## Diffusing-wave spectroscopy from head-like tissue phantoms: influence of a non-scattering layer

Optics Express, Vol. 14, Issue 22, pp. 10181-10194 (2006)

http://dx.doi.org/10.1364/OE.14.010181

Acrobat PDF (386 KB)

### Abstract

We investigate the influence of a non-scattering layer on the temporal field autocorrelation function of multiple scattered light *g*^{(1)}(*
r
*,

*τ*) from a multilayer turbid medium such as the human head. Data from Monte Carlo simulations show very good agreement with the predictions of the correlation-diffusion equation with boundary conditions taking into account non-diffusive light transport within the non-scattering layer. Field autocorrelation functions measured at the surface of a multilayer phantom including a non-scattering layer agree well with theory and simulations when the source-receiver distance is significantly larger than the depth and the thickness of the non-scattering layer. Our results show that for source-receiver distances large enough to probe the dynamics in the human cortex, the cortical diffusion coefficient obtained by analyzing field autocorrelation functions neglecting the presence of the non-scattering cerebrospinal fluid layer is underestimated by about 40% in situations representative of the human head.

© 2006 Optical Society of America

## 1. Introduction

*k*

_{0}= 2

*π*/

*λ*is the wavenumber of light in the medium,

*l*

^{*}is the transport mean free path length and

*P*(

*,*

**r***s*) is the normalized distribution of photon path lengths

*s*at the position

*r*for a source located at the origin. The latter quantity can be directly measured by a time-of-flight experiment.

*g*

^{(1)}(

*,*

**r***τ*) during contralateral stimulation of the somatomotor cortex by a finger opposition protocol [5

5. T. Durduran, G. Yu, M. G. Burnett, J. A. Detre, J. H. Greenberg, J. Wang, C. Zhou, and A. G. Yodh, “Diffuse optical measurement of blood flow, blood oxygenation, and metabolism in a human brain during sensorimotor cortex activation,” Opt. Lett. **29**, 1766–1768 (2004). [CrossRef] [PubMed]

6. J. Li, G. Dietsche, D. Iftime, S. E. Skipetrov, G. Maret, T. Elbert, B. Rockstroh, and T. Gisler, “Non-Invasive Detection of Functional Brain Activity with Near-Infrared Diffusing-Wave Spectroscopy,” J. Biomed. Opt. **10**, 044002-1–12 (2005). [CrossRef]

5. T. Durduran, G. Yu, M. G. Burnett, J. A. Detre, J. H. Greenberg, J. Wang, C. Zhou, and A. G. Yodh, “Diffuse optical measurement of blood flow, blood oxygenation, and metabolism in a human brain during sensorimotor cortex activation,” Opt. Lett. **29**, 1766–1768 (2004). [CrossRef] [PubMed]

6. J. Li, G. Dietsche, D. Iftime, S. E. Skipetrov, G. Maret, T. Elbert, B. Rockstroh, and T. Gisler, “Non-Invasive Detection of Functional Brain Activity with Near-Infrared Diffusing-Wave Spectroscopy,” J. Biomed. Opt. **10**, 044002-1–12 (2005). [CrossRef]

7. H. Ito, K. Takahashi, J. Hatazawa, S.-G. Kim, and I. Kanno, “Changes in Human Regional Cerebral Blood Flow and Cerebral Blood Volume During Visual Stimulation Measured by Positron Emission Tomography,” J. Cereb. Blood Flow Metab. **21**, 608–612 (2001). [CrossRef] [PubMed]

8. M. Wolf, U. Wolf, V. Toronov, A. Michalos, L. A. Paunescu, J. H. Choi, and E. Gratton, “Different Time Evolution of Oxyhemoglobin and Deoxyhemoglobin Concentration Changes in the Visual and Motor Cortices during Functional Stimulation: A Near-Infrared Spectroscopy Study,” NeuroImage **16**, 704–712 (2002). [CrossRef] [PubMed]

