Blood-Flow Measurements With a Small Number of Scattering Events

Blood-flow measurementswith a small number of scattering events

Pavel Starukhin, Sergey Ulyanov, Ekaterina Galanzha, and Valery Tuchin

Results of simulations of the diffraction of a laser beam by a small blood vessel imbedded in scatteringtissue are presented. The form of the spectra of biospeckle intensity fluctuations is analyzed. TheDoppler shift of intensity fluctuations of scattered light is investigated as a function of the laser beamradius, the radius of the blood vessel, the depth of the vessel in the tissue, and the scattering charac-teristics of flowing blood. A formula that serves as the basis for a method of absolute measurements ofblood-flow velocity is derived. © 2000 Optical Society of America

OCIS codes: 170.3340, 290.0290, 170.0170, 290.4210.

1. Introduction

At present, Doppler methods for measuring blood mi-crocirculation in capillary nets of tissues have beendeveloped in detail. However, often the investiga-tion of the microflow in a narrow single vessel, ratherthan in the entire capillary net, is important in thediagnosis or efficient treatment of many diseases.

The diffraction of focused coherent beams from athin capillary has been extensively studied. Thissubject has a very long history. A laser Doppler mi-croscope was first applied for blood-flow measure-ments in single retinal vessel in the beginning of the1970’s.1,2 A. Priezzhev and his co-workers studiedlaser light scattering from a wide class of bioflows byusing the differential scheme interferometer; see, forexample, Ref. 3. Methods of speckle interferometryhave been applied for problems in measurements ofeye blood microcirculation.4,5 The original Dopplerophthalmoscope, utilizing bidirectional registrationof light scattered from the retinal vessel, also shouldbe mentioned in this context.6,7

T. Asakura and co-workers considered statisticalproperties of biospeckles scattered from a single bloodvessel.8,9 Papers10,11 have been dedicated to the in-vestigation of statistical characteristics of statisti-cally inhomogeneous biospeckles that are formed by

The authors are with the Saratov State University, Astrakhan-skaya 83, Saratov 410026, Russia. S. Ulyanov’s e-mail address [email protected].

Received 23 July 1999; revised manuscript received 3 January2000.

0003-6935y00y162823-08$15.00y0© 2000 Optical Society of America

diffraction of focused laser beams. The formation ofspeckled biospeckles and their statistics were consid-ered in Ref. 12. In particular, in Ref. 13 it wasshown that the spectrum of the fluctuations of thescattered intensity may contain a high-frequencypeak, even in the absence of a specially organizedreference wave. Moreover, the frequency position ofthis peak depends on the number of scatterers in theprobing volume.

Thus at the present time two main techniques forlaser diagnostics of blood flow in a single vessel havebeen developed: Doppler and speckle techniques.In Refs. 14 and 15 it is stated that these techniquesare practically identical. In our opinion, the Dopplereffect and speckle dynamics are two different phe-nomena of coherent light scattering. The interrela-tion of the Doppler and the speckle techniques, theirdifferences, and subtleties in interpretation of themeasuring signal were discussed thoroughly in pa-pers by T. Asakura and others; see Ref. 16, for exam-ple.

Specific features of these two methods concern theconditions for registration of scattered light. For thelaser Doppler technique the size of the aperture is setto be small to define the direction of the scatteredlight with little ambiguity. For the speckle tech-nique a large number of speckles are integrated bythe aperture of a photoreceiver. However, when thefocused laser beam is diffracted by a blood vessel witha small number of scattering events, the size of eachspeckle in the Fraunhofer zone is essentially largerthan the size of the photoreceiver aperture. So, onlya small fraction of the scattered irradiation is re-ceived over the detection area. Thus there is no im-portant difference between speckle and Doppler

1 June 2000 y Vol. 39, No. 16 y APPLIED OPTICS 2823

ThatDm

teOdt

tIsi1kpv

dficpb

Ttf

s

Ts

w

n~cc

2

methods in this particular case of high-resolution la-ser diagnostics of microvessels. Both methods ana-lyze the temporal fluctuations of scattered intensitywithin a small detection angle.

