## Technique for handling wave propagation specific effects in biological tissue: Mapping of the photon transport equation to Maxwell’s equations

Optics Express, Vol. 16, Issue 22, pp. 17792-17807 (2008)

http://dx.doi.org/10.1364/OE.16.017792

Acrobat PDF (1066 KB)

### Abstract

A novel algorithm for mapping the photon transport equation (PTE) to Maxwell’s equations is presented. Owing to its accuracy, wave propagation through biological tissue is modeled using the PTE. The mapping of the PTE to Maxwell’s equations is required to model wave propagation through foreign structures implanted in biological tissue for sensing and characterization of tissue properties. The PTE solves for only the magnitude of the intensity but Maxwell’s equations require the phase information as well. However, it is possible to construct the phase information approximately by solving the transport of intensity equation (TIE) using the full multigrid algorithm.

© 2008 Optical Society of America

## 1. Introduction

1. S. Kumar, K. Mitra, and Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. **35**, 3372–3378 (1996). [CrossRef] [PubMed]

2. M. Premaratne, E. Premaratne, and A. J. Lowery, “The photon transport equation for turbid biological media with spatially varying isotropic refractive index,” Opt. Express **13**, 389–399 (2005). [CrossRef] [PubMed]

3. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. **14**, 105–112 (2008). [CrossRef]

## 2. Formulation: construction of phase information from a radiance profile

## 2.1. Modeling light propagation in tissue

3. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. **14**, 105–112 (2008). [CrossRef]

*I*

*(*

_{PTE}*z*,

*u*,

*ϕ*,

*t*) is the radiance (units:

*W*.

*m*

^{-2}.

*sr*

^{-1}.

*Hz*

^{-1}), (

*z*,

*θ*,

*ϕ*) are the standard spherical coordinates,

*u*=cos

*θ*,

*t*is the time variable,

*σ*

*and*

_{t}*σ*

*are attenuation and scattering coefficients, respectively, and*

_{s}*σ*

*=*

_{t}*σ*

*+*

_{s}*σ*

*, where σ*

_{a}*is the absorption coefficient. The speed of light in the medium is denoted by*

_{a}*v*,

*P*(

*u*

^{′},

*ϕ*

^{′};

*u*,

*ϕ*) is the phase function and

*F*(

*z*,

*u*,

*ϕ*,

*t*) refers to the source term.

3. C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. **14**, 105–112 (2008). [CrossRef]

*I*

*. The LRKF method reduces the transient PTE, given by Eq.(1), to an ordinary differential equation of only one independent variable,*

_{PTE}*z*, and solves it numerically. In the LRKF algorithm the discrete ordinate method [6] is used to discretize the azimuthal and zenith angles (

*ϕ*and

*θ*), a Laguerre expansion [3

**14**, 105–112 (2008). [CrossRef]

*z*.

## 2.2. Derivation of the transport-of-intensity equation for phase construction

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

*ε*is the electric permittivity of the medium, µ

_{0}is the permeability of free space,

**E**and

**H**denote the electric field and the magnetic field, respectively. From Eq.(2) and Eq.(3), since there is no mixing between any of the components of the electric and the magnetic field vectors, we can move on to a scalar theory [8

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

*x*,

*y*,

*z*,

*t*) describes the electromagnetic field and it is complex. Using the Fourier integral Ψ(

*x*,

*y*,

*z*,

*t*) can be expressed as [8

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

*k*

_{0}=

*ω*/

*c*is the wave number in free space. Then, identifying

*ε*

*(*

_{ω}*x*,

*y*,

*z*)

*µ*

_{0}

*c*

^{2}as the square of the position-dependent refractive index,

*n*

*(*

_{ω}*x*,

*y*,

*z*), of the medium, we can re-write Eq.(6) as

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

*Ψ*

*(*

_{ω}*x*,

*y*,

*z*) in Eq.(7) as a perturbed plane wave [8

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

*e*

*represents the unscattered plane wave and*

^{jkz}*Ψ*̃

*(*

_{n}*x*,

*y*,

*z*) represents the complex envelope [9]. That is, we have considered the paraxial condition where the rays are not exactly parallel to each other; or in other words, a field with perturbed wave fronts. Using Eq.(8) in Eq.(7), we obtain

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

*Ψ*̃

*(*

_{ω}*x*,

*y*,

*z*), to be “beam-like” so that its second derivative in the z direction is much smaller in magnitude than its second derivative in the

*x*and

*y*directions. Therefore, the

*I*represents the irradiance (units:

*W*.