6. J. Li, G. Dietsche, D. Iftime, S. E. Skipetrov, G. Maret, T. Elbert, B. Rockstroh, and T. Gisler, “Non-Invasive Detection of Functional Brain Activity with Near-Infrared Diffusing-Wave Spectroscopy,” J. Biomed. Opt. **10**, 044002-1–12 (2005). [CrossRef]

*P*(

*,*

**r***s*) breaks down in the presence of a extended non-scattering inclusion representing, e.g., the cerebrospinal fluid (CSF) layer surrounding the cortex [9

9. M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. **41**, 767–783 (1996). [CrossRef] [PubMed]

10. E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. **36**, 21–31 (1997). [CrossRef] [PubMed]

11. H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. **44**, 2897–2906 (1999). [CrossRef]

*P*(

*,*

**r***s*), the presence of the cerebrospinal fluid layer might thus affect the shape of the field autocorrelation function

*g*

^{(1)}(

*,*

**r***τ*) and the determination of parameters characterizing the cortical dynamics, such as the cortical diffusion coefficient, from experimental data.

*g*

^{(1)}(

*,*

**r***τ*) from head-like multilayer systems, using experimental data from a tissue phantom, Monte Carlo simulation and an extension of the analytical correlation-diffusion equation to non-diffusive boundaries in the absence of refractive index mismatches. The comparison of simulations with analytical theory on phantoms with optical parameters representative for the human head shows very good agreement provided the modified boundary conditions are taken into account. While well described by the analytical theory for large source-receiver distances, experimental data from the multilayer phantom show slight disagreement at short source-receiver distances which might arise from (i) refractive index mismatches or (ii) low-order scattering not taken into account by the diffusion theory. The analysis of simulated DWS data from a (2+1)-layer model (2 diffusive and 1 non-scattering layers) agrees very well with the analytical (2+1)-layer theory over a wide range of diffusion coefficients characterizing the cortical dynamics, allowing to retrieve cortical diffusion coefficients to within 20%. The simple 2-layer diffusion theory neglecting the non-scattering CSF layer agrees reasonably well with the simulated (2+1)-layer data, but underestimates the cortical diffusion coefficient by about 40%.

*g*

^{(1)}(

*,*

**r***τ*) for the backscattering geometry with both point-like source and receiver in the presence of a non-scattering layer located between two scattering layers and briefly describe the Monte Carlo (MC) simulation procedure used to obtain

*g*

^{(1)}(

*,*

**r***τ*) for a multilayer medium. Experimental details are given in Section 2.2. The influence of the thickness and absorption coefficient of the non-scattering layer on

*g*

^{(1)}(

*,*

**r***τ*) is studied with simulation data and theory in Section 3.1, and experiments on a multilayer phantom are compared with theory in Section 3.2.

## 2. Materials and methods

### 2.1. Theory and simulations

#### A. Diffusion model

*ρ*. The diffusion approximation allows to model the propagation of light in a multilayer medium as shown in Fig. 1. A clear, non-scattering layer with thickness

*d*is located between two scattering layers (layer 1, thickness Δ

_{1}, and layer 2 with thickness Δ

_{2}).