In spite of results achieved in the field of high-resolution measurement of bioflows, the most impor-tant question is still unanswered: Is it possible tocarry out an absolute measurement of blood-flow ve-locity in a vessel with a diameter larger than 40 mm?

he scattering in microvessels with such diameters isigh. As a result, the well-defined Doppler peak dis-ppears in the spectrum of the fluctuations of scat-ered intensity; see, for instance, Ref. 17. Thusoppler diagnostics cannot be applied for absoluteeasurements.Certainly the bandwidth dF of the spectrum of in-

ensity fluctuations of dynamic biospeckles has a lin-ar dependence on the average velocity of the flow.n the other hand, evidently the value of dF alsoepends on the diameter of the vessel and the scat-ering characteristics of the flowing bioliquid.16

Some attempts to calibrate the measurement systemby using phantoms of blood flows in glass capillarieswere described in Ref. 9. The methods of direct cal-ibration are not reliable, because the properties offlows in vitro are essentially different from charac-eristics of blood flows in native capillaries in vivo.n Ref. 18 the dependence of the Doppler spectrumhape on the properties of the scattering flows wasnvestigated by a Monte Carlo method. Reference8 refers to vessels of large diameter only. To ournowledge, there were no efforts to analyze this de-endence as it applies to diagnostics in small bloodessels.Our purpose in this paper is to study the depen-

ence of the bandwidths of Doppler spectra and theirorms on the condition of perfusion of a microvessel,ts radius, the size of the probing beam, and opticalharacteristics of blood. A formula allowing one toerform absolute measurements of the velocity oflood flow in vivo is derived.

2. Tissue Model and Calculation Procedure

The model of tissue used in the simulations consistsof a dermis that surrounds a horizontal vessel. Theprofile of velocity in a blood vessel has a parabolicshape19 ~a so-called Poiseuille’s flow!. The vessel isembedded in the tissue at a depth h; see Fig. 1. Tobe more specific about the flow, assume that maxi-mum blood velocity in the center of the vessel equals1 mmys.

Optical parameters of the medium components aregiven in Table 1 for the wavelength l 5 633 nm.These data are close to the values for optical proper-ties of biotissue published in Refs. 20 and 21.

A semi-infinite model of tissue is used. AHenyey–Greenstein function20 is applied as a phase-scattering function of biotissue. The factor of anisot-ropy g for blood is varied over the range 0.9–0.995.

he detector is placed on the axis of the optical sys-em, which is perpendicular to the tissue layer sur-ace. A standard method of Monte Carlo

824 APPLIED OPTICS y Vol. 39, No. 16 y 1 June 2000

imulations is used.22 As is well known, when ran-dom samples of size N are taken from a random pop-ulation with standard deviation s, the distribution ofthe sample means is also normal with a standarddeviation ~called the standard error23! of

se 5 syÎN. (1)

he sampling distribution of the average Dopplerhift and the required number N of photons that

should reach the detector are estimated before calcu-lation. This number is chosen under the condition24

that the standard error be no more than 5%. Usu-ally N is approximately 2000. Prevenient statisticalevaluation of the required number of photons mini-mizes the computing time.

All the simulated photons that reach the detectorare sorted into a frequency distribution. Such a dis-tribution represents the number of photons withDoppler shifts in the predefined frequency inter-vals.25

The spectra of intensity fluctuations S1~v! is givenby26

SI~v! 5 *2`

`

H~j!H~j 2 v!dj, (2)

here H~j! is the histogram of frequency shifts ofphotons scattered from the medium.

3. Experimental Setup and Measuring Procedure

The measuring system for the investigation of bio-flows is presented in Fig. 2. A He–Ne laser ~l 5 633

m, 1 mW power!, focuses into a spot of small radiusw0 5 1.5 mm! in the investigated microvessel. Aonventional optical microscope equipped with a TVamera and videorecorder permits the visual obser-

Fig. 1. Optical model of the vessel under consideration.

Table 1. Optical Parameters of Biological Tissue Used in Monte CarloSimulations

Component ma ~cm21! ms ~cm21! g n

Dermis 1.8 187 0.81 1.3Blood 13 509 0.9–0.995 1.35

w

nd

mclwsTacfl

vation of the lymph or blood flow in a vessel. Acomputer image analyzer processes frames of thevideo images. One of the images of a blood vessel ofa white rat mesentery is shown in Fig. 3.

