Depth Resolution in Coherent Hemodynamics Spectroscopy
Angelo Sassaroli, Xuan Zang, Kristen T. Tgavalekos and Sergio Fantini
Department of Biomedical Engineering, Tufts University, 4 Colby Street, MA 02155, Medford, U.S.A.
Keywords: Near-infrared Spectroscopy, Diffuse Optics, Diffusion Theory, Functional Optical Imaging, Cerebral
Hemodynamics.
Abstract: Coherent hemodynamics spectroscopy (CHS) is a novel method based on the frequency-resolved study of
induced hemodynamic oscillations in living tissues. Approaches to induce hemodynamic oscillations in
human subjects include paced breathing and cyclic thigh cuff inflation. Such induced hemodynamic
oscillations result in coherent oscillations of oxy-, deoxy-, and total hemoglobin concentrations in tissue,
which can be measured with near-infrared spectroscopy (NIRS). The novel aspect of CHS is to induce
hemodynamic oscillations at multiple frequencies in order to obtain frequency-resolved spectra of coherent
hemodynamics. A dedicated mathematical model recently developed by our group, can translate the phase
and amplitude spectra of these hemodynamic oscillations into physiological parameters such as capillary
and venous transit times, and the autoregulation cutoff frequency. A typical method used in near-infrared
tissue spectroscopy to measure oscillations of hemoglobin concentrations is based on the modified Beer-
Lambert law, which does not allow for the discrimination of hemodynamic oscillations occurring in the
scalp from those occurring in the brain cortex. In this work, we show preliminary results obtained by using
diffusion theory for a two-layered medium, so that the hemodynamic oscillations obtained for the first and
second layer are assigned to hemodynamic oscillations occurring in the scalp/skull and brain cortex tissues,
respectively.
1 INTRODUCTION
Most neuroimaging techniques, with the exception
of electroencephalography (EEG) and
electrophysiological techniques, do not measure
directly neural activation but rather measure some
associated hemodynamic responses. Among these
neuroimaging techniques we mention: functional
magnetic resonance imaging (fMRI), positron
emission tomography (PET), single-photon emission
computed tomography (SPECT) and near infrared
spectroscopy (NIRS). Therefore, in order to obtain
an assessment of brain function by these
neuroimaging techniques, it is of the utmost
importance to understand the relationship between
neural activation and hemodynamic changes
(neurovascular coupling). Several hemodynamic
models have been proposed in the literature, among
which we mention the oxygen diffusion limitation
model (Buxton and Frank, 1997), and the
Windkessel model (Mandeville et al., 1999). The
former was introduced in order to understand the
large imbalance (observed in PET studies) between
blood flow and oxygen consumption changes
associated with brain activation. The latter was
introduced to understand the relationship between
the dynamics of blood flow and blood volume
(measured by fMRI) during forepaw stimulation in
rats. These general models can be used and adapted
to each neuroimaging technique, which is sensitive
to different physiological parameters.
Near-infrared spectroscopy (NIRS) is a non-
invasive optical method which relies on the so-called
diagnostic window of tissue transparency in the
wavelength range 600-900 nm. In this wavelength
range, the main absorbers in tissue are oxy-
hemoglobin, deoxy-hemoglobin, water, and lipids.
Recently, there has been a rapid growth of
applications where NIRS is used, mainly due to its
portability and continuous monitoring of tissue,
which are unique assets of this optical method.
Many hemodynamic models have been proposed
also in NIRS, most of them requiring the solution of
complex system of differential equations with many
unknown parameters (Huppert et al., 2007; Diamond
et al., 2009; Boas et al., 2008).
Sassaroli, A., Zang, X., Tgavalekos, K. and Fantini, S.
Depth Resolution in Coherent Hemodynamics Spectroscopy.