12. F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E **63**, 061404-1–11 (2001). [CrossRef]

*α*

^{2}(

*τ*) = 3

*μ*

_{a}

*μ*′

_{s}+

*μ*

*r*

^{2}(

*τ*)〉, with

*μ*

_{a}the absorption coefficient and

*μ*

_{s}′ = 1/

*l*

^{*}the reduced scattering coefficient. The source located at

*′ = {*

**r***′ =*

**ρ***,*

**0***z*′} inside the first layer at depth

*z*′ = 1/

*μ*′

_{s 1}has a time-independent strength

*s*

_{0}. Eq. (3) is conveniently solved using the Fourier transform of

*G*(

*,*

**r***τ*) with respect to the transverse coordinate

**ρ***m*th layer is [6

**10**, 044002-1–12 (2005). [CrossRef]

*,*

**q***τ*) =

*τ*) +

**q**^{2}. The constants

*A*

_{m}and

*B*

_{m}are determined by the boundary conditions. Near the surface (

*z*= 0) the boundary conditions are [6

**10**, 044002-1–12 (2005). [CrossRef]

*z*

_{m}is the extrapolation length of layer

*m*[6

**10**, 044002-1–12 (2005). [CrossRef]

*z*<

*z*′ and layer 1 for

*z*′ <

*z*< Δ

_{1}. Layer 2 (

*z*> Δ

_{1}+

*d*) is separated from layer 1 by the non-scattering layer of thickness

*d*. Light which is scattered from layer 1 into the non-scattering layer may travel ballistically and will thus be converted to a diffusing photon in layer 2 with an angle-dependent transfer probability 0 ≤

*f*(

*q*= |

*|) ≤ 1. The same holds for the transfer of photons from layer 2 to layer 1. The boundary conditions including the transfer probability*

**q***f*(

*q*) can thus be written as

_{2}→ ∞, i.e. a semi-infinite layer 2,

*f*(

*q*), we use the formalism given in [13

13. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A **17**, 1671–1681 (2000). [CrossRef]

*J*

_{0}(

*x*) is the zeroth order Bessel function of the first kind and

*μ*

_{a}the absorption coefficient of the non-scattering layer. In the case of a non-scattering layer without absorption, Eq. (14) simplifies to

*K*

_{1}(

*x*) is the modified Bessel function of the second kind. Note that for vanishing thickness of the non-scattering layer (

*d*= 0),

*f*(

*q*) = 1 and the diffusive boundary conditions of [14

14. S. E. Skipetrov and R. Maynard, “Dynamic multiple scattering of light in multilayer turbid media,” Phys. Lett. A **217**, 181–185 (1996). [CrossRef]

*G*

_{0}(

*,*

**r***τ*) at the position

*= {*

**r***,*

**ρ***z*= 0} at the surface of the sample is then obtained by the inverse Fourier transform of

*Ĝ*

_{0}(

*,*

**q***z*= 0,

*τ*) [6

**10**, 044002-1–12 (2005). [CrossRef]

*g*

^{(1)}(

*,*

**r***τ*) is then obtained by normalization of

*G*

_{0}(

*,*

**r***τ*) with its value at

*τ*= 0.

*z*

_{m}= 2/(3μ′

_{sm}). The next section briefly reviews the basic scheme of the Monte Carlo simulation procedure.

#### B. Monte Carlo simulations

*g*

^{(1)}(

*,*

**r***τ*) were performed by simulating the light propagation [15

15. L. V. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Progr. Biomed. **47**, 131–146 (1995). [CrossRef]

*P*= 3 layers and a pixel size of 0.05cm). Each layer is characterized by an absorption coefficient

*μ*

_{a}, a reduced scattering coefficient

*μ*′

_{s}, and its thickness. The refractive index of the layers was chosen to be uniform,

*n*

_{med}= 1.33. At each scattering event, the photon direction is randomly chosen such that the Henyey-Greenstein phase function with anisotropy factor

*g*= 0 is sampled for all layers. Photons entering or leaving the non-scattering layer pass through the layer without being refracted or scattered. Simulations were carried out with

*N*= 4 × 10

^{6}photons.

*= {*

**r***,*

**ρ***z*= 0} can be written as

*w*

_{α}(

*) is the weight of photon path*

**r***α*(with

*M*

_{α}steps) starting at the origin and ending at the position

*, determined by the absorption between successive scattering events [16*

**r**16. D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A **14**, 192–215 (1997). [CrossRef]

*τ*) is the field autocorrelation function for photon path

*α*which is given by

*q*

_{αl}is the magnitude of the scattering vector of the

*l*th scattering event in path

*α*and 〈Δ

*τ*)〉 is the mean-square displacement within time

*τ*in layer

*j*. The weighting factor

*h*

_{αlj}is unity if the scattering event

*l*of path

*α*is within layer

*j*, and zero otherwise. The motion of the scatterers is modelled by Brownian diffusion, 〈Δ

*D*

_{j}

*τ*, characterized by a diffusion coefficient

*D*

_{j}for each layer.

