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Use of low-coherence speckled speckles forbioflow measurements

Sergey S. Ulyanov and Valery V. Tuchin

Formation of speckled speckles in the case of biflow perfusion by partially coherent light was considered.Dependencies of statistical characteristics of low-coherence biospeckles with a small number of scattererson the scattering properties of the flow and on the coherence length of incident light were analyzed. Itwas shown that the value of the Doppler bandwidth in the scattered light essentially depends on the ratiobetween the coherence length and the average size of the flow’s inhomogeneities. A procedure forreconstructing velocity distribution in a single blood vessel was suggested. © 2000 Optical Society ofAmerica

OCIS codes: 170.0170, 290.0290.

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1. Introduction

At present, optical Doppler tomography ~ODT! is aromising technique for the diagnostics of randomow.1–3 In general, analysis of the physical principles

of ODT measurements is reduced to consideration ofthe process of dynamic scattering of low-coherence ra-diation in a random object. Regretfully, at the mo-ment, the exact theory of diffraction of low-coherencelight in statistically inhomogeneous media has not yetbeen developed. However, properties of speckle pat-terns produced under partially coherent light illumi-nation ~or, in other words, under the perfusion of ascattering object by speckles! have been extensivelystudied. Speckled speckles, or doubly scattered speck-les, are formed in this case. Evidently, the scatteringof low-coherence light and the diffraction of partiallycoherent irradiation have the same nature.

Two important and unsolved problems should beoutlined:

1. At the scattering of focused coherent beams,specific speckles,4,5 so-called speckles with a smallnumber of scatterers, are formed. Their specialproperties should be taken into account during anal-

The authors are with the Department of Optics, Saratov StateUniversity, Saratov 410026, Russia. S. Ulyanov’s e-mail addressis ulianov@sgu.ssu.runnet.ru.

Received 22 March 2000; revised manuscript received 22 August2000.

0003-6935y00y346385-05$15.00y0© 2000 Optical Society of America

ysis of measurement signal formation by a high-resolution system, with low-coherence irradiation.

2. As is well known, initial irradiation becomespartially coherent when the coherent light is dif-fracted from a multilayered random medium or by ahighly scattering flow. This means that dynamicspeckled speckles6–8 appear inside of the medium.

similar phenomenon also occurs in the case of thecattering of low-coherence light, but it was ignoredn ODT measurements of random flows.

So, to the author’s knowledge, for practical pur-oses there are no bridges between the theory of ODTeasurements and the theory of speckled speckle

ormation. This brief report partially fills this gap.

2. Dynamics of Coherence Speckled Speckles

Blood or lymph flow in a narrow native vessel may beimitated with a set of random screens.8,9 As hasalready been mentioned, speckled speckles areformed when a coherent beam scatters from a cascadeof screens. In Ref. 8 the correlation function of in-tensity fluctuations of statistically inhomogeneousspeckled speckles was studied with regard to blood-flow measurement.

Let us consider the diffraction of a focused Gauss-ian beam from several moving equidistant screens.Correlation function Gu

~n!~t! of the amplitude of scat-tered light behind the nth screen is given by8

Gu~n!~t! 5 expF2A~n!Snnt

l D2G , (1)

1 December 2000 y Vol. 39, No. 34 y APPLIED OPTICS 6385

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where

A~n! 5

A~n21!HF l

W~n21!G4

1 C~n!2 J

2A~n21!F l

W~n21!G2

1 F l

W~n21!G4

1 C~n!2

14p2s2

Lc2 ,

(2a)

W~n! 5 32A~n21!F l

W~n21!G2

1 F l

W~n21!G4

1 C~n!2

F2p2l2

DZ~n!2 GF l

W~n21!G2 4

0.5

,

(2b)

C~n! 5 2 3 pZ~n!

l

Z~n!2

l2 1 p2 W04

l4

1pl

DZ~n!4 , (2c)

n is the number of the screen, Z~n! is the distancebetween the nth screen and the beam-waist plane,DZ~n! is the distance between the nth and the ~n 11!th screens, W0 is the beam-waist radius, and l ishe wavelength of irradiation. All the screens pos-ess identical statistical properties: s2 is the mean-

square deviation of the phase screen surface fromflatness, and Lc is the correlation length of the screenprofile. Each nth screen moves with the velocity nncorrespondingly. Screens move in the planes thatare transverse to the direction of light beam propa-gation. The process of coherent beam scatteringfrom two moving screens is shown in Fig. 1~a!.

3. Scattering of Focused Low-Coherence Beam from aRandom Flow

The real bioflow is a continuous medium. However,in some partial cases, the discrete analog of such amedium could be applied.8,9 A set of moving randomphase screens was recently found to provide an ex-cellent optical model for a microvessel with flowingbiological liquid.

