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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3921_Библиотеки_им_академика_М_И_Перельмана
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5.2 Theory of Laser Doppler Spectrum Decomposition 93
As can be observed in Figure 5.2, the DSPDs are scaled along the frequency
axis for different speeds of the particles. On the other hand, the DSPDs are
scaled along the amplitude axis with concentration of moving particles. Thus, the
spectrum of the laser Doppler signal S
f
can be expressed as a superposition of
DSPDs calculated for different speeds h(V , f ). This superposition is schematically
illustrated in Figure 5.4 for an assumed distribution of speeds of particles in the
medium.
The superposition can be expressed by the following system of linear equations:
S
f
1
= c
(
V
1
)
h
V
1
, f
1
+ c
(
V
2
)
h
V
2
, f
1
+···+c
(
V
N
)
h
V
N
, f
1
S
f
2
= c
(
V
1
)
h
V
1
, f
2
+ c
(
V
2
)
h
V
2
, f
2
+···+c
(
V
N
)
h
V
N
, f
2
,
.
.
.
S
f
M
= c
(
V
1
)
h
V
1
, f
M
+ c
(
V
2
)
h
V
2
, f
M
+···+c
(
V
N
)
h
V
N
, f
M
(5.6)
which can be rewritten as matrix equation
S
T
=
c
T
h (5.7)
where row vector c contains concentrations c(V) of particles moving with speed V.
The spectrum S represents the measured data normalized by its integral. The
DSPDs, h, can be calculated for assumed g of the scattering anisotropy of the
0
10
−4
10
−3
10
−2
10
−1
h(V
3
, ∆f ) (a.u.)
10
−4
10
−3
10
−2
10
−1
h(V
2
, ∆f ) (a.u.)
10
−3
10
−2
10
−1
h(V
1
, ∆f ) (a.u.)
10
−4
0 5 10 15 20
∆ f (kHz)
10
−3
10
−2
10
−1
S(∆f ) (a.u.)
V
1
V
2
V
3
0.5 0.4 0.25 0.1 0
Concentration (a.u.)
1
2
3
4
5
Speed (mm s
−1
)
0
∆f
max
h(V
3
, ∆f )d∆f = 0.25
0
∆f
max
h(V
2
, ∆f )d∆f = 0.5
0
∆f
max
h(V
1
, ∆f )d∆f = 0.25
0
∆f
max
S(∆f )d∆ f = 1
Figure 5.4 Schematic representation of Laser Doppler spec-
trum S
f
as a superposition of Doppler shift probability
distributions calculated for different speeds h(V,f ).
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94 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
particles, refractive index n, and light wavelength λ. If the spectrum is sampled
at M frequencies, the DSPDs are estimated at 3N speeds and sampled at 3M
frequencies. Furthermore, the unknown moving particles speed distribution c is
sampled at 3N speeds. Finally, the laser Doppler spectrum S corresponds to first M
elements of 1 × 3M size row vector c
T
h. This system of equations can be solved for
N ≤ M. However, because of the similarity of the DSPDs, the problem is ill-posed
and the solving process is rather difficult.
Moving particles speed distribution c = c(V) is expressed in arbitrary units. Inte-
gral (zeroth moment) of the laser Doppler spectrum is assumed to be proportional
to the total concentration of the moving particles [12]. Typically, measured spec-
trum S and DSPDs matrix h are normalized by their integrals. Thus, the integral
of the distribution c is equal 1 when the error of solution of Eq. (5.7) approaches
0. When series of spectra collected during in vivo measurement are acquired,
the normalization of the spectra can be carried out, considering the integral of
selected (or averaged) spectrum from the rest flow period as a reference spectrum.
This procedure allows monitoring changes in concentration of moving particles in
arbitrary units.