*m*

^{-2}.

*Hz*

^{-1}) and

*ϕ*represents the phase. Using Eq.(13) in Eq.(12) and separating the imaginary part we get the following relationship [8

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

11. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

## 2.3. Construction of phase information from the irradiance profile

*ϕ*(

*x*,

*y*,

*z*) numerically using a suitable technique such as the full multigrid algorithm [12

12. T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D: Appl. Phys. **32**, 563–567 (1999). [CrossRef]

13. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. **199**, 65–75 (2001). [CrossRef]

11. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

14. T. E. Gureyev and K.A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A **13**, 1670–1682 (1996). [CrossRef]

15. T. E. Gureyev and K.A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. **133**, 339–346 (1997). [CrossRef]

12. T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D: Appl. Phys. **32**, 563–567 (1999). [CrossRef]

13. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. **199**, 65–75 (2001). [CrossRef]

*I*(

*x*,

*y*,

*z*)>0 over a simply-connected planar region [17

17. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport of intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A **12**, 1942–1946 (1995). [CrossRef]

12. T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D: Appl. Phys. **32**, 563–567 (1999). [CrossRef]

13. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. **199**, 65–75 (2001). [CrossRef]

*i*=1,…,

*M*,

*j*=1,…,

*M*for

*M*×

*M*grid points. Also,

*I*

*,*

_{i}*=*

_{j}*I*(

*x*

*,*

_{i}*y*

*,*

_{j}*z*),

*ϕ*

*,*

_{i}*=*

_{j}*ϕ*(

*x*

*,*

_{i}*y*

*,*

_{j}*z*), Δ=

*x*

*+1-*

_{i}*x*

*=*

_{i}*y*

*+1-*

_{j}*y*

*and*

_{j}*z*=

*z*and

*z*=

*z*+

*δz*, we obtain two intensity profiles. Thus, we can use the following approximations in Eq.(17).

*θ*is the zenith angle used in Eq.(1) and

*d*

*ω*̃ is an infinitesimal solid angle [18, 19

19. R. Ramamoorthi and P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object,” J. Opt. Soc. Am. A **18**, 2448–2459 (2001). [CrossRef]

*x*,

*y*and

*z*can be approximately calculated from the two intensity profiles, as shown in Eq.(17), Eq.(18) and Eq.(19), the only unknown in Eq.(17) is

*ϕ*

*,*

_{i}*. Hence, we can use the full multigrid algorithm to solve Eq.(17) for*

_{j}*ϕ*

*,*

_{i}*and thus the phase can be retrieved on each grid point.*

_{j}*h*as

*is a linear operator, the error satisfies*

_{h}*in order to find*

_{h}*v*

*. Classical iteration methods, such as Jacobi or Gauss-Seidel can be used to do this.*

_{h}*of Γ*

_{H}*on a coarser grid with mesh size*

_{h}*H*. Then the residual equation, Eq.(24), is approximated by

*has smaller dimension, Eq.(26) is easier to solve than Eq.(24). In the full multigrid algorithm, the first approximation is obtained by interpolating from a coarse-grid solution and at the coarsest level we start with the exact solution [16]. Using the full multigrid algorithm as detailed above we can solve Eq.(16) for*

_{H}*u*. Thus, the phase at each grid point,

*ϕ*

*,*

_{i}*is retrieved.*

_{j}11. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

21. D. Paganin and K. A. Nugent, “Non-interferometric phase imaging with partially coherent light,” Phys. Rev. Lett. **80**, 2586–2589 (1998). [CrossRef]

22. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. **23**, 817–819 (1998). [CrossRef]

23. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III: The effects of noise,” J. Microscopy **214**, 51–61 (2004). [CrossRef]

*δz*(cf. Eq.(19)), leads to a blurring of the retrieved phase which becomes negligibly small if

*δz*tends to zero from above. Reference [23

23. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III: The effects of noise,” J. Microscopy **214**, 51–61 (2004). [CrossRef]

*δz*in the presence of a given level of noise, this optimal defocus distance being proportional to the cube root of the standard deviation of the noise. Typical reconstruction accuracies from experiments involving TIE-based phase retrieval are on the order of 1–5% [8

8. D. M. Paganin, *Coherent X-Ray Optics*, (Oxford University Press, New York, 2006). [CrossRef]

22. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. **23**, 817–819 (1998). [CrossRef]

24. W. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. **26**, 2166–2185 (1990). [CrossRef]

24. W. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. **26**, 2166–2185 (1990). [CrossRef]

24. W. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. **26**, 2166–2185 (1990). [CrossRef]

26. S. A. Prahl, J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. **32**, 559–568 (1993). [CrossRef] [PubMed]

27. C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Trans. **43**, 2009–2020 (2000). [CrossRef]

28. A. E. Profio, “Light transport in tissue,” Appl. Opt.28, 2216–2222 (1989). [CrossRef] [PubMed]

29. M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue 1. Models of radiation transport and their application,” Lasers in Medical Science **6**, 155–168, (1991). [CrossRef]

28. A. E. Profio, “Light transport in tissue,” Appl. Opt.28, 2216–2222 (1989). [CrossRef] [PubMed]

29. M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue 1. Models of radiation transport and their application,” Lasers in Medical Science **6**, 155–168, (1991). [CrossRef]

## 3. Example: wave propagation through a slit in a metal screen implanted in tissue

*z*=

*z*

*and then converting these to the corresponding electric and magnetic fields, so that the field due to the slit in the metal screen can be modeled. Then, at the exit of the metal screen, the electric and magnetic fields can be converted back to the intensity profile so that the tissue layer beyond this plane can be modeled by solving the PTE.*

_{A}*F*(

*z*,

*u*,

*ϕ*,

*t*)=0 in Eq. (1).

**14**, 105–112 (2008). [CrossRef]

*z*=0 to

*z*=

*z*

*-. Thus, we can obtain the radiance profile at the plane just before the tissue-metal screen interface (i.e. at z=zA-). However, in order to model the propagation of the laser pulse beyond this plane, Maxwell’s equations should be used. Maxwell’s equations require the phase of the field in addition to the magnitude. Thus, the phase information of the field at*

_{A}*z*=

*z*

*- should be retrieved in order to model the light propagation through the slit in the metal screen.*

_{A}*I*

*obtained by solving Eq.(1) should be converted to an irradiance profile*

_{PTE}*I*, using the relationship in Eq.(20). Thus, the irradiance at

*z*=

*z*

*-,*

_{A}*I*

*-, and at*

_{A}*z*=

*z*

*--*

_{A}

_{δ}*,*

_{ z}*I*

*--*

_{A}

_{δ}*, can be obtained by solving the PTE for radiance and integrating over the hemisphere. Then, we use the approximations in Eq.(18) and Eq.(19) in order to solve the TIE. That is,*

_{z}**32**, 563–567 (1999). [CrossRef]

**199**, 65–75 (2001). [CrossRef]

*ϕ*(

*x*,

*y*,

*z*). Thus, the phase at the tissue-metal screen interface is retrieved using the intensity values at two infinitesimally separated planes.

