## Finite element method for diffusive light propagations in index-mismatched media

Optics Express, Vol. 12, Issue 8, pp. 1727-1740 (2004)

http://dx.doi.org/10.1364/OPEX.12.001727

Acrobat PDF (2576 KB)

### Abstract

Near-infrared (NIR) light propagations in strongly scattering tissue have been studied in the past few decades and diffusion approximations (DA) have been extensively used under the assumption that the refractive index is constant throughout the medium. When the index is varying, the discontinuity of the fluence rate arises at the index-mismatched interface. We introduce the finite element method (FEM) incorporating the refractive index mismatch at the interface between the diffusive media without any approximations. Intensity, mean time, and mean optical path length were computed by FEM and by Monte Carlo (MC) simulations for a two-layer slab model and a good agreement between the data from FEM and from MC was found. The absorption sensitivity of intensity and mean time measurements was also analyzed by FEM. We have shown that mean time and absorption sensitivity functions vary significantly as the refractive index mismatch develops at the interface between the two layers.

© 2004 Optical Society of America

## 1. Introduction

*ϕ*is the fluence rate,

**J**is the diffuse power flux,

*D*(

**r**)=[3(

*µ*

_{a}(

**r**)+(1-

*g*)

*µ*

_{s}(

**r**))]

^{-1}is the diffusion coefficient,

*µ*

_{a}and

*µ*

_{s}are the absorption and the scattering coefficients, and

*q*

_{0}is an isotropic source distribution. The anisotropy factor

*g*is typically of the order of 0.9 for biological tissue, indicating strongly forward biased scattering. The term

*µ′*

_{s}=(1-

*g*)

*µ*

_{s}is called the reduced scattering coefficient that incorporates the anisotropic scattering effect into the isotropic diffusion equation. For brevity,

*ϕ*(

**r**,

*ω*) has been written to

*ϕ*(

**r**) in Eq. (1).

5. J. Ripoll and M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces,” J. Opt. Soc. Am. A **16**, 1947–1957 (1999). [CrossRef]

**n̂**is a unit outward surface normal vector at the tissue surface. ζ is a coefficient that accounts for the Fresnel reflections at the boundary and is given by

*R*

_{ϕ}and

*R*

_{J}are diffuse reflectances for the fluence rate

*ϕ*and the normal flux

*J*

_{n}=

**n̂**·

**J**, respectively.

*R*

_{ϕ}and

*R*

_{J}are given by

*θ*is the angle of incidence, and

*R*

_{p}is the Fresnel reflection coefficient for unpolarized light. Most works on NIR imaging of tissue have assumed that the refractive index is constant throughout the medium. In that case, both

*ϕ*and

*J*

_{n}are continuous for the overall domain. When there is an index mismatched internal boundary inside the medium,

*ϕ*is discontinuous across the boundary although

*J*

_{n}is still continuous across it [4

4. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A **12**, 2532–2539 (1995). [CrossRef]

6. G. W. Faris, “Diffusion equation boundary conditions for the interface between turbid media: a comment,” J. Opt. Soc. Am. A **19**, 519–520 (2002) [CrossRef]

*i*,

*j*)=(1, 2) and (2, 1) where

*ϕ*

^{(i)}and (

*i*.

**n̂**

_{1}is a unit surface normal vector directed from the medium 1 to the medium 2 at the interface and vice versa for

**n̂**

_{2}.

5. J. Ripoll and M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces,” J. Opt. Soc. Am. A **16**, 1947–1957 (1999). [CrossRef]

*J*

_{n}in Eq. (6) and the amplitude and the phase of the diffuse photon density wave (DPDW) were numerically investigated by FEM and by MC simulations for a two-layer slab model [7

7. H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. **48**, 2713–2727 (2003) [CrossRef] [PubMed]

## 2. Finite element method

_{1}and Ω

_{2}at the interface

*S*with refractive index mismatch. The exterior boundary Γ of Ω is also split into Γ

_{1}and Γ

_{2}as shown in Fig. 1. After splitting the domain, we have

_{L}for

*L*=1, 2. To meet the discontinuity condition given by Eq. (6), we should duplicate each node on

