1874
J. Opt. Soc. Am. A / Vol. 22, No. 9 / September 2005
R. B. Saager and A. J. Berger
Direct characterization and removal of interfering
absorption trends in two-layer turbid media
Rolf B. Saager and Andrew J. Berger
The Institute of Optics, University of Rochester, Rochester, New York 14627
Received January 19, 2005; accepted March 6, 2005
We propose a method to isolate absorption trends confined to the lower layer of a two-layer turbid medium, as
is desired in near-infrared spectroscopy (NIRS) of cerebral hemodynamics. Several two-layer Monte Carlo
simulations of NIRS time series were generated using a physiologically relevant range of optical properties and
varying the absorption coefficients due to bottom-layer, top-layer, and/or global fluctuations. Initial results
showed that by measuring absorption trends at two source–detector separations and performing a leastsquares fit of one to the other, processed signals strongly resemble the simulated bottom-layer absorption properties. Through this approach, it was demonstrated that fitting coefficients can be estimated within less than
⫾2% of the ideal value without any a priori knowledge of the optical properties present in the model. An analytical approximation for the least-squares coefficient provides physical insight into the nature of errors and
suggests ways to reduce them. © 2005 Optical Society of America
OCIS codes: 170.3660, 170.5280, 170.7050, 170.1470.
1. INTRODUCTION
Noninvasive monitoring of the functional activity of the
brain is a growing field. Several mature methods already
exist for the investigation of functional brain activity:
electroencephalography (EEG), positron emission tomography (PET), magnetoencephalography (MEG), and functional magnetic resonance imaging (fMRI). Each method
has advantages and drawbacks. EEG and MEG are directly sensitive to neural activation (EEG measures electric fields generated from neural activity, whereas MEG
measures the resultant induced magnetic field fluctuations) and have correspondingly high temporal resolution,
but their spatial resolution is low (of the order of tens of
centimeters). PET, which detects positron emission produced by the decay of radioactive isotopes introduced into
the blood stream, and fMRI, which exploits the paramagnetic effect of deoxygenated hemoglobin on resonant magnetic fields, can offer submillimeter resolution, but have
poor temporal resolution (typically seconds for fMRI and
longer for PET), high purchase and operating expenses,
and an indirect correlation with neural activity through
the invoked hemodynamics.
Another modality being explored is near-infrared spectroscopy (NIRS). It works by sensing changes in optical
absorption of hemoglobin and is thus an indirect modality
like PET and fMRI. Unlike those methods, NIRS measures changes in both oxygenated and deoxygenated hemoglobin concentration; a more complete description of
the hemodynamics is therefore obtained. Both time resolution (milliseconds to seconds, limited by the hemodynamics) and spatial resolution (subcentimeter to fewcentimeter, limited by the diffusive nature of photon
transport) lie between the extremes of the four methods
previously mentioned. NIRS systems are also substantially less expensive and more compact than PET and
fMRI systems.
One significant limitation of NIRS is that the light
1084-7529/05/091874-9/$15.00
must interact with blood in regions other than the cerebral cortex. The most elementary NIRS measurement
consists of delivering light at one surface location and collecting remitted light at another point some distance
away. For the light to reach the brain during its travels, it
must pass through all intervening layers of the head.
These layers include the scalp, which has its own vasculature that influences the optical measurements.
Even with this issue of multiple layers, most NIRS
work is based on single source–detector combinations, referred to here as single-detector optodes (SDOs). Even
when grids of sources and detectors are used for mapping,
each pair of nearest neighbors is typically treated as an
independent SDO. Because only one distance is used, the
modeling must be correspondingly simplistic, treating the
underlying tissue as a single homogeneous layer with a
given absorption coefficient ␮a. Changes in ␮a are derived
from the changes in the measured diffuse reflectance signal S共t兲:
冉 冊
⌬A共t兲 = − ln
S共t兲
S0
= ⌬␮a共t兲 · 具L典,
共1兲
where ⌬A共t兲 is the calculated absorbance change and 具L典
is the average travel distance of the detected photons.
This approximate proportionality is often called the modified Beer-Lambert law (MBLL). It is assumed that 具L典, although dependent on ␮a共t兲, does not change significantly
with time.
Despite the simplicity of SDOs and the homogeneous
model, they have been useful in many proof-of-principle
studies. Single-optode investigations of visual1 and motor
and visual stimulation2 have shown increases of oxygenated hemoglobin during intervals of cognitive activation.
Several studies of simple cognitive tasks, such as finger
tapping tasks and passive knee movement, have been per© 2005 Optical Society of America
R. B. Saager and A. J. Berger
Vol. 22, No. 9 / September 2005 / J. Opt. Soc. Am. A
formed using multiple SDOs to demonstrate its potential
as a brain-mapping–localization technique.3,4
In part because a SDO cannot distinguish between
scalp and cerebral hemodynamics, studies such as these
use multiple trials and block averaging to suppress uncorrelated trends. In some cases, hundreds to thousands of
trials are required.5 The severity of the interfering trends
varies strongly with the physiology of the individual.
