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4.4 Monte Carlo and LDF 83
Table 4.1
Measurement depth in millimeters for muscle,
liver, as well as gray and white matter for the wavelengths
543, 633, and 780 nm.
LDPM fiber
separation/LDPI
beam ∅ (mm)
Muscle Liver Gray matter White matter
543 633 780 543 633 780 543 633 780 543 633 780
LDPM 0.0 0.10 0.18 0.19 0.046 0.11 0.13 0.088 0.15 0.16 0.064 0.092 0.099
LDPM 0.25 0.23 0.48 0.55 0.11 0.31 0.40 0.21 0.43 0.48 0.12 0.19 0.20
LDPM 0.50 0.29 0.62 0.73 0.14 0.40 0.52 0.27 0.58 0.64 0.17 0.27 0.28
LDPM 1.2 0.40 0.83 0.96 0.19 0.56 0.71 0.37 0.79 0.86 0.26 0.45 0.46
LDPI 0.5 0.19 0.60 0.79 0.053 0.23 0.53 0.17 0.48 0.68 0.10 0.17 0.20
LDPI 2.0 0.24 0.73 0.92 0.054 0.32 0.66 0.21 0.59 0.81 0.12 0.22 0.26
Table 4.2 Measurement depth in millimeters for skin on
index finger pulp and volar side of the forearm both at rest
and at 44
◦
C heat provocation.
LDPM fiber Index finger Forearm unprovoked Forearm provoked
separation/LDPI
beam ∅ (mm)
543 633 780 543 633 780 543 633 780
LDPM 0.0 0.31 0.33 0.35 0.43 0.48 0.52 0.52 0.60 0.65
LDPM 0.25 0.32 0.35 0.41 0.44 0.49 0.53 0.53 0.61 0.66
LDPM 0.50 0.34 0.36 0.50 0.45 0.50 0.55 0.54 0.61 0.67
LDPM 1.2 0.40 0.53 0.85 0.50 0.56 0.70 0.58 0.66 0.73
LDPI 0.5 0.33 0.36 0.53 0.44 0.50 0.56 0.53 0.62 0.67
LDPI 2.0 0.34 0.39 0.68 0.46 0.52 0.63 0.55 0.64 0.70
4.4
Monte Carlo and LDF
Although the Dopplereffectis a light wavephenomenon, the Monte Carlotechnique
that is based on the particle representation of light has been used throughout this
chapter to quantify the Doppler power spectrum from a simulated medium. This
may seem contradictory. However, each individual Doppler shift is calculated
using the wave representation of light and the Doppler shift of each photon is
then stored as an extra attribute of the photon in the Monte Carlo simulation [34].
The calculation of the Doppler power spectrum is based on the optical Doppler
spectrum, which is given from the distribution of photons with different total
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84 4 Laser Doppler Flowmetry
Doppler shifts. In summary, the photons in the Monte Carlo simulations behave
as information carriers, but both the individual Doppler shifts and the resulting
Doppler power spectra are calculated using light wave theory, using input from the
information-carrying photons.
It has frequently been shown that Doppler power spectra calculated with the
aid of Monte Carlo simulations correlate with ex vivo measurements for LDPM
[47, 48, 50, 51]. Recently, it has been shown that they correlate well with in vivo
measurements [37]. For LDPI, on the other hand, speckle effects also have to be
considered [40].
