fast development of this device and optical techniques associated with it, this principle has led over the last 35 years to the measurement of blood ßow in a number of tissues of the body [1]. In 1972, the retina provided the Þrst measurements of blood velocity obtained by means of the Doppler shift [2]. LDV measurement of blood velocity in human retinal vessels was published 2 years later [3]. Subsequent to the pioneering work of Stern [4], who assessed blood ßow in the tissue of the skin, Riva et al. [5] described a method to measure blood velocity in the human optic nerve microcirculation. This method was then extended to the measurement of blood ßow in the vascular bed of the cat optic nerve head (ONH) [6], the human ONH [7, 8], and the subfoveal choroid [9]. In 1995, Michelson and Schmauss Þrst reported mapping of blood ßow in the microcirculation of the human retina by means of scanning LDF [10].

The present chapter is by no means an exhaustive description of LDV and LDF. Regrettably, due to limitation of space, a choice of papers and reviews among the extensive laser Doppler literature related to the eye had to be made. Part of the material of this chapter has been reproduced with permission from previous publications [11Ð14].

7.2Retinal Laser Doppler Velocimetry

7.2.1The Doppler Effect

The basis of LDV and LDF is the Doppler effect, Þrst described in 1842 by the Austrian physicist Christian Doppler in an article entitled ÒOn the Colored Light of Double Stars and Some Other Heavenly Bodies.Ó Doppler describes the frequency shift that a sound or a light wave undergoes when emitted from an object that is moving away or towards an observer. It manifests itself, for example, in the increase in the pitch of the siren of an approaching ambulance.

Consider a particle, such as a red blood cell (RBC), moving at velocity V (Fig. 7.1). A laser

fo + f

fo

ai

aS

Fig. 7.1 The Doppler effect: The frequency of the light scattered by the particle moving at the velocity V in the direction deÞned by as is shifted in frequency by an amount Df compared to that of the incident light (frequency f0 and direction deÞned by ai with V)

beam of single frequency, f0, is incident on this particle. This beam, which is deÞned by the wave vector Ki, will be scattered by the particle in vari-

ous directions. The beam scattered in the direc-

tion of the detector, deÞned by Ks , is shifted in frequency by an amount:

refractive index of the medium and λ is the wavelength in vacuo of the incident laser light.

In order to determine Df , the scattered light must be detected by a sensor, which transforms the electric Þeld of the scattered light incident upon it into a current. This current is in turn processed to determine its frequency content. The technique by which Df is determined from the scattered Þeld is known as Òoptical mixing spectroscopy (OMS)Ó or Òlight beating spectroscopyÓ [15, 16].

Understanding the application of OMS to the measurement of RBC velocity in retinal vessels requires a description of the electric Þeld of the scattered light, followed by that of the current at the output of the detector. This description will be Þrst carried out in the simpliÞed case of singly scattering particles moving in a glass capillary tube and then extended to multiply scattering RBCs.

Fig. 7.2 (a) Schematic representation of a volume of a

vector Ki . The light scattered by the particles moving in the tube and the vessel wall is deÞned by Ks . (b) Assumed parabolic velocity proÞle of the particles in the capillary tube. The cylindrical shell DR contains particles moving at velocity V, the value of which is given by PoiseuilleÕs law (Adapted from Riva and Feke [13] with permission from the Publisher)

E0 (Ki‚t ) cosw0t

Es (Ks‚t ) a

b

v

R

R

7.2.2Electric Field Scattered by Singly Scattering Particles Moving in a Capillary Tube

direction Ks at time t by a dilute solution of singly scattering particles moving at constant velocity through a glass capillary tube (Fig. 7.2) consists of light scattered by both the particles and the glass

wall. The incident monochromatic beam (single

frequency, f0), Eo cosωot (ωo = 2π / f0 ), is assumed to illuminate uniformly the entire cross section of the tube.

If the particles are much smaller than the cap-

Vmax is the centerline (maximum) velocity of the particles. Due to this distribution of velocities, the scattered light consists of a range of Dopplershifted frequencies.

With Poiseuille ßow, if one divides the range of velocities (0 to Vmax) into N equal increments, DV, (Fig. 7.2) the particles with speeds between V and V+DV are located in a shell of radial thick-

ness DR, where from Eq. 7.2, DR=R02DV/2RVmax. Furthermore, if the concentration of particles across

the tube is uniform, there will be equal numbers of

particles ßowing at each velocity range [2].