As blood or lymph flows through the vessel, thestrongly focused laser beam is modulated in the waistplane. This leads to the formation of a dynamicspeckle pattern in the far zone of diffraction. Thetemporal fluctuations in scattered intensity are de-tected by a photoreceiver in the paraxial region.The speckle-interferometric signal is amplified, re-corded on the audiotape, and processed further bycomputer. A photodetected signal of the measuringsystem, obtained from a blood vessel with diameter of12 mm, is shown in Fig. 4. The results of a moredetailed experimental investigation of bioflows by

Fig. 2. Optical scheme of the measuring system: 1, laser; 2, 4,microobjectives; 3, beam splitter; 5, stage; 6, rat; 7, mirror; 8, lamp;9, optical system of conventional microscope; 10, camera; 11, imageanalyzer.

Fig. 3. Image of a small blood vessel.

means of speckle microscopy were presented earlierin Refs. 27 and 28. Usually the normalized spectralmoments are used in Doppler flowmetry to charac-terize the velocity of bioflow29:

Mn 5

*0

`

vnSI~v!dv

*0

`

SI~v!dv

, (3)

here SI~v! is a power spectrum obtained from in-tensity fluctuations in dynamic speckles; SI~v! ismeasured in volts per hertz. The value of the firstspectral moment M1 is assumed to be directly pro-portional to the average velocity V0 of a random flow:

Vo 5 kM1, (4)

where k is the coefficient of proportionality, which isdefined by the scattering characteristics of the me-dium. The value of V0 is the maximum velocity ofblood in the center of the vessel, measured in milli-meters per second. Equation ~4! is purely emperical;

evertheless, it forms the foundation for the proce-ure for evaluation of blood-flow velocity.In some cases measurements of absolute velocityay be performed directly with a conventional mi-

roscope. Looking at the small vessels with radii ofess than 5 mm, one may sometimes observe a singlehite blood cell. Such a cell looks like a white bright

pot that separates the unceasing flow; see Fig. 5.he rectangular box in this plot marks such a cell asleukocyte. Clearly, the velocity of the white blood

ell is exactly the same as the velocity of the bloodow,19 and the velocity of this moving white lumen

may be determined by analysis of frames of vesselimages grabbed into the computer.

4. Shapes of Doppler Spectra of Scattered Light

We obtained 30 histograms of frequency shifts of scat-tered photons and carried out their approximationsby several fitting functions. By definition, the resid-ual error is the difference, measured vertically, be-tween the data point and the calculated fittingcurve.23 Dividing the residual errors by the corre-sponding estimated values gives the normalized re-siduals. The values of standard deviations of

Fig. 4. Output signal of the measuring system.


attpo

m

2

normalized residuals are presented in Table 2. Ascan be seen from Table 2, the standard deviation issmaller when the approximation uses a Lorentzianfunction.

Substitution of the Lorentzian function into Eq. ~2!yields the spectra of intensity fluctuations. Again,the best fit for Doppler spectra of the scattered inten-sity is given by the Lorentzian function ~see Fig. 6!.

5. Dependence of Doppler Shift on Conditions ofPerfusion of Blood Vessel

A. Dependence of Doppler Shift on Beam Radius

The average Doppler shift30 DF as a function of thewaist beam radius has been investigated. The re-sults of Monte Carlo simulations for three differentvalues of the radius of the blood vessel are presentedin Fig. 7. The Doppler shift decreases as the beamsize increases. We have approximated the obtaineddependence by the function f ~x! 5 1ys. In this casethe normalized standard deviation of residual pointsfrom the fitted curve does not exceed 5%. The de-pendence of the Doppler shift on the value of thewaist beam radius is not essential in the range W0 [

Fig. 5. Image of a small blood vessel containing a moving whiteblood cell. The rectangular box marks the leukocyte.

Table 2. Standard Deviation of Normalized Residuals for Three Valuesof Vessel Radius

Type of Fitting Function

Vessel Radius~mm! Lorentzian Exponential Gaussian

20 5.871 3 1022 1.358 3 1021 6.572 3 1022

40 3.859 3 1022 1.365 3 1021 1.537 3 1021

60 5.986 3 1022 1.434 3 1021 1.305 3 1021


@10 mm; 100 mm#. The relative variation of DF is lessthen 20%.

