## Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head

Optics Express, Vol. 10, Issue 3, pp. 159-170 (2002)

http://dx.doi.org/10.1364/OE.10.000159

Acrobat PDF (974 KB)

### Abstract

We describe a novel Monte Carlo code for photon migration through 3D media with spatially varying optical properties. The code is validated against analytic solutions of the photon diffusion equation for semi-infinite homogeneous media. The code is also cross-validated for photon migration through a slab with an absorbing heterogeneity. A demonstration of the utility of the code is provided by showing time-resolved photon migration through a human head. This code, known as ‘tMCimg’, is available on the web and can serve as a resource for solving the forward problem for complex 3D structural data obtained by MRI or CT.

© Optical Society of America

## 1. Introduction

1. A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. **20**, 435–442 (1997). [CrossRef] [PubMed]

2. B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. **23**, 1716–1718 (1998). [CrossRef]

1. A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. **20**, 435–442 (1997). [CrossRef] [PubMed]

5. D. A. Benaron, W. F. Cheong, and D. K. Stevenson, “Tissue Optics,” Science **276**, 2002–2003 (1997). [CrossRef] [PubMed]

6. D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab **20**, 469–77 (2000). [CrossRef] [PubMed]

7. B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology **218**, 261–6. (2001). [PubMed]

6. D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab **20**, 469–77 (2000). [CrossRef] [PubMed]

10. M. A. Franceschini, V. Toronov, M. Filiaci, E. Gratton, and S. Fanini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express **6**, 49–57 (2000). http://www.opticsexpress.org/oearchive/source/18957.htm [CrossRef] [PubMed]

6. D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab **20**, 469–77 (2000). [CrossRef] [PubMed]

13. C. S. Robertson, S. P. Gopinath, and B. Chance, “A new application for near-infrared spectroscopy: Detection of delayed intracranial hematomas after head injury,” Journal of Neurotrauma **12**, 591–600 (1995). [CrossRef] [PubMed]

14. S. R. Hintz, W. F. Cheong, J. P. van Houten, D. K. Stevenson, and D. A. Benaron, “Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging,” Pediatr Res **45**, 54–9. (1999). [CrossRef] [PubMed]

15. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. **20**, 426–428 (1995). [CrossRef]

16. J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,“ Appl. Opt. **34**, 8038–8047 (1995). [CrossRef] [PubMed]

2. B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. **23**, 1716–1718 (1998). [CrossRef]

17. M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys Med Biol **44**, 2703–21 (1999). [CrossRef] [PubMed]

18. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. **26**, 1698–707. (1999). [CrossRef] [PubMed]

20. E. Okada, M. Schweiger, S. R. Arridge, M. Firbank, and D. T. Delpy, “Experimental validation of Monte Carlo and finite-element methods of estimation of the optical path length in inhomogeneous tissue,” Appl. Opt. **35**, 3362–3371 (1996). [CrossRef] [PubMed]

21. 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**, 767–83. (1996). [CrossRef] [PubMed]

24. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys Med Biol **43**, 1285–302. (1998). [CrossRef] [PubMed]

25. S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. **27**, 252–64. (2000). [CrossRef] [PubMed]

26. 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 Opt Image Sci Vis **17**, 1671–81. (2000). [CrossRef] [PubMed]

27. O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems **14**, 1107–1130 (1998). [CrossRef]

^{2}detector with 10

^{8}photons propagated within 5–10 hours of computer time on a Pentium III 1000 MHz CPU. The method is validated against an accepted analytic solution for a semi-infinite medium, and cross-validated for a slab geometry with an absorbing inclusion. We then illustrate the utility of the Monte Carlo method with a novel simulation and movie of time-resolved photon migration through a human head and deduce the depth sensitivity of different measurement types. The human head model was obtained from a 3D segmented anatomical MRI. The ability to perform such simulations in a medium with arbitrary boundaries and spatial variation in the optical properties, in our publicly available code ‘tMCimg’ [28

28. J. J. Stott and D. A. Boas, tMCimg: Monte Carlo code for photon migration through general 3D Media. http://www.nmr.mgh.harvard.edu/DOT

29. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine **47**, 131–146 (1995). [CrossRef] [PubMed]

## 2. Method

29. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine **47**, 131–146 (1995). [CrossRef] [PubMed]

^{-1}(NA).

