Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes

INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 51 (2006) 311–333 doi:10.1088/0031-9155/51/2/009

Absorption and scattering coefficient dependence oflaser-Doppler flowmetry models for large tissuevolumes

T Binzoni1,2,4, T S Leung3, D Rufenacht2 and D T Delpy3

1 Departement de Neurosciences Fondamentales, Faculty of Medicine, University of Geneva,Switzerland2 Department de Radiologie et Informatique Medicale, University Hospital, Geneva, Switzerland3 Department of Medical Physics and Bioengineering, University College London, UK

E-mail: [email protected]

Received 18 May 2005, in final form 7 November 2005Published 4 January 2006Online at stacks.iop.org/PMB/51/311

AbstractBased on quasi-elastic scattering theory (and random walk on a latticeapproach), a model of laser-Doppler flowmetry (LDF) has been derivedwhich can be applied to measurements in large tissue volumes (e.g. whenthe interoptode distance is >30 mm). The model holds for a semi-infinitemedium and takes into account the transport-corrected scattering coefficientand the absorption coefficient of the tissue, and the scattering coefficientof the red blood cells. The model holds for anisotropic scattering and formultiple scattering of the photons by the moving scatterers of finite size. Inparticular, it has also been possible to take into account the simultaneouspresence of both Brownian and pure translational movements. An analyticaland simplified version of the model has also been derived and its validityinvestigated, for the case of measurements in human skeletal muscle tissue.It is shown that at large optode spacing it is possible to use the simplifiedmodel, taking into account only a ‘mean’ light pathlength, to predict the bloodflow related parameters. It is also demonstrated that the ‘classical’ bloodvolume parameter, derived from LDF instruments, may not represent the actualblood volume variations when the investigated tissue volume is large. Thesimplified model does not need knowledge of the tissue optical parametersand thus should allow the development of very simple and cost-effective LDFhardware.

4 Address for correspondence: Departement de Neurosciences Fondamentales, Centre Medical Universitaire, 1,rue Michel-Servet, 1211 Geneva 4, Switzerland.

0031-9155/06/020311+23$30.00 © 2006 IOP Publishing Ltd Printed in the UK 311

312 T Binzoni et al

1. Introduction

Laser-Doppler flowmetry (LDF) has been used to non-invasively evaluate tissue blood flowand/or speed for nearly 35 years (for a review see e.g. Leahy et al (1999) and Briers (2001)).Blood flow related parameters are obtained from the normalized autocorrelation function orfrom the power spectrum of the detector’s photoelectric output current. The majority ofcommercially available LDF use the power spectrum approach (Leahy et al 1999), while newprototypes developed in research laboratories often also use the autocorrelation function. Bothapproaches have allowed the development of laser-Doppler 2D and 3D imagers (Briers 2001,Durduran 2004). LDF images can be intuitively seen as a set of one-point measurements andthus the heart of the underlying theory is always based on the autocorrelation function or onanalysis of the power spectrum.

Theoretically, thanks to the Wiener-Khintchine theorem, the normalized autocorrelationfunction and the power spectrum approaches are completely equivalent. This means that,theoretically, one can see an underlying unity to the various different approaches that havehistorically appeared under different names such as light beating spectroscopy, intensityfluctuation spectroscopy, photon correlation spectroscopy (Cummins and Swinney 1970,Cummins and Pike 1974, 1977), diffuse wave spectroscopy, diffuse correlation spectroscopy(Pine et al 1988, Durduran 2004) or laser-Doppler flowmetry (Bonner and Nossal 1981,Shepherd and Oberg 1990), etc. However, with the recent development of experimental LDFinstruments working at large interoptode spacing (>∼20 mm) and/or at high haemoglobinconcentrations (Boas et al 1995, Lohwasser and Soelkner 1999, Kienle 2001, Kolkman et al2001, Binzoni et al 2002, 2003, 2004, Durduran 2004), the choice of the method becomescritical. In fact, when the interoptode spacing is increased, the power spectrum becomes ‘flat’because the energy is distributed over a large frequency range (Shepherd and Oberg 1990).An increase in red blood cells’ speed also has the tendency to flatten the power spectrum.Thus, even with sufficient laser power, at ∼30 mm spacing and at high blood flow (e.g.during postischaemic hyperaemia in a skeletal muscle) one may have the impression thatthere is no signal because the power spectrum is ‘drowned’ in the background noise. Thismeans that the standard approach of estimating the blood flow through the computation of themoments of the power spectra (Bonner and Nossal 1981) may become unusable. Moreover, atlarge optode spacings or at high tissue blood concentration, the appearance of non-linearitiesmakes the interpretation of the moments in terms of blood flow more difficult (Binzoni et al2004). To overcome in part these problems, we have previously proposed to directly ‘fit’ thepower spectrum with an analytical model (Binzoni et al 2003, 2004). This approach allowsone to obtain absolute values for the red blood cells’ speed and also to work at interoptodespacing of ∼15 mm. Unfortunately, even in this case, the method fails when the interoptodespacing is increased to ∼30 mm (e.g. the minimum reasonable distance enabling one toinvestigate the human brain through skull), because the fitting procedure becomes unstable.Moreover, the analytical model describing the power spectrum is represented by a very complexinfinite sum of terms (Binzoni et al 2004) and even if one has a good signal-to-noise ratio, atspacings >30 mm an unreasonably large number of terms of the series have to be taken intoaccount to obtain a sufficient precision, and in practice this method becomes impossible toapply.

Fortunately, the mathematical analogue of the Heisenberg uncertainty principle tells usthat if the power spectrum is poorly localized in the frequency domain (i.e. ‘flat’, and thushidden by noise), then the normalized autocorrelation function is well localized in the timedomain and is theoretically more easily detectable. In this case, the normalized autocorrelationfunction approach appears to be a better strategy when investigating large tissue volumes

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 313

(Durduran et al 2004). Thus, the choice of the power spectrum or of the autocorrelationfunction approach depends on the specific application.

Given the increasing interest in making experimental measurements in large tissuevolumes, in the present paper we derive an exact analytical expression for the normalizedautocorrelation function. We show that contrary to its power spectrum counterpart, theproposed mathematical model can be expressed by a finite number of mathematical terms. Themodel specifically takes into account both the Brownian and the net translational componentsof the red blood cells’ speed (Zhong et al 1998, Binzoni et al 2004). In particular, themodel considers that the moving particles have finite size, i.e. they are not considered to bepoint-like scatterers such as in other approaches, e.g., correlation diffusion (Dougherty et al1994). However, if the size of the moving particles is reduced to zero (point-like) thenthe expressions derived in the model simplify to reproduce the classical results foundusing correlation diffusion. Moreover, we also explicitly analyse the dependence of theautocorrelation function on the many different pathways covered by the light as it travelsthrough the tissue (this problem being negligible at small spacing ∼1 mm; Nossal et al 1989).More specifically, we show that there are a range of optical parameters, where even at largeoptode spacing and/or at high haemoglobin concentrations, a model taking into account onlyone ‘mean pathlength’ for the light is sufficient for the derivation of the flow-related parameters.It is hoped that the present theoretical work will be useful in the future implementation of newLDF instruments based on the autocorrelation function approach.

2. The mathematical theory

In this section we develop the analytical model for the normalized autocorrelation function.The model will hold for multiple scattering of the photons with the moving particles (i.e.,in our case red blood cells) and in particular it takes into account both the Brownian andthe translational components of the particles’ movement as well as their concentration. Theinfluence of the simultaneous presence of different light pathlengths is also introduced in theanalysis. The pathlengths for photon migration in the investigated tissue are obtained froma random-walk photon migration model based on a discrete time lattice. The validity of thisapproach for large optode spacing has already been demonstrated elsewhere (Bonner et al1987, Weiss et al 1998). The random-walk approach can also implicitly take into accountthe presence of anisotropic scattering (Gandjbakhche et al 1992, 1993) and has the furtheradvantage that it can easily be introduced into the laser-Doppler theory. The derivation of thepresent model is based on well-known hypothesis already discussed elsewhere (Bonner andNossal 1981, Nossal et al 1989, Binzoni et al 2004).

