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4.1 Theory 73

and the solid angle  between the area of illumination and each point on the

detector [39]

≈



(4.16)

Since the distance D between the detector and the ﬁber tip or the surface area on

the tissue from which the backscattered light illuminates the detector is usually

much larger than the radius r of the ﬁber tip or the tissue area, the solid angle can

be approximated to

 ≈

πr

(4.17)

In total

det

(4.18)

coherence areas ﬁt on the detector, which for the case of the simulated setup

used for Figure 4.2 gives n

≈ 8. Each coherence area can be considered as an

independent realization of the same stochastic process, and i

DC,c

, i

AC,c

, and P

D,c

( f )

derived above are thus valid for each single coherence area. The static current

generated over the whole detector is simply summarized as



DC,c

= n

DC,c

, (4.19)

whereas, due to the random phases between the various i

AC,c

, the time-varying part

of the whole detector is summarized as



AC,c

√

AC,c

(4.20)

In analogy with Eq. (4.4), the optical Doppler spectrum G(β) over the whole detector

is normalized so that



∞

G(β)dβ = i

(4.21)

It then follows from Eqs. (4.4), (4.19), and (4.21) that

G(β) = n

(β) (4.22)

and then it follows, by combining Eqs. (4.10), (4.11), (4.20), and (4.22), that the

Doppler power spectrum generated by the whole detector is expressed as

( f ) =



G)( f )for f

0forf ≤ 0

(4.23)

In Figure 4.3, the effect of an increased velocity (panel a) and an increased RBC

tissue fraction (panel b) on the Doppler power spectrum is shown. It can be seen

that an increased velocity broadens the spectrum, whereas an increased RBC tissue

fraction ﬁrst increases the power for all frequencies

0 Hz, and for high tissue

fractions, also causes a signiﬁcant broadening because of multiple Doppler shifts.

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74 4 Laser Doppler Flowmetry

0 5 10 15 20

−8

−6

−4

−2

Frequency (kHz)

Doppler power spectrum (Hz

−1

)

Doppler power spectrum (Hz

−1

)

1.25 mm s

−1

2.5 mm s

−1

5.0 mm s

−1

0 5 10 15 20

−8

−6

−4

−2

Frequency (kHz)

0.01%

0.1%

(a)

(b)

Figure 4.3 Effect on the Doppler power spectrum for

increased velocity (a) and increased RBC tissue fraction (b).

These spectra are based on Monte Carlo simulations of a static homogeneous

medium with µ

= 0.1 mm

−1

, µ

= 13 mm

−1

, and g = 0.85 containing RBCs in

blood (435 g RBC per l blood, hematocrit (HCT) = 42%) with µ

= 0.5 mm

−1

=222mm

−1

, and g = 0.991 (a Gegenbauer kernel phase function with α

=1.0

and g

= 0.948). The optical Doppler spectra G(β) were given by the simulation,

and the Doppler power spectra calculated using Eq. (4.13). In Figure 4.3a, the RBC

tissue fraction is 0.1g RBC/100g tissue (or percentage) and the RBC velocityshown

on the x-axis is the mean velocity of a rectangular velocity distribution between

zero and twice that mean velocity. In Figure 4.3b, the velocity of the RBCs is evenly

distributed between 0 and 5 mm s

−1

. The ﬁber separation is 0.25 mm, and both

emitting and receiving ﬁbers have a core diameter of 0.125 mm and a numerical

aperture (NA) = 0.37.

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4.2 Conventional Measures 75

4.2

Conventional Measures

The conventional perfusion measure is calculated as the ﬁrst moment of the

Doppler power spectrum normalized with i

[7]

perf

∞

−∞

( f )df

(4.24)

Note that P

( f ) = 0forf ≤ 0 (Eq.(4.15)). Similarly, an RBC tissue fraction estimate

is calculated as

CMBC

∞

−∞

( f )df

(4.25)

where CMBC is a commonly used abbreviation for concentration of moving RBCs.