_{a}L) where μ

_{a}is the absorption coefficient and L is the length traveled by the photon [29

29. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine **47**, 131–146 (1995). [CrossRef] [PubMed]

**47**, 131–146 (1995). [CrossRef] [PubMed]

**47**, 131–146 (1995). [CrossRef] [PubMed]

31. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am A **11**, 2727–2741 (1994). [CrossRef]

*μ*

_{s}is old and new scattering coefficient, as described in Jacques and Wang [32]. If the absorption coefficient has changed, then the photon weight is decreased with the new absorption coefficient [32]. The scattering angle is determined by the value of the scattering anisotropy factor,

*g*, within the particular voxel containing the scattering event.

*J*

_{out}(

**r**), is divided by the number of simulated photons. Normalizing the photon fluence,

*Φ*(

**r**), is more involved. To conserve energy, the exiting photon flux plus the number of photons absorbed in the medium must equal the number of simulated photons, which we normalize to 1. The number of photons absorbed at a given point is

*Φ*(

**r**) μ

_{a}(

**r**). Therefore,

**r**

_{i}indicates the position of each surface and volume voxel,

*V*

_{voxel}is the voxel volume, and

*A*

_{i}is the area of the surface element at position

**r**

_{i}. The normalization factor for

*Φ*(

**r**) is determined from this relation.

33. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. **26**, 1335–1337 (2001). [CrossRef]

## 3. Solutions of the Diffusion Equation for Comparison with Monte Carlo

31. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am A **11**, 2727–2741 (1994). [CrossRef]

34. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. **28**, 2331–2336 (1989). [CrossRef] [PubMed]

35. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. **19**, 879–888 (1992). [CrossRef] [PubMed]

36. A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from semi-infinite turbid medium,” Journal of the Optical Society of America **14**, 246–254 (1997). [CrossRef] [PubMed]

34. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. **28**, 2331–2336 (1989). [CrossRef] [PubMed]

36. A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from semi-infinite turbid medium,” Journal of the Optical Society of America **14**, 246–254 (1997). [CrossRef] [PubMed]

**r**

_{s},

**r**

_{d},

*t*) is the photon fluence at the detector position

**r**

_{d}at time t, generated by a point source of amplitude

*S*at position

**r**

_{s}.

*D*=

*v*/(

*3μ*

_{s}

*’*) is the photon diffusion coefficient [37

37. K. Furutsu and Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys.Rev.E **50**, 3634 (1994). [CrossRef]

38. T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am A **14**, 3358–3365 (1997). [CrossRef]

*μ*

_{s}

*’*is the reduced scattering coefficient,

*μ*

_{a}is the absorption coefficient, and

*v*is the speed of light in the medium. The semi-infinite boundary condition is satisfied by the method of images [31

31. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am A **11**, 2727–2741 (1994). [CrossRef]

36. A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from semi-infinite turbid medium,” Journal of the Optical Society of America **14**, 246–254 (1997). [CrossRef] [PubMed]

**r**

_{s,i}.

**r**

_{s}and

**r**

_{d}are the position of the source and detector respectively,

*G*is the Greens function of the photon diffusion equation for the background optical properties given the boundary conditions. For the Born approximation, Φ

_{pert}is not normalized by Φ

_{o}(

**r**

_{s},

**r**

_{d}) If the background is homogeneous then Φ

_{o}and

*G*can be expressed analytically in some simple geometries [34

34. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. **28**, 2331–2336 (1989). [CrossRef] [PubMed]

40. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys Med Biol **37**, 1531–60 (1992). [CrossRef] [PubMed]

_{o}and

*G*is amplitude factor

*S*(see eq. (4)).

### Validation of the Monte Carlo Code in a Homogeneous Semi-Infinite Medium

**47**, 131–146 (1995). [CrossRef] [PubMed]

**14**, 246–254 (1997). [CrossRef] [PubMed]

^{-1}, a scattering anisotropy of

*g*= 0.01, and an absorption coefficient of 0.005 mm

^{-1}. The medium had dimensions of 60 × 60 × 60 mm with the source positioned at (x,y,z) = (30,30,1) mm. All boundaries were treated as index matched. The source was sufficiently far from the edges so that the medium is effectively semi-infinite. A simulation was executed with 10

^{8}photons which took approximately 6 hours on a 1 GHz Pentium 3.