2.1. General expression for the normalized autocorrelation function

The normalized temporal autocorrelation function, g(2)(τ, r) (where τ is the correlation delaytime), describing the interaction between the photons generated by the laser of the LDF systemwith the biological tissue and its vascular network can be written as (Bonner and Nossal 1981,Boas 1996)

g(2)(τ, r) = 1 + β(io − isc)

2 + 2(io − isc)isc|I (τ, r)| + i2sc|I (τ, r)|2

i2o

, (1)

where 0 < β < 1 is an instrumental factor which depends upon the optical coherence ofthe signal at the detector surface (Cummins and Swinney 1970). The total light intensityfalling upon the photodetector gives rise to a current (dc) which is designated as io and the

314 T Binzoni et al

intensity of that portion which arises from photons that have interacted (scattered) with movingcells is designated as isc = io(1 − P0(r)) (Shepherd and Oberg 1990), where Pm(r) is theprobability that a photon makes m collisions (m = 0, 1, 2, . . .) with moving erythrocytesbefore emerging from the tissue at a distance r (mm) from the laser source. I (τ, r) is thenormalized intermediate scattering function of the Doppler shifted light and may be expressedas (Shepherd and Oberg 1990)

I (τ, r) =∞∑

m=1

Pm(r)[I1(τ )]m

1 − P0(r), (2)

where I1(τ ) is the contribution of the intermediate scattering function from photons thatexperience only one collision (Bonner and Nossal 1981) and will be derived below. This meansthat I (τ, r), holding for multiple scattering, can be expressed in terms of single scatteringevents as demonstrated by Shepherd and Oberg (1990). The function I1(τ ) contains all theinformation concerning the red blood cells’ speed and does not depend on the macroscopic‘geometry’ of the investigated tissue (e.g. semi-infinite medium, finite slab, etc). From this,by substituting for isc, g

(2)(τ, r) can be written as

g(2)(τ, r) = 1 + β(P0(r) + (1 − P0(r))I (τ, r))2. (3)

It is now necessary to express P0(r) as a function of the tissue optical parameters and theprinciples for this derivation, based on a random-walk approach, may be found in Nossal et al(1989). In summary,

Pm(r) =∑

n

p(m|n)P(n, r), (4)

where p(m|n) is the conditional probability that a photon interacts m times with a movingerythrocyte, given that it experiences n scattering events in total before being detected, i.e.

p(m|n) = (m(n))m e−m(n)

m!, (5)

where m(n) is the ‘lambda factor’ of the Poisson distribution and depends on n. The remainingfunction P(n, r) in equation (4) is the probability that a photon makes n collisions before itreaches the detecting optode at a distance r from the laser source on the tissue surface and isa complex function of the tissue absorption and scattering coefficients (see e.g. Bonner andNossal 1990). The function P(n, r) also depends on the geometry of the investigated tissueand in the present paper we will consider the case of a semi-infinite medium. By substitutingequations (2) and (4) in equation (3) with the explicit formulation for p(m|n),P(n, r) andI1(τ ), it is possible to derive g(2)(τ, r). This is the aim of the following sections.

2.2. The parameter m(n)

The definition of the parameter m(n) is an important issue in the construction of the LDF modelbecause it allows one to obtain a more or less complex formulation for I (τ, r). Moreover,m(n) depends upon the red blood cell concentration and thus a change in m(n) may (for a givenspeed of the moving particles) mean a variation in the blood flow (see following sections).It must be noted that in their original work Bossal and Nossal (see e.g. Shepherd and Oberg1990) have considered m(n) to be proportional to n, i.e. m(n) ≡ κn, where κ is a constant.This assumption can be derived theoretically for instance by observing that in a random walkthe probability of having at least one collision with a moving particle along a path of length Lmay be expressed as 1−e−µs,rbcL, where µs,rbc (mm−1) is the macroscopic scattering coefficientfor the red blood cells alone (i.e. taking into account the influence on scattering of all the red

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 315

blood cells in the investigated tissue volume but not the influence of the background tissue).Now, in the random-walk theory, the distance L can typically be expressed in terms of unitmean steps as

L = µ′−1s n, (6)

where µ′s (mm−1) is the macroscopic transport scattering coefficient of the tissue (i.e.

scattering from all scatterers inside the tissue, including the moving particle). Thus, byusing equation (5) we can now write 1 − e−µs,rbcL in an equivalent way as

∞∑m−1

p(m|n) = 1 − e−µs,rbcL (7)

(note that the index m in the sum starts from 1). In fact, the right- and left-hand sides ofequation (7) both express the same probability of having at least one collision with a movingparticle along a path of length L. By substituting from equation (6) and after some manipulation,one effectively finds

m(n) = µs,rbc

µ′s

n ≡ κn. (8)

The important point here is that µs,rbc (and thus κ) depends on the red blood cell concentration(per unit tissue), a parameter of physiological interest. In fact, it has been shown that one canreasonably express µs,rbc for a biological tissue as (Twersky 1970, Hammer et al 1998)

µs,rbc = H(1 − H)

Vrbcσs,rbc, (9)

where H is the volume fraction of the red blood cells in the tissue and σs,rbc is the single redblood cell scattering cross section. Thus, equations (8) and (9) show that a change in the tissueblood volume induces a variation in the κ value.

If H is small, then equation (9) can be approximated as

µs,rbc ≈ H

Vrbcσs,rbc = 1

Vrbc

nrbcVrbc

Vσs,rbc = nrbc

Vσs,rbc, (10)

where V is the total investigated tissue volume and nrbc is the number of erythrocytes containedin V. In this case, equation (8) gives

κ ≈ nrbcσs,rbc

µ′sV

, (11)

and this corresponds to the classical relationship proposed by Bonner and Nossal (Shepherdand Oberg 1990, Nossal et al 1989). In this case, the parameter κ varies linearly as a functionof the number of the red blood cells, nrbc (for a given V ), or of their number concentration,nrbcV

. It is clear that for the exact expressions in equation (9) this rule is not valid.

2.3. Brownian and translational movements

As highlighted in the above sections, all the information concerning the red blood cells’speed is contained in the function I1(τ ) (see equation (2)). In the present work, we willconsider the specific case where the red blood cells have both a Brownian,

⟨V 2

Brown

⟩, and a pure

(global) translational speed component, Vtrans. More precisely, the term⟨V 2

Brown

⟩represents

the second moment of the speed distribution function in the absence of bulk translationalmovement (Binzoni et al 2004). The Brownian component of the speed distribution functionis represented in the present case by a normal distribution because it suitably describes abiological tissue (Bonner and Nossal 1981, Cheung et al 2001, Durduran 2004). This means

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that 〈VBrown〉 (mean speed, different from 〈V 2Brown〉) is nil by definition. The Vtrans component

explains biologically the real input–output blood flow through the tissue, i.e. going from thearterial to the venous side. When no net flow is present, e.g. as a first approximation duringarterial occlusion, we have only a random movement of the red blood cells but with no netdisplacement. It must be noted that the present model is more general than that usually utilizedin LDF (Bonner and Nossal 1981). However, the latter model can be easily retrieved again ifnecessary, just by considering Vtrans = 0.

In a previous paper (Binzoni et al 2004), we have demonstrated that the function I1(τ )

holding for the conditions described above can be written as

I1(τ ) = 1

π

12ξa2

12ξa2 +⟨V 2

Brown

⟩τ 2

∫ π

0exp

(−3

2

V 2trans sin(θ ′)2τ 2

12ξa2 +⟨V 2

Brown

⟩τ 2

)dθ ′, (12)

where ξ = 0.1 in our case and a is the ‘mean radius’ of the red blood cell (e.g. a ≈ 2.75 µm).In summary, the parameter ξ comes from the fact that the structure factor of the consideredmoving scatterers (the red blood cells) can be expressed as exp(−2ξ (Qa)2) (Van de Hulst1959, Ishimaru 1978, Bonner and Nossal 1981), where Q is the Bragg scattering vectorusually appearing in the laser-Doppler theory. The parameter θ ′ is the angle existing betweenthe wave vector of the light coming out from the investigated tissue and Vtrans. By developingequation (12) as a power series for Vtrans, one gets

I1(τ ) = 12ξa2∞∑

N=0

(−1)N(

32

)N (2NN

)τ 2N

4N�(N + 1)(12ξa2 +

⟨V 2

Brown

⟩τ 2

)N+1 V 2Ntrans, (13)

where � is the gamma function and the terms(

2NN

)are the binomial coefficients. By taking

the limit of the sum in equation (13), one finally obtains

I1(τ ) =12ξa2IBessel

0

(3 V 2

transτ2

4(

12ξa2+〈V 2Brown〉τ 2

))exp

(− 3 V 2

transτ2

4(

12ξa2+⟨V 2

Brown

⟩τ 2

))12ξa2 +

⟨V 2

Brown

⟩τ 2

, (14)

where IBessel0 (x) is in this case the modified Bessel function of the first kind.

