## Monte Carlo algorithm for efficient simulation of time-resolved fluorescence in layered turbid media

Optics Express, Vol. 16, Issue 17, pp. 13188-13202 (2008)

http://dx.doi.org/10.1364/OE.16.013188

Acrobat PDF (417 KB)

### Abstract

We present an efficient Monte Carlo algorithm for simulation of time-resolved fluorescence in a layered turbid medium. It is based on the propagation of excitation and fluorescence photon bundles and the assumption of equal reduced scattering coefficients at the excitation and emission wavelengths. In addition to distributions of times of arrival of fluorescence photons at the detector, 3-D spatial generation probabilities were calculated. The algorithm was validated by comparison with the analytical solution of the diffusion equation for time-resolved fluorescence from a homogeneous semi-infinite turbid medium. It was applied to a two-layered model mimicking intra- and extracerebral compartments of the adult human head.

© 2008 Optical Society of America

## 1. Introduction

*in vivo*imaging of human tissue was mainly restricted to superficially accessible tissues like skin [2

2. K. Svanberg, I. Wang, S. Colleen, I. Idvall, C. Ingvar, R. Rydell, D. Jocham, H. Diddens, S. Bown, G. Gregory, S. Montan, S. Andersson-Engels, and S. Svanberg, “Clinical multi-colour fluorescence imaging of malignant tumours--initial experience,” Acta Radiol. **39**, 2–9 (1998). [PubMed]

3. M. Ortner, B. Ebert, E. Hein, K. Zumbusch, D. Nolte, U. Sukowski, J. Weber-Eibel, B. Fleige, M. Dietel, M. Stolte, G. Oberhuber, R. Porschen, B. Klump, H. Hortnagl, H. Lochs, and H. Rinneberg, “Time gated fluorescence spectroscopy in Barrett’s oesophagus,” Gut **52**, 28–33 (2003). [CrossRef]

*vivo*imaging of fluorescence-labeled molecular probes was reported for small animals [4

4. V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. **8**, 757–760 (2002). [CrossRef] [PubMed]

5. A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, and C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. **19**, 327–331 (2001). [CrossRef] [PubMed]

6. R. Weissleder, “Scaling down imaging: Molecular mapping of cancer in mice,” Nat. Rev. Cancer **2**, 11–18 (2002). [CrossRef] [PubMed]

7. R. Weersink, M. S. Patterson, K. Diamond, S. Silver, and N. Padgett, “Noninvasive measurement of fluorophore concentration in turbid media with a simple fluorescence/reflectance ratio technique,” Appl. Opt. **40**, 6389–6395 (2001). [CrossRef]

8. K. R. Diamond, T. J. Farrell, and M. S. Patterson, “Measurement of fluorophore concentrations and fluorescence quantum yield in tissue-simulating phantoms using three diffusion models of steady-state spatially resolved fluorescence,” Phys. Med. Biol. **48**, 4135–4149 (2003). [CrossRef]

9. D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially resolved fluorescence from depth-dependent fluorophore concentrations,” Phys. Med. Biol. **46**, 369–383 (2001). [CrossRef] [PubMed]

10. D. Stasic, T. J. Farrell, and M. S. Patterson, “The use of spatially resolved fluorescence and reflectance to determine interface depth in layered fluorophore distributions,” Phys. Med. Biol. **48**, 3459–3474 (2003). [CrossRef] [PubMed]

12. J. Wu, J. Wang, L. Perelman, I. Itzkan, R. Dasari, and F. Ms, “Time-resolved multichannel imaging of fluorescent objects embedded in turbid media,” Opt. Lett. **20**, 489–491 (1995). [CrossRef] [PubMed]

13. R. H. Mayer, J. S. Reynolds, and E. N. Sevick-Muraca, “Measurement of the fluorescence lifetime in scattering media lay frequency-domain photon migration,” Appl. Opt. **38**, 4930–4938 (1999). [CrossRef]

17. J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, and E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. **70**, 87–94 (1999). [CrossRef] [PubMed]

18. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express **15**, 6696–6716 (2007). [CrossRef] [PubMed]

19. A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Moller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Non-invasive detection of fluorescence from exogenous chromophores in the adult human brain,” Neuroimage **31**, 600–608 (2006). [CrossRef] [PubMed]

21. A. J. Welch, C. Gardner, R. Richard-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers. Surg. Med. **21**, 166–178 (1997). [CrossRef] [PubMed]

22. R. J. Crilly, W. F. Cheong, B. Wilson, and J. R. Spears, “Forward-adjoint fluorescence model: Monte Carlo integration and experimental validation,” Appl. Opt. **36**, 6513–6519 (1997). [CrossRef]

