Download Laser Doppler Flowmetry - a Theoretical Framework

Laser Doppler Flowmetry
- a Theoretical Framework
Ingemar Fredriksson, Carina Fors and Johannes Johansson
Department of Biomedical Engineering, Linköping University
Abstract
Laser Doppler Flowmetry (LDF) is a non-invasive method to estimate the blood perfusion in
the microcirculation. The method was first introduced over 30 years ago and has undergone a
continuous development since. This work presents a thorough theoretical framework of the
technique. It is described in detail how the laser light interacts with the tissue and moving red
blood cells, how the light forms a speckle pattern on the detector and how the frequency
content of the formed speckle pattern can be used to estimate the blood perfusion. Hopefully,
this will increase the understanding of the technique, leading to better perfusion estimates and
new areas of application.
Cite this article as: Fredriksson I, Fors C and Johansson J, "Laser Doppler Flowmetry - a
Theoretical Framework", Department of Biomedical Engineering, Linköping University
(2007), www.imt.liu.se/bit/ldf/ldfmain.html
Contents
1.
2.
3.
4.
Introduction ....................................................................................................................2
Light source....................................................................................................................3
Doppler shift and light interaction with tissue .................................................................4
From E-fields to Doppler power spectrum ......................................................................6
4.1 E-field simulation .........................................................................................................6
4.2 The signal on the whole detector...................................................................................8
5. The CMBC and Perf estimates ........................................................................................9
6. Practical considerations ................................................................................................11
6.1 Measurement volume..................................................................................................11
6.2 Heterogeneities ...........................................................................................................11
6.3 Biological zero............................................................................................................12
6.4 Movement artefacts.....................................................................................................12
6.5 Band-pass filtering of the perfusion signal ..................................................................13
6.6 Practical tips ...............................................................................................................13
References............................................................................................................................14
Appendix A – Calculation of a single Doppler shift ..............................................................15
Appendix B – Theoretical derivation of the Doppler power spectrum ...................................17
Appendix C – Theoretical justification of the CMBC and Perf estimates...............................19
1
1. Introduction
Laser Doppler flowmetry (LDF) is a method for assessment of microvascular blood perfusion.
“Doppler” refers to the frequency shift that arises in light that has been scattered by moving
red blood cells. This fundamental principle constitutes the basics of LDF theory. In other
words, by illuminating a tissue sample with single-frequency light and processing the
frequency distribution of the backscattered light an estimate of the blood perfusion can be
achieved.
The first experimental studies using LDF were performed in the 1970’s [1, 2]. Various
instrumentation setups have been used through the years and the original method have been
developed and improved [3]. In the early 1980’s the fibre-optic based laser Doppler perfusion
monitoring (LDPM) technique was introduced. LDPM allows for temporal perfusion
measurements in a small sampling volume. In order to enable spatial measurements, the laser
Doppler perfusion imaging (LDPI) technique was developed. The surface of the tissue is
scanned by a freely impinging laser beam, thus generating an image of the spatial variability
of blood perfusion. An alternative method, utilizing a CMOS array, has been proposed by
Serov et al, in order to speed up the imaging procedure [4]. Today both LDPM and LDPI
instruments are commercially available.
Currently, LDF does not give an absolute measure of blood perfusion. In the clinical setting
this is a limiting factor and the reason why LDF instruments are not routinely used in health
care. However, LDF has found its use in research. Among the applications are
pharmacological trials, allergy patch testing, wound healing, physiological assessments and
skin disease research [3]. The skin is probably the most studied organ but also internal organs
such as kidneys, liver, muscles, intestines, brain and heart have been investigated in a number
of studies. Figure 1.1 shows an intramuscular LDPM probe inserted in the heart wall and an
LDPI image of a hand.
Figure 1.1 Left: LDPM measurements on the heart muscle during surgery. Right: LDPI image of a
hand.
2
The aim of this article is to present the theory behind laser Doppler flowmetry. Figure 1.2
gives an overview of the contents. We will begin with the light source and its properties, and
then proceed to light interaction with tissue and derive the expression for Doppler shifts. In
the next step we explain how the backscattered light forms an interference pattern on the
detector surface and how the electric field of the light, E(t), is linked to the photocurrent, i(t),
and also how the frequency spectrum of the light, I( ), is linked to the power spectral density
of the detector current, P( ). Finally, we describe how the perfusion signal, Perf, and the
concentration of moving red blood cells, CMBC, are estimated from P( ). We finish this
article by discussing some practical considerations.
Figure 1.2 Overview of the theory behind laser Doppler flowmetry.
2. Light source
The light used in LDF must be monochromatic, i.e., consist of a single frequency. The
frequency shifts caused by Doppler scattering will then result in a frequency broadening of
the originally monochromatic light.
In section 4 we will see that the backscattered light forms an interference pattern on the
photodetector and that the fluctuations in this pattern carry information about the Doppler
shifts. In order for interference to occur, the coherence length of the laser must be much
longer than the difference in path length for light waves that were emitted at the same
moment. The coherence length is the interval between phase jumps in the laser light, see
figure 2.1.