DOI: 10.5220/0005792101850191
In Proceedings of the 4th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2016), pages 187-193
ISBN: 978-989-758-174-8
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
187
Recently, we proposed a novel technique to
study tissue hemodynamics, named coherent
hemodynamics spectroscopy (CHS) (Fantini, 2014a;
Fantini, 2014b). The technique is based on inducing
stable hemodynamic oscillations by introducing
controlled, periodic perturbations (e.g. by paced
breathing, cyclic thigh cuff inflation/deflation, etc.)
at multiple frequencies. We also developed a
mathematical hemodynamic model to translate the
hemoglobin oscillations caused by the periodic
perturbations, as measured by NIRS, into blood
flow, blood volume, and oxidative metabolism
dynamics. The mathematical model is analytical,
and therefore simpler and less computationally
expensive than other models proposed in the NIRS
literature. The output variables of our analytical
model are four: 1) the phase difference of deoxy-
and oxy-hemoglobin oscillations; 2) the phase
difference of oxy- and total hemoglobin oscillations;
3) the ratio of amplitudes of deoxy- and oxy-
hemoglobin oscillations; 4) the ratio of amplitudes
of oxy- and total hemoglobin oscillations. The
model depends on six unknown parameters which
are fitted for when the model is applied to
experimental data. Among these unknown
parameters, there are the capillary and venous blood
transit times, and the autoregulation cutoff
frequency, which provides a quantitative measure of
cerebral autoregulation efficiency. In fact, we
remind that most studies, both in transcranial
Doppler ultrasound (TCD) and NIRS, treat the
process of cerebral autoregulation as a high-pass
filter, where blood pressure is the input and cerebral
blood flow is the output of the filter. Our analytical
model has been tested in several experimental
studies on healthy human subjects (Pierro et al.,
2014a; Kainerstorfer et al., 2015) and on
hemodialysis patients (Pierro et al., 2014b).
In NIRS, the oscillations of oxy- and deoxy-
hemoglobin concentrations are usually derived by
applying the modified Beer-Lambert law (mBLL) to
the intensity changes measured at two (or more)
wavelengths. The mBLL, unlike the diffusion
equation, which is also used to study photon
migration in tissues, does not require the medium to
be highly scattering (Sassaroli and Fantini, 2004).
However, the mBLL is based on two assumptions:
1) the changes of optical intensity are due only to
absorption changes; 2) the changes in absorption are
homogeneously distributed in the tissue probed by
light. While the first assumption is reasonable, the
second one is more questionable. For example, it is
well known that brain activation and subsequent
hemodynamic changes are focal in nature.
Therefore, it would be important to release the
assumption of homogeneous absorption changes and
use a model of light propagation which takes into
account (at least partially) the more complex
geometry of the human head and of the
heterogeneous hemodynamic changes associated
with brain perfusion. The human head is a layered
biological medium with scalp, skull, subarachnoid
space, and the brain cortex (grey matter) as its
“layers.” In the literature, a two-layered diffusion
model has been used for measuring the baseline
optical properties of two “effective” layers of the
head, where the first layer is representative of the
scalp and the skull, lumped in one layer, and the
second layer represents the brain (Choi et al, 2004;
Gagnon et al, 2008; Hallacoglu et al., 2013). In this
work, we have obtained the dynamics of oxy- and
deoxy-hemoglobin concentrations by applying the
solution of the diffusion equation in the frequency
domain for a two-layer model medium. To the best
of our knowledge, it is the first time that such a
model is applied to calculate the dynamics of oxy-
and deoxy-hemoglobin concentrations. The
oscillations of oxy- and deoxy-hemoglobin
concentrations induced at a frequency of 0.067 Hz,
by means of a cyclic thigh cuff occlusion/release
protocol, are calculated by using both the mBLL at
three different source-detector separations and the
two-layer diffusion model. In particular, we show
that the phase difference between deoxy- and oxy-
hemoglobin oscillations depends on the source-
detector separation when the mBLL is used, and is
different than the phase values retrieved by the two-
layer model in the top and bottom layers.