Most groups discover that certain people are “good activators” while other subjects cannot produce detectable signals.
The need for multiple trials and the low success rate
among subjects have impeded the progress of the NIRS
method. Several groups have discussed the issue of interfering blood flow and partially correlated effects such as
blood pulsatility, sympathetic vessel dilation, and Mayer
waves, along with coupling issues of blood volume versus
blood flow artifacts.6,7 Auxiliary biometric devices, such as
pulse oximeters, respiration meters, and laser Doppler
flowmeters can be incorporated to help discriminate such
effects from localized cerebral activations. Multivariate
processing techniques, such as principal component
analysis, can be used in conjunction with multiple optodes
to remove trends that are common to different regions of
the head.8
A slightly fancier optode is one with a second detector
located at some point between the source and the first detector. We call this a dual-detector optode (DDO). Such optodes have already been employed in other studies to
demonstrate detection of depth-dependent changes in
absorption.9,10 By analogy with the discussion above, the
DDO geometry now permits the head to be modeled as a
system of two homogeneous layers with different ␮a
values.11 Isolation of the contribution of a particular layer
is now possible.12 The generalization of Eq. (1) becomes
⌬AN共t兲 = ⌬␮a1共t兲 · 具L1典N + ⌬␮a2共t兲 · 具L2典N ,
⌬AF共t兲 = ⌬␮a1共t兲 · 具L1典F + ⌬␮a2共t兲 · 具L2典F ,
共2兲
where subscripts 1 and 2 refer to the upper and lower layers, respectively, and N and F refer to the near and far
detectors, respectively. Increasingly complex systems
[with lateral heterogeneities and/or more than two layers]
can be modeled using correspondingly more sources and
detectors; such three-dimensionally resolved calculations
of ⌬␮a are usually called diffuse optical tomography
(DOT).13
Following Fabbri et al.,11 all influence of ␮a1共t兲 can be
removed from the time course ⌬AF共t兲 by subtracting a
multiple of ⌬AN共t兲:
R共t兲 ⬅ ⌬AF共t兲 − K⌬AN共t兲 = ⌬␮a1共t兲共具L1典F − K具L1典N兲 + ⌬␮a2共t兲
⫻共具L2典F − K具L2典N兲.
共3兲
It is clear that we can eliminate ⌬␮a1共t兲 and make R共t兲
have the same shape as ⌬␮a2共t兲 by choosing
K=
具L1典F
具L1典N
,
共4兲
i.e., the ratio of the average path length traveled in the
top layer by photons reaching the far and near detectors.
1875
Unfortunately, these average path lengths are not known
a priori and cannot generally be calculated from cw DDO
measurements. Fabbri et al. have pointed out that K can
be measured experimentally by obtaining the ratio
⌬AF共t兲 / ⌬AN共t兲 during a time interval when ␮a2 is relatively static.11 This approach is valid, but requiring static
hemodynamics in the cerebral layer is a strong constraint
that is not under the complete control of the experimenter
or the subject.
In this paper, we propose a new method of subtracting
the two measurements to isolate useful information about
hemodynamics in the lower, cerebral layer. The approach
is to compute a least-squares fitting coefficient of ⌬AN共t兲
to ⌬AF共t兲 using the entire time course of data. We will
show that this approach correctly gives K in the special
case of ⌬␮a1 and ⌬␮a2 being uncorrelated in time. More
generally, the method isolates the component of ⌬␮a2共t兲
that is uncorrelated with ⌬␮a1共t兲. We will argue that this
quantity is relevant for identifying localized cerebral activation signals. The method derives robustness from its
use of the entire time course, and it requires no presumption of inactivity in one layer.
In the following sections, we will (1) introduce the
theory that not only describes the two-layer system we
are considering, but motivates the employment of leastsquares as a robust method of estimating and removing
interfering effects; (2) describe the Monte Carlo simulations used to generate data used in our analysis; (3) demonstrate the application of our new method across a range
of physiologic parameters and scenarios; and finally (4)
discuss the results, limitations, and potential refinements
to this approach.
2. THEORY
A. Physiology
As noted above, simplified multilayer models of the head
have proven useful in interpreting the way it is probed by
near-infrared light. The bottom layer is the layer of interest (brain) and all layers above it (scalp, bone, cerebrospinal
fluid,
etc.)
are
potential
sources
of
interference.14–16 Since hemoglobin is the dominant chromophore in the near-infrared, the only layers with timevarying optical properties are the two experiencing blood
flow, namely scalp and brain. Scattering changes are considered to be negligible in relation to the magnitude of absorption changes, and the remaining optical properties
present in the tissue are assumed to be constant and
homogeneous.17
Under these conditions it is reasonable to adopt a twolayer model to describe the optically active properties of
the head. Here, the top layer contains the average static
optical properties assigned to scalp, bone, and cerebrospinal fluid (CSF) as well as the active absorption properties
of blood in the scalp. The bottom layer then contains the
static and dynamic optical properties assigned to the
outer regions of the brain.