*S*. If the original node is chosen to be possessed by Ω

_{1}, the duplicated node must be assigned to Ω

_{2}and vice versa. We expanded the fluence rate

*ϕ*

^{(L)}(

**r**) by

*ϕ*

^{(L)}(

**r**)=

**r**) using the piece-wise linear nodal basis functions {

**r**)} for Ω

_{L}, where

*N*[

*L*] is the total number of nodes assigned to Ω

_{L}. The FEM discretization of Eq. (8) gives

*L*=1 and 2 where Λ

^{(L)}is given by

**n̂**

_{1}and

**n̂**

_{2}are the unit vectors directed toward Ω

_{2}and Ω

_{1}, respectively, on

*S*as illustrated in Fig. 1 and ζ

^{(L)}is ζ at Γ

_{L}determined by Eq. (3). From Eq. (6), we have

**n̂**

_{1}·

**J**=ζ′

^{(1)}

*ϕ*

^{(1)}-ζϕ

^{(2)}ϕ

^{(2)}and

**n̂**

_{2}·

**J**=ζ′

^{(2)}ϕ

^{(2)}-ζϕ

^{(1)}ϕ

^{(1)}and their substitutions to Eq. (10) result in the following form:

**F**

^{(L)}=

**K**

^{(L)}+

**C**

^{(L)}-

*iω*

**B**

^{(L)}+ζ

^{(L)}

**A**

^{(L)}for

*L*=1, 2. The entries of the matrices are given by

*L*=1, 2 and

*M*=1, 2. In Eq. (14),

*n*

^{(L)}is the refractive index of Ω

_{L}and

*c*

_{0}is the light speed in the vacuum.

5. J. Ripoll and M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces,” J. Opt. Soc. Am. A **16**, 1947–1957 (1999). [CrossRef]

**16**, 1947–1957 (1999). [CrossRef]

## 3. Extraction of time domain measurement data

7. H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. **48**, 2713–2727 (2003) [CrossRef] [PubMed]

8. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. **34**, 8026–8037 (1995) [CrossRef] [PubMed]

*ϕ*

_{n}(

**r**)≡

*t*

^{n}

*(*ϕ ˜

**r**,

*t*)d

*t*where

*(*ϕ ˜

**r**,

*t*) is the fluence rate in the time domain. Because

*ϕ*(

**r**,

*ω*)=

*(*ϕ ˜

**r**,

*t*)exp(

*iωt*)d

*t*we have

*ϕ*

_{n}(

**r**)=(-

*i∂*/

*∂ω*)

^{n}

*ϕ*(

**r**,

*ω*)|

_{ω=0}which is the

*n*-th order derivative of

*ϕ*(

**r**,

*ω*) with respect to

*ω*at

*ω*=0.

*ϕ*

_{0}(

**r**) is nothing else but the DC fluence rate. It is well established that the integrated intensity E(

**r**) and the mean time of flight <

*t*(

**r**)> are given by E(

**r**)=ζ

*ϕ*

_{0}(

**r**) and <

*t*(

**r**)>=

*ϕ*

_{1}(

**r**)/

*ϕ*

_{0}(

**r**), respectively. In the finite element domain, we have

**E**=ζ

**Φ**

_{0}and <

**t**>=

**Φ**

_{1}/

**Φ**

_{0}where

**Φ**

_{0}and

**Φ**

_{1}are the DC and the first temporal moment solution vectors for the fluence rate, respectively. If we rewrite Eq. (11) to

**Φ**

_{0}satisfies

**Φ**

_{1}is derived from the differentiation of Eq. (18) with respect to

*ω*at

*ω*=0 and it becomes

9. 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]

*c*

_{i}<

*t*

_{i}>(

**r**)=[-

*∂ϕ*

_{0}(

**r**)/

*∂µ*

_{a,i}]/

*ϕ*

_{0}(

**r**) where

*c*

_{i}is the speed of light,

*µ*

_{a,i}is the absorption coefficient, and <

*t*

_{i}(

**r**)> is the mean time of flight, respectively, in the

*i*-th layer. When we denote the term -

*∂ϕ*

_{0}/

*∂µ*

_{a,i}as

**Φ**

_{1}by differentiating Eq. (21) with respect to

*µ*

_{a,i}. If one finds the solution vector,

**t**

_{i}> and the partial mean optical path length vector

*c*

_{i}<

**t**

_{i}> for the

*i*-th layer are determined from

*c*

_{i}<

**t**

_{i}>=

**Φ**

_{0}.