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87
5
Toward Assessment of Speed Distribution of Red Blood Cells
in Microcirculation
Adam Liebert, Stanislaw Wojtkiewicz, and Roman Maniewski
5.1
Introduction
Laser Doppler flowmetry remains a prospective technique of microcirculation
assessment. Despite serious advancements of the instrumentation [1, 2] and
successes of clinical tests [3, 4] of the methodology, the laser Doppler systems are
mostly used for research applications and not for routine clinical examinations. In
recent decades, modern optical, electronic, and fiber optic techniques combined
with advanced signal processing were applied in order to provide depth-resolved
perfusion assessment [5–8] and imaging of perfusion of large tissue area [9 – 11]
based on the laser Doppler technique. Since the microvasculature is a complex
system that depends on multiple physical and physiological factors (such as
hematocrit, psychological state of the subject, ambient temperature, humidity,
etc.) the procedures for measurement of perfusion need standardization. In most
clinical studies, stimulation tests are applied, which allow the changes of the relative
signals measured to be obtained. However, even the interindividual repeatability of
such measurements carried out in clinical conditions may be rather low because of
the influence of the above-mentioned factors. This effect, together with the lack of
absolute calibration of the signals, represents obstacles for acceptance of the laser
Doppler perfusion assessment technique in clinical practice.
The perfusion assessed by laser Doppler flowmetry represents a product of speed
and concentration of the moving red blood cells averaged in the volume interro-
gated by the photons traveling from the source to the detector [12]. It was shown
that the perfusion signal depends on the optical properties of the medium under
investigation [13]. It was also proposed that the signal can be normalized by a quan-
tity related to the volume through which the laser light passes. Such a procedure
may lead to strict dependence of the signals on perfusion of the tissue [14, 15].
Classical signal processing used in laser Doppler perfusion measurements is
based on an evaluation of the first moment of the AC component of the power spec-
tral density of the optical signal remitted from the tissue. It was shown that the first
moment is linearly related to the perfusion of the tissue [16]. However, the absolute
value of the speed and concentration of the red blood cells remains unknown. The
Microcirculation Imaging, First Edition. Edited by Martin J. Leahy.
2012 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2012 by Wiley-VCH Verlag GmbH & Co. KGaA.
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88 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
spectrum of the laser Doppler signal contains information on the speed distribution
of the particles moving in the volume interrogated by the photons traveling from
the source to the detector [17], and this information can be extracted by appropriate
processing [18–20]. The processing technique is based on spectral decomposition
using the Doppler frequency shift probability distribution obtained for a single
Doppler scattering event. Considering a single Doppler scattering event, it can be
shown that this probability depends only on distributions of the scattering angle and
the angle between direction of the photon and the particle. In consequence, assum-
ing that the last angle is random and the scattering angle probability distribution
can be estimated by the Henyey –Greenstein phase function, the Doppler frequency
shift probability distribution for a single Doppler scattering event can be calculated
by Monte Carlo simulations. It can also be derived analytically, which makes the
procedure much faster. The measured laser Doppler spectrum represents superpo-
sition of spectra formed by the distribution of Doppler frequency shifts scaled along
frequency axis for different speeds of the moving particles. The system of linear
equations formed by summing these distributions can be solved in order to assess
the distribution of speedof the moving particles.Moreover, a theoretical basisfor the
evaluation of spectra obtained in a multiple scattering regime was proposed. How-
ever, the inverse solution problem becomes ill-posed, and the system of equations is
very difficult to solvefor the multiple scattering regime. The method of laser Doppler
spectrum decomposition was validated using Monte Carlo simulations by genera-
tion of laser Doppler spectra for assumed distributions of speed of moving particles.
The method was also tested in phantom experiments. Finally, the speed distribu-
tions were obtained for red blood cells moving in a microvascular network during
venous and arterial occlusion as well as during thermal stimulation of the tissue.
5.2
Theory of Laser Doppler Spectrum Decomposition
5.2.1
Single Doppler Scattering
The theory described below is based on several assumptions that were used
previously to show that the laser Doppler perfusion signals are linearly related
to the concentration of red blood cells and their speed [16, 21]. The most critical
assumption is that the concentration of red blood cells is small and, in consequence,
that the probability that a single photon will undergo more than one Doppler
scattering event during its travel between source and detector is very small. This
assumption leads to the observation that the first moment of the spectrum of the
laser Doppler signal is related to the perfusion of the tissue. With this assumption,
we consider the spectrum of the laser Doppler signal, which can be measured for
a medium filled with particles moving at single speed, but with random direction
of movement. The geometry of a single Doppler scattering event is presented in
Figure 5.1.