2 MATERIALS AND METHODS
2.1 Experimental Setup
The NIRS measurements were performed with a
commercial frequency-domain tissue spectrometer
(OxiplexTS, ISS Inc., Champaign, IL). The laser
intensity outputs were modulated at a frequency of
110 MHz. An optical probe connected to the
spectrometer by optical fibers delivered light at two
wavelengths, 690 and 830 nm, at six different
locations, separated by a single collection optical
fiber by: 7.3, 12.3, 17.6, 27.5, 32.8, 38.1 mm (690
nm) and 8.0, 13.2, 18.1, 28.1, 33.4, 37.3 mm (830
nm).
The optical probe was placed against the left
side of the subject’s forehead and secured with a
flexible headband. The optical instrument was
calibrated by using an optical phantom of known
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
188
optical properties (Fantini et al., 1995). Pneumatic
thigh cuffs were wrapped around both subject’s
thighs and connected to an automated cuff inflation
system (E-20 Rapid Cuff Inflation System, D.E.
Hokanson, Inc., Bellevue, WA). The air pressure in
the thigh cuffs was continuously monitored with a
digital manometer (Series 626 Pressure Transmitter,
Dwyer Instruments, Inc., Michigan City, IN).
Analog outputs of the thigh cuff pressure monitor
were fed to auxiliary inputs of the NIRS instrument
for concurrent recordings with the NIRS data. The
subject underwent a protocol of eight cycles, where
in each cycle the thigh cuff was inflated at a pressure
of 200 mmHg for 8 s and released for 7 s. The
method of analytic signal (Boashash, 1992; Pierro et
al., 2012) was used to associate an average phase
and amplitude to the oscillations of oxy-, deoxy-,
and total hemoglobin concentrations. In other words,
the oscillations of oxy-, deoxy-, and total
hemoglobin concentrations are described with
phasors, polar vectors whose amplitude and phase
fully describe sinusoidal oscillations at a given
frequency.
2.2 Hemodynamic Model
The hemodynamic model links the phasors
representing the oscillations of blood flow, blood
volume, and metabolic rate of oxygen to the phasors
representing the oscillations of oxy-, deoxy-, and
total hemoglobin concentrations. In the following
Eqs. (1)-(3), O(), D(), T() are the phasors that
describe the oscillations of oxy-, deoxy-, and total
hemoglobin concentrations, which are functions of
the oscillation angular frequency . Also, cbv(),
cbf(), and cmro
2
() are the phasors that describe
the oscillations of cerebral blood volume, blood
flow, and metabolic rate of oxygen, respectively.
The model equations are as follows:
ω
ctHb
CBV
ω
CBV
ω
ctHb
〈
〉
〈
〉
Ƒ
CBV
ω
CBV
ω
ω
ω
(1)
ω
ctHb1
CBV
ω
1
CBV
ω
ctHb
〈
〉
〈
〉
Ƒ
CBV
ω
CBV
ω
ω
ω
(2)
ω
ctHbCBV
ω
CBV
ω
(3)
In Eqs. (1)-(3), ctHb is the hemoglobin
concentration in blood, Ƒ
is the Fåhraeus factor
(ratio of capillary-to-large vessel hematocrit), and
the superscripts (a), (c), and (v) indicate partial
contributions from the arterial, capillary, and venous
compartments, respectively, to the baseline blood
volume (CBV
0
) and the oscillatory blood volume
(cbv). The total baseline blood volume is defined by:
CBV
CBV
Ƒ
CBV
CBV
. Also, S
(a)
,
〈
〉
, and S
(v)
are the arterial, mean capillary, and
venous saturation. The mean capillary and venous
saturations are given by
〈
〉
1
/α
and
where is
the rate constant of oxygen diffusion and
is the
mean transit time of blood in the capillaries. The
capillary (
ω
) and venous (
ω
)
transfer functions are represented by the following
Eq. (4) and Eq. (5), respectively:
ω
1
1
ω
(4)
ω
.
.
(5)
where t
(v)
in Eq. (5) is the blood venous transit time.