B. Parametrization of Absorption Changes
For compactness, we denote all time-series vectors such
as ⌬AN共t兲 and ⌬AF共t兲 henceforth in bold type, i.e., as ⌬AN
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J. Opt. Soc. Am. A / Vol. 22, No. 9 / September 2005
R. B. Saager and A. J. Berger
and ⌬AF, respectively. We parametrize the changes in absorption in both layers in the following manner:
⌬␮a1 = G,
⌬␮a2 = ␥G + B,
共5兲
where B (bottom) is an absorption time course that appears only in the lower layer and G (global) is another one
that can appear in both layers with different magnitudes,
depending on the scaling parameter ␥. Without loss of
generality, we can define B to be the part of ⌬␮a2 uncorrelated with G in the limit of infinitely long measurement; i.e.,
B·G
lim
t→⬁ G
·G
共6兲
= 0.
By hypothesis the goal of the optical measurements here
is to extract not ⌬␮a2 (the total cerebral hemodynamic
time course), but B (the signal that is intrinsic to the cerebral layer only).
By including ␥, we allow for the possibility that the interfering trend, G, need not be exclusively confined to the
top layer. Since bone and CSF confine vascular beds to
scalp or brain layers, cerebral hemodynamics are confined
to the bottom layer; however, that does not place any requirement that interfering processes need be confined to
the top layer. Interference may be global in nature as
well.
Even for a two-layer model the parametrization of Eq.
(5) does not describe the most general case. There could
be additional activity present in the scalp, described by a
third time course T (top) that is independent of both B
and G. Under these circumstances, an exact solution for
B cannot be determined with only two measurements. We
include such a case in our simulations below in an effort
to study the performance of this approach even in underdetermined circumstances.
D. Least-Squares Solution
The possibility of this “ideal” extraction always exists in
this model, regardless of choice of N and F. The value for
␣I, however, cannot be directly determined by the proposed measurements since the various 具L典 lengths and ␥
are not directly measurable in a cw system.
A practical alternative is to fit ⌬AN to ⌬AF using a
least-squares criterion. Such a fit will create a residual R
that is uncorrelated with ⌬AN. If ⌬AN is similar in shape
to G, this will cause R to resemble B, as desired. Unlike
␣I [Eq. (7)] a least-squares coefficient ␣LS can always be
calculated, independent of any assumptions. We therefore
are motivated to explore the notion of least-squares fitting
in more detail.
As just noted, least-squares fitting will be most successful if ⌬AN is similar in shape to G. Within the limits of the
model, this can be ensured by placing the N detector sufficiently close to the source that very few of the detected
photons explore the bottom layer. Mathematically, we define
⑀=
␣I =
具L1典F + ␥具L2典F
具L1典N + ␥具L2典N
共7兲
,
(subscript I for ideal) then the residual becomes
冋 冉
R = 具L2典F 1 −
具L2典N 具L1典F + ␥具L2典F
具L2典F 具L1典N + ␥具L2典N
冊册
B,
共8兲
where the term in front of B is simply a constant.
The parameter ␣I explicitly extracts B, whereas K extracts ⌬␮a2, which in general is not the same time course.
We note that Eq. (7) reduces to Eq. (4) in the limit of ␥
= 0, i.e., in the special case where ⌬␮a2 is identical to B
with no interference from G.
共9兲
,
E. Derivation
Fitting the N absorbance measurement to the F measurement by least-squares yields a fitting coefficient ␣LS,
where
␣LS =
⌬AN · ⌬AF
⌬AN · ⌬AN
共10兲
.
Substituting the definitions from Eqs. (2) and (5) it can
be shown (see Appendix A) that
−2
具L2典F B · G
·
+⑀
具L1典N G · G
再
G · 共B + ␥G兲
G·G
再冋
冎册
2
共B + ␥G兲 · 共B + ␥G兲
具L2典F
具L1典N
G·G
−
G · 共B + ␥G兲 具L1典F
G·G
⬅ ␣ˆ LS .
具L1典N
冎
共11兲
The hat on ␣LS emphasizes that it is an analytical estimate of the experimental value ␣LS. Terms to second order in ⑀ are neglected in accordance with the assumption
of small ⑀. In this same limit the 具L2典N can be dropped
from Eq. (7), yielding
␣I ⬇
−1
具L1典N
and say that this is the ⑀ Ⰶ 1 limit. At the end of this paper, we discuss the practical limitations of this approximation.
␣LS ⬇ ␣I +
C. Ideal Solution
Measurements of reflectance at two detectors N and F
permit the calculation of ⌬AN and ⌬AF [see Eq. (2)]. Now
if we introduce a scaling parameter ␣ and take a weighted
difference of the two equations, assuming the parametrization of Eq. (5), we find the residual signal R has the
same temporal shape as B, provided the appropriate
weighting is employed. Specifically, if we choose
具L2典N
具L1典F + ␥具L2典F
具L1典N
.