The meaning of these measures was ﬁrst evaluated by Bonner and Nossal [6].

The evaluation in this section is based on Larsson et al. [36] and has been partly

presented earlier in [38]. A homogeneous medium is assumed.

First, Beer–Lambert’s law lets us estimate the fraction of non-Doppler-shifted

photons

≈ e

−dζ

RBC

s,RBC

(4.26)

where d is the average path length, ζ

RBC

is the tissue fraction of RBCs, and µ

s,RBC

is the scattering coefﬁcient of the RBCs. Since we really have a distribution of

path lengths, this is only an approximation that introduces signiﬁcant errors; but

it is enough for the examination in this section. We also deﬁne the fraction of

Doppler-shifted photons ζ

= 1 − ζ

and introduce

H(β) =

G(β)

(

1 −δ

(

β −β

))

(4.27)

where δ is Dirac’s delta function. Note that

∞

−∞

H(β)dβ = 1. Furthermore, G(β)

is divided into one non-Doppler-shifted part G

(β) and one Doppler-shifted part

(β) as in (Eq. (4.4) )



(β) = i

(

β −β

)

(β) = i

H(β)

(4.28)

By using Eq. (4.23), P

( f ) can then be rewritten as

( f ) = 2n

−1

G)( f )

= 2n

−1



(

+ G

)

(

+ G

)



( f )

= 2n

−1











( f )



 

stationary





( f )



 

heterodyne





( f )



 

homodyne







(4.29)

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76 4 Laser Doppler Flowmetry

for f

0. The stationary, heterodyne, and homodyne terms can be rewritten as





( f ) =



∞

−∞

(β)G

(β +f )dβ

= i



∞

−∞

(

β −β

)



β −β

+ f



dβ

= i

δ( f )

= 0forf

0 (4.30)





( f ) =



∞

−∞

(β)G

(β +f )dβ

= i



∞

−∞

δ(β − β

)H(β + f )dβ

= i



+ f



(4.31)





( f ) =



∞

−∞

(β)G

(β +f )dβ

= i



∞

−∞

H(β)H(β +f )dβ (4.32)

Inserting Eqs. (4.30–4.32) into Eq. (4.29) gives

( f ) = 2n

−1

2ζ



+ f



+ ζ



∞

−∞

H(β)H(β +f )dβ

(4.33)

for f

0. Inserting this result into Eq. (4.25) yields

CMBC

∞

−∞

( f )df

2ζ



∞

−∞



+ f



+ζ



∞

−∞



∞

−∞

H(β)H(β +f )dβdf

2ζ

+ ζ



∞

−∞

H(β)



∞

−∞

H(β + f )df dβ

2ζ

+ ζ



∞

−∞

H(β)dβ



2ζ

+ ζ





1 −ζ



≈





1 −e

−2dc

RBC

s,RBC





 

(4.34)

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4.2 Conventional Measures 77

and similarly when inserted into Eq. (4.24)

perf

∞

−∞

( f )df

=f 

∞

−∞

( f )df

≈f 





1 −e

−2dc

RBC

s,RBC





 

(4.35)

where f  is the average frequency of P

( f ). To understand the meaning of s

CMBC

and s

perf

, each of the terms A, B, and C are now analyzed. Term A is constant

as long as n

is constant, which may not be the case in an LDPI system (see

Section 4.3). In practice, i

and i

are often processed separately, giving two

different ampliﬁcations of the signals, which implies that the term A may rather



#



Term B can be rewritten using the MacLaurin series expansion

of e, that is, e

= 1 + x

1! +x

2! +···For low RBC tissue fractions, B can thus

be approximated to 2dc

RBC

s,RBC

, which is linear to the RBC tissue fraction. For

higher RBC tissue fractions, higher order moments of the MacLaurin series have

to be considered, introducing nonlinearities. This leads to an underestimation of

the CMBC and perfusion values for higher RBC tissue fractions. Furthermore,

the path length generally decreases when the RBC tissue fraction increases [40],

making the underestimation even worse.