In the case where Vtrans = 0, then

I1(τ )|Vtrans=0 = 12ξa2

12ξa2 +⟨V 2

Brown

⟩τ 2

, (15)

which is the classical Bonner and Nossal (1981) model for I1(τ ).

2.4. Function g(2)(τ, r) for a semi-infinite medium with Brownian andtranslational movements

To obtain the normalized autocorrelation function g(2)(τ, r) appearing in equation (3), itis necessary to know the functions P0(r) and I (τ, r). These functions can be derived byexploiting the mathematical results originally proposed by Bonner and Nossal (Shepherdand Oberg 1990) and Binzoni et al (2004). In this case, by substituting equation (8) inequation (5),

p(m|n) = (κn)m e−(κn)

m!. (16)

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 317

The probability distribution function P(n, r), appearing in equation (4), for a semi-infinitemedium and expressed in terms of the usual dimensionless random-walk theory parameters(where for practical reasons we force the notation to be P(n, r) ≡ P(n, ρ)) is

P(n, ρ) =√

32πn3

(1 − exp

(− 6n

))(exp

(− 3ρ2

2n− µn

))ρ√

ρ2 + 4√ρ2 + 4 exp(−ρ

√6µ) − ρ exp

(−√ρ2 + 4

√6µ

) , (17)

where the correspondence with the real physical parameters is given by

ρ = rµ′s√2

, n = µ′sct, µ = µa

µ′s

. (18)

The parameter µa is the macroscopic absorption coefficient of the tissue, c is the speed of lightin the tissue and t is the time. By substituting equations (16) and (17) in equation (4) and byapproximating the sum by an integral over n (Bonner et al 1987),

P0(ρ) ≈√

ρ2 + 4 exp(−ρ

√6(κ + µ)

)+ ρ exp

(−√ρ2 + 4

√6(κ + µ)

)√

ρ2 + 4 exp(−ρ√

6µ) − ρ exp(−√

ρ2 + 4√

6µ) , (19)

where again one uses the notation P0(r) ≡ P0(ρ).Thus, the function I (τ, r) ≡ I (τ, ρ) can be derived from equations (2), (4), (17) and (19)

giving the expression

I (τ, ρ) = N−1

exp(−ρ

√6µ

√1 + κ

µ(1 − I1(τ ))

) − exp(−ρ

√6µ

√1 + κ

µ

)ρ

−exp

(−√ρ2 + 4

√6µ

√1 + κ

µ(1 − I1(τ ))

) − exp(−√

ρ2 + 4√

6µ√

1 + κµ

)√

ρ2 + 4

,

(20)

where

N =

exp(−ρ√

6µ) − exp(−ρ

√6µ

√1 + κ

µ

)ρ

−exp(−

√ρ2 + 4

√6µ) − exp

(−√ρ2 + 4

√6µ

√1 + κ

µ

)√

ρ2 + 4

. (21)

By substituting equations (19) and (20) in equation (3), it is possible to explicitly computeg(2)(τ, r) ≡ g(2)(τ, ρ) for any optical parameter value, i.e.

g(2)(τ, ρ)

= 1 + β

(√ρ2 + 4 exp(−ρ

√6(µ + κ(1 −I1(τ ))))−ρ exp(−

√ρ2 + 4

√6(µ + κ(1 −I1(τ ))))√

ρ2 + 4 exp(−ρ√

6µ) − ρ exp(−√

ρ2 + 4√

6µ)

)2

,

(22)

where I1(τ ) is given by equation (14). To our knowledge, equation (22) has not been derivedbefore for random-walk theory and for I1(τ ) taking into account both Brownian, translationalmovements and multiple scattering including components relating to higher order scatteringprocesses, i.e. those which involve multiple convoluted spectral shifts. It must be noted thatequation (22) holds for multiple scattering even if in equation (22) one can only see the term

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I1(τ ), i.e. the intermediate scattering function for photons that experience only one collision.This result is explained by the fact that equation (22) allows one to express I (τ, ρ) in termsof I1(τ ). Equation (22) holds for a semi-infinite medium and can be expressed in ordinaryphysical parameters by substituting ρ and µ from equation (18). In fitting real LDF datato equation (22), the fitted parameters would be Vtrans,

⟨V 2

Brown

⟩(contained in I1(τ )) and κ .

However, equation (22) is probably not very practical for experimental applications, becauseit implicitly contains too many unknown parameters (see section 5). For this reason, we derivebelow a simplified version of the model.

2.5. The simplified autocorrelation function g(2)(τ, r)

The previous result for g(2)(τ, r) is very general and takes into account the most importantfeatures of our biological system, such as the red blood cell concentration, the absorptionand scattering coefficients, etc. Unfortunately, this model cannot be easily applied to ‘fit’the experimental data obtained from an LDF instrument, and thus to evaluate the absolutespeed and flow values (Binzoni et al 2003), because there are too many unknowns to estimate(see section 5). For this reason, LDF instruments (whether working in the time or frequencydomain) often implement simplified models which consider that the light covers only onepossible ‘mean’ pathlength in the tissue (Nossal et al 1989). This approach greatly simplifiesthe calculations and in particular it allows one to obtain an analytical expression that is easy toimplement in the LDF instrument. For this reason, in this section we will derive a ‘simplified’g(2)(τ, r), by taking into account only one ‘mean’ pathlength. Following this, in the subsequentsections we try to understand the physical meaning of the parameters of this simplified modelby comparing it with the ‘exact’ solution, g(2)(τ, r), presented in the previous section, the aimbeing to see if the simplified model can predict the main variables (e.g. blood speed) containedin g(2)(τ, r).

A simplified model, g(2)(τ, r), can be derived (Shepherd and Oberg 1990) by redefiningthe probability distribution function P(n, r) appearing in equation (4). This is done byassuming that P(n, r) is sharply peaked around a ‘mean’ 〈n〉r , i.e. all the photon paths areapproximately of equal length. Of course, 〈n〉r depends on r (the index r is used to denotethis) because the larger the interoptode distance, the greater must be the ‘mean’ number ofpossible photon collisions. Thus, a new approximated probability function analogue to Pm(r)may be defined as (Bonner and Nossal 1981)

Pm(r) ≈ P m(r) = 〈m〉mr e−〈m〉r

m!, (23)

where the bar over P m is to denote that this is a simplified model over a single mean photonpathlength. It must be noted that this approximation (equation (23)) was originally defined(Bonner and Nossal 1981) for very short interoptode spacings and that we do not know forthe moment if it also holds for large interoptode spacing. The validity of this ‘approximation’will be discussed in the following sections and is one of the aims of the present work. Bydefinition, the function P m allows one to estimate the mean number of collisions of a photonwith a moving particle along a unique ‘mean’ path as

∞∑m=0

mP m(r) = 〈m〉r . (24)

Equation (23) represents the central assumption allowing one to define the simplified modeland with this definition the general equation (2) becomes

I (τ, r) ≈ I (τ, r) = exp(−〈m〉r (1 − I1(τ ))) − e−〈m〉r

1 − e−〈m〉r , (25)

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 319

and thus

g(2)(τ, r) ≈ g(2)(τ, r) = 1 + β(P 0(r) + (1 − P 0(r))I (τ, r))2. (26)

By substituting equations (23) and (25) in equation (26),

g(2)(τ, r) = 1 + β exp(−2〈m〉r (1 − I1(τ ))) (27)

and by utilizing the definition for I1(τ ) (equation (14)),

g(2)(τ, r) = 1 + β

× exp

−2〈m〉r

12ξa2+〈V 2Brown〉τ 2−12ξa2IBessel

0

3 V 2

transτ2

4

(12ξa2+〈V 2

Brown〉τ2)

exp

− 3 V 2

transτ2

4

(12ξa2+〈V 2

Brown〉τ2)

12ξa2+〈V 2

Brown〉τ 2

.