5.2.2
Multiple Scattering
In classical laser Doppler measurements performed on human skin, the combi-
nation of optical properties, source –detector separation, and low concentration
of moving particles (red blood cells) assure that most of Doppler-shifted photons
reaching the detector undergo only a single Doppler scattering event. Higher-order
10
0
Normalized power (dB)
10
1
10
2
10
3
10
4
10
0
10
1
10
2
10
3
r = 2 mm
c = 1%
c = 10%
c = 30%
r = 3 mm
r = 4 mm
r = 5 mm
10
4
0 5 10 15 20
Frequency (kHz)(a)
0 5 10 15 2
0
Frequency (kHz)(b)
Figure 5.5 Laser Doppler spectra obtained for different
source– detector separations (a) and for different concentra-
tions of moving particles (b) with the assumption that the
single speed of moving particles corresponds to f
max
/2.
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5.2 Theory of Laser Doppler Spectrum Decomposition 95
Doppler scattering events lead to appearance of higher frequencies in measured
spectra and cause the spectra to become more flat. This effect is shown in
Figure 5.5, where spectra for different concentrations of moving particles and
different source – detector separations were presented. The theory of laser Doppler
spectrum decomposition for multiple Doppler scattering regimes was proposed to
utilize the decomposition method in more general cases.
The theory of multiple Doppler scattering events represents an extension of the
method described above. The DSPDs characterizing double-scattering h
2
(V, f )
can be expressed as convolutions of single-scattering DSPDs h
1
(V, f ) according
to the following formula
h
2
V, f
= h
1
V, f
∗ h
1
V, f
(5.8)
The results of convolution of DSPDs expressed by Eq. (5.8) for two assumed
different speeds of moving particles (V
1
and V
2
) yields the results presented in
Figure 5.6.
Consequently, the DSPDs for Doppler scattering of order s can be calculated with
the use of iterative approach
h
s
V, f
= h
s−1
V, f
∗ h
1
V, f
(5.9)
0
h
1
(V
1
, ∆f )
h
1
(V
2
, ∆f )
h
2
(V
1
, ∆f ) = h
1
(V
1
, ∆f ) ∗ h
1
(V
1
, ∆f )
h
2
(V
1
, V
2
, ∆f ) = h
1
(V
1
, ∆f ) ∗ h
1
(V
2
, ∆f )
h
2
(V
2
, ∆f ) = h
1
(V
2
, ∆f ) ∗ h
1
(V
2
, ∆f )
0 2 4 6 8 101214161820
Frequency ∆f (kHz)
0.005
0.01
0.015
0.02
0.025
Normalize power (a.u.)
Figure 5.6 DSPDs obtained for a double-Doppler scat-
tering regime by convolutions calculated according to
the Eq. (5.8) for two assumed speeds of moving parti-
cles V
1
and V
2
, which correspond to f
max
/4 = 5kHz and
f
max
/2 = 10 kHz, respectively.
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96 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
h =
h
1
(V
1
, ∆f )
h
1
(V
2
, ∆f )
h
1
(V
N
, ∆f )
Figure 5.7 Illustration of the calculation of Doppler shift
probability distributions matrix for the multiple scattering
regime.
Unfortunately, when more than one scattering event is considered, the number
of DSPDs that must be included in the analysis grows very quickly. In this way
also the number of unknowns in the system of equations increases. All possi-
ble variations of convolutions of DSPDs obtained for different speeds must be
included. The number of all variations is equal to N
s
. However, commutativity
of convolution allows one to reduce this number. The convolution procedure is
illustrated in Figure 5.7. For two scattering events, the number of derived DSPDs
that must be included in the system of equations is given by Q = N(N + 3)/2
(where N is the number of speeds considered). More generally, in case of s scatter-
ing events, the system of linear equations should cover all possible combinations,
with repetitions (multicombinations) from a set of particles moving with N dif-
ferent speeds summed up for the considered number of scattering events from
1tos. Thus, the number of DSPDs Q , which should be included in the system of
equations, is given by
Q =
s
k=1
N + k −1
k
(5.10)
Table 5.1 Number of unknowns for different number of
speeds and Doppler scattering events taken into considera-
tion during decomposition process.