**E**

_{0}, as shown in Figure 1, the electric field at

*z*=

*z*

^{+}

*can thus be written as*

_{A}*z*=

*z*

^{+}

*can be obtained from*

_{A}5. S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, “Enhanced transmission versus localization of a light pulse by a subwavelength metal slit,” Phys. Rev. B. **70**, 195428 (2004). [CrossRef]

5. S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, “Enhanced transmission versus localization of a light pulse by a subwavelength metal slit,” Phys. Rev. B. **70**, 195428 (2004). [CrossRef]

**e**

*is the unit vector in*

_{y}*y*direction. Since the time variation of

*U*(

*x*,

*z*,

*t*) is very slow, it approximately satisfies the Helmholtz equation. Hence,

*j*=1,2,3 and

*k*

*is the wave number in region*

_{j}*j*. The field in region 1 can be decomposed into three components:

*U*

*represents the incident field,*

^{i}*U*

^{r}represents the field that would be reflected if there were no slit in the screen and

*U*

*represents the diffracted field in region 1 due to the presence of the slit [5*

^{d}5. S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, “Enhanced transmission versus localization of a light pulse by a subwavelength metal slit,” Phys. Rev. B. **70**, 195428 (2004). [CrossRef]

**70**, 195428 (2004). [CrossRef]

**70**, 195428 (2004). [CrossRef]

**70**, 195428 (2004). [CrossRef]

**E**

_{d}_{2}) just after the metal screen is obtained, these can be combined to obtain the intensity (i.e. the irradiance) using the relationship

*v*and

*ε*are the propagation speed and the permittivity in the medium, respectively. Once the irradiance profile on the plane

*z*=

*d*

_{2}is obtained, it should be converted back to a radiance profile so that the PTE can be used to model the light propagation beyond this plane. Figure 4 shows a strategy that can be used for mapping the irradiance profile to the radiance profile, as needed for solving the PTE. In Fig. 4, the axes (

*x*,

*y*,

*z*) represent the global coordinate system used in solving the PTE; also shown is the ray-centred spherical coordinate system used for describing the irradiance-to-radiance mapping forward hemisphere.

*et al.*[19

19. R. Ramamoorthi and P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object,” J. Opt. Soc. Am. A **18**, 2448–2459 (2001). [CrossRef]

*z*=

*d*

_{2}is converted back to a radiance profile at each point, in the forward hemisphere, on the interface, the LRKF method can be used to model the light propagation through the remaining layers of tissue.

## 4. Numerical results and discussion

*I*

*, and thus representing*

_{PTE}*I*

*as*

_{PTE}*I*

*/*

_{PTE}*I*

_{0}does not change the equation. Therefore, we use an arbitrary scale for

*I*

*throughout this paper. The time units are normalized by*

_{PTE}*T*

*, spatial units by*

_{s}*Z*

*and scattering and absorption coefficients by 1/*

_{s}*Z*

*.*

_{s}*T*is the factor determining the width of the input pulse while

*t*

_{0}determines the time at which pulse attains its peak value.

*T*

*=*

_{s}*T*. We choose this normalization factor due to the fact that the Laguerre approximation of the Gaussian pulse is very accurate for pulses with

*T*=1 or greater. Therefore, with this scaling it is possible to obtain very accurate results even for very narrow pulses, which are used in many biomedical applications. For pulses with other shapes, it is recommended that a least square fit is used to obtain a Gaussian approximation, subsequently setting

*T*

*to be the width of that Gaussian pulse. We have set*

_{s}*Z*

*=*

_{s}*v*×

*T*̄. Here,

*T*̄ can be chosen to suit the particular application. However, these scaling factors should be chosen carefully so that the matrices that are used in the LRKF method remain well-conditioned. For the simulations presented in this paper, without loss of generality, we have chosen

*T*/

*T*̄=1 so that

*Z*

*=*

_{s}*v*×

*T*.

*et al.*[19

19. R. Ramamoorthi and P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object,” J. Opt. Soc. Am. A **18**, 2448–2459 (2001). [CrossRef]

*z*=2, on a plane just before the tissue-metal screen interface, obtained by solving the PTE using the afore-mentionedLRKF method. Here, we have modeled the tissue layer with a normalized scattering coefficient of 0.3, normalized absorption coefficient of 0.5 and the Henyey-Greenstein phase function [18] with an asymmetry factor of 0.7. The normalized velocity was taken to be 1 while the refractive index of the tissue layer was assumed to be 1.37. Figure 6 shows how the irradiance profile on the (

*x*,

*y*) grid at

*z*=2 varies with time.