(28)

Equation (28) represents the ‘simplified’ model and is the heart of the present work. If onewants to consider the special Vtrans = 0 case, then

g(2)(τ, r)|Vtrans=0 = 1 + β exp

(−2〈m〉r

⟨V 2

Brown

⟩τ 2

12ξa2 +⟨V 2

Brown

⟩τ 2

). (29)

It must be noted that equation (29) represents the classical model (expressed here in the timedomain) supposed to hold for the majority of existing LDF instruments (and LDF imagers)that use the zeroth and first moments of the power spectrum to estimate blood flow (Bonnerand Nossal 1981). In the classical formulation, the parameter 〈m〉r (found by computing thezeroth moment of the power spectrum) is usually considered to be proportional to the ‘bloodvolume’.

3. Methods

3.1. Probing the validity of the simplified model g(2)(τ, r)

At this point, an interesting issue is to investigate if the simplified model g(2)(τ, r) still givesthe actual values for

⟨V 2

Brown

⟩1/2and Vtrans if fitted to LDF data. In particular, it would also be

important to see what, at large optode spacing, is the physical/physiological meaning of theparameter 〈m〉r . In fact, it is not trivial to intuitively foresee if the estimated values

⟨V 2

Brown

⟩1/2,

Vtrans and 〈 ˆm〉r will always have any physiological meaning (here the ‘hat’, , over the variablesis used to denote the values estimated using the fitting procedure) and if for example theestimated

⟨V 2

Brown

⟩1/2and Vtrans values will follow the true

⟨V 2

Brown

⟩1/2and Vtrans values. To this

end, the exact model for g(2)(τ, r) is extremely useful because it can be utilized as a probeallowing one to investigate the parameters appearing in the simplified model g(2)(τ, r).

In practice, this test has been realized by randomly generating 30 000 datasets containing arange of

{µ′

s, µa, µs,rbc, r,⟨V 2

Brown

⟩1/2, Vtrans

}values (see later). The values of each individual

dataset were then fed into the function g(2)(τ, r) and an equivalent number of syntheticexperimental curves were generated. The simplified model g(2)(τ, r) was then utilized to ‘fit’the synthetic curves (g(2)(τ, r)) and the estimated parameters 〈 ˆm〉r ,

⟨V 2

Brown

⟩1/2and Vtrans were

compared to the exact values, e.g.⟨V 2

Brown

⟩1/2and Vtrans. The fitting has been performed using

a non-linear least-square algorithm (Levenberg–Marquardt) with the prior knowledge that thefitting parameters are positive numbers.

320 T Binzoni et al

For the simulation, we have chosen to cover a range of parameter values correspondingin the present case to typical human skeletal muscle. The range of tissue parameters quotedin the literature (e.g. Zaccanti et al 1995, Torricelli et al 2004) shows a large variation whichdepends on the tissue’s physiological status, the wavelength used, etc. For this reason, a widerange of values have been covered in the simulation. The parameters values were uniformlygenerated in the following ranges: µ′

s ∈ [0.4, 1.2] mm−1 (where [·, ·] is a closed interval), µa ∈[1 × 10−2, 5 × 10−2] mm−1, µs,rbc ∈ [1, 500] mm−1, r ∈ [5, 60] mm, Vtrans ∈ [0, 15] mm s−1,⟨V 2

Brown

⟩1/2 ∈ [0, 5] mm s−1, β = 1, a = 2.75 × 10−3 mm and ξ = 0.1. The timescale τ was formed by 26 logarithmically equally spaced time points in the interval[10−7, 10−1] s. The 30 000 generated g(2)(τ, r) functions were then fitted using the simplifiedmodel, g(2)(τ, r). The estimated parameters were 〈 ˆm〉r ,

⟨V 2

Brown

⟩1/2and Vtrans.

3.2. Mean number of collisions of a photon with moving red blood cells

If one considers the simplified model g(2)(τ, r), one intriguing question to ask is what isthe relationship between the parameter 〈m〉r and the ‘actual’ mean number of collisions of aphoton with a moving particle along the mean path n computed using the exact model (in thiscontext, corresponding to the real ‘experimental value’). In practice, we would like to see ifthe estimated value 〈 ˆm〉r can be approximated as

〈 ˆm〉r ↔ m(n), (30)

where by definition, thanks to equation (17), the exact value for m(n) is

m(n) = κn = κ∑

n

nP(n, r). (31)

In this case, if 〈 ˆm〉r is a good estimation of m(n) then the two parameters must be at least linearlyrelated. It must be noted that the ‘blood flow’ is classically considered to be proportional to〈m〉rVtrans, and this is the reason why 〈 ˆm〉r Vtrans and m(n)Vtrans have also been compared (seesection 5).

4. Results

Figure 1 shows the estimated 〈 ˆm〉r ,⟨V 2

Brown

⟩1/2, Vtrans and 〈 ˆm〉r Vtrans parameters obtained by

fitting the simplified model g(2)(τ, r) (equations (14) and (28)) to the synthetic data pointsgenerated numerically with the ‘exact’ model g(2)(τ, r) (equations (8), (14), (18) and (22)).This study has been undertaken for relatively large optode spacings (i.e. r takes values between5 and 60 mm). Each point in the figure corresponds to a unique

{µ′

s, µa, µs,rbc, r,⟨V 2

Brown

⟩1/2,

Vtrans}

dataset. It is clear that⟨V 2

Brown

⟩1/2and Vtrans are very poorly estimated by the simplified

model while the ‘blood volume’ and ‘flow’ represented by 〈 ˆm〉r and 〈 ˆm〉r Vtrans seem to betterreproduce the theoretical values m(n) and m(n)Vtrans, especially for higher values.

Figure 2 shows the same data as in figure 1, but only the points having µs,rbc ∈ [10,

60] mm−1 and r ∈ [40, 41] mm are presented. The interesting observation in this case isthat the estimated values predict fairly well the actual parameters m(n),

⟨V 2

Brown

⟩1/2, Vtrans

and m(n)Vtrans. The mean percentage errors in the estimation of m(n),⟨V 2

Brown

⟩1/2, Vtrans and

m(n)Vtrans (i.e. 100(Vtrans − Vtrans)/Vtrans) are −0.08 ± 0.18%, −0.56 ± 1.99%, −1.60 ±1.16% and −1.68 ± 1.1%, respectively. The quality of the correlations are emphasized bythe R-square statistics appearing in each panel. More specifically, these results confirm thehypothesis that 〈 ˆm〉r ≈ m(n) (for µs,rbc ∈ [10, 60] mm−1 and r ∈ [40, 41] mm). The points

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 321

0 5 10 150

5

10

15

20

Vtr

ans

Vtrans

1 2 3 4 50

2

4

6

8

10

<V

Bro

wn

2>

1/2

<VBrown2 >1/2

0 0.5 1 1.5 2 2.5

x 105

0

0.5

1

1.5

2

2.5x 10

5

m(n)

<m

>r

0 1 2 3

x 106

0

1

2

3x 10

6

m(n) Vtrans

<m

>r V

tran

s

_

_^

^

^^

_ _^

_ _

Figure 1. The parameters 〈m〉r , 〈V 2Brown〉1/2 and Vtrans are estimated with the simplified model

g(2)(τ, r) while m(r), 〈V 2Brown〉1/2 and Vtrans are the actual values (see section 3).