According to our hemodynamic model, only the
capillary and venous compartments contribute to
hemoglobin oscillations caused by oscillations of
capillary blood flow and cerebral metabolic rate of
oxygen. On the contrary, because capillary
recruitment in the brain is negligible (Villringer,
2012) only the arterial and venous compartments can
contribute to hemoglobin oscillations caused by
blood volume oscillations. In particular, our model
Depth Resolution in Coherent Hemodynamics Spectroscopy
189
predicts that the capillary and venous compartments
behave as low pass filters having ,
as
inputs, and O, D, T as outputs. Finally, the
oscillations of blood flow are related to those of
blood volume through a high-pass relationship
typical of the autoregulation process:
ω
ω
ω
ω
ω
ω
(6)
In Eq. (6), k is the inverse of the modified Grubb’s
exponent and
ω
is the RC high pass
filter having as input the weighted average of arterial
and venous blood volume oscillations, and as output
the capillary blood flow oscillations. The expression
of
is as follows:
ω
(7)
In Eq. (7), ω
2π
and
is the autoregulation cutoff frequency.
2.3 Solution of the Diffusion Equation
in a Two-Layer Cylindrical
Geometry
A two-layer diffusive medium is divided into a top
region (first layer) and bottom region (second layer)
that are separated by a planar surface. The two
media have different optical properties, namely the
absorption coefficient μ
and the reduced
scattering coefficient μ
. The diffusion equation in
the frequency domain (FD) is written as:
)(δ)(
),()(μ)],()([
0
r
rrrr
AC
a
P
v
iD
(8)
In Eq.(8) is the fluence rate, i.e. the power per unit
area impinging from all directions (units: W/m
2
) at
an arbitrary field point inside the medium (r);
1/3μ
is the diffusion coefficient, v is the speed of
light in the medium, is the angular modulation
frequency of the light intensity (in this study /(2
is 110 MHz), is the Dirac delta, and P
AC
is the
source power. For the solution of Eq. (8) in a two-
layered cylindrical medium, we have followed the
approach of Liemert and Kienle (Liemert and
Kienle, 2010). For a point source incident at the
center of a layered cylindrical medium, the general
solution of the two-layer diffusion equation in
cylindrical coordinates (r=(,,z); z is the direction
of the cylinder’s axis pointing inside the medium) is
given by (Liemert and Kienle, 2010):
)()ρ(),,(
π
)(
),(
1
2
10
2
n
n
nnk
AC
k
saJsJzsG
a
P
r
(9)
where
k
is the fluence rate in the k
th
layer of the
medium (k = 1,2), s
n
are the positive roots of the 0
th
-
order Bessel function of the first kind divided by
a
'
= a + z
b
, (where a is the radius of the cylinder),
and J
m
is the Bessel function of the first kind of
order m. Also, z
b
is the distance between the
extrapolated and the real boundary,
z
b
= 2D
01
(1+R
eff
)/(1-R
eff
), R
eff
is the fraction of
photons that are internally diffusely reflected at the
cylinder boundary and D
01
is the diffusion
coefficient in the first layer. Here, we report the
solution for G
k
only for the first layer (k = 1), since it
is the layer where the reflectance is calculated. For
the first layer (the one illuminated by the light
source), G
1
is given by the following expression:
)](
1
[αsinh
2
α
02
)](
1
[αcosh
1
α
01
2
α
021
α
01
)](
1
αexp[
1
α
01
)](
1
[αsinh)]
0
(
1
[αsinh
101
2
)]2
0
(
1
αexp[)
01
αexp(
),,(
1
b
zLD
b
zLD
DD
b
zLD
b
zz
b
zz
D
b
zzzzz
z
n
sG
(10)
where L is the thickness of the first layer, and
k
is
given by:
2,1
μ
α
0
2
0
k
vD
i
s
D
k
n
k
ak
k
(11)
Equation (10) is obtained for the limiting case when
the second layer is infinite in the z direction and for
the case of two layers having the same refractive
indices (Liemert and Kienle, 2010). In Eq. (11), D
0k
and
ak
are the diffusion coefficient and the
absorption coefficient of the k
th
layer, respectively.