Wavelengths commonly used in LDF are 633 nm (red) and 780 nm (near-infrared). Output
power is usually about 1 mW.
3
Figure 2.1 The interval length between two phase jumps is the coherence length of the laser. (Note
that this picture only illustrates the principle. The coherence length usually contains far more than 4-5
periods.)
3. Doppler shift and light interaction with tissue
When light is scattered by a moving object it will be frequency shifted depending on the
movement of the object, the direction of the incoming light and the direction of the scattered
light. Let ki describe the propagation vector (rad/m) of an incoming photon hitting a scattering
particle with velocity v (m/s) and ks the propagation vector of the photon after being scattered,
figure 3.1. The angular frequency shift, βD (rad/s), will then be:
β D = −v ⋅ q = −v ⋅ (ki − ks ) = −
4π
λ
v sin (θ / 2 ) cos(ϕ ) cos(α ).
(3.1)
Here denotes the wavelength (m) of the photon in the surrounding medium, the scattering
angle between ki and ks, the angle between v and the plane of scattering and the angle
between the projection of v in the plane of scattering and (ki – ks). The difference between ki
and ks is often denoted the scattering vector q. A theoretical derivation of equation 3.1 can be
found in Appendix A.
v
ks
v//
v//
α
ks
q
ki
ki
ki
q
Figure 3.1 A photon with propagation vector ki is scattered by a red blood cell with velocity v and gets
a new propagation vector ks after scattering. The velocity component in the plane of scattering is
denoted v// and the difference between ki and ks, the scattering vector, is denoted q. The angle between
v and the plane of scattering is denoted , the angle between ki and ks, is denoted , the angle between
ki and v// is denoted and the angle between q and v// is denoted .
4
In equation 3.1, it can be seen that larger scattering angles will generally give a larger
frequency shift, with the largest possible shift for photons reflected straight back against the
incoming direction ( = rad). Biological tissue is a turbid medium where photons are
multiple scattered. However, if the tissue is not moving, flowing red blood cells will mainly
be the cause of noticeable frequency shifts. Due to the numerous scattering events before
hitting a red blood cell, the direction of the incoming propagation vector, ki, can usually be
regarded as random compared to the direction of the blood flow. Laser Doppler measurements
will thus, unlike ultrasound Doppler, be insensitive to the direction of blood flow. The
propagation vector of the scattered photon, ks, is on the other hand far from independent on ki
for red blood cells, as the scattering angle, , is usually very small. The frequency shift is thus
generally much smaller than if all scattering angles had equal probability. Any individual
photon may on the other hand be scattered multiple times by moving red blood cells, making
the total shift larger on an average.
An interesting feature of equation 3.1 is that the frequency shift for small scattering angles
(
0 rad) will be largest, i.e. q and v parallel, for movement of the red blood cell orthogonal
to the propagation vector of the incoming light in the plane of scattering. Light parallel to v
will thus in general give very small frequency shifts for strongly forward-scattering objects
such as red blood cells. This is rather counter-intuitive as Doppler shifts from the usually
more familiar case of direct reflections (
rad) are largest for movement parallel to the
wave propagation and smallest for movement orthogonal to the wave propagation. The latter
is also on average the case for isotropic scattering ( completely random), see figure 3.2.
Figure 3.3, right panel, illustrates the probability of various Doppler shifts for strongly
forward scattering objects, such as red blood cells, which is the result of the frequency
probability for isotropic scattering (figure 3.2) weighted with the scattering phase function
(figure 3.3, left panel). The scattering phase function used here is the Henyey-Greenstein
function with <cos > = 0.99.
0
π
π
π/2
ψ
π/2
0
-π/2
θ
-π 0
Figure 3.2 Frequency shift as a function of and the angle between the incoming light and the velocity
component of the scattering particle in the plane of scattering, according to equation 3.1.
5
104
103
102
p(θ)
101
0
1
π/2
10-1
10-2
-π/8
π/32
π
-π/16
0
θ (rad)
π/16
ψ
π/8
π/64
0
-π/2
θ
-π
0
Figure 3.3 Left: Approximate scattering phase function for the scattering angle in a red blood cell
(Henyey-Greenstein,<cos >=0.99 rad, logarithmic scale). Right: Frequency shift times the
approximated scattering phase function as a function of and the angle between the incoming light
and the velocity component of the red blood cell in the plane of scattering, . The probability for
scattering angles larger than /32 rad is very low and not included.
4. From E-fields to Doppler power spectrum
In 1961, Forrester presented a theoretical framework on how the optical Doppler spectrum
(the frequency content of the detected Doppler shifted light), I( ), is linked to the Doppler
power spectrum (the power spectral density of the detector current), P( ) [5]. His work was
revised by Cummins and Swinney [6] and extended by Larsson [7]. This chapter starts with a
description of a simulation of E-fields with a variety of frequencies, and it is shown how the
power spectral density of the detector current generated by these E-fields is related to the
frequency distribution of the E-fields. That relation is then theoretically examined, based on
the three mentioned works, first for one single point on the detector and then generalized for a
whole detector.