0 5 10 150

5

10

15

R2 stat.: 1.0000

Vtr

ans

Vtrans

1 2 3 4 51

2

3

4

5

R2 stat.: 0.9995

<V

Bro

wn

2>

1/2

<VBrown2 >1/2

0 0.5 1 1.5 2

x 104

0

0.5

1

1.5

2x 10

4

R2 stat.: 1.0000

m(n)

<m

>r

0 0.5 1 1.5 2

x 105

0

0.5

1

1.5

2x 10

5

R2 stat.: 0.9999

m(n) Vtrans

<m

>r V

tran

s

_

_^

^

^^

_ _

^

__

Figure 2. As in figure 1 but using only the points having µs,rbc and r in the intervals [10, 60] mm−1

and [40, 41] mm, respectively. The dashed line is the identity line. R2 is the R-square statistics.

for µs,rbc values falling in the interval [1, 10) ∪ (60, 500] mm−1 may of course be representedby deleting the points appearing in figure 2 from figure 1 (for each panel).

322 T Binzoni et al

0.4 0.6 0.8 1 1.2−30

−20

−10

0

10

20

30

µs’ (mm−1)

% e

rror

(V

tran

s)

0.4 0.6 0.8 1 1.2−30

−20

−10

0

10

20

30

µs’ (mm−1)

% e

rror

(<

VB

row

n2

>1/

2 )0.4 0.6 0.8 1 1.2

−30

−20

−10

0

10

20

30

µs’ (mm−1)

% e

rror

(m

(r))

0.4 0.6 0.8 1 1.2−30

−20

−10

0

10

20

30

µs’ (mm−1)

% e

rror

(m

(r)

Vtr

ans)

Figure 3. The points appearing in figure 1 have been divided into 20 groups (each of the fourpanels separately) by dividing first the interval µ′

s = [0.4, 1.2] mm−1 into 20 equally spacedsubintervals and then by choosing the corresponding points. The percentage error of the estimatedvalues compared to the exact theoretical value has then been computed. The circles, trianglesand bars correspond to the means, medians and standard deviations of the errors inside eachsubinterval, plotted as a function of µ′

s. This analysis has been performed for each pair of variablesVtrans–Vtrans, 〈V 2

Brown〉1/2–〈V 2Brown〉1/2, ˆm(r)–m(r) and ˆm(r)Vtrans–m(r)Vtrans with the results

shown in four separate panels.

From the latter result, it seems that a more restricted range of µs,rbc values and a preciser improve the quality of the predictions obtained with the simplified model (g(2)(τ, r)). Tobetter understand this observation, we have systematically investigated the influence of eachparameter µ′

s, r, µs,rbc and µa, independently, on the ‘goodness’ of the estimation of m(n),⟨V 2

Brown

⟩1/2, Vtrans and m(n)Vtrans when using the simplified model g(2)(τ, r). In essence, the

interval defined in figure 1 for µ′s (i.e. [0.4, 1.2] mm−1) was subdivided into 20 equally spaced

subintervals. Then, the Vtrans–Vtrans couples (i.e. a point in figure 1, first panel) having aµ′

s falling in the first subinterval were chosen. The percentage error of the estimated valuescompared to the exact theoretical value was then calculated (i.e., 100(Vtrans−Vtrans)/Vtrans). Themean, the median and the standard deviation of the obtained errors were then computed. Thisoperation was then repeated for each of the 20 subintervals. The resulting 20 means, mediansand standard deviations were finally plotted as a function of the mean µ′

s subinterval (figure 3,

first panel). The above operation for Vtrans–Vtrans was then repeated for⟨V 2

Brown

⟩1/2–⟨V 2

Brown

⟩1/2,

〈 ˆm〉r–m(n) and 〈 ˆm〉r Vtrans–m(n)Vtrans to obtain the remaining three panels in figure 3.Figures 4–6 represent results of the same procedure as used for figure 3 but this time to

study the dependency of the ‘goodness’ of the estimated values Vtrans,⟨V 2

Brown

⟩1/2, 〈 ˆm〉r and

〈 ˆm〉r Vtrans on the parameters r, µs,rbc and µa, respectively.Figure 7 shows a set of g(2)(τ, r) curves (from equation (28)) in the time-domain

representation and in the frequency-domain representation (‘the power spectrum’, S(2πν)).

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 323

0 20 40 60−30

−20

−10

0

10

20

30

r (mm)

% e

rror

(V

tran

s)

0 20 40 60−30

−20

−10

0

10

20

30

r (mm)

% e

rror

(<

VB

row

n2

>1/

2 )0 20 40 60

−30

−20

−10

0

10

20

30

r (mm)

% e

rror

(m

(r))

0 20 40 60−30

−20

−10

0

10

20

30

r (mm)%

err

or (

m(r

) V

tran

s)

Figure 4. The points appearing in figure 1 have been divided into 20 groups (each of the fourpanels separately) by dividing first the interval r = [30, 50] mm into 20 equally spaced subintervalsand then by choosing the corresponding points. The percentage error of the estimated valuescompared to the exact theoretical value has then been computed. The circles, triangles and barscorrespond to the means, medians and standard deviations of the errors inside each subinterval,plotted as a function of r. This analysis has been performed for each pair of variables Vtrans–Vtrans,〈V 2

Brown〉1/2–〈V 2Brown〉1/2, ˆm(r)–m(r) and ˆm(r)Vtrans–m(r)Vtrans with the results shown in the four

separate panels.

The power spectra, S(2πν), were computed numerically by using (Cummins and Swinney1970)

S(2πν) = 1

π

∫ +∞

−∞ei2πν(g(2)(τ, r) − 1) dτ . (32)

The parameters used to generate the eight g(2)(τ, r) curves are all the combinationsof the values 〈m〉r = {1200, 2500} (typical for r ≈ 20 mm−1),

⟨V 2

Brown

⟩1/2 = {2, 5} andVtrans = {0, 10}. As explained in the introduction, these two representations correspond tothe two possible main kinds of existing LDF hardwares (those directly generating g(2)(τ, r)

and those directly generating S(2πν)). It must be noted that in the S(2πν) representationthe dashed curve is found by using the minimum values for each 〈m〉r ,

⟨V 2

Brown

⟩1/2and Vtrans

parameter, whereas the dash-dotted curve is found using the maximum values.

5. Discussion and conclusions

5.1. General discussion

Based on the quasi-elastic scattering theory, the present work has allowed us to analyticallyderive an LDF model, g(2)(τ, r), holding for a semi-infinite medium and taking into account

324 T Binzoni et al

0 100 200 300 400 500−30

−20

−10

0

10

20

30

µs,rbc

(mm−1)

% e

rror

(V

tran

s)

0 100 200 300 400 500−30

−20

−10

0

10

20

30

µs,rbc

(mm−1)

% e

rror

(<

VB

row

n2

>1/

2 )0 100 200 300 400 500

−30

−20

−10

0

10

20

30

µs,rbc

(mm−1)

% e

rror

(m

(r))

0 100 200 300 400 500−30

−20

−10

0

10

20

30

µs,rbc

(mm−1)

% e

rror

(m

(r)

Vtr

ans)

Figure 5. The points appearing in figure 1 have been divided into 20 groups (each of the fourpanels separately) by dividing first the interval µs,rbc = [1, 500] mm−1 into 20 equally spacedsubintervals and then by choosing the corresponding points. The percentage error of the estimatedvalues compared to the exact theoretical value has then been computed. The circles, trianglesand bars correspond to the means, medians and standard deviations of the errors inside eachsubinterval, plotted as a function of µs,rbc. This analysis has been performed for each pairof variables Vtrans–Vtrans, 〈V 2

Brown〉1/2–〈V 2Brown〉1/2, ˆm(r)–m(r) and ˆm(r)Vtrans–m(r)Vtrans with the

results shown in the four separate panels.

the main optical parameters of the tissue, i.e., µ′s, µa and µs,rbc. By its construction, g(2)(τ, r)

holds for both anisotropic scattering and multiple scattering of the photons with movingparticles having a finite size. In particular, it has also been possible to introduce the conceptsof both Brownian and pure translational movements of the red blood cells, an approach that iscloser to physiological reality. However, it is always possible to obtain the ‘classical’ modeljust by putting Vtrans = 0 in g(2)(τ, r) and the present results will of course remain validbecause the theory and the tests hold for any Vtrans ∈ [0, 15].