The Þeld Es (K,t) is the sum of the Þeld scattered by the particles and the Þeld scattered by the vessel wall:

ELO (K,t) is the Þeld of the light scattered from the tube wall. It has the same frequency as the incident light since the tube is not moving and acts as a reference beam, conventionally called Òlocal oscillator (LO).Ó

Under the conditions of both uniform concen-

i=0

The power spectral density of the scattered

magnitude ALO A/2 from f0 to f0 + Dfmax and zero

ranges between 0 and @ 5´104 kHz , the required resolution (Df/f0) for its detection, namely <<l0−10, was far beyond the capability of spectroscopic techniques available until @ 1965 . However, with OMS, the information contained in the optical spectrum could be translated from the high frequencies of light waves down to sufÞciently low frequencies, where instruments operate with adequate resolution to resolve the spectrum of Doppler shifts.

Dfmax is deÞned as the cutoff frequency of Pi (Df ) .e

is the charge of the electron, and ScohLO is the coherence area of the local oscillator at the detec-

tor surface.

The third term of Pi (Df ) in Eq. 7.9 is a squared replica of the second term of PEs(f) in Eq. 7.7, but shifted down towards low frequencies, so that the particle velocities can be determined from measurements of Pi (Df ). Thus, the process of mixing at the detector surface of the light scattered by the particles with the light from the glass wall (local oscillator) results in the downshifting of the spectrum from optical frequencies to frequencies in the kHz range (Fig. 7.3), where high-resolution spectral analy-

(DSPS) of the Photocurrent

In the application of OMS, Es (K,t) is directed

onto the surface (S) of a square wave photodetec-

tor. This Þeld generates a photocurrent, i(r,t) , where r is a point on S. The time average of this current is:

S

b is the responsivity of S. All the information on the particles velocities can be extracted from i(t) . This extraction, which is complex and beyond the scope of this chapter, leads to the following expression for the Doppler shift power

intense than the Þeld scattered by the particles

power spectral density of the signal term be larger than that of the shot noise term. A measure of this ability is the intrinsic signal-to-noise ratio (SNR), which is:

bSS cohLO A2 LO A2

ebSA2Lo

2

7.2.5Experimental Test of the Bidirectional LDV Technique

With the aim of testing the validity of the bidirectional scheme [11], polystyrene latex spheres (diameter=0.3 mm) suspended in water were passed at known V max through a glass capillary tube (diameter=200 mm) placed in the retinal plane of a Topcon model eye. The incident laser beam (helium-neon, 632.8 nm) was focused on the tube which was placed at different locations of the fundus. The direction of the ßow was also varied stepwise from 0¡ (tube horizontal) to 180¡. Measurements were performed for an emmetropic and an axially ametropic (± 4 diopters) eye. DSPS were recorded for two scattering directions by means of a two-channel digital signal spectrum analyzer (Fig. 7.4, bottom). These DSPS are characterized by sharp cutoffs from

which Dfmax,1 and Dfmax,2 can be accurately determined. The value of Vmax calculated using Eq. 7.11

was found to be in good agreement (<3% difference) with the value obtained from the known ßow through the tube and the cross section of the tube. The difference was attributed to errors in Dα and D* fmax . Feke et al. demonstrated similar results and

geometry ( Dα ) [19].

The increase in spectral power in the region of the cutoff frequencies for the DSPS shown in Fig. 7.4 bottom is probably due to a nonuniform illumination of the tube by the laser beam, which was smaller than the tube internal diameter. As a

Fig. 7.4 Top: Fundus-camera-based scheme for the detection of light scattered from blood moving with velocity Vmax along two directions K1 and K2 deÞned by apertures A1 and A2 in the plane of the eye pupil. The beam through A2 is focused at P2 after deßection by a prism in the collecting optics. The direction P1 −P2 is aligned with that of A1 −A2. The direction A1 −A2 can be rotated in the plane of the pupil

for alignment with the direction of the vessel (Adapted from Riva et al. [25] with permission from the Publisher). Bottom: DSPS obtained simultaneously along two directions of the scattered light from a suspension of polystyrene spheres in water through a 200-mm-internal-diameter glass capillary tube mounted in the ÒretinalÓ plane of a model eye (Adapted from Riva et al. [18] with permission from the Publisher)

result, the faster particles received more light relative to the slower ones located close to the vessel wall. The effects of laser beam size, eccentricity, as well as absorption of light by the ßowing medium, have been reported by Petrig and Follonier [21].

7.2.6The Output Signal-to-Noise Ratio of the DSPS and Its Dependence on the Recording Time

Due to the stochastic nature of the photodetector emission process, the photocurrent is a ßuctuating quantity so that there is an uncertainty in the

measurement of the DSPS when the measurement is performed in a Þnite time. This phenomenon is illustrated by four DSPS recorded in different times from a 0.1 suspension of 1-mm-

expected, the ßuctuations of decrease as the measurement time increase. It can be shown that, based on a previous analysis [13, 16], if