共12兲
As Eq. (11) shows, ␣LS will not exactly equal ␣I. An expression for the expected fractional discrepancy can be
written as
␦␣ˆ =
␣ˆ LS − ␣I
␣I
= ␦␣ˆ 1 + ␦␣ˆ 2 ,
where we identify two separate error terms,
共13兲
R. B. Saager and A. J. Berger
␦␣ˆ 1 =
and
␦␣ˆ 2 = − ⑀
⫻
再冋
冋
B·G
G·G
B·G
Vol. 22, No. 9 / September 2005 / J. Opt. Soc. Am. A
冉
G · G 具L1典F + ␥具L2典F
册冋
册冎
+␥ + ␥
具L2典F
具L2典F
具L1典F + ␥具L2典F
B·G
G·G
冊
冉 冊
B·G
+2
G·G
.
共14兲
2
B·B
−
G·G
册
共15兲
The merit of Eq. (13) is that it estimates in analytical
form the discrepancy between ␣I and ␣LS. The two terms
identified as sources of error in the estimate are governed
by physiologic properties. The first is dependent on the
chance (finite-time) correlation of absorption fluctuation,
whereas the second is additionally dependent on the statistical probe depth of the near detected signal.
We note that in the limit when ⑀ goes to zero (i.e., when
detector N is sufficiently close to the source), ␦␣ˆ 2 vanishes. If in addition the measurement time is arbitrarily
long, then the correlation 共B · G兲 / 共G · G兲 can be made arbitrarily small, at which point ␣LS = ␣I. We thus see that
least-squares fitting extracts the B time course under optimal conditions. Even if multiple events need to be recorded before this correlation can be made negligible, the
resulting residual R provides a corrected version of the
entire time course, allowing variations between stimulus
responses to be inspected. Such opportunities are unavailable when averaging multiple responses to reduce noise
in the SDO paradigm.
3. MONTE CARLO SIMULATIONS
There are several approximations made in the preceding
analysis of light propagation in a two-layer system with
temporally varying absorption:
1. ⑀ Ⰶ 1, i.e., the N detector is only faintly sensitive to
the lower layer;
2. 具exp共−␮aL兲典 ⬇ exp共−␮a具L典兲, i.e., averages of exponentials are equal to exponentials of averages;
3. 具L共t兲典 ⬇ 具L共t = 0兲典, i.e., average path lengths are constant in time.
To test the range of validity of these approximations,
Monte Carlo (MC) simulations were employed. Because
the goal was to simulate a time series with varying ␮a but
fixed scattering properties, the “white MC” could be employed for efficiency.18,19 The white code simulates photon
behavior in two separate stages. In the first stage a pencil
beam of photons is injected into the two-layer model with
all absorptions set to zero (hence white). This simulation
accounts only for physical dimensions, index mismatches,
and scattering properties of the two layers. The end result
of this stage is a large file that contains the complete histories of each photon injected into the system. In particular, all photons that arrive at the N and F detectors (idealized to collect over a full 2␲ steradians) are noted.
The second stage of the routine then applies weights to
all of these photons’ histories based on particular absorption properties specified for each layer. While the first
1877
stage of the ANSI-C-compiled routine takes 40 min to
generate the white histories of one million simulated photons, the second stage runs in ⬇10 s on a generic personal
computer with a 1.5 GHz processor and 512 megabyte
memory. After the first part is completed, the second stage
may be run hundreds of times within half an hour to generate simulated ⌬␮a1 and ⌬␮a2 time courses that lead to
⌬AN and ⌬AF measurements.
The G and B trends were simulated for a 12-epoch
block design experiment and were generated as follows.
Each individual epoch consisted of a series of 100 absorption values. The spacing between data points can be
thought of as arbitrary temporal units. The activation signal B was modeled by a third-order skewed Gaussian
function shaped to simulate a hypothetical cerebral response (see Fig. 1). The interfering G signal was generated by smoothed random numbers to provide uncorrelated fluctuations on time scales at and faster than that of
B, as seen in NIRS cerebral measurements. These functional forms for G and B were chosen to emphasize the
connection to cerebral hemodynamics. Outside of the correlation criterion stated in Eq. (6) the forms do not affect
the generality of the results.
A multidimensional parameter space of optical and
physiologic properties was examined. Selected ranges and
values were based on previously published results from
several sources.3,11,20,21 For this initial investigation the
temporally varying absorption component was induced by
changes in oxygenation only (with fixed blood volume)
without affecting the generality of the method. Twenty
different combinations spanning the ranges of layer thickness, scattering, and blood volume were simulated, and
the time courses G and B were scaled to create an average of 1% fluctuation in oxygenation.
The far-detector distance was set at dF = 35 mm, and
various near-detector measurements were tabulated at
values of dN ranging from 1 to 13 mm. This led to different values of ⑀, the path length ratio. For example in Fig.
1, where dN = 1 mm, ⑀ = 0.029, whereas for 13 mm, ⑀ in-
Fig. 1. Graphical representation of the model described in the
Monte Carlo simulation.(a) dN, dF are the source–detector separations for the near and far detector, respectively, and SN, SF are
the detected signals at those locations. Note that this is merely a
cartoon of the system; items are not drawn to scale. (b) Typical
plots of the fluctuations in absorption properties specific to each
layer. (c) Plots of the detected absorption changes at each
detector.