Number of doppler Number of Number of
scattering events, s speeds, N unknowns, Q
11010
12020
21065
2 20 230
3 10 285
3 20 1770
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5.2 Theory of Laser Doppler Spectrum Decomposition 97
Some examples of the number of unknowns for different combinations of number
of speeds and scattering events are presented in Table 5.1.
It has been shown that for noise-free spectra analyzed for N = 20 and s = 2,
the speed distributions can be estimated. However, for higher numbers of speeds
and/or scattering events, the system of equations is ill-posed and hard to solve.
In case of multiple Doppler scattering, Eq. (5.7) can be rewritten as follows:
S
T
= a
T
h (5.11)
where row vector S represents the Doppler spectrum sampled at M frequencies
and normalized by its integral and h is the matrix of convoluted DSPDs normalized
by their integrals calculated as shown in Figure 5.7. Row vector a corresponds
to the product of unknown speed distribution c and a matrix of coefficients C,
which describes weights of the different speeds in all convolutions of DSPDs. The
repeated variations of convolutions can be omitted, and representation of defined
speed in each convoluted DSPD can be considered. Speed distribution c can be
derived utilizing the vector a, which can be obtained by solving Eq. (5.11) and
matrix C
a
T
= c
T
C
T
(5.12)
Solution of the ill-posed problem of decomposition can be found using a
constraint-based approach. Methods based on pseudoinverison or minimization of
solution error allow the c row vector in
R
N
Euclidean space to be obtained. However,
the speed distribution values should be located in
R
N
+
subspace, considering that
the concentrations of moving particles cannot be negative. This observation applied
to the decomposition problem transforms solving of the set of linear equations to
a convex minimization problem with constraints. The Eqs. (5.11) and (5.12) can be
rewritten as
minimize f
1
(a) =
S
T
− a
T
h
·
S
T
− a
T
h
T
subject to a ≥ 0
minimize f
2
(c) =
a
T
− c
T
C
T
·
a
T
− c
T
C
T
T
subject to c ≥ 0
(5.13)
For typical in vivo laser Doppler measurements, the influence of higher-order
Doppler scattering events may be insignificant for the short source–detector
separations used. However, speed distributions for single and higher-order Doppler
scattering events can provide depth-resolved information about perfusion of the
tissue.
An example of speed distribution decomposition for a medium in which multiple
scattering is present is shown in Figure 5.8. Speed distributions were derived for
laser Doppler spectra simulated with Monte Carlo code assuming that number of
Doppler scattering events was limited to s = 1, 2, and 3. The laser Doppler spectra
are shown in Figure 5.8a. Speed distributions derived by decomposition of the
laser Doppler spectra are shown in Figure 5.8b. These distributions were obtained
from the laser Doppler spectra with the use of only single Doppler scattering
DSPDs, while decomposed spectra are affected by multiple Doppler scattering.
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98 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
−60
0 5 10 15 20
Frequency (kHz)
0
Simulated s = 1
Simulated s = 2
Simulated s = 3
Reconstructed
123 5
Speed (mm s
−1
)
−50
−40
−30
−20
−10
0
Normalized power (dB)
0
0.08
0.06
0.04
0.02
0.1
0.12
0.14
0.16
Concentration (a.u.)
4
Assumed
Decomposed s = 1
Decomposed s = 2
Decomposed s = 3
(a)
(b)
Figure 5.8 Laser Doppler spectrum ob-
tained for semi-infinite medium, source–
detector separation r = 1 mm, laser light
wavelength λ = 780 nm, concentration of
moving particles c = 1%, scattering coeffi-
cient µ
s
= 2mm
−1
, absorption coefficient
µ
a
= 0.01 mm
−1
, anisotropy factor g = 0,
and refractive index n = 1.4. Components of
the spectrum related to the different orders
of Doppler scattering events s are presented
in (a). Speed distributions derived from laser
Doppler spectra shown in (a) were obtained
with assumed single Doppler scattering de-
composition regime and presented in (b).