B. Dependence of Doppler Shift on Vessel Radius

As was mentioned above, the average Doppler shiftdepends not only on the velocity of blood but on thevessel radius as well. Results of the Monte Carlosimulations are presented in Fig. 8. As we can see,the average Doppler shift increases as the blood ves-sel radius r0 increases. This is caused by an in-crease in the mean number of photons scattered bythe moving blood cells. The dependence of DF on r0is nearly linear.

C. Dependence of Doppler Shift on Vessel Depth

Evidently the thickness of the upper layers of tissue~i.e., vessel depth h; see Fig. 1! may vary. It may bessumed that this difference can cause a change inhe average Doppler shift. The results of simula-ions for four different values of the vessel radius areresented in Fig. 9. It can be seen that dependencen the vessel depth is practically negligible. A stan-

Fig. 6. Doppler spectrum of fluctuations of scattered intensity:points, results of Monte Carlo simulations; solid curves, approxi-mation by a Lorentz function. a, r0 5 40 mm; b, r0 5 60 mm; c, r0 580 mm.

Fig. 7. Dependence of the average Doppler shift as a function ofwaist beam radius; squares, results of Monte Carlo simulations;solid curves, approximation by the function f ~x! 5 1yx. a, r0 5 20

m; b, r0 5 40 mm; c, r0 5 60 mm.

b

w^

fmtoaDsDt

dp

a

dard statistical analysis has shown that there is notrend in the curve presented in Fig. 9. This hypoth-esis has been tested against the criterion of inver-sions and the series criterion24 and accepted with aconfidence interval of a 5 0.05. This conclusion isprobably supported by the consideration that thenumber of scattering events inside the vessel variesinsignificantly when the depth h of the vessel iswithin @0; 100 mm#; see Ref. 31.

D. Doppler Shift Dependence on the Blood AnisotropyFactor

Blood is a member of the unique class of biologicalobjects that are characterized by a very high value ofthe factor g ~close to 1!. Thus small alterations ofthe blood anisotropy factor in the vicinity of 1 maycause considerable modifications of the scatteringproperties of the bioflow and, as a result, changes inthe characteristics of the Doppler signal.

In the single-scattering approximation, the Dopp-ler shift is expressed as

F0 52Vl

sinSu

2Dcos~a!, (5)

Fig. 8. Dependence of the bandwidth of the Doppler spectrum onthe radius of the investigated blood vessel.

Fig. 9. Dependence of the bandwidth of the Doppler spectrum onthe depth of the imbedded blood vessel, in the tissue: points,results of Monte Carlo simulations; solid curves, linear approxi-mation. a, r0 5 20 mm; b, r0 5 40 mm; c, r0 5 60 mm; d, r0 5 100mm.

where V is the velocity of the scatterer, u is the angleetween incident k0 and exiting ks wave vectors, and

a is the angle between the vector Dk 5 ks 2 k0 andthe vector V. After averaging over the number ofscattered photons, assuming that u and a are statis-tically independent and that all the scatterers havethe same velocities, and taking into account that g 5^cos~u!&, we obtain the relation for the average Dopp-ler shift:

Df 52Vl

sinSarccosg2D^cos~a!&, (6)

here angle brackets denote ensemble averaging andcos~a!& acts as an unimportant constant.

The dependence of the average Doppler shift as aunction of g for a thin vessel with a diameter of 40m is presented in Fig. 10. The solid curve is ob-

ained from Eq. ~6!. Differences between the resultsf calculations and Monte Carlo simulation are ;5%,nd so are practically negligible. Apparently theoppler shift has the same behavior as the average

cattering angle. As Eq. ~6! predicts, the averageoppler shift decreases as g increases, which causes

he normalized spectral moments to decrease.The results of Monte Carlo simulations for four

ifferent vessel radii for the range g [ @0.9; 0.99# areresented in Fig. 11. As the thickness of the vessel

Fig. 10. Dependence of average Doppler shift on g factor of blood:, analytical curve; b, Monte Carlo simulation.

Fig. 11. Dependence of normalized first spectral moment on gfactor of blood: a, r0 5 40 mm; b, r0 5 60 mm; c, r0 5 80 mm; d, r0 5100 mm.


toi

As

osiebvshlttmf

atmpotminb2Msp

aem

vmtittsiltabu

lst

2

is increased, single scattering is transformed intomultiple scattering. This leads to an increase in theaverage Doppler shift and all the spectral moments.However, all DF curves as function of g remain mono-onic, as in the case of single scattering, for all valuesf radius r0 for vessels within the @40 mm; 100 mm#nterval.