**11**, 2727–2741 (1994). [CrossRef]

## 4. Validating the Monte Carlo code in a slab geometry with an absorbing inclusion

*Φ*

_{o}(

**r**

_{s},

**r**), and from the detector into the medium,

*G*(

**r**,

**r**

_{d}), (see eq. (7), so called adjoint fields [41

41. S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Finite-element-method calculations,” Appl. Opt. **34**, 8026–8037 (1995). [CrossRef] [PubMed]

42. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express **2**, 213–226 (1998). http://www.opticsexpress.org/oearchive/source/4014.htm [CrossRef] [PubMed]

^{-1}, the scattering anisotropy factor was set to 0.01, the absorption coefficient of the background was set to 0.005 mm

^{-1}, while that of the inclusion was increased from 0.005 mm

^{-1}to 0.065 mm

^{-1}. A full Monte Carlo simulation (which traces out the scattering path of every photon) was executed with the different absorption coefficients to calculate the flux of photons received at the detector. These results normalized by the flux detected with no absorbing inclusion are shown by the square symbols in fig. 3b. Notice that as the absorption coefficient initially increases, there is an approximate linear decrease in the intensity which then saturates for higher absorption coefficients. The detected flux of photons for different absorption coefficients can also be calculated from the photon history file produced by a single Monte Carlo simulation, as described above in eq. (2).

## 5. Full 3D Head

43. A. M. Dale, B. Fischl, and M. I. Sereno, “Cortical surface-based analysis. I. Segmentation and surface reconstruction,” Neuroimage **9**, 179–94 (1999). [CrossRef] [PubMed]

^{3}voxels and the Monte Carlo simulation recorded the temporal response and took approximately 10 hours. The continuous-wave and 200 MHz results were obtained from the modulus of Fourier transform of the temporal response. One contour line is shown for each half order of magnitude (10 dB) signal loss, and the contours end after 3 orders of magnitude in loss (60 dB). For the 3D Monte Carlo simulation, we assumed that μ

_{s}’ = 1 mm

^{-1}and μ

_{a}=0.04 mm

^{-1}for the scalp and skull, μ

_{s}’ = 0.01 mm

^{-1}and μ

_{a}=0.001 mm

^{-1}for the CSF, and μ

_{s}’ = 1.25 mm

^{-1}and μ

_{a}=0.025 mm

^{-1}for the gray/white matter. Note how the contours extend several millimeters into the brain tissue, indicating sensitivity to changes in cortical optical properties. The depth penetration difference between the continuous-wave and 200 MHz measurements is difficult to discern. A ratio of the two sensitivity profiles (not shown) shows that the 200 MHz profile is shifted slightly towards the surface. The time-domain sensitivity profiles suggest the possibility of obtaining greater penetration depths in the head from measurements made at longer delay times. This is further supported by the movie of the temporal evolution of a light pulse within the head as shown in fig. 6. This movie illustrates the usefulness of the code for visualizing the temporal evolution of photon migration through a heterogeneous medium.

^{8}photons on the 3D human head, as determined from running 9 independent Monte Carlo simulations with different random number seeds. Fig. 7a shows the flux of photons exiting from the head as a function of distance from the source voxel to each voxel into which photons escaped, i.e. voxels describing the air surrounding the head. This flux is normalized by the number of simulated photons and of course depends on the 3D geometry and spatially varying optical properties. The expected exponential decay of the photon flux with distance is observed. The structure is seen in the data because of the cross-sectional area of different air voxels against the head. Some air voxels adjoined the head on 1, 2, or more surfaces, while some voxels only touched at the corner. The needed correction factor is just the effective cross-sectional area of each air voxel (see eq. (1)). At present, the code does not correct for this cross-sectional area for a curved surface. The noise was determined from the standard deviation of the 9 independent Monte Carlo simulations. This deviation is seen in fig. 7a by the different color circles not perfectly overlapping at larger separations. The signal-to-noise ratio versus separation is shown in fig. 7b.

