## 3D optical tomography in the presence of void regions

Optics Express, Vol. 7, Issue 13, pp. 462-467 (2000)

http://dx.doi.org/10.1364/OE.7.000462

Acrobat PDF (378 KB)

### Abstract

We present an investigation of the effect of a 3D non-scattering gap region on image reconstruction in diffuse optical tomography. The void gap is modelled by the Radiosity-Diffusion method and the inverse problem is solved using the adjoint field method. The case of a sphere with concentric spherical gap is used as an example.

© Optical Society of America

## 1 Introduction

1. S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Problems **15**, R41–R93 (1999). [CrossRef]

1. S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Problems **15**, R41–R93 (1999). [CrossRef]

2. S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and Reconstruction,” Phys. Med. Biol. **42**, 841–853 (1997). [CrossRef] [PubMed]

3. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. **43**, 1285–1302 (1998). [CrossRef] [PubMed]

4. O. Dorn, “A Transport-BackTransport Method for Optical Tomography,” Inverse Problems **14**, 1107–1130 (1998). [CrossRef]

5. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. **26**1698–1707 (1999). [CrossRef] [PubMed]

*Radiosity-Diffusion*model which assumes diffusive regions coupled by non-scattering voids [7

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

8. 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 Non-Scattering Regions,” Med. Phys. **27**, 252–264 (2000). [CrossRef] [PubMed]

*non-local*boundary conditions [9

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

10. H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A **17**, 1659–1670 (2000). [CrossRef]

11. M. Schweiger and S. R. Arridge, “Comparison of 2D and 3D reconstruction algorithms in Optical Tomography,” Appl. Opt. **37**, 7419–7428 (1998). [CrossRef]

## 2 The Radiosity-Diffusion Model

*R*diffusing regions {Ω

_{1}, Ω

_{2}, … Ω

_{R}} and

*V*void regions {Ξ

_{1}, Ξ

_{2},… Ξ

_{V}}. Let Ω

^{d}=

_{i}be the union of all diffusing regions and Ξ

^{d}=

_{i}be the union of all void regions. Thus Ω=Ω

^{d}∪ Ξ

^{d}. Each region has an outer boundary

_{i}=∂

_{k}∂

_{i}=∂

_{k}∂

*∂*Ω.

*r*;

*ω*) at modulation frequency

*ω*satisfies the homogeneous equation

*κ*=1/(3(

*µ*

_{a}+

*µ′*

_{s})) is the diffusion coefficient defined in terms of

*µ*

_{a}(

*r*) and

*µ′*

_{s}(

*r*), the spatially varying absorption and reduced scattering coefficients respectively and with local inhomogeneous boundary conditions

*η*is the incoming flux modelled as a Neumann source [13

13. S. R. Arridge and M. Schweiger, “The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions,” Med. Phys. **22**, 1779–1792, (1995). [CrossRef] [PubMed]

*v*is the surface normal pointing into the void region, and non-local boundary conditions

*h*(

*m*,

*m′*) is unity if

*m*,

*m′*are in line of sight across the void, and zero otherwise. A more exact boundary condition replaces the term Φ(

*m′*;

*ω*)/2

*A*in eq(3)by

*R*

^{(0)},

*R*

^{(1)}are derived from the Fresnel coefficients taken over local coordinates

*R*

^{(0)}=2

*R*(

*ϑ*)sin

*ϑ*cosϑdϑ,

*R*

^{(1)}=3

*R*(ϑ)sin ϑcos

^{2}ϑdϑ

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

*A*=(1-

*R*

^{(1)})/(1-

*R*

^{(0)}). For each Neumann source term

*η*the measureable is

## 3 Inverse Problem

14. F. Natterer and F. Wübbeling, *Mathematical Methods in Image Reconstruction* (SIAM, Philadelphia, 2001). [CrossRef]

*η*

_{j};

*j*=1,…,

*S*}, and given data {

*g*

_{j};

*j*=1,…,

*S*}, and look for the minimisation of the norm

*C*with respect to the data from the

*j*

^{th}source is given by

_{j}the solution to the adjoint diffusion equation

10. H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A **17**, 1659–1670 (2000). [CrossRef]

## 4 Implementation in 3D

15. J. Schoberl, “NetGen”, http://www.sfb013.uni-linz.ac.at/joachim/netgen/

*u*

_{k};

*k*=1…

*D*} with value one at nodal points

*N*

_{k}and zero at all other nodes. For simplicity we consider the case of concentric spherical surfaces which are defined parametrically. The discretisation of eq(1) leads to sparse symmetric matrices, whilst the implementation of the non-local boundary conditions eq(3) leads to a dense

*coupling matrix*whose order,

*D*

_{B}, is the number of nodes in the void boundary. In 3D this number is proportional to the surface area of the void surfaces which is the principle extra overhead to the computational requirements of the foward solver. In the previous 2D treatment [8

8. 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 Non-Scattering Regions,” Med. Phys. **27**, 252–264 (2000). [CrossRef] [PubMed]

*h*(

*m*,

*m′*) as well as the surface normal functions

*directly from the parametric representation of the surfaces which greatly simplifies the setup time of the forward solver. The function*ν ^

*form factor*corresponding to its use in Computer Graphics [16]. In our current implementation we approximate this function by a bilinear expansion over the FEM shape functions on the surface of the void [17]. Between two element faces

*τ*

_{α},

*τ*

_{α′}; ∊

*δ*Ξ containing

*N*,

*N′*nodes respectively the form factor can be approximated by

*n*(

*k*) maps the local node to the global nodes in the complete mesh.