## 4. Extraction of absorption sensitivity functions

8. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. **34**, 8026–8037 (1995) [CrossRef] [PubMed]

10. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15**, R41–R93 (1999) [CrossRef]

*A*=1/ζ. Like Eq. (1),

*ψ*(

**r**,

*ω*) has been written to

*ψ*(

**r**) for brevity. When the interfacial index mismatch exists,

*ψ*(

**r**) must also satisfy the saltus condition given by Eq. (6). The

*n*-th temporal moment of the adjoint solution is given by

*ψ*

_{n}(

**r**)≡

*t*

^{n}

*(*ψ ˜

**r**,

*t*)d

*t*where

*(*ψ ˜

**r**,

*t*) is the adjoint solution in the time domain. Since

*ψ*(

**r**,

*ω*)=

*(*ψ ˜

**r**,

*t*)exp(-

*iωt*)d

*t*, we have

*ψ*

_{n}(

**r**)=(

*i∂*/

*∂ω*)

^{n}

*ψ*(

**r**,

*ω*)|

_{ω=0}. In the finite element domain, the adjoint problem given by Eq. (23), Eq. (24), and Eq. (6) with the replacement of

*ϕ*(

**r**) by

*ψ*(

**r**) becomes

**q**has the entry

*q*

_{i}=

*ζφ*

_{i}(

**m**) and

**Ψ**is the adjoint solution vector. Just as we did for

**Φ**

_{0}and

**Φ**

_{1}, the DC adjoint solution vector

**Ψ**

_{0}and the first temporal moment vector

**Ψ**

_{1}are obtained from the solutions of the following equations:

*J*

^{(E)}in the intensity measurement and the absorption sensitivity functions

*J*

^{(<t>)}in the mean time measurement are given by [8

8. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. **34**, 8026–8037 (1995) [CrossRef] [PubMed]

10. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15**, R41–R93 (1999) [CrossRef]

*J*

^{(T)}is the Jacobian of the first temporal moment and is given by

## 5. Results

### 5. 1. 2D Results

*µ′*

_{s}=1.0-mm and

*µ*

_{a}=0.01mm

^{-1}for the whole computing domains. Four different sets of refractive indices (

*n*

_{1},

*n*

_{2}) were generated by permutation of 1.33 and 1.58 like Ref. 7. We have assumed the collimated pencil beam incidence and created a unit delta function source at one reduced scattering length 1/

*µ′*

_{s}below the irradiated surface as is usually done. The upper layer has 5 mm thickness and infinite width. The lower layer has infinite thickness and width in our model. The computing domain was truncated into a rectangle region of 160 mm width and 80 mm height. 22474 triangle elements and 11496 nodes were initially generated for the domain and they were used as the input mesh for the index-matched case. For the index-mismatched case, 161 nodes on the interface were duplicated and total 11657 nodes were used for FEM computations.

^{8}photons and the isotropic scattering was assumed throughout the domain (i.e.,

*g*=0).

*n*

_{1},

*n*

_{2}) sets are plotted in Fig. 4. One can observe that the correct fluence rates in the semi-infinite homogeneous domain are obtained for

*n*

_{1}=

*n*

_{2}. When the refractive index is mismatched (i.e.,

*n*

_{1}≠

*n*

_{2}) at the interface between the layers, the discontinuity of the fluence rate arises at the interface. The fluence rate jumps for

*n*

_{1}<

*n*

_{2}and it drops for

*n*

_{1}>

*n*

_{2}when it goes across the interface from the medium 1 to the medium 2, which is a feature consistent with that shown in Fig. 2.