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5.2 Theory of Laser Doppler Spectrum Decomposition 89
V
Light wave before
Doppler scattering
event
Moving particle direction
Initial
wave propagation direction
Wave propagation direction
after Doppler scattering event
Light wave
after Doppler scattering event
∆K
K
0
K
s
j
q
Figure 5.1 Geometry of a laser Doppler scattering event occurring at a moving particle
[22].
The frequency shift that is caused by such a scattering event occurring on a
particle moving with speed V is expressed by the following formula:
f =
2nV
λ
sin
θ
2
cos ϕ (5.1)
where ϕ is an angle between directions of the photon and the moving particle, θ
is the scattering angle, λ is the wavelength, and n is the refractive index of the
medium, assuming that
•
the directions of the photons are random
•
the particles move in random directions
•
particles move with speed V
•
photon can undergo only one Doppler shift scattering event
The spectrum of the laser Doppler signal represents a distribution of probability
called Doppler shift probability distribution (DSPD). As can be noted from Eq. (5.1),
this probability distribution depends only on the distributions of the scattering
angle θ and the angle between the direction of the photon and the direction
of the particle ϕ. The first angle depends on anisotropy of the scattering in the
medium. Commonly, the light scattering in the tissue is described by a phase
function proposed by Henyey and Greenstein for the assessment of scattering
angle distribution during light penetration through interstellar clouds [23]. It
was reported that this phase function can be used to estimate the scattering
angle distribution for living tissues [24]. Actually, for tissue optics, the original
Henyey–Greenstein phase functions need a small modification [25–27], that is,
multiplication by sine of scattering angle θ
f
(
θ
)
=
1 −g
2
sin θ
2
1 +g
2
− 2 g cos θ
3
/
2
if θ ∈
[
0;π
]
(5.2)
where g ∈
[
−1;1
]
is an anisotropy factor. This ensures that for isotropic scattering
(g = 0), the number of photons scattered per solid angle remains constant.
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90 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
The Henyey–Greenstein function is heuristic and its parameter g, called the
anisotropy factor, is equal to the mean cosine of the scattering angle
g =
4π
0
f
(
θ
)
cos θ d (5.3)
The second angle ϕ can be assumed to be random, which means that its proba-
bility distribution is uniform. This assumption is rather strong in microvascular
networks. The directions of movement of red blood cells are not random and are
determined by the geometry of the microvessels. However, considering that the
photons are diffusely distributed in the tissue and their travel distances are longer
than inverse of scattering coefficient,the assumption that angles between photons
and moving particles are uniformly distributed is reasonable.
It can be noted from Eq. (5.1) that the maximum frequency shift related to the
Doppler scattering event is
f
max
=
2nV
max
λ
(5.4)
and the amplitude of the Doppler shift for a single photon may have values between
0 and f
max
. The probability distribution of Doppler shifts obtained for a large set
of photons depends only on angles θ and ϕ . It should be noted that for a Doppler
scattering event occurring with high anisotropy of light scattering, the sine term
in Eq. (5.1) is much smaller than one (sin(θ/2) 1). Such a situation is typical for
scattering of light on red blood cells, for which anisotropy factor is estimated to be
about 0.98–0.99 [24]. Thus, for such scattering events, the Doppler frequency shift
is usually lower than f
max
for interactions with particle moving at speed greater
than V
max
. This effect should be taken into consideration when defining maximum
speed of moving particles V
max
= λf
max
/2n (see Eq. (5.4)), which can be observed
in the laser Doppler power density spectra.