4.1 E-field simulation
A simulation was performed with 10 000 moving E-field sources with frequencies varying
between L – 15 000 and L + 15 000 rad/s ( L 2.4 · 1015 rad/s, a 780 nm laser), figure 4.1.
The sources were randomly distributed on the surface of a 0.125 mm diameter circle, and their
interference were logged on a 1 × 1 mm detector, placed 5.0 cm away from the sources. Since
the distance from each source to each point on the detector varies spatially, a smooth speckle
pattern forms on the detector. The sources have different frequencies, and they also move, and
therefore the speckle pattern is non-static. Figure 4.2 shows a short sequence (10 ms) of the
simulated speckle pattern, were the time lap between each frame is 0.1 ms, i.e. a sampling
frequency of 10 kHz. The simulation technique is further described by Rabal et. al. [8].
6
Intensity
10
0
10
-1
10
-2
10
-3
10
-4
βL - 15
βL - 10
βL - 5
βL
βL + 5
Frequency [krad/s]
βL + 10
βL + 15
Figure 4.1 Optical Doppler spectrum, I( ), of the simulated E-fields.
Figure 4.2 Simulated 10 ms sequence of a speckle pattern.
(To see the moving animation, see www.imt.liu.se/bit/ldf/ldfmain.html)
All points on the detector generates a current, i(t), linked to the mixed E-field at that point,
E(t), as
2
i (t ) = k E (t ) = kE (t ) E (t )* ,
(4.1)
where k is a detector specific constant. The total current for the whole detector area varies
over time as plotted in figure 4.3, and the power spectral density, referred to as the Doppler
power spectrum, P( ), of that signal is plotted in figure 4.4. The auto correlation of the
optical Doppler spectrum in figure 4.1 is also plotted in that figure. Both spectra are
normalized. Except for the noise and the offset induced by limited numerical precision at the
highest frequencies, they are equal. Thus, it seems that
P (ω ) ∝ I ( β ) I ( β ),
(4.2)
where denotes the cross correlation. A theoretical derivation of that relation can be found in
Appendix B.
7
2
1.8
Detector current
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.02
0.04
0.06
Time [s]
0.08
0.1
Power
Figure 4.3 The total current generated by the detector during 100 ms. The first 10 ms are shown in
figure 4.2.
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
0
Doppler power spectrum
Auto correlation of
optical Doppler spectrum
5
10
15
20
Frequency [krad/s]
25
30
Figure 4.4 The power spectral density of the detector current, i.e. the Doppler power spectrum, P( ),
and the auto correlation of the optical Doppler spectrum, I (β ) I ( β ) . The Doppler power spectrum
was low pass filtered.
4.2 The signal on the whole detector
The current generated at each point on the detector has one static part
∞
idc = k I ( β )dβ ,
(4.3)
0
and one time varying part, iac, from which the Doppler power spectrum, in analogy with
equation 4.2, can be expressed as
P(ω ) = PSD(iac ) = 2k 2 ⋅ I ( β ) I ( β ).
(4.4)
Appendix B contains a derivation of idc, iac and P( ).
When studying figure 4.2, one discover “blobs” consisting of points with similar behaviour.
Each such “blob” is usually referred to as a speckle area. The size of one speckle area, As, can
be estimated from the wavelength of the light, L, and the solid angle between a point on the
detector and the skin or fibre tip surface, ,
As = λ2L / Ω.
8
(4.5)
Since the distance, D, between the surface of the skin or the fibre tip and the detector is
usually much larger than the radius, rspot, of the fibre tip or the light emitting spot on the skin
surface, the solid angle can be approximated to
Ω≈
2
π ⋅ rspot
D2
(4.6)
.
Introducing Adet as the total area of the detector, ns = Adet / As speckle areas fit on that detector.
For the geometry used in the simulation of figure 4.2, approximately eight speckle areas fit on
the detector. Now, each speckle area can be considered as an independent realization of the
same stochastic process. Therefore, when summing the detector currents over all points of the
detector, idc scales linearly against the number of speckle areas, whereas iac scales to the
square root of the number of speckle areas,
idc ∝ ns
(4.7)
iac ∝ ns .
Thus, for the entire detector,
∞
idc = ns ⋅ k dc I ( β )dβ ,
(4.8)
0
and
P (ω ) = ns ⋅ kac2 ⋅ I ( β ) I ( β ),
ω > 0.
(4.9)
Here, the constant k, introduced in equation 4.1, is replaced by kdc and kac for the dc and ac
part, respectively, since those signals are usually processed completely separately.
Looking at expression 4.7, one can realize that the signal to noise ratio of the iac signal
decreases when the number of speckle areas, ns, increases, arguing that the detector should
contain as few speckle areas as possible. However, since each speckle area is an independent
realization of the same stochastic process, a proper average is only achieved when the detector
covers several speckle areas.