*z*=1.95 was obtained, and these two profiles were used to retrieve the phase of the field at

*z*=2. For phase retrieval, we first translated the code given in [16] for the full multigrid algorithm to Matlab scripting, and then modified it to solve the TIE, which involved slight modifications to a few subroutines. Then, the irradiance values and the phase values were combined according to Eq.(29) to construct the electric field at

*z*=2, which is shown in figure 7.

## 5. Conclusion

30. G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. **23**, 1721–1733 (1987). [CrossRef]

30. G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. **23**, 1721–1733 (1987). [CrossRef]

^{2}per pulse or 1.0 Watt/cm

^{2}for continuous exposure [31]. However, structures such as photonic crystals can be implanted to obtain enhanced signals by properly engineering the photon density of states. The proposed technique can be used to model such foreign structures implanted in tissue. Since our main focus is on how to map the photon transport equation to Maxwell equations, for our simulations we have used a simple foreign structure, a metal screen with a slit, in order to introduce the proposed technique without the additional mathematical complexity of modeling more complicated structures.

## References and links

1. | S. Kumar, K. Mitra, and Y. Yamada, “Hyperbolic damped-wave models for transient light-pulse propagation in scattering media,” Appl. Opt. |

2. | M. Premaratne, E. Premaratne, and A. J. Lowery, “The photon transport equation for turbid biological media with spatially varying isotropic refractive index,” Opt. Express |

3. | C. C. Handapangoda, M. Premaratne, L. Yeo, and J. Friend, “Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue,” IEEE J. Sel. Top. Quantum Electron. |

4. | F. L. Neerhoff and G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. |

5. | S. V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, “Enhanced transmission versus localization of a light pulse by a subwavelength metal slit,” Phys. Rev. B. |

6. | S. Chandrasekhar, |

7. | S. C. Chapra and R. P. Canale, |

8. | D. M. Paganin, |

9. | M. Born and E. Wolf, |

10. | M. R. Spiegel, |

11. | M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. |

12. | T. E. Gureyev, C. Raven, A. Snigireva, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D: Appl. Phys. |

13. | L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. |

14. | T. E. Gureyev and K.A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A |

15. | T. E. Gureyev and K.A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. |

16. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

17. | T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport of intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A |

18. | G. E. Thomas and K. Stamnes, Radiative Transfer In The Atmosphere And Ocean, (Cambridge University Press, 1999). |

19. | R. Ramamoorthi and P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object,” J. Opt. Soc. Am. A |

20. | D. Paganin, K. A. Nugent, and Peter Hawkes (editor), |

21. | D. Paganin and K. A. Nugent, “Non-interferometric phase imaging with partially coherent light,” Phys. Rev. Lett. |

22. | A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. |

23. | D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III: The effects of noise,” J. Microscopy |

24. | W. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. |

25. | M. H. Niemz, |

26. | S. A. Prahl, J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. |

27. | C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Trans. |

28. | A. E. Profio, “Light transport in tissue,” Appl. Opt.28, 2216–2222 (1989). [CrossRef] [PubMed] |

29. | M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue 1. Models of radiation transport and their application,” Lasers in Medical Science |

30. | G. Yoon, A. J. Welch, M. Motamedi, and M. C. J. van Gemert. “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. |

31. | W. D. Burnett, “Evaluation of laser hazards to the eye and the skin,” Amer. Ind. Hyg. Assoc. J. |

**OCIS Codes**

(170.1470) Medical optics and biotechnology : Blood or tissue constituent monitoring

(170.4580) Medical optics and biotechnology : Optical diagnostics for medicine

(170.6930) Medical optics and biotechnology : Tissue

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: September 4, 2008

Revised Manuscript: September 28, 2008

Manuscript Accepted: October 10, 2008

Published: October 17, 2008

**Virtual Issues**

Vol. 3, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Chintha C. Handapangoda, Malin Premaratne, David M. Paganin, and Priyantha R. D. S. Hendahewa, "Technique for handling wave propagation specific effects in biological tissue: Mapping of the photon transport equation to Maxwell’s equations," Opt. Express **16**, 17792-17807 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-22-17792

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