1878
J. Opt. Soc. Am. A / Vol. 22, No. 9 / September 2005
Table 1. Parameter Space of Optical and
Physiologic Properties Examined in Simulations
Parameter
Top Layer
Bottom Layer
␮s⬘
Thickness
Blood volume
Baseline oxygenation
Oxygenation change
7–15 cm−1
7–12 mm
2–3%
80%
⫾1%
7–15 cm−1
—
2–3%
80%
0–1%
creases to 0.42. The ⌬AN and ⌬AF signals in Fig. 1 look
similar, but in fact ⬇40% of the ⌬AF signal is the contribution from bottom-layer fluctuations.
For NIRS techniques to report changes in both oxyhemoglobin and deoxyhemoglobin, at least two wavelengths
in the 650–900 nm range need to be employed. For the
case of these simulations, optical properties related to 690
and 830 nm were examined. Since the Monte Carlo simulations did not model any external noise sources or artifacts, wavelength selection did not influence the quality of
the ␣LS computation. To that end, results presented here
will be only of data generated at 830 nm. Figure 1 shows
a schematic of the simulated Monte Carlo model with
plots of typical absorption fluctuations and detected signals. Because the MC program keeps track of each photon’s history, average path lengths can be calculated, permitting direct evaluation of ␣I and the individual terms in
the expression for ␦␣ˆ [Eq. (13)]. MC simulations were performed given over the range of optical properties given in
Table 1. In all cases the least-squares coefficient ␣LS was
computed and, where applicable, compared to the ideal
value ␣I. Residual time series R = ⌬AF − ␣⌬AN were created using three different values of ␣: ␣LS, the ideal ␣I
[Eq. (7)], and ␣ = 0, corresponding to the unaltered measurement from a single detector.
4. RESULTS
R. B. Saager and A. J. Berger
the absorption changes of the bottom layer used in the
simulations. The second plot shows ⌬AF, the resulting absorbance signal detected at the far detector. This is the
type of signal one would measure using a standard cw
NIRS system with a single source–detector pair. From
this plot, it is evident how strongly the top layer interference affects the measurement. Below this are the two corrected signals R using ␣LS and ␣I. Under these conditions, least-squares performs extremely well in extracting
the individual activation events of the B time course,
whereas these activations can not be discerned in the
single optode measurement (⌬AF, corresponding to R0,
the residual computed using ␣ = 0).
Figure 2(b) shows the result of event-averaging the different R time courses, as is commonly done in SDO. The
average for R0, the single optode signal, begins to reveal
the activation sequence of B, but significant interference
from G remains. The average for RLS, in contrast, resembles B nearly perfectly.
Overall the least-squares method was very accurate at
estimating ␣I under these conditions. For all combinations of optical parameters with ␥ = 0, using 12 epochs and
dN = 1 mm, the standard deviation between ␣LS and ␣I
was 1.9%.
With this set of simulations, we can evaluate the accuracy of Eq. (11), the analytical estimate of ␣LS. For dN
= 1 mm the standard deviation between the two is only
0.04% over all simulations, with a bias of ⫹0.03%. Errors
increase when the near detector is farther from the
source, because this increases ⑀ [see the final term in Eq.
(11)]; however, even at dN = 13 mm, the standard deviation and bias increase to only 1.2% and 2.2%, respectively.
The formula is therefore useful in estimating ␣LS from
fundmental properties of the optical system.
We therefore proceed to examine the two error terms in
Eq. (13), the estimate of the discrepancy between ␣LS and
␣I. The first error term, ␦␣ˆ 1, scales as the correlation between B and G, with no dependence on the location of the
The main purpose of the MC simulations was to determine how accurately ␣LS estimates the target value ␣I,
along with any consistent bias. Conceivably, the standard
deviation ␴LS in the ␣LS values might depend on the timeindependent quantities that were varied from run to run:
layer thickness, scattering coefficient, and blood volume
percentage. As a check, each simulated time course was
subdivided into its 12 epochs, and ␣LS was computed 12
times, once for each epoch. We found that ␴LS remained
the same to within less than 2% across the range of layer
thicknesses, scattering coefficients, and blood volume percentages simulated (see Table 1).
Because changes in these baseline parameters did not
noticeably influence the stability of ␣LS estimation, we
can draw general conclusions from particular simulations. In the following we report in detail on representative cases from three scenarios.
A. Scenario 1: Top Layer Interference
Figure 2 shows results from the simplest nontrivial situation possible: the interfering absorption changes G are
exclusively located in the top layer. In terms of our model,
␥ is equal to zero. The top plot in Fig. 2(a) displays ⌬␮a2,
Fig. 2. Results for simulations with no interfering effects in the
bottom layer 共␥ = 0兲. (a) From top to bottom, these plots show the
individual ⌬␮a2 in the bottom layer, the detected signal at the far
detector alone (␣ = 0, i.e., a SDO configuration), the extracted signal from the least-squares approach, and the ideal extraction calculated from average photon pathlengths. (b) Plots of the block
average of 12 individual events under the same conditions.