5. The CMBC and Perf estimates
The concentration of moving red blood cells (CMBC) and the perfusion (Perf) can be
estimated from the Doppler power spectrum. As previously mentioned, these estimates are not
absolute measures but, in a given sample with low red blood cell (RBC) concentration, CMBC
and Perf scale linearly with the RBC concentration and the tissue perfusion, respectively. In
LDF, perfusion is defined as the RBC concentration times the average RBC speed. Thus,
CMBC ∝ C RBC
Perf ∝ C RBC v RBC
(5.1)
where CRBC is the RBC concentration and v RBC the average RBC speed. For high
concentrations of RBCs the CMBC and Perf estimates varies nonlinear with CRBC, while the
relationship between perfusion and the average RBC velocity is entirely linear, provided that
CRBC is constant and the frequency distribution of the photocurrent is within the bandwidth of
the system, figure 5.1.
9
Perf
Perf
CMBC
CRBC
<vRBC>
CRBC
Figure 5.1 Approximate relationships for CMBC and CRBC (left), Perf and CRBC (middle) and Perf and
<vRBC> (right), when all other variables are kept constant.
The CMBC is proportional to the integral of the Doppler power spectrum and can be written
in the form
∞
CMBC ∝ P(ω )dω.
(5.2)
0
The perfusion is proportional to the integral of the frequency-weighted Doppler power
spectrum and can be written in the form
∞
Perf ∝ ωP(ω )dω.
(5.3)
0
P( ) in equation 5.2 and 5.3 is the power spectrum for only the frequency-shifted part of the
light, i.e., P( ) = 0 when = 0 (see also the derivation of P( ) in Appendix B).
The CMBC and Perf estimates in equation 5.2 and 5.3 depend on the total light intensity, or
the static part of the photocurrent, idc (see equation 4.8). Therefore, in the practical situation
the estimates should be normalised with a factor that compensates for this dependency. In
LDPM devices the estimates are usually normalised with idc2 while the normalisation factor in
LDPI devices usually is idc. A theoretical justification of the normalised CMBC and Perf
estimates is given in Appendix C.
0 when the light has been solely static scattered.
In the ideal case P( ) should be 0 for
However, because of detector noise, P( ) and thus the CMBC and Perf estimates will be > 0
even when the light is not Doppler broadened. In order to compensate for the influence of
noise, calibration measurements should be performed. By determining the noise level (i.e., the
CMBC/Perf) for static scattered light of different intensities the noise can be subtracted from
the CMBC and Perf estimates.
If we include total light intensity and noise compensation, the CMBC and Perf estimates can,
for LDPM devices, be calculated according to
∞
CMBC =
idc2
∞
Perf =
P (ω )dω
0
− nCMBC (idc )
(5.4)
− n Perf (idc )
(5.5)
ωP(ω )dω
0
idc2
where n(idc) is the noise.
10
When measuring with LDF on tissues with a high concentration of red blood cells, multiple
scattering and a large homodyne term (see the theoretical justification in Appendix C) lead to
a nonlinear relationship between CRBC and the CMBC and Perf estimates, figure 5.1. A
method to compensate for the nonlinearities has been proposed by Nilsson [9].
6. Practical considerations
6.1 Measurement volume
For most in vivo optical measurement techniques, such as LDF, pulse oximetry,
photoplethysmography (PPG) and diffuse reflectance spectroscopy, it is very difficult to
define the measurement volume. In all of these techniques, a small part of the signal
originates from interactions far away from the measurement probe, whereas most of the signal
content originates from a volume close to the probe. The aim of the techniques is to measure
something which affects the tissue optical properties (such as the blood content for PPG).
Since the measurement volume is affected by the tissue optical properties, the measurement
volume actually changes with changes of the desired quantity.
To predict the measurement volume for LDF, the best tool is the Monte Carlo technique,
which is able to simulate LDF measurements from tissue with complex geometries and
various vascular beds, and to log the positions of all Doppler shifts. Studies of the
measurement volume in LDF using the Monte Carlo technique has previously been performed
by others [10, 11]. Although they concluded that most of the reflected light had only been
scattered close to the detector, a large contribution of the LDF perfusion signal originates
from deeper situated blood vessels of higher blood velocity. For most biological tissues, the
measurement volume is in the area of one cubic millimetre, generally decreasing with
increased absorption and scattering, and for optical fibre systems, increasing with increasing
fibre separation.
6.2 Heterogeneities
Since LDF has a very small sampling volume it becomes sensitive to spatial variations in
blood perfusion and optical properties of the tissue. The perfusion signal can change a lot if
the measurement site is changed, which can be a problem if changes in blood perfusion are
studied through LDPM. The heterogeneity is partly caused by the heterogeneous distribution
of superficial veins and arterioles. For example, in a study on dorsal forearm skin the
perfusion signal could differ by about one magnitude for positions 2.5 mm apart [12]. The
sensitivity to variations depends on LDF-system and probe configuration.