### 2.2. Experimental

*T*= 295 K on a phantom consisting of a cylindrical vat made of stainless steel (diameter 15 cm) whose 2 fluid compartments were separated by a glass window with thickness

*d*= 0.05cm and refractive index ≈ 1.4 (see Fig. 2). Fluid layers consisted of turbid aqueous suspensions (refractive index

*n*

_{med}= 1.33) of polystyrene latex spheres. Particle diffusion coefficients, as measured by quasi-elastic light scattering on dilute suspensions, were

*D*

_{1}= 9.2 × 10

^{-9}cm

^{2}/s for layer 1 (representing scalp and skull) and

*D*

_{2}= 1.26 × 10

^{-8}cm

^{2}/s for layer 2 (representing the cortex). Modelling the dynamics within the skull by Brownian motion is motivated by the observation that DWS data from humans probing scalp and skull show no sign of static scattering (detectable by a reduction of the intercept of the intensity autocorrelation function [17

17. P. N. Pusey and W. van Megen, “Dynamic Light Scattering by Nonergodic Media,” Physica A **157**, 705–741 (1989). [CrossRef]

_{1}= 0.88 cm and Δ

_{2}= 7 cm, respectively. Layer 2 is sufficiently thick that for the source-receiver distances used here, it can be assumed to be semi-infinite. Light from a diode laser with wavelength

*λ*

_{0}= 802 nm (Toptica TA-100) was delivered into a point-like source by a multimode fiber. Multiple scattered light was collected at distances 0.5cm ≤

*ρ*≤ 4 cm from the source by a 6-mode fiber [18

18. T. Gisler, H. Rüger, S. U. Egelhaaf, J. Tschumi, P. Schurtenberger, and J. Rička, “Mode-selective dynamic light scattering: theory versus experimental realization,” Appl. Opt. **34**, 3546–3553 (1995). [CrossRef] [PubMed]

*g*

^{(2)}(

*,*

**r***τ*) of the measured photon count rate was computed by a correlator (ALV-5000E). In order to eliminate artefacts, the top liquid layer was covered with a rigid black plastic sheet serving both to reduce internal reflections at the free liquid surface, and to suppress surface fluctuations. Through small holes in the cover, the source and receiver fibers were immersed about 1–2 mm into the liquid in order to ensure stable optical coupling.

*g*

^{(1)}(

*,*

**r***τ*) were analyzed using the solution of the diffusion equation for the infinite geometry [19

19. S. Feng, F. Zeng, and B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. **34**, 3826–3837 (1995). [CrossRef] [PubMed]

*D*

_{1,2}and assuming the absorption coefficient

*μ*

_{a1,2}= 0.0223 cm

^{-1}of water [20

20. L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-*μ*m spectral range,” Appl. Opt. **32**, 3531–3540 (1993). [CrossRef] [PubMed]

*μ*′

_{s 1}= 8cm

^{-1}and

*μ*′

_{s 2}= 17.24cm

^{-1}agreed to within 4% with calculations based on Mie theory [21]. Anisotropy factors

*g*

_{1}= 0.94 and

*g*

_{2}= 0.92 for layers 1 and 2, respectively, were calculated using Mie theory [21].