We know from Eq. (4.1) that the size of each Doppler shift is linearly dependent

on the velocity of the moving RBC. Therefore, term C will be linearly dependent

on the RMS velocity of RBCs [6], under the assumption that everything else

is constant and the bandwidth of the system is wide enough. However, the

average frequency increases when the photons are Doppler shifted multiple times,

which they are to a greater extent when the RBC tissue fraction is high. This

compensates the nonlinearities introduced in term B to some extent, but not

entirely.

In conclusion, CMBC reﬂects the concentration (tissue fraction) of moving RBCs

and the perfusion value reﬂects the tissue fraction of RBCs times their average

velocity. It can also be noted that since s

CMBC

and s

perf

are normalized with i

,they

are independent of the total detected light intensity. The relationship between the

tissue fraction of RBCs in a homogeneous medium and simulated s

CMBC

and s

perf

are shown in Figure 4.4. The simulated setup was the same as described at the end

of Section 4.1.2. The velocity of the RBCs in the medium was evenly distributed

between 0 and 5 mm s

−1

. In this ﬁgure, the nonlinear behavior of not only s

CMBC

but also s

perf

becomes evident. It must be emphasized that the nonlinearity is

strongly dependent on the geometry and optical properties of the medium and

therefore cannot be easily accounted for, as suggested in [7].

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78 4 Laser Doppler Flowmetry

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

grams RBC / 100 g tissue

0 0.2 0.4 0.6 0.8 1

grams RBC / 100 g tissue

CMBC (a.u.)

100

120

140

Perfusion (a.u.)

(a)

(b)

Figure 4.4 Relationship between RBC tissue fraction and s

CMBC

(a) and s

perf

(b).

4.3

Hardware Realizations

As stated at the beginning of this chapter, both single point (LDPM) and imaging

(LDPI) LDF systems exist. In this section, the special characteristics of these

systems, as well as a newer type of imaging system based on fast CMOS cameras,

are described.Noise reduction and typical calibration procedures are also described.

4.3.1

LDPM

In single point systems, optical ﬁbers are typically used to guide the light from

the laser source to the tissue and from the tissue to the detector. Most often,

separate ﬁbers are used, but a single ﬁber can also be used when a very small

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4.3 Hardware Realizations 79

(b) (c)(a)

Figure 4.5 Schematic illustrations of the various LDF

modalities, LDPM (a), LDPI (b), and CMOS imager (c).

measurement volume is wanted [40]. The most common source–detector distance

in probes provided by the two leading manufacturers of LDF instruments (Perimed

AB, J

arf

alla, Sweden, and Moor Instruments Ltd., Axminster, UK) is 0.25 mm.

Generally, short source –detector distances have been preferred to limit the mea-

surement depth, to improve signal to noise and to avoid a high degree of multiple

Doppler shifts leading to severe nonlinearities in the conventional CMBC and

perfusion measures.

In a ﬁber-based system, the detector ﬁber illuminates the detector at a certain

distance between the ﬁber tip and the detector (Figure 4.5a). The illuminating

angle is constant and determined from the numerical aperture of the ﬁber, under

the reasonable assumption that the light impinging on the ﬁber has a uniform

angle distribution. During these conditions, the number of coherence areas on the

detector is constant (Eq. (4.18)), and the DC-squared normalized Doppler power

spectrum P

( f )/i

will only depend on the optical spectrum G(β) (Eq. (4.23)).

Another type of LDPM system, not based on ﬁbers, also exists, where the optics

and electronics are built on a single probe or chip that is in direct contact with the

tissue [41]. These systems are less sensitive to movements and can therefore be

used underless-controlled situations, but at low perfusion levels, the signal-to-noise

ratio is lower due to the shorter coherence length of the lasers used.