6. Absolute Measurements of Flow Velocity

A simple regression analysis32 based on the results ofMonte Carlo simulations allows us to find an analyt-ical equation for the relationship between the value ofthe blood-flow velocity and the characteristics of thevessel:

V0 5M1yM0

171.8 2 3333.3~g 2 0.98) 1 1.055(ro 2 60!.

(7)

s before, r0 is the radius of the blood vessel ~mea-ured in micrometers!; g is the anisotropy factor; M0

and M1 are the zeroth and the first spectral moments,respectively, which are calculated in accordance withEq. ~2! by use of the experimental spectra of theDoppler signal. This formula is valid for, at least,the ranges g [ @0.95; 0.995# and r0 [ @40 mm; 100mm#. The relative error of approximation by Eq. ~7!does not exceed 20%.

Equation ~7! is the main practical result of thispaper. When the diameter of the microvessel ismeasured with a conventional microscope and the gfactor is known, then Eq. ~7! yields the absolute valueof the maximum blood velocity.

7. Discussion of Results

Measurements of blood-flow velocities in the mi-crovessels were carried out by two methods. Thetheoretical velocity was calculated by means of Eq.~7!, which uses measured spectral moments of theDoppler signal. The experimental value of the ve-locity was defined by the processing of the sequencesof video images, which were recorded with a conven-tional microscope. The results of these investiga-tions are shown in Fig. 12.

It should be noted that the comparison of theo-retical results with experimental data could not bepresented in an explicit form. The point is thatdirect measurements of velocity are possible onlyfor vessels with radii of less than 5 mm. In vesselsf larger diameters, leukocytes cannot be seen vi-ually, so the blood flow does not look spatiallyntermittent. On the other hand, the transportquation ~solved by Monte Carlo methods! cannote used for the study of light-scattering processes inessels of very small diameters. A Monte Carloimulation may be applied only for the analysis ofighly scattering media for which the transport

ength of photon migration is comparable with thehickness of the vessel. Thus theoretical data ob-ained in the present paper relate to a vessel of ;50m in diameter; experimental data may be obtained

rom frame-by-frame analysis for vessels with di-


meters at least five times smaller. This is whyhe values of velocities that are measured experi-entally inherently differ from their theoretically

redicted values. As can be seen from Fig. 12, the-retical values exceed the experimental ones by 40imes. The other possible reason for this abnor-ally high discrepancy between theory and exper-

ment is that the Henyey–Greenstein function doesot correctly describe the scattering properties oflood microvessels. As was demonstrated in Ref.0, the Gegenbauer kernel phase function or theie scattering phase function is preferable for the

tudy of light-scattering phenomena in a thin sam-le with a high ~.0.95! anisotropy factor.Experimental points are dispersed and scattered

round the theoretical curve. The correlation co-fficient between the theoretical and the experi-ental data is also not high, Rc 5 0.4. This fact

may have a simple explanation. Results of inves-tigations distinctly demonstrate that blood flow isinherently not constant. Therefore four directmeasurements of velocity were carried out for onesingle vessel with a 4-mm radius. The measuredalues of the velocity are 160, 358, 400, and 533mys. As can be seen, the normalized variation of

he velocity exceeds 100%. Because of the variabil-ty of the flow, the dependence of the flow velocity onhe vessel radius could not be reliably found. Ashe results of a linear regression analysis show, thelope of velocity dependence on the vessel diameters close to 0. The coefficient of the mutual corre-ation of the mentioned characteristics is also rela-ively small; it equals 0.7. The hypothesis of thebsence of correlation between the velocity of thelood and the radius of the vessel has been tested byse of the test for zero intercept23 and is accepted at

a confidence level of a 5 0.01.Examine now the dependence of the blood-flow ve-

ocity on the first spectral moment of the Dopplerignal of the measuring system. As follows from sta-istical analysis of residual points, the Durbin–

Fig. 12. Relation between the theoretical and the experimentalresults of measurements of the velocity of blood in a vessel ofdiameter 7.5 mm; solid line, linear regression.

i0

sslm

oN

laser speckles based on the photon correlation,” Opt. Commun.
Watson value ~for more details see Ref. 23! is close to2. This means that residuals are not correlated.~Strictly speaking, the criterion of Durbin–Watsonequals 1.7, which is less than 2; a value of 2 indicatesa small positive adjacent correlation!. Thus there isa linear correlation between the values of the spectralmoments of the Doppler signal and the velocity ofblood in a microvessel. This illustrates the quanti-tative correspondence between the experimental datafor absolute velocity measurements and the theoret-ical results.
8. Summary

Usually an exponential Doppler spectrum appearsin the Monte Carlo simulations, but the spectrum offluctuations of the intensity of light that is scatteredfrom a narrow blood vessel has a Lorentzian shape.