7. H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. **48**, 2713–2727 (2003) [CrossRef] [PubMed]

*n*

_{1}=

*n*

_{2}, the mean time of flight increases for

*n*

_{1}<

*n*

_{2}and it decreases for

*n*

_{1}>

*n*

_{2}. We found the excellent agreements of the mean time between the FEM data and the MC data. The partial mean optical path length [9

9. 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]

**34**, 8026–8037 (1995) [CrossRef] [PubMed]

*n*

_{1}=

*n*

_{2}, the sensitivity distributions are essentially identical for the index of 1.33 and 1.58 and only the absolute magnitudes differ between the two. For the index-mismatched case, the discontinuities of the sensitivity functions occur for both the intensity and the mean time of flight measurements. In comparison with the case of

*n*

_{1}=

*n*

_{2}, the sensitivity in the layer 2 increases for

*n*

_{1}<

*n*

_{2}and it decreases for

*n*

_{1}>

*n*

_{2}below the interface. This indicates that the interfacial index mismatch will have significant influences on the quality of the tomographic reconstruction of the absorption coefficients in the layered structure often encountered in real tissues. The suppression of the sensitivity in the deeper tissue layer (i.e. layer 2) for

*n*

_{1}>

*n*

_{2}implies that the measured intensity and mean time data will be less altered by the absorption changes there. The enhancement of sensitivity in the deeper tissue layer for

*n*

_{1}<

*n*

_{2}implies that the measured data will be greatly altered by the absorption changes there in comparion with the index-matched case. Therefore, the absorption inhomogeneities in the deeper layer will be easy or hard to reconstruct depending upon the refractive index mismatch at the interface.

### 5.2. 3D Results

## 6. Discussions

**48**, 2713–2727 (2003) [CrossRef] [PubMed]

*n*

_{2}varies, the mean time increases for

*n*

_{2}>

*n*

_{1}and decreases for

*n*

_{2}<

*n*

_{1}in comparison with the case for

*n*

_{2}=

*n*

_{1}. This phenomenon comes partly from the variation of the light speed. But the investigation of the optical path length shows that the distance the average photon travels in each layer is significantly altered by the Fresnel reflections at the internal boundary. For

*n*

_{2}>

*n*

_{1}, if photons entered the layer 2 from the layer 1 they have difficulties in coming back to the layer 1 due to the total internal reflections and tend to reside in the layer 2; as a result, the absorption sensitivity in the layer 2 increases. For

*n*

_{2}<

*n*

_{1}, the transmission of light into the layer 2 from the layer 1 tends to be prohibited again due to the total internal reflections and photons tend to reside in the layer 1; in this case, the absorption sensitivity in the layer 2 decreases. For

*n*

_{2}=

*n*

_{1}, there are no restrictions for the photons to move across the interface. The unbalance of the incoming and the outgoing photons across the interface due to the index mismatch is responsible for the path length variations, for the mean time variations and also for the measurement sensitivity variations.

*ϕ*/3|

*J*

_{n}|. In order for the DA to be valid,

*ϕ*/3|

*J*

_{n}| should be large compared to 1. The air/tissue boundary condition given by Eq. (2) is often called the partial current boundary condition (PCBC) and the existence of such boundary strains the DA [3

3. R. C. Haskell, L. O. Svaasand, T. -T. Tsay, T. -C. 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]

*ϕ*/3|

*J*

_{n}| is deterministic by the boundary and depends only on the refractive index of the underlying tissue. For the tissue/tissue boundary, however,

*ϕ*/3|

*J*

_{n}| is not deterministic. It is generally dependent upon the positions along the interface and the geometry of the interface. Rearranging the terms in Eq. (6) gives rise to

*C*

_{1}≡1/ζ′

^{(1)}and

*J*

_{n}≡

**n̂**

_{1}·

**J**, where

*ϕ*

_{1}and

*ϕ*

_{1}are the fluence rate in the layer 1 and layer 2, respectively, at the interface. In principle, any set of (

*ϕ*

_{1},

*ϕ*

_{2},

*J*

_{n}) is allowed under the constraint imposed by Eq. (31). We have collected (

*ϕ*

_{1},

*ϕ*

_{2},

*J*

_{n}) values at the nodes along the interface for the 2D mesh shown in Fig. 3 and plotted

*ϕ*

_{1}/3

*J*

_{n}with respect to

*ϕ*

_{2}/3

*J*

_{n}in Fig. 12. As shown in Fig. 12, (

*ϕ*

_{1}/3

*J*

_{n},

*ϕ*

_{2}/3

*J*

_{n}) pairs are located at diverse positions on the locus defined by Eq. (31) with