4.3.2

LDPI

In a conventional imaging system (Figure 4.5b), a laser beam is scanned over the

tissue of interest to form an image. In each scanning position, the perfusion value

is calculated and transformed into a pixel value, and the scanning procedure lasts

for a couple of minutes to form an entire image. The detector is typically placed

some 20–30 cm above the tissue surface. In some systems, a lens focuses the light

on the detector.

The backscattered light from the impinging laser beam escapes the tissue from

a spot centered on the point of injection, with a radially decaying intensity. The

slope of the decay, and thus the apparent spot size, is strongly dependent on the

absorption properties of the tissue. The spot size, by itself, reduces the calculated

https://t.me/medicina_free

80 4 Laser Doppler Flowmetry

perfusion value since it affects the solid angle to the detector points. Since an

increased RBC tissue fraction increases the total absorption of the tissue and thus

reduces the spot size, the increase in the estimated CMBC and perfusion will be

underestimated. The effect, which is explained below, is similar for both systems

with and without a light-focusing lens in front of the detector [42].

For simplicity, assume that the backscattered light exits the tissue in a homo-

genous spot. For a system without a lens in front of the detector, the solid angle

between the spot and each point on the detector varies with the square of the

radius r of the spot (Eq. (4.17)). As given by Eq. (4.35), the perfusion value is

inversely proportional to the number of coherence areas n

, which in turn is

linearly dependent on r

(Eqs. 4.16–4.18). Therefore, the calculated perfusion value

is inversely proportional to r

for a system without a lens.

For a system with a light-focusing lens, the solid angle is determined by the

dimension and position of the lens and is thus independent of the spot size.

However, as the lens focuses the light on the detector, the illuminated part of the

detector will directly depend on the size of the spot, under the assumption that

the entire spot is imaged on the detector. Therefore, the ‘‘active’’ detector area will

depend on r

and the calculated perfusion value is inversely proportional to r

also

for a system with a lens.

4.3.3

CMOS Imager

A conventional LDPI system has the drawbacks that it is slow and that the

estimated perfusion value is affected by the spot size of backreﬂected light. Both

these drawbacks are addressed with the new generation of LDPI systems, which

are based on fast CMOS cameras [43, 49]. In these systems, the backscattered light

is focused on a detector array, where every pixel in the array is used to calculate

a perfusion value from the part of the tissue that the pixel sees (Figure 4.5c). By

simultaneously illuminating the whole tissue area of interest, the whole image is

acquired simultaneously, which is naturally faster than scanning the area point

by point. Every pixel in the array also has a constant solid angle (based on the

lens), and they can be assumed to be evenly illuminated, so the perfusion value is

independent of the spot size.

The major drawback with these systems is that the cameras used are not yet

fast enough. A normal Doppler power spectrum has frequency components of

signiﬁcant power up to about 15 kHz. The sample frequency thus has to be at

least 30 kHz to collect the entire frequency content of the Doppler signal and avoid

aliasing. The system presented in [43] operated with a sampling frequency of only

8 kHz, which will have the effect that when the velocity of the RBCs is increased,

a substantial part of the energy in the Doppler power spectrum will exceed 4 kHz

and the expected increase in the perfusion value will be underestimated, or the

perfusion value may even decrease.

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4.3 Hardware Realizations 81

4.3.4

Noise

Various types of noise inﬂuence the Doppler power spectrum. Some of this noise

originates from variations in background light intensity and common mode noise.

To get rid of this latter type of noise, a differential detector technique is often used

for i

, which simply calculates the difference between the light falling on two

adjacent detectors. By calculating the difference, common noise is cancelled out,

whereas the statistical properties of the two separate realizations of i

remain [5].

However, other types of noise are not removed by this technique. The amplitude

of shot noise, which is a variation in photoelectron emission rate of the detector,

is linearly dependent on the total detector current i

, whereas thermal noise

is constant as long as the temperature is held constant [44]. Consequently, an

expression of these types of noise can be established

noise



f , i



= k

noise

( f )i

+ l

noise

( f ) (4.36)

where the parameter k

noise

relates to the shot noise and l

noise

to the thermal noise.