The average Doppler shift of the spectrum de-pends only slightly on the waist beam radius. Thedepth of the imbedded vessel in the tissue has anegligible influence on the statistical characteris-tics of the Doppler signal. The value of the Dopp-ler shift is linearly proportional to the radius of theblood vessel, at least in the range r0 [ @20 mm; 150mm#. The dependence of DF on the scattering an-sotropy factor g is nearly linear within the @0.9;.99# interval.Theoretical and experimental investigations

how that the first spectral moment of the Dopplerignal is linearly proportional to the blood-flow ve-ocity. This provides a basis for absolute velocity

easurements.

This work is supported by the Russian Foundationf Basic Research through grants N96-15-96389,98-02-17997, and N99-15-96040.

References1. C. E. Riva, B. Ross, and G. B. Benedek, “Laser Doppler mea-

surements of blood flow in capillary tubes and retinal arteries,”Invest. Ophthalmol 11, 936–944 ~1972!.

2. H. Mishina, T. Asakura, and S. Nagai, “A laser Doppler mi-croscope,” Opt. Commun. 11, 99–102 ~1974!.

3. B. A. Levenko, A. V. Priezzhev, S. G. Proskurin, and N. B.Savchenko, “Laser Doppler microscopy of biological objectswith various optical properties,” Bull. Russ. Acad. Sci. Phys.59, 1070–1075 ~1995!.

4. Y. Aizu, T. Asakura, and A. Kujimn, “Compensation of eyemovements in retinal speckle flowmetry using flexible corre-lation analysis based on the specific variance,” J. Biomed. Opt.3, 227–236 ~1998!.

5. Y. Aizu, K. Ogino, T. Sugita, T. Yamamoto, N. Takai, and T.Asakura, “Evaluation of blood flow at ocular fundus by usinglaser speckle,” Appl. Opt. 31, 3020–3029 ~1992!.

6. C. E. Riva, J. E. Grunwald, S. H. Sinclair, and K. O’Keefe,“Fundus camera based retinal LDV,” Appl. Opt. 20, 117–120~1981!.

7. G. T. Feke, A. Yoshida, and C. L. Schepens, “Laser basedinstruments for occular blood flow assessment,” J. Biomed.Opt. 3, 415–422 ~1998!.

8. Y. Aizu and T. Asakura, “Bio-speckle phenomena and theirapplication to the evaluation of blood flow,” Opt. Laser Tech-nol. 23, 205–219 ~1991!.

9. Y. Aizu, H. Ambar, T. Yamamoto, and T. Asakura, “Measure-ments of flow velocity in a microscopic region using dynamic

72, 269–273 ~1989!.10. S. S. Ul’yanov, “The peculiarities of manifestation of the Dopp-

ler effect at the scattering of focused Gaussian beams on mov-ing random inhomogeneous media,” Bull. Russ. Acad. Sci.Phys. 59, 133–137 ~1995!.

11. S. S. Ul’yanov, “A new type of manifestation of Doppler effect.“An application to blood and lymph flow measurements,” Opt.Eng. 34, 2850–2855 ~1995!.

12. S. S. Ulyanov, “Speckled speckles statistics with a small num-ber of scatterers, an implication for blood flow measurements,”J. Biomed. Opt. 3, 237–245 ~1998!.

13. S. S. Ul’yanov, “Dynamic of statistically inhomogeneous speck-les: a new type of manifestation of Doppler effect,” Opt. Lett.20, 1313–1315 ~1995!.

14. J. D. Briers and S. Webster, “Laser speckle contrast analysis~LASCA!: a nonscanning, full-field technique for monitoringcapillary blood flow,” J. Biomed. Opt. 1, 174–179 ~1996!.