For the calculation of the reflectance (R), one may
apply Fick’s law:
0
),(),(
101
z
,z
z
DR
(12)
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
190
From the expression of the reflectance (which is a
complex function), we can calculate the AC(,)
and phase (,) as follows: AC(,) = |R(,)|,
(,) = Arg[R(,)]. The solution of the diffusion
equation presented in this section was embedded in
an inversion procedure which used the Levenberg-
Marquardt method and that recovers the optical
properties (
μ
,μ
) of the two layers and the
thickness of the first layer from measured AC and
phase ( data at six source detector separations
(Hallacoglu et al., 2013). The inversion procedure is
run at each time point. By combining the dynamics
of the absorption coefficients retrieved by the
inversion procedure at two wavelengths, we were
able to calculate the dynamics of oxy- and deoxy-
hemoglobin concentrations in the first and second
layer.
3 RESULTS
Figure 1 shows the thigh cuff pressure oscillations
(top panel) and the oscillations of oxy- (O) and
deoxy-hemoglobin (D) concentrations induced by
the cuff at three different source-detector
separations. More precisely, the oscillations of oxy-
and deoxy-hemoglobin concentrations were
calculated according to the modified Beer-Lambert
law (mBLL) by using the combined intensity
oscillations at the first, third, and sixth source-
detector separations (~8, 18, 38 mm, respectively).
The temporal trends of O and D plotted in Fig. 1
are related to the phasors O() and D() (see
Eqs. (1) and (2)) by the relationships:
∆
Re
|
|
, ∆Re
|
|
where is the angular frequency of the cuff cyclic
inflation. The phase shift between deoxy- and oxy-
hemoglobin oscillations (Arg(D)-Arg(O)) are:
106°±8°, 74°±8° and 68°±1° at the first, third, and
sixth source-detector separations, respectively.
In Fig. 2, we report the thigh cuff pressure
oscillations (top panel) and the oscillations of oxy-
(O) and deoxy-hemoglobin (D) concentrations
calculated by using the two-layer model. The phase
shifts between deoxy- and oxy-hemoglobin
oscillations (Arg(D)-Arg(O)) are: 24°±7° and
34°±9° in the top and bottom layers, respectively.
Figure1: Oscillations of the cuff pressure (top panel) and
oscillations of O and D at the first (~8 mm: second
panel) third (~18 mm: third panel) and sixth (~38 mm:
bottom panel) source-detector separations, calculated by
using the modified Beer-Lambert law (mBLL). The phase
differences between deoxy- and oxy-hemoglobin
oscillations (Arg(D)-Arg(O)) are 106°±8°, 74°±8° and
68°±1° at the first, third, and sixth source-detector
separations, respectively.
Figure 2: Oscillations of the cuff pressure (top panel) and
oscillations of O and D in the top (second panel) and
bottom (third panel) layers, calculated by using the two-
layer model. The phase differences between deoxy- and
oxy-hemoglobin oscillations (Arg(D)-Arg(O)) are 24°±7°
and 34°±9° in the top and bottom layers, respectively.