R. B. Saager and A. J. Berger
Vol. 22, No. 9 / September 2005 / J. Opt. Soc. Am. A
1879
portion of it that is unique to the bottom layer, i.e., B, as
t → ⬁.
Fig. 3. Characterization of error sources in the least-squares estimate. (a) Mean and distribution of error values described by
␦␣ˆ 1 (stars) and ␦␣ˆ 2 (circles) as a function of epochs recorded (i.e.,
measurement time); while ␦␣ˆ 1 is independent of dN, ␦␣ˆ 2 is shown
here for dN = 1 mm. (b) Mean and distribution of error values as a
function of dN; all data shown were calculated from a 12-epoch
measurement.
near detector (i.e., on ⑀). Therefore, increasing the integration time should reduce this error term, as chance correlations from finite sampling are reduced. No bias is expected. Figure 3(a) shows the mean and standard
deviation of ␦␣ˆ 1 from many simulations, plotted for various numbers of epochs. As predicted, there is no significant bias, and the error amplitude drops as measurement
time increases. The second error term ␦␣ˆ 2 is more complex, containing correlation terms, but also a dependence
on ⑀ and other terms that are always positive. As shown
in Fig. 3(a), the magnitude and spread in ␦␣ˆ 2 errors were
much smaller than for ␦␣ˆ 1, indicating the primary problem was chance correlations between finite samplings of
B and G. Figure 3(b) takes the final data points from Fig.
3(a) and shows how the errors increase as dN increases
from 1 to 13 mm. At the largest separations ␦␣ˆ 2 introduces a significant bias in the estimation of ␣I, and the
standard deviation becomes comparable to that introduced by ␦␣ˆ 1.
B. Scenario 2: Global Interference
For this case we allow for the possibility that interference
occurs globally. Now the interfering absorption changes
occur in both layers but in differing amounts. In terms of
our model, ␥ is equal to 0.5, with the amplitudes of the B
and G time courses set equal. Figure 4 displays a summary of typical results from simulations run under this
scenario. Yet again, RLS strongly resembles B, whereas
the single optode result R0 is even further degraded by
the overall increase in interference.
In this case the standard deviation between ␣LS and ␣I
(again for 12 epochs and dN = 1 mm) was 1.3%. The
method estimates the correct fitting coefficient accurately,
while being insensitive to the presence of the interference
G in the lower layer. In contrast, the SDO performance
R0, suffers more when the interference is present in the
lower layer. The higher the value of ␥, the more interference there is and the greater the amount of averaging
needed to obtain a useful SDO signal. Again we emphasize that the least-squares approach not only significantly
improves the event averaged signal, but provides an entire RLS time course from which information pertaining to
single events can be extracted. We also stress that the extracted RLS time course is not ⌬␮a2, but rather only the
C. Scenario 3: Independent Top Layer and Global
Interference
As noted earlier, a T signal could also be present in the
top layer, in which case the problem is underdetermined
theoretically. This was simulated with the G and T signals contributing equally to the total fluctuation in the
top layer, while the bottom layer absorption changes were
30% G and 70% B.
As Fig. 5 shows, there is degradation in the signal extraction in all methods, even including the extraction us-
Fig. 4. Results for simulations with equal contributions of interference and activation in the bottom layer 共␥ = 0.5兲. (a) From
top to bottom, these plots show the individual ⌬␮a2 in the bottom
layer, the detected signal at the far detector alone (␣ = 0, i.e., a
SDO configuration), the extracted signal from the least-squares
approach, and the ideal extraction calculated from average photon path lengths. (b) Plots of the block average of 12 individual
events under the same conditions.
Fig. 5. Results for simulations with both top layer and global
interference terms 共␥ = 0.3兲. (a) From top to bottom, these plots
show the individual ⌬␮a2 in the bottom layer, the detected signal
at the far detector alone (␣ = 0, i.e., a SDO configuration), the extracted signal from the least-squares approach, and the ideal extraction calculated from average photon path lengths. (b) Plots of
the block average of 12 individual events under the same
conditions.
1880
J. Opt. Soc. Am. A / Vol. 22, No. 9 / September 2005
ing the ideal scaling parameter. This is expected, since
the theoretical model used to solve for this ideal value is
now invalid. It is encouraging, however, that leastsquares can still offer significant improvement toward the
isolation of the underlying activation signal. Remarkably,
the block-averaged RLS resembles the activation more
than the block average of ⌬␮a2 itself.
5. DISCUSSION
Removing uncorrelated features from NIRS data is an important goal. Any technical improvement in eliminating
these features directly, rather than by signal averaging,
offers several payoffs. The benefits could include shorter
measurement times, a larger population of eligible subjects, and the ability to inspect responses to individual
stimuli rather than ensemble averages. Techniques that
solve one part of the problem may eventually be incorporated into a global solution.
The DDO–least-squares approach investigated here
suppresses two sources of interference: top-layer-only
fluctuations (i.e., from the scalp) and global fluctuations
(i.e., systemic responses, such as respiration). It does not
reduce the interference from uncorrelated cerebral activity; measurements of two separate cerebral regions could
be employed for this purpose, as has been suggested by
others.8
The results reported here indicate that the technique is
a useful building block for improving NIRS cerebral hemodynamic monitoring. The targeted sources of interference were dramatically suppressed in all simulated cases,
including ones where the accompanying theoretical model
was strongly violated.