23. J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Andersson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. A **20**, 714–727 (2003). [CrossRef]

19. A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Moller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Non-invasive detection of fluorescence from exogenous chromophores in the adult human brain,” Neuroimage **31**, 600–608 (2006). [CrossRef] [PubMed]

## 2. Monte Carlo simulations

### 2.1 Theoretical background

*r*. The absorption and reduced scattering coefficients of the layered medium can be defined independently for each layer, indexed by

*j*. For modeling absorption, we applied a variance reduction technique, i.e. we follow the propagation of a photon bundle of gradually decreasing weight at the excitation wavelength

*λ*[24

_{x}24. M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. **38**, 1859–1876 (1993). [CrossRef] [PubMed]

25. J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. **46**, 879–896 (2001). [CrossRef] [PubMed]

*λ*) the weights of which decrease due to absorption at

_{m}*λ*.

_{m}- Scattering is isotropic and can be characterized by a single reduced scattering coefficient. For interoptode distances of a few centimeters as considered here, a completely randomized scattering direction is a good approximation.
- The reduced scattering coefficients of each layer
*j*are equal at the excitation (*λ*) and emission (_{x}*λ*) wavelengths, i.e._{m}*µ*′_{sx,j}≈*µ*′_{sm,j}≈*µ*′_{s,j}. This assumption is justified since excitation and emission wavelengths usually differ by a few 10 nm only. In the near-infrared region the reduced scattering coefficient of tissues typically depends only slightly on wavelength, i.e. for such small differences in wavelength the reduced scattering coefficient differs by less than 10% [26].26. J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, and T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt.

**37**, 3586–3593 (1998). [CrossRef] - All fluorescence photon bundles generated along the path of a particular bundle of excitation photons follow this very trajectory. This assumption is justified since the ensembles of all excitation and fluorescence trajectories from a given location within the medium to the detector are virtually the same provided scattering is isotropic and equal at
*λ*and_{x}*λ*._{m}

*R*is a random number within the interval (0,1). When the path between two subsequent scattering events crosses the boundary between two layers with

*j*=

*A*and

*j*=

*B*respectively, the portion of the free path length in medium

*B*is recalculated as

*L*is the corresponding value before considering different scattering coefficients. Fresnel reflections at the boundaries between the layers as well as at the surface of the tissue are neglected to avoid prolongation of the computation time. After each scattering event

_{B}*q*the cumulative path lengths

*l*traveled by the photon bundle

^{(q)}_{ij}*i*within the layers

*j*up to this scattering event

*q*are stored for further processing after completion of the trajectory calculation. At the point of detection

*e*, i.e. upon crossing the surface of the tissue, the total path lengths for this bundle in all layers

*l*are evaluated. Based on

^{(e)}_{ij}*l*and

^{(q)}_{ij}*l*, multiple sets of absorption parameters can be applied to the same trajectory as will be discussed below.

^{(e)}_{ij}*j*-th tissue layer are denoted by

*µ*and

_{ax,j}*µ*at the excitation and emission wavelengths, respectively. The fluorescent dye (fluorophore) that is distributed in the tissue contributes to absorption which is described by the concentrationdependent absorption coefficients

_{am,j}*µ*and

_{afx,j}*µ*at

_{afm,j}*λ*and

_{x}*λ*, respectively. The coefficient

_{m}*µ*describes reabsorption of fluorescence photons by the fluorophore. It should be noted, however, that a possible reemission process after reabsorption is omitted. This assumption is justified as long as the fluorescence quantum yield of the fluorophore is small which is true for a dye like indocyanine green (ICG). Moreover, considering the coefficients

_{afm,j}*µ*and

_{am,j}*µ*at a fixed

_{afm,j}*λ*only, we neglect that fluorescence photons in different parts of the fluorescence spectrum might experience different reabsorption by the fluorophore as well as absorption by the tissue.

_{m}*P*

^{(q)}

_{c,i}along the path between two consecutive scattering events numbered by

*q*-1 and

*q*is given by the probability of absorption by the fluorophore and its fluorescence quantum yield

*Φ*.

*Φ*=

_{j}*Φ*for all layers, i.e. the same microenvironment in all compartments. For

*q*=1, i.e. on the path between source and first scattering event,

*l*

^{(q-1)}

_{ij}equals 0.