The derivation of g(2)(τ, r) was performed with the particular purpose of applying thetheory of LDF measurements made at large interoptode spacings (e.g. >30 mm) and this is thereason why the optical parameters µ′

s, µa and µs,rbc have been taken into account. In reality, atlarge interoptode spacings, the number of possible ‘pathways’ taken by the photons inside thetissue before they reach the detector is enormously increased and a more precise descriptionof this phenomenon can be obtained only through the introduction of µ′

s, µa and µs,rbc. Itmust be noted that if one wants to apply the exact model for very short times, as in the case ofsmall interoptode spacings (e.g. <1 mm), then the present random-walk theory on a constantlattice is no longer valid and a continuous-time random-walk description becomes necessary.It has been previously demonstrated (Weiss et al 1998) that the theory applied in the presentwork is the limit of the continuous-time random-walk description for a large number of steps(e.g. for large optode spacings).

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 325

0.01 0.02 0.03 0.04 0.05−30

−20

−10

0

10

20

30

µa (mm−1)

% e

rror

(V

tran

s)

0.01 0.02 0.03 0.04 0.05−30

−20

−10

0

10

20

30

µa (mm−1)

% e

rror

(<

VB

row

n2

>1/

2 )0.01 0.02 0.03 0.04 0.05

−30

−20

−10

0

10

20

30

µa (mm−1)

% e

rror

(m

(r))

0.01 0.02 0.03 0.04 0.05−30

−20

−10

0

10

20

30

µa (mm−1)

% e

rror

(m

(r)

Vtr

ans)

Figure 6. The points appearing in figure 1 have been divided into 20 groups (each of the fourpanels separately) by dividing first the interval µa = [1 × 10−2, 5 × 10−2] mm−1 into 20 equallyspaced subintervals and then by choosing the corresponding points. The percentage error of theestimated values compared to the exact theoretical value has then been computed. The circles,triangles and bars correspond to the means, medians and standard deviations of the errors insideeach subinterval, plotted as a function of µa. This analysis has been performed for each pairof variables Vtrans–Vtrans, 〈V 2

Brown〉1/2–〈V 2Brown〉1/2, ˆm(r)–m(r) and ˆm(r)Vtrans–m(r)Vtrans with the

results shown in the four separate panels.

The Monte Carlo tests previously performed to validate the random-walk approach (e.g.Bonner et al 1987, Nossal et al 1989, Gandjbakhche et al 1992, 1993) remain valid becausethey concern only the diffusive part of the model. The inclusion of the Doppler frequency shiftsin the Monte Carlo simulation (Soelkner et al 1997) is not essential in the present context butwill certainly be a matter for further studies. The main point here is that we were obliged to usethe random-walk approach because the aim was to compare the ‘exact’ model g(2)(τ, r) withthe simplified model g(2)(τ, r) (equation (28)). To do this, it was necessary to generate a verylarge (30 000) number of g(2)(τ, r) datasets, followed by a fitting with g(2)(τ, r). This wouldhave been an impractical task to perform by Monte Carlo simulation. In this sense, random-walk theory represents a very powerful tool, giving the possibility to have a fast ‘opticallytunable’ synthetic tissue, allowing one to test various LDF algorithms as in the present case.

In summary, it has been demonstrated that the ‘simplified’ model g(2)(τ, r) can be used,in place of the exact model g(2)(τ, r), to assess the physiological parameters if the interoptodespacing is large and if the µs,rbc changes are not too large (see below). It has previously beenshown that, contrary to g(2)(τ, r), the simplified model g(2)(τ, r) can also be used for smallinteroptode spacings (i.e. the classical applications; Bonner and Nossal 1981, Bonner et al1987). However, Larsson et al (2003) have shown that there are some situations for smalloptode spacings where this may not be true. Actually, this can be explained by observing thatg(2)(τ, r) or g(2)(τ, r) are built using a random-walk approach that is equivalent to a diffusion

326 T Binzoni et al

0 5000 10000 150000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x 10−5

ν (Hz)

S(2

πν)

(s)

10−6

10−4

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

τ (s)

g(2

) (τ,r

)_

Figure 7. Simplified autocorrelation functions, g(2)(τ, r), as a function of time, τ , and thecorresponding power spectra S(2πν) (equation (32)) as a function of the frequency ν. Each curvewas obtained using different values for m(r), 〈V 2

Brown〉1/2 and Vtrans (see section 4). The dashedcurve represents the lowest values for each parameter, the dash-dotted curve the highest.

approximation of the photon transport in the tissues. It is well known that the diffusionapproximation model breaks down when the interoptode spacing is too small. So, in the futureit remains to be demonstrated whether g(2)(τ, r) also holds (or not) in the small optode spacingrange. This however will have to be done using a continuous-time random-walk descriptionor an equivalent approach.

It is noted that g(2)(τ, r) does not depend explicitly on the optical parameters µ′s, µa and

µs,rbc because they are now implicitly contained in the global fitting parameter 〈m〉r . Thismeans that an LDF instrumentation based on g(2)(τ, r) can be quite simple, even for applicationin large tissue volumes. In fact, the experimental assessment of µ′

s and µa (for example, thehybrid system described in Durduran (2004)) requires the use of sophisticated instrumentation(e.g. time of flight, frequency modulation, etc) which increases both the complexity and theprice of the hardware. These were the key reasons why the validity of g(2)(τ, r) has beenstudied over a large set of conditions even if we would have preferred to deal only with the‘exact’ model g(2)(τ, r). Of course, if one has access to a complex hardware allowing oneto obtain all the optical parameters, such as was the case for the work published by Yu et al(2005a), then it is possible to apply directly the ‘exact’ model g(2)(τ, r). The advantage of thisapproach is the access to absolute LDF-related values and thus the possibility to extract morephysiological information from the experimental LDF data and the extremely useful ability tocompare different subjects without the need to make percentage normalizations. The problemof the ‘biological zero’ would also be solved by separating the Brownian and the translationalcomponents. It must be noted that the present model holds for real moving particles withfinite size and is thus nearer to the biological reality. However, if one takes the limit for small

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 327

particle sizes (i.e. point-like), and by setting Vtrans = 0, one can obtain the results of the modelused by Yu et al (2005a) (see section 6.3).

5.2. The simplified model g(2)(τ, r)

The main result of this investigation is presented in figure 1, where one can see for a typicalhuman skeletal muscle, the predictions for 〈 ˆm〉r ,

⟨V 2

Brown

⟩1/2, Vtrans and 〈 ˆm〉r Vtrans, obtained

using the simplified model g(2)(τ, r) for all 30 000 datasets. This appears to show that theexpected values for the speed,

⟨V 2

Brown

⟩1/2and Vtrans, do not accurately reproduce the actual

values⟨V 2

Brown

⟩1/2and Vtrans. The results for 〈 ˆm〉r and 〈 ˆm〉r Vtrans are a little better, but the model

still does not have a sufficient precision to produce acceptable results for an LDF instrument(especially if one considers that real measurements ‘move’ only on a small part of for instancethe 〈 ˆm〉r values). However, one must realize that the results reported in figure 1 represent avery large range of optical and geometrical parameters. This means for example that figure 1represents simultaneously the data for any interoptode distance, r ∈ [5, 60], whereas inpractice during a real experiment one chooses only a fixed distance r.

For the above reasons, figure 2 presents only the points from figure 1 having r ∈ [40, 41]mm and µs,rbc ∈ [10, 60] mm−1. In this case, we obtain a very good correlation, alsodemonstrated by the R2 statistics, between the estimated and the actual parameter values.Under these conditions, the simplified g(2)(τ, r) model appears to be a very good predictorof the flow-related LDF variables. The choice of the simulation range µs,rbc ∈ [1, 500] wasmade to ensure we had taken into account all reasonable possibilities. However, this rangeprobably covers more than the physiologically possible values for the human skeletal muscle.In fact, a rough estimation of µs,rbc for a haematocrit ranging from 0.37 to 0.57, and a rangeof 5–10% for the blood content of the skeletal muscle (Todo et al 1986), gives the interval ofvalues µs,rbc ∈ [10, 60] utilized in figure 2.

Thus, figure 2 highlights the fact that under some conditions g(2)(τ, r) can be a goodmodel, but it is not clear from this figure what the actual limits of the optical parameters arefor which this observation remains true. Figures 3–6 try to address this problem.