LDF techniques are usually claimed to measure microcirculation on the basis of having a very
small sampling volume where only microscopic blood vessels are present. This is not correct
if the sampling volume actually contains larger vessels, as these tend to have both much
higher blood volume and flow speed than microscopic vessels and can thus dominate the
perfusion signal. To further complicate things, a larger vessel may cause a larger sampling
volume, as fewer photons are needed in order to cause noticeable impact on the signal if they
are more likely to get a larger frequency shift. On the other hand, a larger vessel is more likely
to absorb a photon travelling through it.
11
6.3 Biological zero
For tissue at rest, scattering objects other than red blood cells are usually considered to be
static in LDF. This is however not entirely true as thermal motion of scattering bodies actually
do add to the perfusion signal and this signal component has been found to be proportional to
the temperature in the range 15 °C to 37 °C. Thus the perfusion signal will not be zero even if
there is no perfusion in the tissue. The residual signal component when the perfusion of the
tissue is halted is called the biological zero. There may also be some non-thermal components
to the biological zero such as motion in vessel walls or incomplete halting of the perfusion.
Temperature-induced changes in actual blood perfusion dominate the perfusion signal in skin
around body temperature but the biological zero can make up a considerable part of the
CMBC signal if LDF is used on tissue with low perfusion or if the temperature is greatly
increased through e.g. thermal treatment of the tissue. [13]
6.4 Movement artefacts
Since LDF measures the motion of scattering objects rather than blood perfusion it is
important that blood perfusion actually dominates the perfusion signal. This may not be the
case if the measured tissue moves as this movement then can cause a much larger perfusion
signal than the perfusion itself. Motion of the tissue should thus be avoided during LDF. If
this cannot be avoided completely the signal should be presented with high time resolution in
order to detect spikes from movements [14]. The affinity for movement artefacts causes
difficulties if LDF is used on moving organs such as the heart or if measurements are to be
done on a subject doing some physical exercise.
Pure rigid body translation will not cause as large disturbances as could be expected. A
velocity component that is the same for all scattering bodies in a tissue volume will cause a
total Doppler shift as if the light had been reflected in the tissue’s surface, figure 6.1 (left). If
the laser source and detector is sufficiently far away from the tissue surface the effective
scattering vector, q = kin – kout, will be very similar for all photons giving a rather small timevarying component, iac, when they mix on the detector surface. A large amount of light may
on the other hand be directly reflected in the tissue surface adding only to the movement
artefact without gaining any frequency shifts from the blood perfusion. This light can be
removed using polarising filters as directly reflected light will keep its polarisation while light
reflected diffusively through the tissue will get a random polarisation. Translation can still
cause large signal variations due to heterogeneity of the tissue though, see corresponding
paragraph above [15]. If the measurements are performed using optical fibres attached to the
tissue translation will cause movement artefacts through deformation of the fibre. Choosing
fibres with small diameters can reduce these artefacts [14].
Non-rigid motion including changes in compression, strain and shear is considerably more
troublesome as such motion will have different velocity components for the scattering bodies
in the tissue, figure 6.1 (right). Static deformations can also disturb the measurement as they
can change the actual blood perfusion in the tissue.
12
N
m=1
− qm ⋅ v = (kout − kin )⋅ v
N
m=1
kin
kout
k2
− qm ⋅ v m =(k 2 − kin )⋅ v1 + .....+ (k out − k N −1 )⋅ v N
kin
kN-1
kout
k2
kN-1
Figure 6.1 Left: Pure rigid-body motion of the tissue will add a frequency shift as if the light was
scattered directly in the tissue surface. Using a detector that only accepts light with very similar
direction as the light illuminating the tissue can dampen this movement artefact. Right: A deforming
motion of the tissue will on the other hand cause a large variation in velocity between the scattering
bodies in the tissue causing a considerably more troublesome movement artefact.
6.5 Band-pass filtering of the perfusion signal
The time-varying part of the photocurrent is band-pass filtered in order to reduce noise. This
filtering can unfortunately also remove a noticeable part of the signal in some special cases. A
too low upper cut-off frequency may dampen frequency components from many multiple
shifts and/or red blood cells with very high velocity, which can reduce the increase in
perfusion signal from increased blood perfusion. In extreme cases the perfusion signal may
even decrease as the perfusion increases. For tissue with very low perfusion a too high lower
cut-off frequency may remove some of the CMBC signal.
6.6 Practical tips
LDF techniques can yet not measure absolute perfusion but can have excellent temporal
resolution. They are therefore best employed to investigate physiological responses to various
stimuli.
Avoid perfusion-affecting stimuli whose response is not of interest in the study. Keep room
temperature stable.
Avoid motion if possible, especially deforming motion.
If a probe is used it is preferable if it does not deform the tissue or cause reactions in it. It may
still need to be attached firmly in order to avoid motion.
13
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1.
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3.
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4.