When an interferometer with a low-coherencesource is used in the flow diagnostics, then its mea-suring signal is formed as a result of light scatteringfrom some selected volume. At the discrete ap-proach this means that the scatterers, which areplaced on the moving screen with number n0 ~and onhe neighboring screens, which are situated in theicinity of the n0th screen in the range of selected

volume!, carry the main contribution to the formationf the Doppler signal. n0 may be interpreted @see

Fig. 1~b!# as a coordinate Dr of the center of the se-lected volume normalized to the value of the profilecorrelation length Lc of the screens, which imitate theflow, i.e.,

n0 < integer~DryLc!. (3a)

The number Dn of screens, which are collected in theselected volume, is approximately defined by the re-

386 APPLIED OPTICS y Vol. 39, No. 34 y 1 December 2000

lation between the coherence length Lcoh of light inthe investigated medium and the distance betweenthe screens. The distance between the screens @seeFig. 1~b!# is assumed to be equal to the value of Lc;thus

Dn < integer~LcohyLc!. (3b)

We assume here that the first optical path ~from theselected screen with number n0 to the output plane ofinterferometer! equals the second optical path of theinterferometer ~which the reference wave goeshrough!. A method of calculating the correlation

Fig. 1. ~a! Illustration for the process of formation of dynamicspeckled speckles. ~b! Explanation of notation: n0 5 DryLc,Dn 5 LcohyLc, Dr 1 r 5 r0.

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function of the intensity fluctuation in the interfer-ence field was presented in Ref. 10. This correlationfunction can be expressed in the form

Gint~t! 5 2 Re@G*u1~t!Gu2~t!#, (4a)

where Gu1~t! and Gu2

~t! are the correlation functions oftemporal fluctuations of complex amplitude of thefirst and the second speckle fields, respectively.

In a low-coherence interferometer the interferencefield is formed as a superposition of the reference fieldand the wave, which comes from the selected volume.Since the amplitude of the reference wave is essen-tially larger than the amplitude of the scattered com-ponent and does not depend on time, Eq. ~4a!mmediately reduces to

Gint~t! } uG2~t!u. (4b)

So, in the considered situation, the correlation func-tion of intensity fluctuation in the interference field isdirectly proportional to the function of uGu

~n!~t!u.The spatial distribution of the velocity of the scat-

terers should be incorporated into the optical model ofthe flow. Let us assume that the velocity profile inblood flow is described by the formula

nn 5 nmax@1 2 ~rnyr0!2# , (5)

where rn is the distance from the nth screen to thevessel axis, r0 is the radius of the vessel, and nmax isthe velocity on the axis of the flow. This is the so-called Poiseuille flow.

As is well known, spectral analysis of temporalfluctuations of scattered light provides detailed in-formation about the dynamics of the random media.Bandwidth DF of the spectrum is often interpreted

s characteristic of the velocity of bioflow. In ac-ordance with the Wiener–Khintchine theorem, wean obtain the power spectrum by using the Fourierransform of the correlation function. The corre-ation function of intensity fluctuations of low-oherence biospeckles is determined formally byelations ~1!, ~2!, and ~4b!, including the velocityistribution Eq. ~5!, and thus can be easily calcu-ated.

Clearly, formulas ~1!, ~2!, and ~4! are not dependentn the parameters Lcoh, Lc, and Dr directly. How-

ever, the mentioned values determine the position ofthe center of the selected volume and the number ofscreens that simulate the considered flow, in accor-dance with relations ~3a! and ~3b!. The results ofDoppler broadening calculations @with relations ~1!–~4!# are presented in Figs. 2~a! and 2~b!. It can beseen that dependence of bandwidth DF of the Dopplerspectrum on the position of the selected volume doesnot correspond exactly to the real velocity distribu-tion in the blood vessel. The function of DF is moresharply peaked about the axis of the vessel than thefunction nn. When the value of coherence length ~or,in our discrete model, the number Dn of screens in theselected volume! is smaller, then the shape of thefunction DF@DryLc# is broader.

It is remarkable that increasing the coherence

length leads to a rising number of scatteringscreens, imitating the flow. However, at the deri-vation of relation ~3b!, it was assumed that Lcoh is

Fig. 2. Dependence of normalized bandwidth of Doppler signalspectrum on the number of selected screens. ~a! Comparison withvelocity profile. Pluses, normalized bandwidth of spectrum; solidcurve, normalized velocity of the screens, W0yLc 5 3. ~b! Depen-

ence for different thickness of selected volume. a, Dn 5 1; b,n 5 3; c, Dn 5 5; d, Dn 5 7.5. ~c! Comparison with experiment.olid curve, velocity profile; dots, experimental data from Ref. 1.

