Laser Doppler Flowmetry - a Theoretical Framework - IMT

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

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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.

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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.

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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 .

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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 = 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.

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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,=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].

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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.

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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.

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(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.

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Perf

Perf

CMBC

CRBC



CRBC

Figure 5.1 Approximate relationships for CMBC and CRBC (left), Perf and CRBC (middle) and Perf and (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.

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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.

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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.

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References 1.

Riva C, Ross B, Benedek GB. Laser Doppler measurements of blood flow in capillary tubes and retinal arteries. Invest Ophthalmol 1972;11(11):936-44.

2.

Stern MD. In vivo evaluation of microcirculation by coherent light scattering. Nature 1975;254(5495):56-8.

3.

Nilsson GE, Salerud EG, Strömberg NOT, Wårdell K. Laser Doppler Perfusion Monitoring and Imaging. In: Vo-Dinh T, editor. Biomedical photonics handbook. Boca Raton, Florida: CRC Press; 2003. p. 15:1-24.

4.

Serov A, Steenbergen W, de Mul F. Laser Doppler perfusion imaging with a complimentary metal oxide semiconductor image sensor. Optics Letters 2002;27(5):300-302.

5.

Forrester AT. Photoelectric Mixing As a Spectroscopic Tool. Journal of the Optical Society of America 1961;51(3):253-259.

6.

Cummins HZ, Swinney HL. Light Beating Spectroscopy. In: Wolf E, editor. Progress in optics. Amsterdam: North-Holland; 1970. p. 135-200.

7.

Larsson M. Influence of optical properties on laser Doppler flowmetry. Linköping, Sweden: Linköping University; 2004.

8.

Rabal HJ, Arizaga R, Cap NL, Grumel E, Trivi M. Numerical model for dynamic speckle: an approach using the movement of the scatterers. Journal of Optics a-Pure and Applied Optics 2003;5(5):S381-S385.

9.

Nilsson GE. Signal processor for laser Doppler tissue flowmeters. Med Biol Eng Comput 1984;22(4):343-8.

10.

Jakobsson A, Nilsson GE. Prediction of sampling depth and photon pathlength in laser Doppler flowmetry. Med Biol Eng Comput 1993;31(3):301-7.

11.

Nilsson H, Nilsson GE. Monte Carlo simulations of light interaction with blood vessels in human skin in the red-wavelength region. In: Priezzhev AV, Asakura T, Briers JD, editors. Optical Diagnostics of Biological Fluids III; 1998: SPIE; 1998. p. 44-53.

12.

Tenland T, Salerud EG, Nilsson GE, Oberg PA. Spatial and temporal variations in human skin blood flow. Int J Microcirc Clin Exp 1983;2(2):81-90.

13.

Kernick DP, Tooke JE, Shore AC. The biological zero signal in laser Doppler fluximetry - origins and practical implications. Pflugers Arch 1999;437(4):624-31.

14.

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.

15.

Karlsson MGD. Movement artefact reduction in laser Doppler blood flowmetry myocardial perfusion applications. Linköping, Sweden: Linköping University; 2005.

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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