## 5 Results

*mm*with concentric sphere radius

*r*

_{inner}and a concentric void gap with outer radius

*r*

_{outer}. 32 sources and detectors are arranged in three rings. We used background parameters

*µ*

_{a}=0.01

*mm*

^{-1},

*µ′*

_{s}=1

*mm*

^{-1}with

*µ*

_{a}=0.005 in the void region. For the sources and detectors we use a cosine weighted patch in the parameters of the surface representation. For the reconstruction basis we use (20×20×20) tricubic interpolated voxels.

*r*

_{inner}=17,16,15

*mm*, and in each case

*r*

_{outer}=20

*mm*. The difference between each void case and the solid sphere emphasises the increased light intensity and decreased mean time in the gap region.

*mm*gap. The blob had a radius of 3

*mm*and an absorption of 0.02

*mm*

^{-1}with its center placed at position (12,0,0). The images shown are at iteration 40 of the conjugate gradient scheme. The reconstruction shows good localisation despite quite a large degree of broadening.

## 6 Discussion and conclusions

10. H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A **17**, 1659–1670 (2000). [CrossRef]

## Acknowledgments

## References and links

1. | S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Problems |

2. | S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and Reconstruction,” Phys. Med. Biol. |

3. | A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. |

4. | O. Dorn, “A Transport-BackTransport Method for Optical Tomography,” Inverse Problems |

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

6. | O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express This issue (2000). |

7. | 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. |

8. | 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 Non-Scattering Regions,” Med. Phys. |

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

10. | H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A |

11. | M. Schweiger and S. R. Arridge, “Comparison of 2D and 3D reconstruction algorithms in Optical Tomography,” Appl. Opt. |

12. | J. Ripoll, Ph.D. thesis, University Autónoma of Madrid, 2000. |

13. | S. R. Arridge and M. Schweiger, “The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions,” Med. Phys. |

14. | F. Natterer and F. Wübbeling, |

15. | J. Schoberl, “NetGen”, http://www.sfb013.uni-linz.ac.at/joachim/netgen/ |

16. | M. F. Cohen and J. R. Wallace, |

17. | H. R. Zatz, Master’s thesis, Cornell University, 1993. |

18. | S. R. Arridge and M. Schweiger, “Photon Measurement Density Functions. Part 2: Finite Element Calculations,” Appl. Opt. |

19. | V. Kolehmainen, M. Vaukhonen, J. P. Kaipio, and S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express This issue (2000). |

**OCIS Codes**

(100.6890) Image processing : Three-dimensional image processing

(100.6950) Image processing : Tomographic image processing

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

**ToC Category:**

Focus Issue: Diffuse optical tomography

**History**

Original Manuscript: October 27, 2000

Published: December 18, 2000

**Citation**

J. Riley, Hamid Dehghani, Martin Schweiger, Simon Arridge, Jorge Ripoll, and Manuel Nieto-Vesperinas, "3D optical tomography in the presence of void regions," Opt. Express **7**, 462-467 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-13-462

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### References

- S. R. Arridge, "Optical Tomography in Medical Imaging," Inverse Problems 15, R41-R93 (1999). [CrossRef]
- S. R. Arridge and J. C. Hebden, "Optical Imaging in Medicine: II. Modelling and Reconstruction," Phys. Med. Biol. 42, 841-853 (1997). [CrossRef] [PubMed]
- A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue," Phys. Med. Biol. 43, 1285-1302 (1998). [CrossRef] [PubMed]
- O. Dorn, "A Transport-BackTransport Method for Optical Tomography," Inverse Problems 14, 1107-1130 (1998). [CrossRef]
- A. D. Klose, A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26 1698-1707 (1999). [CrossRef] [PubMed]
- A. D. Klose, A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26 1698-1707 (1999).
- O. Dorn, "Scattering and absorption transport sensitivity functions for optical tomography," Opt. Express 7, 492-506 (2000), http://www.opticsexpress.org/oearchive/source/26901.htm. [CrossRef] [PubMed]
- M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, "An investigation of light transport through scattering bodies with non-scattering regions," Phys. Med. Biol. 41, 767-783 (1996). [CrossRef] [PubMed]
- 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 Non-Scattering Regions," Med. Phys. 27, 252-264 (2000). [CrossRef]
- J. Ripoll, S. R. Arridge, H. Dehghani, and M. Nieto-Vesperinas, "Boundary conditions for light propagation in diffusive media with nonscattering regions," J. Opt. Soc. Am. A 17, 1671-1681 (2000). [CrossRef]
- H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, "Optical Tomography in the Presence of Void Regions," J. Opt. Soc. Am. A 17, 1659-1670 (2000). [CrossRef]
- M. Schweiger and S. R. Arridge, "Comparison of 2D and 3D reconstruction algorithms in Optical Tomography," Appl. Opt. 37, 7419-7428 (1998).
- J. Ripoll, Ph.D. thesis, University Autonoma of Madrid, 2000. [CrossRef] [PubMed]
- S. R. Arridge and M. Schweiger, "The Finite Element Model for the Propagation of Light In Scattering Media: Boundary and Source Conditions," Med. Phys. 22, 1779-1792, (1995). [CrossRef]
- F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
- J. Schoberl, "NetGen", http://www.sfb013.uni-linz.ac.at/ joachim/netgen/
- M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).
- H. R. Zatz, Master's thesis, Cornell University, 1993. [CrossRef] [PubMed]
- S. R. Arridge and M. Schweiger, "Photon Measurement Density Functions. Part 2: Finite Element Calculations," Appl. Opt. 34, 8026-8037 (1995).
- V. Kolehmainen, M. Vaukhonen, J. P. Kaipio and S. R. Arridge, "Recovery of piecewise constant coefficients in optical diffusion tomography," Opt. Express 7, 468-480 (2000), http://www.opticsexpress.org/oearchive/source/24842.htm

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