15. J. D. Briers, “Laser Doppler and time-varying speckle: a rec-onciliation,” J. Opt. Soc. Am. A 13, 345–350 ~1996!.

16. Y. Aizu and T. Asakura, “Coherent optical techniques for di-agnostics of retinal blood flow,” J. Biomed. Opt. 4, 61–75~1999!.

17. T. Eiju, M. Nagai, K. Matsuda, J. Ohtsubo, K. Homma, and K.Shimizu, “Microscopic laser Doppler velocimeter for blood ve-locity measurements,” Opt. Eng. 32, 15–20 ~1993!.

18. W. Ruetten, T. Gellekum, and K. Jessen, “Investigation oflaser Doppler techniques using the Monte Carlo method,” inPhoton Transport in Highly Scattering Tissue, S. Avrillier, B.Chance, G. J. Mueller, A. V. Priezzhev, and V. V. Tuchin, eds.,Proc. SPIE 2326, 277–288 ~1995!.

19. A. Perlin and T.-K. Hung, “Flow development of a train ofparticles in capillaries,” J. Eng. Mech. Div. EM1, 49–66~1978!.

20. A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldbach, and H. J.Schwarzmaier, “Influence of the scattering phase function ap-proximation on the optical properties of blood determined fromthe integrating sphere measurements,” J. Biomed. Opt. 4,47–53 ~1999!.

21. A. Kiele, R. Hibst, “A new optimal wavelength for treatment ofport wine stains,” Phys. Med. Biol. 40, 1559–1576 ~1995!.

22. V. V. Tuchin, S. R. Utz, and I. V. Yaroslavsky, “Tissue optics,light propagation, and spectroscopy,” Opt. Eng. 33, 3178–3188~1994!.

23. A. D. Aczel, Complete Business Statistics ~Irwin, Homewood,Ill., 1989!.

24. J. S. Bendat and A. G. Piersol, Random Data. Analysis andMeasurements Procedures ~Wiley, New York, 1986!.

25. M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M.Dassel, and J. G. Aarnoudse, “Laser Doppler blood flowmetryusing two waveguides: Monte Carlo simulations and mea-surements,” Appl. Opt. 33, 3549–3558 ~1994!.

26. F. F. de Mul, W. Steenbergen, T. Vonck, and J. Greve, “Co-herence effects in modeling laser-Doppler perfusion flow-metry,” in CIS Selected Papers: Coherence-DomainMethods in Biomedical Optics, V. V. Tuchin, ed., Proc. SPIE2732, 123–133 ~1996!.

27. S. S. Ulyanov, V. V. Tuchin, A. A. Bednov, G. E. Brill, and E. I.Zakharova, “Speckle-interferometric method in application tothe blood and lymph flow monitoring in microvessels,” LasersMed. Sci. 12, 31–41 ~1997!.

28. A. A. Bednov, S. S. Ul’yanov, V. V. Tuchin, G. E. Brill, and E. I.Zakharova, “Investigations of dynamics of lymph flow bymeans of speckle interferometric method,” Appl. Non-linearDyn. 4, 42–51 ~1996!.

29. A. P. Shepherd and P. A. Oberg, eds., Laser Doppler BloodFlowmetry, ~Kluwer Academic, Boston, 1989!.


30. M. H. Koelink, F. F. M. de Mul, J. Greve, R. Graaff, A. C. M.

3

a highly scattering medium,” in Nonlinear Dynamics and

3

2

Dassel, and J. G. Aarnoudse, “Analytical calculations andMonte Carlo simulations of laser Doppler flowmetry using acubic lattice model,” Appl. Opt. 31, 3061–3067 ~1992!.

1. P. Yu, Starukhin, S. S. Ulyanov, and V. V. Tuchin, “Monte-Carlo simulation of Doppler shift for laser light propagation in


Structures in Biology and Medicine: Optical and Laser Tech-nologies: International Workshop, V. V. Tuchin, ed., Proc.SPIE 3053, 42–47 ~1997!.

2. L. Stevens, Applied Multivariate Statistics for the Social Sci-ences ~Erlbaum, Hillsdale, N.J., 1986!.

Blood-Flow Measurements With a Small Number of Scattering Events

Documents

Transcript of Blood-Flow Measurements With a Small Number of Scattering Events