The method and theory described in this paper are also
in agreement with other current work on the DDO approach. In the limit of top-layer-only interference 共␥ = 0兲
and uncorrelated absorption timecourses 共B · G = 0兲, our
theory matches exactly with that described by Fabbri et
al.11 (i.e., lim ␣I = K). We also note that their suggested
method of determining K, namely by using a time interval
during which ⌬␮a2 = 0, is a special case of satisfying the
requirement for uncorrelated time courses.
One particular issue these simulations do not address
is that of tissue heterogeneity. Although efforts were
made to simulate bulk optical properties that are physiologically relevant, blood flow occurs across a complex and
confined network of vessels. The corresponding optical effects, particularly in the small volume probed by the near
detector, can be strongly location specific, thus violating
the assumption of homogeneous layers. This paper provides the groundwork for isolation of cerebral signals, but
future studies must address the issue of heterogeneity by
additional measurements or by judicious choice of source–
detector distance.
This initial consideration of homogeneous layers, however, provides some valuable insights on its own. Equation (13) identifies two primary sources of error in estimating the scale factor ␣. We have demonstrated that this
analytical formulation of error is in fact in good agreement with the actual least-squares calculation, even
though some approximations have been made to simplify
R. B. Saager and A. J. Berger
its form. As noted above, the error terms have straightforward interpretations. In particular, ␦␣ˆ 1 represents the error due to chance correlations between B and G during
the finite measurement time. These chance correlations
can be reduced by increasing the integration time. This is
different from the method proposed by Fabbri et al. in
which K is estimated during a period when no lower-layer
(cerebral) fluctuations occur. In this case increased integration time would presumably lead to greater violations
of this underlying assumption. In the least-squares approach, we are afforded the luxury of using all data collected whether stimulation is occurring or not.
Having gained a good understanding of the principal
factors affecting the performance of our approach in an
idealized homogeneous case, we can then apply these insights in addressing layer heterogeneity. The influence of
heterogeneity on the measurements can be reduced by increasing the distance dN and thus averaging over a larger
tissue volume. However, this increases ⑀, and as the modeling shows, the ␦␣ˆ 2 error increases. The simple model
helps us to analyze this tradeoff. In particular we note
that ␦␣ˆ 2 takes the form of a deterministic bias, since ⑀ and
some of the correlation terms are positive definite [see Eq.
(13)]. Perhaps, then, one way of compensating for upperlayer heterogeneity is to increase dN, simulate the
amount of ␦␣ˆ 2 error that this adds, and then apply a bias
correction to the experimental least-squares estimate of
␣I. The proper value of dN for optimizing this tradeoff
would have to be determined.
The far-detector path length also plays a role in the
magnitude of the first error term in ␦␣ˆ . The error is proportional to 具L2典F / 共具L1典F + ␥具L2典F兲. This functional form
suggests an experimental modification that could reduce
␦␣ˆ . If we reduce the source–detector separation of the far
detector, dF, the detected photons will travel a statistically shorter distance in the bottom layer and hence reduce this error term. This action serendipitously would
increase the 2D mapping resolution were an array of
DDOs to be used. Once again, however, this has a
tradeoff. Reducing the distance also reduces the measured absorbance change, making it more susceptible to
instrumental noise artifacts. Further study is required to
examine the potential benefits of this action. Such a study
would require refinements to the two-layer model with
considerations of instrumentation artifacts and noise,
which were not considered in this initial study. Once
again, the value of dF that optimizes the tradeoff would
have to be determined. The homogeneous model, while
simple, provides the conceptual framework for choosing
optimal separation distances for both detectors.
6. CONCLUSION
This work has investigated the novel combination of a
dual detector optode and a least-squares approach for the
extraction of cerebral hemodynamics signals in nearinfrared spectroscopy. Monte Carlo results based on a
simple two-layer, homogeneous model show great promise. Two major results are that (a) the method removes interference to within 2% of the ideal limit in all cases
R. B. Saager and A. J. Berger
Vol. 22, No. 9 / September 2005 / J. Opt. Soc. Am. A
APPENDIX A
where such a limit exists, and (b) the method substantially removes interference even in more complicated
cases where there is no unique theoretical solution. Further investigations include increasing the detail of the
Monte Carlo models to incorporate, for instance, dedicated layers for bone and CSF; and analyzing measurements from an experimental DDO system currently under
refinement. A mature instrument incorporating this approach should be able to offer greatly improved specificity
for cerebral hemodynamics, thus reducing measurement
time and increasing the range of potential subjects for
both clinical and research purposes.
␣LS =
We derive the analytical approximate form of the leastsquares fitting coefficient ␣ˆ LS [Eq. (11)].