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5.3 Validation of the Spectrum Decomposition Method 99
Thus, the comparison of obtained speed distributions shows influence of the
presence of higher-order Doppler scattering on the results of the decomposition
when only single Doppler scattering DSPDs are used. It can be noted that the
decomposed distributions contain high-speed components resulting from the
simplified procedure utilized.
5.3
Validation of the Spectrum Decomposition Method
5.3.1
Laser Doppler Spectra Generated by Monte Carlo Simulations
The method of laser Doppler spectrum decomposition was validated by the analysis
of spectra obtained in Monte Carlo simulations. The spectra were simulated for
different assumed speed distributions and concentrations of moving particles.
The optical properties of the medium were selected according to the published
characteristics data of living tissues [30, 31]. Results of spectra decompositions for
assumed uniform and Gaussian speed distributions are presented in Figure 5.9.
The decomposition was carried out for the single Doppler scattering regime, and it
can be noted that the method allows for successful estimation of speed distributions
for nonnoisy spectra obtained by Monte Carlo simulations.
Laser Doppler spectrum decomposition for multiple Doppler scattering regimes
requires spectra with high signal-to-noise ratio [18]. For in vivo measured laser
Doppler spectra, the signal-to-noise ratio is lower than the required value and the
decomposition for multiple Doppler scattering regime is not possible. However,
for spectra simulated with the Monte Carlo method, the decomposition is possible.
An example of such decomposition is presented in Figure 5.10 for two Doppler
scattering events. The spectra were generated for two different speed distributions
and two source –detector separations r. It should be noted that the relation between
speed distributions integrals for first (s = 1) and second (s = 2) Doppler scattering
events corresponds to the relation between the total power of detected photons that
undergo only one and those that undergo only two Doppler scattering events.
Furthermore, the system of linear equations (Eq. (5.7)) can be utilized for
simulations of laser Doppler power density spectra when the following parameters
are assumed:
•
moving particles speed distribution;
•
scattering anisotropy;
•
medium refractive index;
•
laser light wavelength;
•
sampling frequency of laser Doppler signal;
•
cut-off frequencies of the filters implemented in the laser Doppler instrument;
•
noise in the measured power density spectra.
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100 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
−60
0
5101520
Frequency (kHz)
0 5 10 15 20
Frequency (kHz)
LD spectra
−50
−40
−30
−20
−10
0
−60
−80
−40
−20
0
Normalized power (dB)
(a) (b)
(c) (d)
Speed distributions
0
024
Speed (mm s
−1
)
024
Speed (mm s
−1
)
0.05
0.1
0.15
0.06
0.08
0.1
0
0.02
0.04
0.12
Concentration (a.u.)
Simulated
Reconstructed
Decomposed
Assumed
Figure 5.9 Results of decompositions of laser Doppler
spectra obtained by Monte Carlo simulations for assumed
uniform (a,c) and Gaussian (b,d) speed distributions.
Moreover, time domain signals could be obtained from the spectrum, assuming
random phase shift of harmonics [32]. The spectrum could be also used for
calculation of perfusion index and simulation of perfusion signals. In other words,
the simulation method can be utilized as a tool in development of the signal
processing algorithms in LD instrumentation [33].
Assuming a homogeneous medium, where the single Doppler scattering regime
dominates, results from Monte Carlo simulations and those obtained from Eq. (5.7)
are similar and the difference tends to zero for an infinite number of simulated
photons. Thus, Eq. (5.7) represents an analytical form of the time-consuming Monte
Carlo algorithm of laser Doppler power density spectra simulation. Comparison of
spectra simulated by the Monte Carlo algorithm and analytically calculated laser
Doppler spectra for single Doppler scattering regime are shown in Figure 5.11.