4.3.5

Calibration

Two different objectives are reached when calibrating an LDF system. First, the

noise parameters k

noise

and l

noise

in Eq. (4.36) are calculated. This is done by

measuring the signal at different DC levels on a static scattering object, that is, an

object without any moving scatterers, such as a piece of white plastic. The noise

perfusion value is then calculated and subtracted from the measured perfusion

values during the normal measurements.

Second, a reference perfusion value is measured. In the Periﬂux 5000 system

(Perimed AB, J

arf

alla, Sweden), this is achieved by placing the measurement

probe in a motility standard such as a beaker containing a certain concentration

of polystyrene microspheres diluted in water. Owing to Brownian motion, these

microspheres, acting as scatterers, move with a certain velocity distribution, which

is dependent on the temperature. Thus, all backscattered light is Doppler shifted,

and in the case of the Periﬂux 5000 system, the perfusion level measured in this

setup is set to 250 pu (perfusion units) at 22

◦

C. Since this is a repeatable procedure,

all other perfusion values can be compared. Other systems have similar calibration

routines.

A slightly different calibration procedure is presented in [45]. First, the noise

spectrum P

noise



f , i



is subtracted directly from the Doppler power spectrum

( f ). Second, the resulting Doppler power spectrum is normalized with the

CMBC from the motility measurement, which is calculated as

CMBC,mot

∞

−∞



mot

( f ) − P

noise



DC,mot

, f



DC,mot

(4.37)

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82 4 Laser Doppler Flowmetry

Thus, the calibrated Doppler power spectrum is calculated as



( f ) =



( f ) − P

noise



, f

#

CMBC,mot

(4.38)

The modiﬁcation of the second calibration step provides a unity CMBC where

all the detected light is Doppler shifted (see Eq. (4.34)). The measured Doppler

power spectra can thus be easily compared to simulated spectra. Another positive

consequence of this is that the second calibration step can be performed with

any scattering liquid in which Brownian motion is present. The calibration is not

limited to expensive motility standard that has to be exchanged from time to time,

and it is independent of the temperature.

4.3.6

Measurement Depth and Volume

For an accuratephysiologicalinterpretation of the perfusionestimate, it is important

to have an estimate of the depth or volume that most of the signal originates from.

It is common to hear that the measurement volume is about 1 mm

or the

measurement depth is in the order of 1 mm, but depending on system properties

as well as optical properties of the measured tissue, blood tissue fraction and ﬂow

distribution in the tissue, the actual measurement volume, or measurement depth

can vary manifold.

Over the years, only two publications have focused on this problem, namely,

Jakobsson and Nilsson 1993 [46] and Fredriksson et al. 2009 [40]. In both these

publications, an important distinction is made between sampling depth/volume

and measurement depth/volume. While the sampling depth/volume reveals where

the detected light has been propagating, the measurement depth/volume reveals

where the perfusion signal has been affected. Since the perfusion signal is to

a major extent only affected by moving RBCs, this is an important distinction

and the measurement depth/volume can differ signiﬁcantly from the sampling

depth/volume.

Both the articles mentioned above are based on Monte Carlo simulations.

Fredriksson et al. accounted for multiple Doppler shifts and speckle effects that

affect the measurement depth/volume in LDPI. In Tables 4.1 and 4.2, typical

measurement depths for some common laser Doppler wavelengths, source–ﬁber

separations (for LDPM), and beam diameters (for LDPI) are shown for some

different tissue types. The measurement depth is deﬁned so that 1 − e

−1

≈ 63%

of the perfusion value originates above that depth. An in depth discussion about

how different system parameters, optical properties, and other tissue properties

affect the measurement depth and volume as well as further examples on both

measurement depth and volume can be found in [40].

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