From figures 3–6 one can clearly see that the best estimated parameter is ˆm(r), andthis is independent of the choice of any particular restricted range. After this, ˆm(r)Vtrans

(the ‘flow’) is the next parameter where the error is small, but in this case a large r value(figure 4) seems to improve the estimation. This is an important point because ˆm(r) (the‘blood volume’) and ˆm(r)Vtrans (the ‘flow’) are the parameters that one usually wants tomeasure. The remaining parameters have relatively large errors, and it appears that it isnecessary to have a simultaneous limited excursion for µs,rbc and r to reach a very low errorlevel (figure 2). This is in fact usually the experimental reality where r is fixed at one valueand µs,rbc does not change significantly (see above).

In practice, it must be noted that the haemoglobin concentration, µs,rbc, µa and µ′s are

intimately related. In fact, a change in haemoglobin concentration is known to change bothµs,rbc (see equation (9)) and µa, whereas a change in µa, e.g. due to a variation in oxygensaturation with constant haemoglobin concentration, does not influence µs,rbc but it willinfluence the pathlength (Klassen et al 2002, Torricelli et al 2004). The relationships existingbetween these parameters may be very complex and further investigations are probably neededto better define this problem. In the present simulations, the parameters µs,rbc, µa and µ′

s wereconsidered as independent and thus the obtained results also hold for the case where somesupplementary constraints are imposed on haemoglobin concentration, µs,rbc, µa and µ′

s. It islikely that these supplementary constraints will reduce the variability in figure 1 but this is amatter for future studies.

328 T Binzoni et al

To conclude this part of the discussion, it is important to stress again the fact that theterms 〈 ˆm〉r Vtrans ≈ m(n)Vtrans have been considered as proportional to the ‘blood flow’ (asis usually assumed in the ‘classical’ LDF literature). In practice, for this hypothesis to betrue, we require that m(n) (or 〈 ˆm〉r ) remain proportional to the blood volume even during theinvestigation of large tissue volumes. However, m(n) represents only the mean number ofphoton collisions with a moving particle, and this is not necessarily proportional to the bloodvolume. This can be seen by inspecting equations (17), (18) and (31) which clearly showthat m(n) depends not only on haemoglobin concentration in a non-linear fashion but alsoon the µs,rbc, µa and µ′

s, parameters that, as we saw above, are intimately interrelated. Thefact that m(n)Vtrans can always been interpreted as proportional to the ‘blood flow’, in all thephysiological situations, has yet to be demonstrated.

5.3. Comparison between the present theory and the correlation diffusion theory

It is instructive at this point to compare the result for g(2)(τ, r) found in the present workand an equivalent expression found using another approach, the correlation diffusion theory(e.g. Boas 1996). In fact, the two approaches are theoretically different but they mustof course at the end describe the same physics. For this purpose, we will analyse asimple case for a semi-infinite medium where Vtrans = 0. In this case, by using the usualSiegert g(2)(τ, r) = 1 + |g(1)(τ, r)|2 relation, the equivalent Boas’ expression (Boas 1996,equation (4.17)) for the autocorrelation function is

g(2)CD(τ, r) = 1 + β

(exp

(− r1l∗

√k2

o〈�r2(τ )〉 + 3µal∗) − exp

(− r2l∗

√k2

o〈�r2(τ )〉 + 3µal∗)

exp(− r1

l∗√

3µal∗) − exp

(− r2l∗

√3µal∗

))2

,

(33)

where ko = 2πnr/λ, λ is the laser wavelength and nr is the index of refraction of the medium.Using the same notation as in the present work, r1 =

√r2 + l∗2, r2 =

√r2 + (l∗ + 2zb)2

and zb = −2/(3µ′s). The speed term appears as 〈�r2(τ )〉 = ⟨

V 2Brown

⟩τ 2. Now, to facilitate

the comparison of g(2)CD(τ, r) with g(2)(τ, r) (equation (22)), we write equation (33) using the

random-walk dimensionless parameters, ρ and µ (equation (18)), and obtain

g(2)CD(τ, r)

= 1 + β

exp

(−(ρ2+

(1√2

)2) 12√

6(µ+

13 k2

o

⟨V 2

Brown

⟩τ 2

))−exp

(−(ρ2+

(1

3√

2

)2) 12√

6(µ+

13 k2

o

⟨V 2

Brown

⟩τ 2

))exp

(−(ρ2+

(1√2

)2) 12 √

6µ

)−exp

(−(ρ2+

(1

3√

2

)2) 12 √

6µ

)

2

.

(34)

(It must be noted that in Boas (1996) the term⟨V 2

Brown

⟩is not called the Brownian but rather

the ‘random’ motion; but this is only a matter of definition (mathematicians often call it aWiener process).) From this result, one can see that g

(2)CD(τ, r) (equation (34)) and g(2)(τ, r)

(equation (22)) are really very similar and this is what we would expect. In fact, theterms containing ρ in equation (34) have their equivalent in equation (22), e.g. ρ2 +

(1

3√

2

)2

corresponds to ρ2 + 1, etc. These ρ-terms comes from the extrapolated zero-boundaryconditions defined when solving the differential diffusion equation for a semi-infinite mediumand they have their equivalent in the present random-walk approach. The fact that in equation(34) they do not appear in front of the exponential, whereas in equation (22) they do, is alsodue to this choice. The important points for the model now are the terms depending on

⟨V 2

Brown

⟩appearing in the exponentials (equation (34)). In fact, to describe the same physical system,

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 329

these terms must be equal in both g(2)CD(τ, r) and g(2)(τ, r). This means that in this case by

definition one must have the condition13k2

o

⟨V 2

Brown

⟩τ 2 ≈ κ(1 − I1(τ )). (35)

For this discussion, it is important now to highlight the fact that g(2)CD(τ, r) (equation (34))

is based on a theory (see e.g. for a general explanation Dougherty et al (1994)) where thescatterers are treated as a point-like particle, i.e. of infinitely small size. In the presentwork, the scatterers are of finite size, a. Thus, to find the original Boas’ term 1

3k2o

⟨V 2

Brown

⟩τ 2

(equation (35)) it is necessary to derive I1(τ ) for a → 0. Now, it has already been demonstratedin the original Bonner and Nossal (1981, equation (18)) work that actually the general,unsimplified, form for I1(τ ) (equation (15)) is

I1(τ ) = 2ξ

2ξ + T 2

{1 − e−2ξL2 − e−L2T 2

1 − e−2ξL2

}, (36)

where L ≡ 2koa and T ≡ ⟨V 2

Brown

⟩1/2τ/(√

6a). If a is of finite size, one simply finds theresults given in the present theory; however, if one takes a → 0 then

I1(τ )|a=0 ≡ lima→0

I1(τ ) = 3(1 − exp

(− 23k2

o

⟨V 2

Brown

⟩τ 2

))2k2

o

⟨V 2

Brown

⟩τ 2

. (37)

Thus, equation (37) represents the ‘exact’ solution for I1(τ ) in the case of point-like particles.To find the Boas’ result, it is necessary to further develop equation (37) as a power series ofk2

o

⟨V 2

Brown

⟩τ 2, i.e.

I1(τ )|a=0 ≈ 1 − 13k2

o

⟨V 2

Brown

⟩τ 2. (38)

In this case, by substituting equation (38) in equation (35) one finds the desired Boas’ termwith the supplementary multiplicative term κ coming from the fact that in the present model(equation (22)) one takes into account the scattering properties of the moving particles,separately from the total tissue scattering. This term can also be naturally introduced into theBoas’ model (see e.g. Cheung et al 2001). Thus, the Boas’ model can be naturally derivedusing the general expression for g(2)(τ, r) (equation (22)) for a point-like particle and for thesituation where k2

o

⟨V 2

Brown

⟩τ 2 is small. This can typically be the case for experimental values

that one can find in biology, e.g. ko = 2π1.4/800 × 10−9 (i.e. for a typical index of refractionand laser wavelength),

⟨V 2

Brown

⟩1/2 = 1 × 10−3 m s−1 and τ is in the range 1 × 10−7 to 5 ×10−4 s.