*q*, the weight of excitation bundle

*i*has decreased to

*λ*. The initial weight of the fluorescence photon bundle emanating from scattering event

_{x}*q*is thus

*W*

^{(q)}

_{x,i}

*P*

^{(q)}

_{c,i}. On its travel to the surface of the tissue it decreases due to absorption at

*λ*by the factor

_{m}*i*to the fluorescence signal is

*r*between the source position and the point of exit at which the bundle

_{i}*i*escapes from the tissue as well as according to its total travel time

*t*. Fluorescence histograms, i.e. distributions of times of arrival of fluorescence photons (DTA), are collected for various detection areas. The total travel time includes the time traveled by the photon bundle at the excitation wavelength up to the point of conversion plus the time traveled by the fluorescence photon bundle from the point of conversion to the point of detection (for simplicity the fluorescence lifetime has been neglected as discussed below). Assuming the same velocity of light

_{i}*c*=

*c*/

_{0}*n*(

*c*=3·10

_{0}^{8}m/s,

*n*- refractive index of the tissue) in all layers and at

*λ*as well as

_{x}*λ*, the total travel time from source to detector of both the excitation and the fluorescence photon bundle

_{m}*i*is

*τ*has been neglected so far. It would lead to an exponentially distributed delay in the arrival of the fluorescence photon bundles at the detector. Consequently, a finite fluorescence lifetime that is the same in all layers can be easily taken into account by convolving the DTA obtained as described before with exp(-

_{F}*t*/

*τ*). Hence, when considering statistical moments of the DTA,

_{F}*τ*simply adds to the first moment of the DTA and

_{F}*τ*

^{2}

_{F}to its variance.

*q*). In fact, the conversion may appear at any location in between the scattering points

*q-1*and

*q*. In case of different total absorption coefficients at

*λ*and

_{x}*λ*, such change in the location of conversion leads to a slight modification of the weight of the fluorescence photon bundle, i.e.

_{m}*l*

^{(q)*}

_{ij}is the distance traveled in

*j*th layer by

*i*th photon bundle between the actual conversion point and

*q*-th scattering event: 0<

*l*

^{(q)*}

_{ij}<(

*l*

^{(q)}

_{ij}-

*l*

^{(q-1)}

_{ij}). The exponential term in Eq. (8) is close to unity provided that

^{-1}at most, and

*µ′*is on the order of 10 cm

_{s}^{-1}. Therefore we decided to calculate

*W*according to Eq. (6).

_{m,i}*λ*when arriving at the detector

_{x}*µ*

_{ax,j}and the dye

*µ*

_{afx,j}and the total path lengths in all layers. These weights are again accumulated according to

*r*and

*t*.

*i*is completed and the values of

*l*

^{(q)}

*and*

_{ij}*l*

^{(e)}

_{ij}are known. Thus multiple sets of absorption coefficients may be applied to the same trajectory. It should be noted that the multiple DTAs and DTOFs obtained in this way are not statistically independent. However, this is not relevant if the number of photon bundles in the simulation is large enough to ensure low statistical uncertainty of the quantities derived. The “white Monte Carlo” approach allows for a formidable saving of computation time. It is applicable if the geometry, i.e. the thickness of layers, as well as the scattering coefficients

*µ*′

_{s, j}in each layer remain constant.

*V*of size

*Δx*,

*Δy*,

*Δz*centered around (

*x,y,z*) by all photon bundles which pass this voxel and have a total travel time within the interval (

*t*-Δ

*t*/2,

*t*+Δ

*t*/2)

*x*,

_{q}*y*and

_{q}*z*are coordinates of the

_{q}*q*th scattering event. The first summation extends over all scattering events

*q*that occur within the given voxel. We call the spatial distribution defined by Eq. (11) “generation probability distribution”. The total spatial integral of the expression in Eq. (11) as a function of time equals the DTA for a given source-detector separation. By restricting the integration to certain compartments (layers) of the medium, their individual contributions to the DTA of fluorescence photons can be calculated.

### 2.2 Monte Carlo code

25. J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. **46**, 879–896 (2001). [CrossRef] [PubMed]

28. A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, “Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons,” Appl. Opt. **43**, 3037–3047 (2004). [CrossRef] [PubMed]

*µ*, the computation time increased by a factor of 10 only.

_{afx}*y*coordinate of the escape point was zero. Equation (11) was evaluated with the trajectories after rotation, i.e. with the new coordinates of all scattering events. Furthermore, for the presentation of the spatially dependent generation probability the values of the 3D voxel matrix were projected onto the source-detector plane (x-z plane) perpendicular to the surface of the medium.