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essentially less than diameter d of the investigatedflow. If Lcoh 3 `, the number of screens Dn tendso its extreme value, which is defined by diameter df the flow, i.e.,

Dn < integer~dyLcoh!. (6)

The case of coherent scattering ~Lcoh 3 `! was con-sidered in Ref. 8.

As calculations show, the value of DF practically doesot depend on the focusing depth of the light beam intohe investigating vessel or on the number of scatterers, ateast within the range of W0yLc [ @0.25; 30#.

Experimental investigations of the structure of mul-tiple scattering were carried out in Ref. 1. 1% In-tralipid ~with ms 5 23 cm21!, which flows through theconduit submerged in a highly scattering phantom wastested with an ODT system. The inner and the outerdiameters of the conduit are 580 and 960 mm, respec-tively. The centerline velocity of the flow is 2 mmys.The dependence of the Doppler spectrum bandwidthon the position of selected volume is shown in Fig. 2~c!.Clearly, the shapes of the curves, which are presentedin Figs. 2~b! and 2~c!, are similar.

4. Reconstruction of Velocity Distribution

The real velocity profile in a blood vessel is distortedas a result of measurements by low-coherence inter-ferometry. Let us assume that the interferometerchose for the analysis the vicinity of the screen withnumber n. Then we can formally write the depen-dence of the Doppler spectrum bandwidth and thevelocity of the selected screen:

DF~n! 5 Rnn~n!, (7a)

where Rn is some proportionality coefficient, depend-ing on ratio rnyr0. This coefficient may be easilycalculated with the presented optical model of thebioflow.

Thus function R~rnyr0! 5 DF~n!yn~n! can be consid-ered to be the calibration function of the measuringsystem. In other words, the real spatial distributionof the velocity may be recovered as follows:

n~n! 5DF~n!

R~rnyr0!, (7b)

where DF~n! is the experimentally measured band-width of the Doppler signal, when the probing beamis focused onto the selected layer of the flow ~corre-sponding to the screen with number n!.

Calculations show that dependence of the shape ofthis function on the diameter of the blood vessel, andon the scattering characteristics of the investigatedflow, is relatively weak. However, the shape of thisfunction essentially depends on the coherence lengthof the irradiation. The form factor of function R~rnyr0! for different values of coherence length is pre-sented in Fig. 3~a!. Remarkably, the shape offunction R~rnyr0! may be asymmetric for large valuesf coherence length @see Fig. 3~a!, curve a#. Equa-ion 7~b! and the function of R ~which should be cal-

388 APPLIED OPTICS y Vol. 39, No. 34 y 1 December 2000

ulated for different diameters of vessels! may serves the basis for measuring absolute velocity of thelood flow with the ODT system.

5. Conclusions

With the experimental results presented in Ref. 1 it ispossible to find the calibration function R~rnyr0!. A

Fig. 3. Calibration functions R~rnyr0!. ~a! Results of calcula-ions: a, Dn 5 7.5; b, Dn 5 5; c, Dn 5 3. ~b! Results of experi-ents. ~c! Interpretation of effect of dynamic speckle diffraction

rom motionless scatterers.

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plot of this experimentally defined function is shownin Fig. 3~b!. The calibration function obtained fromhe measurement results is similar to the theoreticalependence @compare Figs. 3~a! and 3~b!#. It alsoas some asymmetry; the left and the right maximaave different amplitudes. However, these peaksre more localized than was predicted by theory. Inthers aspects experimental and theoretical data aren good agreement.

An interesting peculiarity of the scattering of low-oherence irradiation by the flows surrounded by thecattering shell should be noted. The Signal of theow-coherence speckle-interferometer may be regis-ered not only from moving scatterers but also from thehell. Of course, fluctuations of scattered intensitybtained from this motionless phantom are essentiallyeaker than ones obtained from the random flow @seeig. 2~c!#. Apparently, there is a clear physical expla-

nation for this phenomenon from the viewpoint of theproposed theory. Because of the strong isotropic scat-tering of light by flowing Intralipid, the surroundingmedium is perfused by dynamic low-coherence speck-les. In the considered situation, motionless scatterersbecome the sources of dynamic fluctuations of the scat-tered field. Thus it is not so easy to define preciselythe positions of the flow boundaries.

Let us assume that the diameter of the flow equalsthe outer diameter of the conduit and then formallycalculate the calibration function @Fig. 3~c!#. In thiscase its form will be similar to the shape of the the-oretical dependence @and to an even greater extentthan the calibration function, which is plotted in Fig.3~b!#. So use of the proposed procedure for recoveryof flow velocity requires great caution.

This study was supported by grant N00-02-81014of the Russian Foundation of Basic Researches, grant

N99-15-96040 of the President of the Russian Feder-ation, and by award N REC-006 of the U.S. CivilianResearch and Development Foundation for the Inde-pendent States of the Former Soviet Union.

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