Serov A, Steenbergen W, de Mul F. Laser Doppler perfusion imaging with a
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Kernick DP, Tooke JE, Shore AC. The biological zero signal in laser Doppler
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Leahy MJ, de Mul FF, Nilsson GE, Maniewski R. Principles and practice of the laserDoppler perfusion technique. Technol Health Care 1999;7(2-3):143-62.
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Karlsson MGD. Movement artefact reduction in laser Doppler blood flowmetry myocardial perfusion applications. Linköping, Sweden: Linköping University; 2005.
14
Appendix A – Calculation of a single Doppler shift
Assume a red blood cell moving with the velocity v being hit by a light wave with the
propagation vector ki before scattering and propagation vector ks after. The angle between ki
and the projection of v in the plane of scattering, v//, is denoted , the scattering angle between
ki and ks is denoted and the angle between v and v// is denoted . The vectorial difference
between the propagation vectors for incoming and scattered light, ki – ks, is denoted the
scattering vector q, see figure A.1.
v
ks
v//
v//
α
ks
q
ki
ki
ki
q
Figure A.1 A photon with propagation vector ki scatters on a red blood cell with velocity v and gets a
new propagation vector ks after scattering. The velocity component in the plane of scattering is
denoted v// and the difference between ki and ks, the scattering vector is denoted q. The angle between
v and the plane of scattering is denoted , the angle between ki and ks, is denoted , the angle between
ki and v// is denoted and the angle between q and v// is denoted .
The incoming light wave has an angular frequency i and a wavelength of i = 2 / ki =
2 c/ i, where c is the speed of light in the tissue. Follow a peak of the wave that is located one
wavelength, i, from the cell at time t0 and then reaches the cell at time t1 (figure A.2). At this
time the cell has moved a small distance.
ks
v//
ki
ki
λi
Figure A.2 A photon with wavelength
i
hits a red blood cell with velocity v.
As v = v <<c the total travelling distance between t0 and t1 for the peak can be calculated as
λi + v ⋅ k i λi (t1 − t 0 ) = λi + v(t1 − t 0 ) cos(ψ ) cos(α ) = c(t1 − t 0 )
(t1 − t0 ) =
λi
.
c − v cos(ψ )cos(α )
15
(A.1)
(A.2)
λs
ks
v//
ki
ki
Figure A.3 After scattering the photon will have the wavelength s.
In the meantime the peak that was at the red blood cell at time t0 has scattered with the angle
and moved relative the cell, and thus the other peak, a distance (assuming v <<c)
λs = c(t1 − t 0 ) − v(t1 − t 0 ) cos(ψ − θ ) cos(α )
= [Using equation A.2]
=
(A.3)
λi − λ cos(ψ − θ ) cos(α )
.
1 − cos(ψ ) cos(α )
v
i c
v
c
This distance is the wavelength of the scattered light. The frequency shift, D, can now be
calculated as the difference between frequency of the scattered light, s = 2 c/ s, and the
frequency of the incoming light, i = 2 c/ i:
β D = β s − β i = 2πc
=
≈
1
λs
−
1
λi
=
2πc
λi
1 − vc cos(ψ ) cos(α )
−1
1 − vc cos(ψ − θ ) cos(α )
2πc 1 − vc cos(ψ ) cos(α ) − 1 + vc cos(ψ − θ ) cos(α )
≈ [Using : v << c ] (A.4)
λi
1 − vc cos(ψ − θ ) cos(α )
2πv
λi
(cos(ψ − θ ) − cos(ψ ) )cos(α ).
It can be seen in equation A.4 that s ≈ i, again since v << c, and consequently that ki ≈ ks .
The propagation vectors of the incoming and scattered light and their difference, the
scattering vector q = (ki – ks), thus forms an isosceles triangle with the angles and ( – )/2,
see figure A.4. The angle between ki and v//, , can now be calculated as a function of the
angle between q and v//, , and the scattering angle between ki and ks,
(ψ − θ ) + (ϕ − π − θ ) = π
2
16
ψ = θ − ϕ + 3π .
2
2
(A.5)
ks
v//
π −θ
2
q
π −θ
ks
ki
π −θ
2
q
2
Figure A.4 Relations between the scattering angle , the angle between the projection of the
incoming light (ki) and the velocity of the red blood cell in the plane of scattering (v//) and the angle
between the scattering vector (q) and v//.
Using equation A.5 equation A.4 can be written
βD =
=
2πv
λi
1
2πv
1
sin(−ϕ − θ ) − sin(−ϕ + θ ) cos(α ) =
2 cos (− 2ϕ ) sin (− θ ) cos(α ) (A.6)
2
2
λi
2
2
2πv
λi
=−
θ
3π
θ
3π
cos( − ϕ +
− θ ) − cos( − ϕ + ) cos(α )
2
2
2
2
4π
v cos(ϕ ) sin θ cos(α ) = −q ⋅ v.
2
λi
Appendix B – Theoretical derivation of the Doppler power
spectrum
Assume that we have a vast amount of E-field sources on the tip of a fibre or on a small spot
on a skin surface. Each source has an individual frequency content, originating from the
frequency of the laser and broadened due to Doppler shifts. Each frequency component of
each source has an individual random phase due to various optical path lengths in the tissue.