## 3. Validation of the Monte Carlo code

29. M. Patterson and B. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. **33**, 1963–1974 (1994). [CrossRef] [PubMed]

*r*equal to the mean radius of the corresponding ring. The raw Monte-Carlo DTAs were divided by the area of the corresponding detection rings. Subsequently each of the two data sets of DTAs, i.e. MC and diffusion, was normalized to the sum of the integrals of all DTAs of this set within the time interval from 1 ns to 3.5 ns. As can be seen in Fig. 2, the results of MC simulations match well with the DTAs obtained from the diffusion model with respect to the shape of the DTAs as well as to their relative amplitudes for different source-detector separations.

*N*), first centralized moment, i.e. mean time of arrival of fluorescence photons (<

_{F,tot}*t*>

_{F}) and second centralized moment, i.e. variance of DTA (

*V*). The comparison of integrals was performed after normalization to the sum of all values in the set of simulations.

_{F}^{-1}and 0.655 mm

^{-1}. For each series of simulations only one absorption coefficient was changed whereas all other coefficients were kept fixed to the values shown in Table 1. Results of the comparison are presented in Figs. (3) and (4). It can be seen that the moments of DTAs of fluorescence photons calculated by the MC method match with the moments calculated from Eq. (13). Discrepancies between both models appear at short interoptode distances and high absorption coefficients. This observation can be explained by the fact that the diffusion approximation approaches its limits in these cases.

## 4. Monte Carlo simulations in a two-layered model

*µ*) in the lower layer [Fig. (5b)] or in the upper layer [Fig. (5c)], respectively. The thickness of the upper layer mimicking extracerebral tissue was taken as 10 mm. When the dye of high concentration is distributed in the lower layer most of the fluorescence photons are generated in this compartment [s. Fig. (5b)]. In contrast, for the higher dye concentration in the upper layer, fluorescence photons are preferentially generated in the upper layer [s. Fig. (5c)]. This analysis suggests a dependence of mean time of arrival of fluorescence photons <

_{afx}*t*>

*on the spatial distribution of the fluorophore. In particular, in case of high dye concentration in the lower tissue compartment, <*

_{F}*t*>

*is expected to be larger than if the dye is distributed homogeneously in the whole medium*

_{F}*µ*in the lower and upper layers of the model were calculated. Analysis of DTAs shows that <

_{afx}*t*>

*and*

_{F}*V*reveal specific changes caused by increasing

_{F}*µ*in the lower and upper layers. We observe an initial increase of mean time of arrival and variance of DTAs when

_{afx}*µ*increases in the lower compartment and a decrease of these moments when the absorption by the dye increases in the upper layer. These observations are in agreement with the qualitative analysis of generation probability discussed above. However, for large values of

_{afx}*µ*in the lower layer and large source-detector separations,

_{afx}*V*and <

_{F}*t*>

*decrease again due to saturation effects.*

_{F}*µ*(

_{afx}*T*) in both layers are shown in the left column of panels in Fig. 8. In addition, we modeled a realistic reabsorption of fluorescence by the dye by assuming that

*µ*(

_{afm}*T*)=

*µ*(

_{afx}*T*)/2. The top row of Fig. (8) shows the results for equal maximum dye concentrations in both layers. We found that the patterns of changes in moments calculated by Monte Carlo simulations are very similar to those observed in

*in-vivo*experiments [19

19. A. Liebert, H. Wabnitz, H. Obrig, R. Erdmann, M. Moller, R. Macdonald, H. Rinneberg, A. Villringer, and J. Steinbrink, “Non-invasive detection of fluorescence from exogenous chromophores in the adult human brain,” Neuroimage **31**, 600–608 (2006). [CrossRef] [PubMed]

*µ*are difficult to estimate for the

_{afx}*in-vivo*situation. Therefore, we analyzed how the ratio of amplitudes of changes of

*µ*in the extracerebral and intracerebral compartments influences the observed patterns of moments. Assuming a constant amplitude of the bolus in the lower compartment (brain) we changed the amplitude of the bolus in the upper layer (extracerebral tissue). In general, the observed patterns of changes in each of the moments show a similar behaviour (cf. rows in Fig. 8). Moreover, the patterns of changes of moments observed at both source-detector separations are similar. Specifically, the amplitude of the positive peaks in Δ<

_{afx}*t*>

*as well as Δ*

_{F}*V*seem to be rather invariant for the different extracerebral concentrations and source-detector separations. However, it can be noted that the dynamics of changes of mean time of flight and variance in the transition from positive to negative values is influenced significantly by the ratio of amplitudes of the boli in both layers.

