g/mL
I
I
l
I
0
7
7.5 101
1.5 1017
2.2 1017
17
3.0 1 0'
Fig. 9. Best linear fits to A phase versus w and B ln(m) versus (2
in the lowfrequency regime where WOT < 0.7. Measurements in
10% Intralipid with 1 and 2 p1g/mL of TPPS4absorber; r = 0.5 cm,
= 514 nm.
gions) should display some demodulation when acces
sible frequencies are used, e.g., up to 200 MHz.
Unfortunately, as indicated clearly in Fig. 9, the
quality of our modulation data is generally poorer
0.13
0.1
E 0.07
0.04
0.01
I
2
0
1
3
4
5
[TPPS] (g/mL)
Fig. 10. Absorption coefficient estimated from multifrequency
measurements versus concentration of TPPS4absorber added to
10% Intralipid. Linear regression yields y = 0.022x + 0.025,
where 0.025 cm1= 3 in the absence of added absorber; x2= 2.17.
614 APPLIED OPTICS / Vol. 32, No. 4 / 1 February 1993
A
+ 1 g/mL
2 pg/mL
8
I
.
.
.
.
.
125 0.6
Page 10
Table 3. Comparison of Actual and Fitted Values of p for TPPS4in 10%
Intralipid, A = 514 nm
[TPPS4]
(pLg/mL)
Actual 13
(TPPS4) cm1
Fitted 3
(TPPS4) cm1
a
0
1
2
4
0
0.046
0.092
0.18
0
0.047
0.069
0.11
aFitted a were obtained from Fig. 10. Actual 13 = 2.3EC.
than the phase response. As a result the acceptabil
ity of mIn(m) obtained from linear fits diminishes with
increasing 13. These factors underscore the impor
tance of acquiring multiple modulation frequencies.
One can employ singlefrequency measurements
to calculate 13 by recognizing that m, =
Mln(m) = ln(m)/
Owing primarily to variations
in the precision of m, however, the reliability of
singlefrequency determinations may not be as high
as multifrequency calculations. Obviously small vari
ations in ln(m)/o2can lead to large errors in the
estimation of 13; thus it is desirable to acquire k and m
for as many frequencies as possible. In addition, by
looking at the entire frequency response, one can
rapidly determine whether it is appropriate to apply
the dispersion relations to calculate optical proper
ties.
This work was performed with support from the
University of California Cancer Research Coordinat
ing Committee, The George Hewitt Foundation for
Medical Research, the Office of Naval Research grant
N0001491C0134, and the Department of Energy
grant DEFG0391ER61227. The authors thank
Khai Vu, Eric Cho, and Matthew McAdams for their
outstanding contributions.
/(o and
2.
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