## 3. Results

### 3.1. Comparison between simulation and theory

*ρ*= 10 mm and

*ρ*= 20 mm. We note the very good agreement between theory and simulations both with and without the (non-absorbing) non-scattering layer between layers 1 and 2, provided the modified boundary conditions Eqs. (9)–(10) and the expression Eq. (15) for the transfer probability

*f*(

*q*) are used. Similar to the DWS signal in transmission geometry, the decay of

*g*

^{(1)}(

*,*

**r***τ*) is shifted towards shorter times as the source-receiver distance is increased, reflecting the increasing contribution of long photon paths to the DWS signal. The presence of a non-scattering layer separating the turbid layers 1 and 2 results in a slowing-down of the field autocorrelation function. This is due to the fact that when photons are scattered into the non-scattering layer, there is a finite probability that they travel ballistically far outside the banana-shaped acceptance volume spanned by source and receiver [19

19. S. Feng, F. Zeng, and B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. **34**, 3826–3837 (1995). [CrossRef] [PubMed]

*g*

^{(1)}(

*,*

**r***τ*) at the shortest source-receiver distance

*ρ*= 10mm where the typical depth probed by DWS,

*ρ*/2 = 5 mm, is smaller than the thickness of the top layer, Δ

_{1}= 8.8mm.

*g*

^{(1)}(

*,*

**r***τ*) is increasingly slowed down, reflecting the larger probability for the photon to escape the acceptance volume by being scattered into the non-scattering layer, which results in a reduced contribution of long paths to

*g*

^{(1)}(

*,*

**r***τ*).

*d*= 0.1 cm, calculating the transfer probability

*f*(

*q*) by Eq. (14). As would be expected for increased absorption loss, the decay of

*g*

^{(1)}(

*,*

**r***τ*) is slowed down as the absorption coefficient of the non-scattering layer is increased. This effect is particularly pronounced for source-receiver distances large enough that the acceptance volume intersects the non-scattering layer. Absorption within the non-scattering layer thus counteracts the ballistic propagation of photons within the non-scattering layer and tends to restore diffusive behavior.

### 3.2. Comparison between theory and experiments

*ρ*, the decay of the field autocorrelation function shifts monotonically towards shorter times, as would be expected for the increasing weight of long photon paths to

*g*

^{(1)}(

*,*

**r***τ*). We find that the presence of the opaque cover and the immersion of the fibers into the top layer is essential for observing the increasingly fast decay of

*g*

^{(1)}(

*,*

**r***τ*) with increasing source-receiver distance. While the agreement between theory and experiment is excellent for

*ρ*> 1.5 cm, it is less so at shorter distances, in particular for long times (

*τ*> 10

^{-4}s). We think that the observed discrepancies arise from contributions of low-order scattering which are not correctly accounted for by the diffusion theory.

## 4. Discussion

*τ*(corresponding to short photon path lengths

*s*) and increasing thickness of the non-scattering layer. On the other hand, when the absorption coefficient of the non-scattering layer is finite, the agreement of theory with simulation is significantly improved, reflecting the fact that diffusive propagation is restored by suppression of long ballistic photon paths within the non-scattering layer.

*ρ*= 1.5cm is remarkable given that the theory curves contain no adjustable parameters. This indicates that for source-receiver distances large enough to probe the cortex, the refractive index mismatch between layers 1 and 2 and the intervening non-scattering layer (which is present in the experiment, but neglected in the theory) only marginally affects the accuracy of the diffusion theory, provided the presence of the non-scattering layer is accounted for by the transfer probability

*f*(

*q*). The discrepancy between theory and experiment for short source-receiver distances (

*ρ*≤ 1cm) and long times is due to the breakdown of the diffusion approximation for short photon paths. Enhanced agreement between experiment and theory for short photon paths could be achieved either by simulating the light transport with realistic values of the anisotropy factors, or by solving the radiative transfer equation [22

22. R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the Diffusing-Wave Spectroscopy Model for the Temporal Fluctuations of Scattered Light,” Phys. Rev. Lett. **92**, 213903-1–4 (2004). [CrossRef] [PubMed]