To summarize, this result highlights the fact that mathematically, even if we assumethat the moving particles have ‘point-like’ dimensions, the exact solution is represented byequation (37) and that equation (38) only appears to be a good approximation ofequation (37) for a specific speed range. Moreover, we must not forget that equations (37)and (38) considered in the above example hold only for Vtrans = 0, i.e. as might be the caseduring an arterial occlusion where, as a first approximation, only a random movement ofthe moving red blood cells is present. This means that when one measures a tissue with anon-zero blood flow, it is necessary to take into account Vtrans in equation (37) or to use themore general model given by equation (14) depending on whether one assumes that a → 0 ora � 0, respectively. In the present work, the fact that a cannot be neglected from the generalformulation of I1(τ ), i.e. equation (14), derives from the imposed real physical values of a (e.g.∼2.75 µm) and the wavelength used (e.g. ∼800 nm). In practice, these values do not allowone to apply the ‘point-like’ approximation and thus, the presence of a is simply a mandatory‘mathematical’ consequence. If one wants a ‘point-like’ model as in the above example, then

330 T Binzoni et al

we must ‘force’ a → 0 against the physical reality. It should also be noted that the finite size ofthe moving particles becomes important for instance in the single/few scattering regime. Evenin the case of multiple scattering, one must not forget that the intermediate scattering functionfor multiple scattering (I (τ, r), equation (2)) is expressed as a function of the intermediatescattering function for single scattering (I1(τ )) which requires a to have finite size. The choiceof the assumptions on the a parameter needs to be further investigated.

The present discussion has clearly shown that the correlation diffusion approach or therandom-walk approach gives globally a similar autocorrelation function. The only differenceis given by the term I1(τ ) which takes into account the ‘laser-Doppler’ phenomenon. Fromthis point of view, the random-walk approach has the advantage of explicitly deriving thegeneral I1(τ ) term whereas the present versions of the correlation diffusion theory requireone to define it. In this sense, the results obtained with random walk concerning I1(τ ) mightbe seen as a tool allowing further improvement in the correlation diffusion algorithms. It isalso true that the random-walk theory allows one in principle to generalize the model (i.e.g(2)(τ, r)) for other geometrical constraints such as slabs, etc; however, there is no doubt thatthe correlation diffusion theory is more flexible from this point of view. In fact, in correlationdiffusion, the model g(2)(τ, r) is explicitly a solution of a diffusion equation and in this caseone can use all the mathematical results accumulated in the diffusion literature. In this sense,it would be useful to see these two approaches not in ‘competition’ but as synergistic, i.e. oneapproach will give information difficult to obtain with the other and vice versa.

Finally, there is one more point concerning the potential validity of I1(τ ) as expressedin equation (38). It has been demonstrated experimentally (Cheung et al 2001, Yu et al2005b) that g(1)(τ, r) must demonstrate a τ dependence instead of a τ 2 dependence. Giventhat g(1)(τ, r) and g(2)(τ, r) are simply related by the Siegert relation (see above), it is easyto see from equation (22) that this is true even for the present model. Thus, for small movingparticles, the correlation diffusion model and the model described here give the same result.In the case of the present mathematical model, the theory directly generates and confirms theexperimental findings of a τ dependence instead of a τ 2 dependence.

5.4. Time-domain and frequency-domain description

The last figure in this work, figure 7, shows some examples of the g(2)(τ, r) functions and therelative power spectra representation, S(2πν), in the frequency domain. The aim here wasto show that the choice of the best measurement approach depends on the tissues parameters.In fact, if S(2πν) becomes too ‘narrow’ (the dashed line), the spectrum may no longer beobserved due to the lower frequency cut-off that is usually present in LDF instruments. Inthis case, the corresponding dashed signal appearing in the g(2)(τ, r) representation may bemore easily detected. On the other hand, if S(2πν) is too ‘flat’ then the features of the curvewill be hidden by the background noise. Another problem may arise when calculating themoments for S(2πν) in order to obtain information concerning the blood flow parameters. Infact, it can be seen in figure 7 that S(2πν) (e.g. the dash-dotted line) does not always go tozero for ν values that are inside the integration window of the LDF instrument. This meansthat the calculated moments may experience large errors of estimation. In this case, a bettersolution would probably be to perform a direct fitting of S(2πν), but this requires one to havean analytical solution in the frequency domain (but for the case where Vtrans �= 0 this has notyet been obtained). In conclusion, by using the general, g(2)(τ, r), or the simplified g(2)(τ, r)

model, it will be very easy to investigate these kinds of problems before developing an LDFinstrument for a particular application.

Absorption and scattering coefficient dependence of laser-Doppler flowmetry models for large tissue volumes 331

Acknowledgment

This work was funded by a grant from the Swiss National Science Foundation (#31-58759.99).

Glossary

β instrumental factor which depends upon the optical coherence of the signal atthe detector surface

κ constant (κ ≡ µs,rbc/µ′s)

ξ constant. Comes from the fact that the structure factor of the considered scatterers(red blood cells) can be expressed as exp(−2ξ(Qa)2)

λ LDF laser wavelengthµ dimensionless random-walk theory absorption (µ ≡ µa/µ

′s)

µa absorption coefficient of the tissue including the red blood cellsµ′

s macroscopic transport scattering coefficient of the tissue including red bloodcells

µs,rbc macroscopic scattering coefficient for the red blood cells alone (without thetissue)

ν frequency variable of S(2πν)

ρ dimensionless random-walk theory distance (ρ ≡ rµ′s/

√2)

σ rbcs single red blood cell scattering cross section

θ ′ angle existing between the wave vector of the light coming out from theinvestigated tissue and trans

τ correlation delay timea ‘mean radius’ of a red blood cellc the speed of light in the investigated tissueg(2)(τ, r) normalized temporal autocorrelation function holding for a semi-infinite medium

and multiple scattering including components relating to higher order scatteringprocesses, i.e. those which involve multiply convoluted spectral shifts. It alsotakes into account the Brownian and the translational movements of the red bloodcells and the optical parameters of the tissue

g(2)(τ, r) simplified, normalized temporal autocorrelation where only a ‘mean’ opticalpathlength is considered. Contains a parameter representing the mean numberof collision of a photon with moving red blood cells

g(2)CD(τ, r) g(2)(τ, r) for a model where Vtrans = 0, however derived using the coherence

diffusion theoryH volume fraction of the red blood cells in the tissueI (τ, r) normalized intermediate scattering function of the Doppler shifted lightI1(τ ) contribution of the normalized intermediate scattering function, I (τ, r), for

photons that experience only one collisionio the constant (dc) current produced by the total light intensity falling upon the

photodetector of the LDF instrumentisc portion of io produced by the light falling upon the photodetector of the LDF that

has interacted (i.e. been scattered) from moving cellsko ko = 2πnr/λ

L represents the length of the path covered by a photon inside the tissue before itis detected

〈m〉r mean number of collisions of a photon with a moving particle along a unique‘mean’ path (the laser source and the detector are separated by a distance r onthe tissue surface)

332 T Binzoni et al

m(n) mean number of interactions of a photon with a moving red blood cell afterhaving been scattered n times with the tissue or red blood cells

n number of scattering events for a photon with the tissue or red blood cells aftera time t

nr index of refraction of the mediumnrbc number of erythrocytes contained in VP m same as Pm(r) but where only one possible ‘mean’ path is consideredPm(r) probability that a photon makes m collisions with a moving erythrocyte before

emerging from the tissue at a distance r from the laser source (depends on thetotal number of collisions experienced with the tissue)

P(n, r) probability that a photon makes n collisions before reaching the detection optodeat a distance r from the laser source on the tissue surface

p(m|n) conditional probability that a photon interacts m times with a moving erythrocyte,given that it experiences n scattering events in total before being detected

Q the Bragg scattering vector usually appearing in the laser-Doppler theoryr distance between the point of measurement and the laser source on the

investigated tissue surface (i.e. the interoptode distance)S(2πν) power spectrum of the ac component of the fluctuating light intensity coming

out from the tissue normalized by i2o

V total investigated tissue volumeVrbc volume of a single red blood cell⟨V 2

Brown

⟩second moment of the velocity distribution for the Brownian component of thered blood cells

Vtrans pure translational speed of the red blood cellsˆ the ‘hat’ over the variables is used to denote the values estimated using the fitting

procedure

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