4 DISCUSSIONS AND
CONCLUSIONS
We have presented some preliminary results about
the possibility of using a two-layer diffusion model
0 20 40 60 80 100 120
0
100
200
(mmHg)
Thigh cuff
0 20 40 60 80 100 120
-0.5
0
0.5
First Distance
0 20 40 60 80 100 120
-0.5
0
0.5
Third Distance
0 20 40 60 80 100 120
-0.5
0
0.5
Sixth Distance
Time [s]
O and
D (
M)
O
D
0 20 40 60 80 100 120
0
100
200
Thigh Cuff
(mmHg)
0 20 40 60 80 100 120
-0.4
-0.2
0
0.2
Top layer
0 20 40 60 80 100 120
-0.4
-0.2
0
0.2
Time (s)
O and
D (
M)
Bottom layer
O
D
Depth Resolution in Coherent Hemodynamics Spectroscopy
191
to provide depth resolution to measured oscillations
of oxy- and deoxy-hemoglobin concentrations. In
the literature, the two-layer diffusion model has
already been used for the calculation of the baseline
optical properties and hemoglobin concentrations in
the two “effective” head tissue layers. To the best of
our knowledge, this is the first time that such a
model is used for the calculation of hemoglobin
concentration oscillations. Typically, in NIRS the
dynamics of hemoglobin species are calculated by
using the modified Beer-Lambert law (mBLL),
which assumes homogeneous absorption changes in
the tissue probed by light. Since this hypothesis may
not be strictly correct in a variety of conditions, it
would be of the utmost importance to discriminate
the hemodynamic oscillations occurring in the
extracerebral layers (scalp and skull) from those
occurring in the brain. For this reason, we have
considered a more realistic model of the head which
comprises two distinct layers. The solution of the
diffusion equation in the FD for a two-layer
geometry was implemented in an inversion
procedure which recovers the dynamics of the
absorption and reduced scattering coefficients of
both layers, as well as the thickness of the first layer,
from measured AC and phase data at six source-
detector separations. We have calculated the phase
shifts between oxy- and deoxy-hemoglobin
oscillations with the two-layer model, and compared
the results with those obtained with the mBLL at
three different source detector separations. Our
hemodynamic model (Eqs. (1)-(3)) shows that the
phase shift between deoxy- and oxy-hemoglobin
concentration is one of the parameters related to the
dynamics of blood volume, blood flow and oxygen
consumption and to the underlying physiological
quantities of diagnostic and functional value (i.e. the
capillary and venous blood transit times, the
autoregulation cutoff frequency, etc.). Therefore,
different phase shifts indicate different dynamics of
the underlying vascular, metabolic, and
physiological parameters.
In this work, we have used a thigh cuff
occlusion/release at a single frequency of 0.067 Hz.
We found that the phase shifts between deoxy- and
oxy-hemoglobin concentrations calculated with the
mBLL at the first source-detector separation
(~8 mm) differ from those calculated at the third
(~18 mm) and sixth (~38 mm) source-detector
separations. Since the first channel mostly probes
the superficial scalp and skull layers, while the
second and the third channels are more sensitive to
the brain hemodynamics, these results indicate that
the vascular dynamics (blood flow, blood volume),
the physiological parameters (capillary and venous
transit times etc.), or both are different between the
scalp and the brain. For the two-layer model, we
have found equal phase shifts (within errors)
between oxy- and deoxy-hemoglobin oscillations for
the top and bottom layers. However, this observation
is not enough to conclude that also the vascular
dynamics and physiological parameters are the same
for the top and bottom layers. In fact, we remind that
CHS makes use of spectra, i.e. of phase shifts and
amplitudes ratios between hemoglobin species
calculated at different frequencies, therefore the
observations we can derive by using only one
frequency (as it was done in this work) are not
conclusive.
Another comment concerns the apparent
discrepancy between the two models used for the
calculations of hemoglobin oscillations. One would
expect that the hemoglobin oscillations of the top
layer retrieved by a two-layer model would reflect
only the oscillations of scalp and skull, and should
be similar to the oscillations measured by mBLL at
the shortest source-detector separations (up to
around 1-1.5 cm). On the contrary, this study could
not confirm this point and further investigations are
needed. One possible explanation is that the
hemoglobin oscillations induced by the forcing
mechanism have a highly heterogeneous nature and
cannot be considered uniform in each head layer (i.e.
the absorption changes are not “layered-like”). In
these conditions, by using the two-layer model we
would most probably obtain a weighted average of
the oscillations occurring in different tissue regions
probed by different channels.
In summary, this work represents the first step
toward the development of depth-resolved CHS,
which ultimately can result in a powerful non-
invasive optical method for the non-invasive
assessment of cerebral perfusion and autoregulation.
ACKNOWLEDGEMENTS
This research is supported by the US National
Institutes of Health (Grant no. R01-CA154774).
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