*C*

_{1}=0.431 and 0.608 for (

*n*

_{1},

*n*

_{2})=(1.33, 1.58) and (1.58, 1.33), respectively. For (

*n*

_{1},

*n*

_{2})=(1.33, 1.58), The minimum value of

*ϕ*

_{1}/3|

*J*

_{n}| and

*ϕ*

_{2}/3|

*J*

_{n}| is 2.463 and 3.275, respectively, and such minima occur at x=0. When (

*ϕ*

_{1}/

*ϕ*

_{2})/(

*n*

_{1}/

*n*

_{2})

^{2}becomes close to 1,

*ϕ*

_{1}/3|

*J*

_{n}| and

*ϕ*

_{2}/3|

*J*

_{n}| reaches the maximum value over hundreds. At a large x,

*ϕ*

_{1}/3|

*J*

_{n}| and

*ϕ*

_{2}/3|

*J*

_{n}| tend to saturate to roughly 7.5 and 10.8, respectively. For (

*n*

_{1},

*n*

_{2})=(1.58, 1.33), the mimimum value of

*ϕ*

_{1}/3|

*J*

_{n}| and

*ϕ*

_{2}/3|

*J*

_{n}| is 4.915 and 3.337, respectively, and their overall variation patterns with respect to x are qualitatively identical to the previous case. Even the index-matched case shows the similar patterns. As shown in Fig. 12, (

*ϕ*

_{1}/

*ϕ*

_{2})/(

*n*

_{1}/

*n*

_{2})

^{2}is close to 1 in the vicinity of x=11 mm, where the sign of

*J*

_{n}changes and the maximum value of

*ϕ*

_{1,2}/3|

*J*

_{n}| appears.

*ϕ*

_{1,2}/3|

*J*

_{n}| may reduce and possibly the DA is strained at the positions around which the minimum occurs. But the strain of the DA in this case occurs not because of the presence of the interface but because of the location of the interface near the source.

## 7. Conclusions

## Acknowledgments

## References and links

1. | A. Ishimaru, |

2. | M. Keizer, M. Star, and P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. |

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

4. | R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A |

5. | J. Ripoll and M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces,” J. Opt. Soc. Am. A |

6. | G. W. Faris, “Diffusion equation boundary conditions for the interface between turbid media: a comment,” J. Opt. Soc. Am. A |

7. | H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. |

8. | S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. |

9. | 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. |

10. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

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

(170.7050) Medical optics and biotechnology : Turbid media

(290.1990) Scattering : Diffusion

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 18, 2004

Revised Manuscript: April 2, 2004

Published: April 19, 2004

**Citation**

Jae Hoon Lee, Seunghwan Kim, and Youn Kim, "Finite element method for diffusive light propagations in index-mismatched media," Opt. Express **12**, 1727-1740 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1727

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### References

- A. Ishimaru, Wave propagation and scattering in random media (Academic, 1978), Chap 7-9.
- M. Keizer, M. Star, and P. R. M. Storchi, �??Optical diffusion in layered media,�?? Appl. Opt. 27, 1820-1824 (1988). [CrossRef]
- R. C. Haskell, L. O. Svaasand, T. �??T. Tsay, T. �??C. 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]
- R. Aronson, "Boundary conditions for diffusion of light,�?? J. Opt. Soc. Am. A 12, 2532-2539 (1995). [CrossRef]
- J. Ripoll and M. Nieto-Vesperinas, �??Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces,�?? J. Opt. Soc. Am. A 16, 1947-1957 (1999). [CrossRef]
- G. W. Faris, �??Diffusion equation boundary conditions for the interface between turbid media: a comment,�?? J. Opt. Soc. Am. A 19, 519-520 (2002) [CrossRef]
- H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. Paulsen, �??The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,�?? Phys. Med. Biol. 48, 2713-2727 (2003) [CrossRef] [PubMed]
- S. R. Arridge and M. Schweiger, �??Photon measurement density functions. Part 2: Finite-element-method calculations,�?? Appl. Opt. 34, 8026-8037 (1995) [CrossRef] [PubMed]
- 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]
- S. R. Arridge, �??Optical tomography in medical imaging,�?? Inverse Problems 15, R41-R93 (1999) [CrossRef]

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