_{F}## 5. Discussion and conclusions

*µ*′

*with wavelength is fairly small [26*

_{s}26. J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, and T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. **37**, 3586–3593 (1998). [CrossRef]

23. J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Andersson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. A **20**, 714–727 (2003). [CrossRef]

34. Q. Liu and N. Ramanujam, “Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media,” J. Opt. Soc. Am. A **24**, 1011–1025 (2007). [CrossRef]

*µ*and

_{afx}*µ*, i.e. depending on whether excitation or fluorescence photons are more likely to be absorbed on the trajectory from source to detector. Finally, the spatial distribution of generation probability is time-dependent, i.e. the location of maximum fluorescence generation depends on the total time of flight of the photon bundles.

_{afm}*in-vivo*fluorescence detection of ICG boli in the adult human brain, e.g. for perfusion monitoring in stroke patients, and the need to understand the observed signals. We started from a two-layered model simulating intra- and extracerebral tissue and different time courses of boli of the fluorophore in both layers. In this way we could explain the most significant features of the observed in-vivo signals of moments of time-of-flight distributions. The initial increase of mean time of flight and variance is connected with the faster inflow of the dye to the intracerebral compared to the extracerebral compartment. By performing various Monte-Carlo simulations for realistic time courses of dye boli in the brain and overlying tissue, we studied the influence of the ratio of fluorophore concentrations in both compartments as well as of the source-detector separation on the signals detected. By comparing fluorescence signals with signals of diffuse reflectance we found that the magnitudes of changes in moments of DTAs of fluorescence photons are in general larger than the magnitudes of changes of moments of DTOFs of diffusely reflected photons. In addition, mean time of arrival of fluorescence photons <

*t*>

*exhibits more specific dynamic features than mean time of flight of diffusely reflected photons <*

_{F}*t*>

*. These result matches well with previous*

_{R}*in-vivo*results [19

**31**, 600–608 (2006). [CrossRef] [PubMed]

35. J. Steinbrink, A. Liebert, H. Wabnitz, R. Macdonald, H. Obrig, A. Wunder, R. Bourayou, T. Betz, J. Klohs, U. Lindauer, U. Dirnagl, and A. Villringer, “Towards non-invasive molecular fluorescence imaging of the human brain,” Neurodegenerative **5**, 296–303 (2008). [CrossRef]

## Acknowledgment

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26. | J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, and T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. |

27. | A. Pifferi, R. Berg, P. Taroni, and S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in |

28. | A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, “Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons,” Appl. Opt. |

29. | M. Patterson and B. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. |

30. | A. Liebert, H. Wabnitz, J. Steinbrink, M. Moller, R. Macdonald, H. Rinneberg, A. Villringer, and H. Obrig, “Bed-side assessment of cerebral perfusion in stroke patients based on optical monitoring of a dye bolus by time-resolved diffuse reflectance,” Neuroimage |

31. | H. Wabnitz, M. Moeller, A. Liebert, A. Walter, R. Erdmann, O. Raitza, C. Drenckhahn, J. P. Dreier, H. Obrig, J. Steinbrink, and R. Macdonald, “A time-domain NIR brain imager applied in functional stimulation experiments,” in |

32. | M. Kohl-Bareis, H. Obrig, K. Steinbrink, K. Malak, K. Uludag, and A. Villringer, “Noninvasive monitoring of cerebral blood flow by a dye bolus method: Separation of brain from skin and skull signals,” J. Biomed. Opt. |

33. | T. S. Leung, I. Tachtsidis, M. Tisdall, M. Smith, D. T. Delpy, and C. E. Elwell, “Theoretical investigation of measuring cerebral blood flow in the adult human head using bolus Indocyanine Green injection and near-infrared spectroscopy,” Appl. Opt. |

34. | Q. Liu and N. Ramanujam, “Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media,” J. Opt. Soc. Am. A |

35. | J. Steinbrink, A. Liebert, H. Wabnitz, R. Macdonald, H. Obrig, A. Wunder, R. Bourayou, T. Betz, J. Klohs, U. Lindauer, U. Dirnagl, and A. Villringer, “Towards non-invasive molecular fluorescence imaging of the human brain,” Neurodegenerative |

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: June 4, 2008

Revised Manuscript: July 31, 2008

Manuscript Accepted: July 31, 2008

Published: August 13, 2008

**Virtual Issues**

Vol. 3, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

A. Liebert, H. Wabnitz, N. Zolek, and R. Macdonald, "Monte Carlo algorithm for efficient simulation of time-resolved fluorescence in layered turbid media," Opt. Express **16**, 13188-13202 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13188

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