The dependence of DSPDs on speed V and concentration of moving particles
can be assessed with the use of Monte Carlo simulations (see Figure 5.2). It can
be clearly noted that f
max
is related to V and that the amplitude of the DSPDs
depends on the concentration of particles c. Spectra shown in Figure 5.2b contain
photons that undergo only single Doppler scattering event. The other photons
undergoing multiple Doppler scattering were omitted. For large concentration of
moving particles, the influence of the number of photons that undergo higher-order
Doppler scattering events is significant. Thus, it should be noted that in this case,
there is no linear relation between the integral of detected spectra and concentration
of moving particles.
The shape of the distribution depends on shape of the scattering angle phase
function. The phase function for moving particles can be approximated by
Henyey–Greenstein phase function [24], Gegenbauer kernel phase function [28], or
Mie theory [29]. In Figure 5.3, DSPD shapes are presented for different anisotropy
factors. These shapes were calculated for a Henyey–Greenstein phase function
given in Eq. (5.2). It can be noted that for high g values, which are typical for a living
tissue [24], the DSPDs become very steep, whereas for low g values, the distribution
is rather flat. With the assumption that the Henyey –Greenstein phase function
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5.2 Theory of Laser Doppler Spectrum Decomposition 91
0
× 10
4
× 10
4
0 5 10 15 20
Frequency (kHz)
0 5 10 15 20
Frequency (kHz)
0.5
1
1.5
2
Number of photons
0
0.5
1
1.5
2
2.5
Number of photons
c = 1%
c = 5%
c = 10%
1/4V
max
1/2V
max
3/4V
max
V
max
(a)
(b)
Figure 5.2 Dependence of Doppler shift
probability distribution on (a) speed and
(b) concentration of moving particles in a
highly scattering medium with few moving
particles, that is, the number of scattering
events is much greater than the number
of Doppler shift scattering events. Results
of Monte Carlo simulations performed for
semi-infinite medium, source–detector sepa-
ration r = 1 mm, and laser light wavelength
λ = 780 nm and optical properties of the
medium: scattering coefficient µ
s
= 2mm
−1
,
absorption coefficient µ
a
= 0.01 mm
−1
,
anisotropy factor g = 0, and refractive index
n = 1.4.
defined in Eq. (5.2) can be applied to the estimation of scattering angle distribution,
the DSPDs denoted by h for assumed particle speed V can be calculated analytically
[19]. It can be assumed that anisotropy of light scattering on moving particles
is high, which leads to small scattering angle θ. As mentioned above, in such
cases, the scattering events occurring on particles moving with maximal speed
may significantly influence the observed power density spectra. Thus, DSPDs h
should be calculated for maximum frequency shift higher than f
max
, and the first
part of the h distribution up to f
max
should be taken for further calculations. For
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92 5 Toward Assessment of Speed Distribution of Red Blood Cells in Microcirculation
10
−8
0
g = 0.99
g = 0.9
g = 0.5
g = 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Normalized power (a.u.)
∆f
∆f
max
(a.u.)
MC simulation
Analytical form
Figure 5.3 Shape of the Doppler shift probability distribu-
tions for different values of anisotropy factor g of the scat-
tering on moving particles.
example, assuming calculation of h for 3f
max
, the DSPD for assumed anisotropy
factor of moving particles g ∈ [ −1, 1], moving particles speed V ∈ [0, 3V
max
], tissue
refractive index n, and laser light wavelength λ can be represented as follows:
h
V, f
=
f
Z
λ
2nV
f
if f ∈
0;
2nV
λ
0iff ∈
2nV
λ
;3f
max
f
Z
(
x
)
=
4
π
·
1 −g
√
1 −x
2
1 −g
2
+ 4gx
2
if x ∈
[
0;1
]
(5.5)
where f
Z
(x) is a probability distribution function of random variable x ∈ [0;1].
Potentially, analytical functions of DSPDs can be calculated for different scat-
tering phase functions. However, the solutions for more complex phase functions
become impossible to derive without numerical approaches. It should be noted
that the Doppler frequency shift probability distributions can be calculated using
numerical methods for any assumed phase function.
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