Under the assumption that the E-field sources are not moving spatially, the mixed E-field on
each point of the detector, constituted by superposition of each E-field source, is described for
each frequency component m as
Em (t ) = I m ∆β e j ( ∆βmt +φ m ) ,
(B.1)
where I m ∆β = I ( β m )∆β is the intensity of the m:th frequency interval of the mixed E-field
and φm is a random phase. Summing over all frequencies, the current generated by each point
on the detector can be expressed as
i (t ) = kE (t ) E (t )*
∞
= ∆β k
m=0
= ∆β k
∞
∞
m = 0n = 0
= ∆β k
∞
m=0
∞
I m e j ( ∆βmt +φ m ) ⋅
m=0
I m e − j ( ∆βmt +φ m )
I m I n e j (( m − n ) ∆βt + (φ m −φ n ))
I m + 2∆βk
∞
∞
m = 0 n = m +1
(B.2)
I m I n cos((m − n)∆β t + (φm − φn )).
The first term in this equation is static and denoted idc. The second term varies over time and
is denoted iac. By introducing b = n – m and = (n – m) , iac is written as
17
iac (t ) = 2∆βk
∞
∞
m = 0 b =1
I m I m + b cos(ωt + (φm + b − φm )).
(B.3)
Note that > 0. Now let iac be a function of both frequency and time, i.e. dropping the sum
over b in equation B.3,
iac (ω , t ) = 2∆βk
where
m,b
∞
m =0
I m I m + b cos(ωt + ψ m, b ),
(B.4)
= φm+b – φm. From this expression, the power spectral density, P( ) is calculated as
P(ω ) =
iac (ω , t ) 2
∆β
= 4∆β k
∞
2
m=0
∞
= 4∆β k 2
m=0
∞
+
∞
∞
m=0
+
I m I m + b cos(ωt + ψ m , b )
I m I m + b cos 2 (ωt + ψ m, b )
I m I m + b I n I n + b cos(ωt + ψ m, b ) cos(ωt + ψ n, b )
m=0 n =0
n≠m
= 4∆β k 2
2
I m I m + b cos 2 (ωt + ψ m, b )
∞
∞
1
I m I m + b I n I n + b cos(2ωt + ψ m, b + ψ n, b )
2 m=0 n =0
n≠ m
+
∞
∞
1
I m I m + b I n I n + b cos(ψ m, b − ψ n, b ) ,
2 m =0n = 0
(B.5)
n≠m
where
...
is the
time average. In the
last
step the
trigonometric identity
cos(a ) cos(b) = 1 / 2(cos(a + b) + cos(a − b)) was employed. The time averages for the three
cosine terms are:
cos 2 (ωt + ψ m, b = 1 / 2
cos(2ωt + ψ m, b + ψ n, b ) = 0
(B.6)
cos(ψ m, b − ψ n, b ) = cos(ψ m, b − ψ n, b ),
and the power spectral density can therefore be rewritten as
P(ω ) = 2∆βk 2
∞
m=0
I m I m + b + 2 ∆β k 2
∞
∞
m =0n = 0
n≠ m
18
I m I m + b I n I n + b cos(ψ m, b − ψ n, b ). (B.7)
The double sum can not be assumed to be zero for any individual frequency. However, the
expectation value of the term is zero, and thus, over any small frequency interval of a normal
continuous optical Doppler spectrum, the term averages to zero. Therefore, we ignore that
double sum and write the Doppler power spectrum as
∞
P(ω ) = 2∆βk 2
m=0
= 2k
2
∞
m =1
Im Im+b
(B.8)
I ( β m ) I ( β m + ω )∆β .
By letting ∆β → 0 we can rewrite the sum to an integral,
∞
P(ω ) = 2k 2 I ( β ) I ( β + ω )dβ
0
(B.9)
= 2k 2 ⋅ I ( β ) I ( β ),
where
denotes the cross correlation. In analogy, the static current, idc, is rewritten as
idc = k
∞
m=0
I ( β m )∆β
(B.10)
∞
∆β → 0
= k I ( β ) dβ .
0
Appendix C – Theoretical justification of the CMBC and
Perf estimates
Larsson has explained the expressions for the CMBC and Perf estimates in the frequency
domain [7]. An alternative derivation in the time domain is given by Nilsson et al [3]. In this
appendix we present a justification based on Larsson’s work.
Let us start by establishing a relationship between the red blood cell concentration CRBC and
the fraction of Doppler-shifted photons f. A simplified expression is given by Beer-Lambert’s
law where the fraction of non-shifted photons can be approximated to
1− f ≈ e
− l p C RBC µS, RBC
where lp is the photon path length in tissue and
S,RBC
(C.1)
is the scattering coefficient for blood.