## 6. Summary

## References and Links

1. | A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. |

2. | B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. |

3. | Q. Zhu, T. Durduran, V. Ntziachristos, M. Holboke, and A. G. Yodh, “Imager that combines near-infrared diffusive light and ultrasound,” Opt. Lett. |

4. | M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, Kidney D., J. Butler, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical mammography with ultrasound localization in a human subject,” J Biomed Opt |

5. | D. A. Benaron, W. F. Cheong, and D. K. Stevenson, “Tissue Optics,” Science |

6. | D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J Cereb Blood Flow Metab |

7. | B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology |

8. | V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res |

9. | V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc Natl Acad Sci U S A |

10. | M. A. Franceschini, V. Toronov, M. Filiaci, E. Gratton, and S. Fanini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express |

11. | S. R. Hintz, D. A. Benaron, A. M. Siegel, A. Zourabian, D. K. Stevenson, and D. A. Boas, “Bedside functional imaging of the premature infant brain during passive motor activation,” J Perinat Med |

12. | A. Bluestone, G. Abdoulaev, C. Schmitz, R. Barbour, and A. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express |

13. | C. S. Robertson, S. P. Gopinath, and B. Chance, “A new application for near-infrared spectroscopy: Detection of delayed intracranial hematomas after head injury,” Journal of Neurotrauma |

14. | S. R. Hintz, W. F. Cheong, J. P. van Houten, D. K. Stevenson, and D. A. Benaron, “Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging,” Pediatr Res |

15. | M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. |

16. | J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,“ Appl. Opt. |

17. | M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys Med Biol |

18. | A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. |

19. | A. Ishimaru, |

20. | E. Okada, M. Schweiger, S. R. Arridge, M. Firbank, and D. T. Delpy, “Experimental validation of Monte Carlo and finite-element methods of estimation of the optical path length in inhomogeneous tissue,” Appl. Opt. |

21. | 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 |

22. | 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. |

23. | M. Firbank, Okada E., and D. T. Delpy, “A theoretical study of the signal contribution of regions of the adult head to near-infrared spectroscopy studies of visual evoked responses,” Neuroimage |

24. | A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys Med Biol |

25. | S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. |

26. | 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 Opt Image Sci Vis |

27. | O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems |

28. | J. J. Stott and D. A. Boas, tMCimg: Monte Carlo code for photon migration through general 3D Media. http://www.nmr.mgh.harvard.edu/DOT |

29. | L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine |

30. | S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in tissues” in |

31. | R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am A |

32. | S. L. Jacques and L. Wang, “Monte-Caro Modeling of Light Transport in Tissues” in |

33. | C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. |

34. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. |

35. | T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. |

36. | A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from semi-infinite turbid medium,” Journal of the Optical Society of America |

37. | K. Furutsu and Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys.Rev.E |

38. | T. Durduran, B. Chance, A. G. Yodh, and D. A. Boas, “Does the photon diffusion coefficient depend on absorption?,” J. Opt. Soc. Am A |

39. | A. C. Kak and M. Slaney, |

40. | S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys Med Biol |

41. | S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Finite-element-method calculations,” Appl. Opt. |

42. | S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express |

43. | A. M. Dale, B. Fischl, and M. I. Sereno, “Cortical surface-based analysis. I. Segmentation and surface reconstruction,” Neuroimage |

**OCIS Codes**

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

(170.5280) Medical optics and biotechnology : Photon migration

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 10, 2001

Revised Manuscript: January 25, 2002

Published: February 11, 2002

**Citation**

David Boas, J. Culver, J. Stott, and A. Dunn, "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head," Opt. Express **10**, 159-170 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-3-159

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### References

- A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends Neurosci. 20, 435-442 (1997). [CrossRef] [PubMed]
- B. W. Pogue and K. D. Paulsen, "High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information," Opt. Lett. 23, 1716-1718 (1998). [CrossRef]
- Q. Zhu, T. Durduran, V. Ntziachristos, M. Holboke and A. G. Yodh, "Imager that combines near-infrared diffusive light and ultrasound," Opt. Lett. 24, 1050-1052 (1999). [CrossRef]
- M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance and A. G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-47. (2000). [CrossRef] [PubMed]
- D. A. Benaron, W. F. Cheong and D. K. Stevenson, "Tissue Optics," Science 276, 2002-2003 (1997). [CrossRef] [PubMed]
- D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong and D. K. Stevenson, "Noninvasive functional imaging of human brain using light," J. Cereb. Blood Flow Metab. 20, 469-77 (2000). [CrossRef] [PubMed]
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