By definition the least-squares fitting coefficient between the absorption time courses is
关⌬␮a1共t兲 · 具L1典F + ⌬␮a2共t兲 · 具L2典F兴 · 关⌬␮a1共t兲 · 具L1典N + ⌬␮a2共t兲 · 具L2典N兴
关⌬␮a1共t兲 · 具L1典N + ⌬␮a2共t兲 · 具L2典N兴 · 关⌬␮a1共t兲 · 具L1典N + ⌬␮a2共t兲 · 具L2典N兴
再
⬇
具L1典F
具L1典N
+
冋
⫻ 1 − 2⑀
␮12 ⬅ ⌬␮␣1共t兲 · ⌬␮␣2共t兲,
␮22 ⬅ ⌬␮␣2共t兲 · ⌬␮␣2共t兲.
具L1典F
⬇
具L1典N
Then Eq. (A2) becomes
=
具L1典F
+ ␮12
具L1典N
冋
具L1典F具L2典N
具L1典N具L1典N
␮11 + 2␮12
具L2典N
+
具L2典F
具L1典N
+ ␮22
具L1典N
册
−2
+ ␮22
具L2典F具L2典N
具L1典N具L1典N
具L2典N具L2典N
.
=
具L1典F
具L1典N
+ ␮12
冋
具L1典F
具L1典N
⑀+
具L2典F
具L1典N
␮11 + 2␮12⑀ + ␮22⑀
册
+ ␮22
具L2典F
具L1典N
2
具L1典N
+⑀
⑀
. 共A4兲
Assuming ⑀ Ⰶ 1, ⑀ → 0, then
␮11
␣LS ⬇
具L1典N
冋
具L1典N
⑀+
具L2典F
具L1典N
册
+ ␮22
具L2典F
具L1典N
⬇
⑀
␮11 + 2␮12⑀
再
⬇ ␮11
⫻
+ ␮12
具L1典F
具L1典F
具L1典N
1
␮11
冋
+ ␮12
1 − 2⑀
冋
册
␮12
␮11
具L1典F
具L1典N
⑀+
具L2典F
具L1典N
冉
册
+ ␮22
具L1典N
⑀
冎
␮12
␮11
␮12 具L2典F
␮11 具L1典N
␮12
具L2典F
2
␮11
具L1典N
⑀+
+⑀
再
具L2典F
具L1典N
册
␮22 具L2典F
␮11 具L1典N
+
−
␮22 具L2典F
␮11 具L1典N
⑀
冎
␮12 具L1典F
␮11 具L1典N
+ O共⑀2兲.
冋
共A5兲
−2
冋
G · 共B + ␥G兲 具L2典F
具L1典N
G·G
具L1典N
G·G
G · 共B + ␥G兲
具L1典N
再
+
共B + ␥G兲 · 共B + ␥G兲 具L2典F
具L1典F
+⑀
具L2典F
再
−2
2
具L1典F
␮11 具L1典N
冋 册 冎
具L1典F
␣LS ⬇
Let [see Eq. (9)] ⑀ = 具L2典N / 具L1典N
␮11
+
冋
册
␮12 具L1典F
共A2兲
.
Now, introduce the parametrizations for ⌬␮a1, ⌬␮a2 in
terms of time courses B and G; relation (A5) becomes
具L1典N具L1典N
共A3兲
共A1兲
.
⌬AN · ⌬AN
Expanding the changes in absorption in terms of the
MBLL, Eq. (A1) becomes
␮11 ⬅ ⌬␮␣1共t兲 · ⌬␮␣1共t兲,
␮11
⌬AN · ⌬AF
␣LS =
For simplicity, let
␣LS
1881
具L2典F
具L1典N
2
G·G
具L1典N
具L2典F
冊
+
G · B 具L2典F
G · G 具L1典N
共B + ␥G兲 · 共B + ␥G兲 具L2典F
G·G
G · 共B + ␥G兲
G·G
G · 共B + ␥G兲 具L1典F
具L1典N
G·G
+␥
册 冎
−
具L1典N
册 冎
2
具L2典F
具L1典N
.
−
G · 共B + ␥G兲 具L1典F
G·G
具L1典N
共A6兲
Noting that the first two terms are equal to the ideal coefficient ␣I, we have finally
1882
J. Opt. Soc. Am. A / Vol. 22, No. 9 / September 2005
␣LS ⬇ ␣I +
−2
再
具L2典F B · G
·
+⑀
具L1典N G · G
G · 共B + ␥G兲
G·G
再冋
冎册
2
⬅ ␣ˆ LS ,
R. B. Saager and A. J. Berger
共B + ␥G兲 · 共B + ␥G兲
具L2典F
具L1典F
8.
G·G
−
G · 共B + ␥G兲 具L1典F
G·G
具L1典N
冎
共A7兲
which is Eq. (11).
ACKNOWLEDGMENTS
We thank Vasan Venugopalan and Carole Hayakawa,
University of California (Irvine), for allowing us access to
their Monte Carlo code, which was modified for use in
these simulations, as well as Richard Aslin and Andrea
Gebhart, Department of Brain and Cognitive Sciences,
University of Rochester, for their many insights and support.
Corresponding author Andrew Berger can be reached
by email at [email protected].
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