Next step is to insert f in the expression for the optical Doppler spectrum I( ). First, we write
I( ) in the form
I (β ) = I 0 (β ) + I d (β )
(C.2)
where I0( ) is the intensity of non-shifted photons and Id( ) is the intensity of Doppler-shifted
photons. Let Itot denote the total intensity, i.e.,
I tot =
∞
I ( β ) dβ ,
−∞
I ( β ) = 0, β < 0.
Then I0( ) and Id( ) can be written in the form
19
(C.3)
I 0 ( β ) = I tot (1 − f ) ( β − β L )
(C.4)
I d ( β ) = I tot fh( β )
where is the Dirac delta function, L the laser frequency and h( ) the normalised frequency
distribution of the Doppler-shifted light, i.e.,
∞
h ( β ) dβ = 1 .
(C.5)
−∞
Now we can express idc and P( ) in terms of f, Itot and h( ). From equation 4.8 we know that
I( ) is related to the stationary part of the detector current idc as
∞
idc = ns k dc I ( β ) dβ
(C.6)
−∞
where ns is the number of speckle areas and kdc is an instrument dependent constant. Inserting
equation C.3 in equation C.6 yields
idc = ns k dc I tot .
(C.7)
P (ω ) = n s 2k ac2 I ( β ) I ( β ), ω > 0
(C.8)
P( ) is given in equation 4.9 as
which can be written as
P(ω ) = 2ns k ac2 I ( β ) I ( β )
= 2ns k ac2 (I 0 ( β ) + I d ( β ) ) (I 0 ( β ) + I d ( β ) )
= 2ns k ac2 I 0 ( β ) I 0 ( β ) + 2 I 0 ( β ) I d ( β ) + I d ( β ) I d ( β ) ,
stationary
heterodyne
(C.9)
homodyne
P( w) = 0, ω ≤ 0.
> 0 in
We only include the alternating part of the photocurrent in P( ), and therefore
equation C.9. This means that the stationary term equals zero. The heterodyne term expands
to
I 0 (β ) I d (β ) =
∞
I 0 ( β ) I d ( β + ω ) dβ
−∞
2
= I tot
(1 − f ) f
∞
( β − β L ) h ( β + ω ) dβ
(C.10)
−∞
2
= I tot
(1 − f ) fh( β L + ω )
and the homodyne term can be written as
I d (β ) I d (β ) =
∞
I d ( β ) I d ( β + ω ) dβ
−∞
=I
2
tot
f
2
∞
(C.11)
h ( β ) h ( β + ω ) dβ .
−∞
Inserting equation C.10 and C.11 in equation C.9 yields
20
P(ω ) = 2n s k I
2 2
ac tot
f 2(1 − f )h( β L + ω ) + f
∞
h( β )h( β + ω )dβ .
(C.12)
−∞
Finally, by using equation C.12 we can now express the normalised CMBC estimate as
∞
CMBC =
P(ω )dω
0
idc2
∞
∞
0
−∞
= since P(ω ) = 0 for ω ≤ 0, P(ω )dω =
P(ω )dω
∞
∞ ∞
2k ac2
=
f 2(1 − f ) h( β L + ω )dω + f
h ( β ) h ( β + ω ) dβ dω
ns k dc2
−∞
− ∞− ∞
=
(C.13)
2k ac2
f (2 − f )
ns k dc2
2k ac2
− 2l C
µ
≈
(1 − e p RBC S, RBC ).
2
ns k dc
Similarly, the normalised Perf estimate can be written as
∞
Perf =
ωP(ω )dω
0
idc2
∞
P(ω )dω
= ω
0
≈ ω
2k ac2
− 2l C
µ
(1 − e p RBC S, RBC )
ns k dc2
idc2
(C.14)
where ω is the power-weighted average frequency. For a single Doppler shift we know
from equation 3.1 that D, and accordingly (see chapter 4), is proportional to the speed of
the blood cells, vRBC. However, photons that are scattered in tissue are usually multiple
shifted, which results in a wider optical spectrum, i.e., ω depends on the amount of multiple
shifts. The amount of multiple shifts, in turn, increases with the concentration of red blood
cells. Accordingly, ω usually depends both on the average red blood cell speed and the
concentration of red blood cells, yielding
ω = v RBC ⋅ g (C RBC )
(C.15)
where g(CRBC) describes the relation between CRBC and the amount of multiple shifts. Perf can
now be written as
Perf ≈ v RBC g (C RBC )
2k ac2
−2l C
µ
(1 − e p RBC S, RBC )
2
ns k dc
21
(C.16)
The Perf estimate is thus proportional to the average red blood cell speed, provided that CRBC
is constant.
Note that neither CMBC (equation C.13) nor Perf (equation C.16) depend on the total light
intensity Itot, when normalised with idc2.
For low concentrations of red blood cells equation C.13 and C.16 can be approximated to
2k ac2
CMBC ≈
2l p µ S, RBC C RBC
ns k dc2
Perf ≈ v RBC
2k 2
g (C RBC ) ac2 2l p µ S, RBC C RBC
ns k dc
using the Maclaurin expansion.
22
(C.17)