^{-9}cm

^{2}/s ≤

*D*

_{2}≤ 10

^{-6}cm

^{2}/s for the optical and geometrical parameters of the phantom experiment (representative of the situation in the human head). From the simulated data we retrieved the cortical diffusion coefficient

*D*

_{2}by fitting the diffusion theory to the MC data, using a Levenberg-Marquardt optimization routine. As shown in Fig. 7, the analysis of the MC data with a 2-layer model, neglecting the presence of the non-scattering layer, underestimates the cortical diffusion coefficient by about 40%. When the (2+1)-layer diffusion theory is used, the agreement with the simulation is enhanced: the error of the retrieved cortical diffusion coefficient

*D*

_{2}increases from very small values at

*D*

_{2}≤ 5 × 10

^{-8}cm

^{2}/s to about 20 % at

*D*

_{2}= 10

^{-6}cm

^{2}/s. The underestimation of the cortical diffusion coefficient is mainly due to a discrepancy between simulation and theory at long times due to the breakdown of the diffusion approximation. Using only the short-time data with

*g*

^{(1)}(

*,*

**r***τ*) > 0.1 for the fitting, the error in the retrieved

*D*

_{2}can be made very small (data not shown). However, this cutoff is arbitrary and should be used with caution.

12. F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E **63**, 061404-1–11 (2001). [CrossRef]

*f*(

*q*) < 1 and by absorption, while the propagation of detected photons diffusing on short paths through layer 1 only is only marginally modified. In the transmission geometry, on the other hand,

*g*

^{(1)}(

*,*

**r***τ*) is dominated by photon paths close to the line of sight between source and receiver, resulting in an only minor distortion of the path length distribution function by a non-scattering layer.

## 5. Conclusions

*g*

^{(1)}(

*,*

**r***τ*) of multiple scattered light by means of a diffusion theory, Monte Carlo simulations and experiments for the backscattering geometry used in DWS experiments on the human head. For a configuration with a point source and a point receiver, boundary conditions for the correlation-diffusion equation were introduced accounting for the presence of the non-scattering layer. Field autocorrelation functions

*g*

^{(1)}(

*,*

**r***τ*) predicted by this modified diffusion theory are in very good agreement with Monte Carlo simulations in the absence of refractive index mismatches between the layers. The comparison with experimental data indicates that contributions from low-order scattering and, possibly, refractive index mismatches between the layers play a role for source-receiver distances comparable to about twice the distance between surface and the cerebrospinal fluid layer. The proper treatment of low-order scattering and refractive index mismatches might thus be important for the quantitative analysis of DWS data from the human head measured at short source-receiver distances. Data measured at larger source-receiver distance which are able to probe the cortex are, on the other hand, rather insensitive to refractive index mismatches. The analysis by the diffusion theory qualitatively reproduces the simulated DWS data for the optical and dynamical properties used here even when the presence of the non-scattering layer is neglected. Nevertheless, the coupling between superficial and deep tissue layers in the DWS signal and the ensuing complexity of the DWS signal for a multilayer system makes it difficult to chart the region in parameter space where the diffusion theory satisfyingly describes the simulations for other combinations of

*μ*′

_{s},

*μ*

_{a},

*D*

_{1}and

*D*

_{2}. Methods for selectively probing deep tissue layers by DWS might ultimately help obviate these complications.

## Appendix: Boundary conditions

13. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A **17**, 1671–1681 (2000). [CrossRef]

*G*(

*,*

**r***τ*) instead of the average intensity

*U*(

*). In our case, since we have no light sources at the boundary (the right-hand side of Eq. (24) of Ref. [13*

**r**13. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A **17**, 1671–1681 (2000). [CrossRef]

*𝓓*′

_{m}= 3(

*μ*

_{am}+

*μ*′

_{sm})]