5.3.2
Laser Doppler Spectra Measured on the Phantom
The decomposition method described was validated in a series of experiments
carried out on a phantom, based on a glass tube (1 mm inner diameter) filled
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5.3 Validation of the Spectrum Decomposition Method 101
0
0.5
1
Speed (mm s
−1
)
r = 2 mm r = 3 mm
12345
0
0.5
1
12345
0
0.5
1
12345
s = 1
s = 2
12345
0
0.5
1
Concentration (a.u.)
(a)
(b)
(c) (d)
Figure 5.10 (a– d) Results of speed distri-
butions calculation for spectra simulated by
Monte Carlo method, obtained by decompo-
sition for two Doppler scattering events (s =
1 and 2) and emitter–detector separations
r = 2 and 3 mm. (a,b) results obtained for
assumed Gaussian speed distribution with
center speed corresponding to f
max
/2and
standard deviation 1 mm s
−1
. For (c,d), the
single speed of moving particles correspond-
ing to f
max
/2 was assumed.
with a suspension containing moving particles (intralipid). The tube was mounted
in an optically turbid medium (silicone rubber), and the laser Doppler probe
was positioned above the tube, covered by 1 mm thick silicon rubber layer, with
source and detection fibers (200 µm core multimode fibers, 250 µm separation)
fixed perpendicularly to the axis of the tube. The solution was pumped at different
speeds achieved by gravitation in order to simulate various perfusion conditions.
Actual mean solution speed was assessed measuring volume flowing by the tube
in time, with knowledge of the tube diameter. Measurements were performed
by using a self-constructed laser Doppler instrument with laser wavelength of
780 nm. The AC components of the signal from photodetector were recorded with
a sampling frequency of 40 kHz.
The spectra of laser Doppler signal and derived distributions of speed of moving
particles obtained for different speeds of the suspension in the phantom are
presented in Figure 5.12. It can be observed that the estimated speed (at which
the concentration reaches its maximum) increases gradually with the speed of the
suspension pumped through the phantom.
The mean speed of moving particles can be estimated from the speed distribution
by calculation of its first moment. First moments of speed distribution related to
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102 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
−80
0 5 10 15 20
Fre
q
uenc
y(
kHz
)
g = 0.0 g = 0.8
0 5 10 15 20
Fre
q
uenc
y(
kHz
)
−70
−60
−50
−40
−30
0
−10
−20
Normalized power (dB)
(
a
)(
b
)
MC simulation
Analytical form
Figure 5.11 Comparison of spectra sim-
ulated by the Monte Carlo algorithm and
analytically calculated laser Doppler spectra
for the single Doppler scattering regime. Re-
sults obtained after simulation of 30 × 10
8
photons for two different anisotropy factors
g = 0.0 (a) and g = 0.8 (b). Analytically ob-
tained spectra were normalized by integrals
of Monte-Carlo-simulated spectra.
the mean intralipid flow speed are presented in Figure 5.13. It can be noted that
the relationship is linear (correlation r = 0.96) with slight saturation observed for
high speeds. However, the absolute values of speed derived by proposed method
are underestimated in comparison to the actual speed of the suspension in the tube
of the phantom.
Gradual increase of speed connected with the maximum concentration in the
distribution was reported in the studies carried out by the Link
¨
oping group [20, 34]
for a suspension with polystyrene particles.
5.4
In Vivo Measurements of Speed Distribution of Red Blood Cells in
the Microvascular Network
The technique of laser Doppler spectrum decomposition facilitated recording speed
distributions of red blood cells in the microvascular network during stimulation
tests. The testswere carried outon healthy volunteersexamined in sittingposition at
room temperature (22
◦
C). Measurements were performed with a self-constructed
laser Doppler instrument operating at laser wavelength of 780 nm and with
sampling frequency of 40 kHz. The probe (200 µm core multimode fibers, 250 µm
source–detector separation) was fixed on the tip of the middle finger of the left hand
using self-adhesive tape. A pressure cuff fixed on the forearm was used to block the
blood flow. Results of arterial and venous occlusions are presented in Figures 5.14
and 5.15, respectively. During artery occlusion, the cuff was manually pumped to
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