^{-1}is the reduced photon diffusion coefficient of the

*m*th layer, and

*is the unit vector normal to the interface. Eq. (36) of Ref [13*

**m****17**, 1671–1681 (2000). [CrossRef]

*G*(

*,*

**r***τ*) as

*S*is the surface the irradiation originates from, which in our case is the

*xy*plane,

*C*

_{m}= (2 -

*R*

_{J})/

*R*

_{U}= 2 since

*R*

_{U}=

*R*

_{0→1}(

*θ*)|]

^{2}cos

*θ*

*d*(cos

*θ*) = 1/2 and

*R*

_{J}= 3

*R*

_{0→1}(

*θ*)|]

^{2}cos

^{2}

*θ*

*d*(cos

*θ*) = 1 in the case without refractive index mismatch (

*R*

_{0→1}(

*θ*), the reflection coefficient with incidence angle

*θ*from medium 0 to medium 1, is equal to 1) from Eq. (30) of Ref [13

**17**, 1671–1681 (2000). [CrossRef]

*z*= Δ

_{1}are

*z*= Δ

_{1}+

*d*:

*θ*

*-*

**ρ***′) is given by Eq. (18) of Ref. [13*

**ρ****17**, 1671–1681 (2000). [CrossRef]

*μ*

_{a}is the absorption coefficient of the non-scattering layer. Defining

*ρ*= |

*-*

**ρ***′| then*

**ρ**## Acknowledgments

## References and links

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6. | J. Li, G. Dietsche, D. Iftime, S. E. Skipetrov, G. Maret, T. Elbert, B. Rockstroh, and T. Gisler, “Non-Invasive Detection of Functional Brain Activity with Near-Infrared Diffusing-Wave Spectroscopy,” J. Biomed. Opt. |

7. | H. Ito, K. Takahashi, J. Hatazawa, S.-G. Kim, and I. Kanno, “Changes in Human Regional Cerebral Blood Flow and Cerebral Blood Volume During Visual Stimulation Measured by Positron Emission Tomography,” J. Cereb. Blood Flow Metab. |

8. | M. Wolf, U. Wolf, V. Toronov, A. Michalos, L. A. Paunescu, J. H. Choi, and E. Gratton, “Different Time Evolution of Oxyhemoglobin and Deoxyhemoglobin Concentration Changes in the Visual and Motor Cortices during Functional Stimulation: A Near-Infrared Spectroscopy Study,” NeuroImage |

9. | M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. |

10. | E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. |

11. | H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. |

12. | F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing-wave spectroscopy of nonergodic media,” Phys. Rev. E |

13. | J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A |

14. | S. E. Skipetrov and R. Maynard, “Dynamic multiple scattering of light in multilayer turbid media,” Phys. Lett. A |

15. | L. V. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Progr. Biomed. |

16. | D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A |

17. | P. N. Pusey and W. van Megen, “Dynamic Light Scattering by Nonergodic Media,” Physica A |

18. | T. Gisler, H. Rüger, S. U. Egelhaaf, J. Tschumi, P. Schurtenberger, and J. Rička, “Mode-selective dynamic light scattering: theory versus experimental realization,” Appl. Opt. |

19. | S. Feng, F. Zeng, and B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. |

20. | L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5- |

21. | A. Ishimaru, |

22. | R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the Diffusing-Wave Spectroscopy Model for the Temporal Fluctuations of Scattered Light,” Phys. Rev. Lett. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(170.5280) Medical optics and biotechnology : Photon migration

(290.1350) Scattering : Backscattering

(290.1990) Scattering : Diffusion

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 27, 2006

Revised Manuscript: August 18, 2006

Manuscript Accepted: September 26, 2006

Published: October 30, 2006

**Virtual Issues**

Vol. 1, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Franck Jaillon, Sergey E. Skipetrov, Jun Li, Gregor Dietsche, Georg Maret, and Thomas Gisler, "Diffusing-wave spectroscopy from head-like tissue phantoms: influence of a non-scattering layer," Opt. Express **14**, 10181-10194 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-22-10181

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### References

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