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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 7, Iss. 13 — Dec. 18, 2000
  • pp: 462–467
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3D optical tomography in the presence of void regions

J Riley, H Dehghani, M Schweiger, S R Arridge, J Ripoll, and M Nieto-Vesperinas  »View Author Affiliations


Optics Express, Vol. 7, Issue 13, pp. 462-467 (2000)
http://dx.doi.org/10.1364/OE.7.000462


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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

Optical Tomography is an example of a non-linear illposed inverse problem. A complete treatment of image reconstruction therefore requires either implicitly or explicitly the optimization of an objective function with respect to the parameters of a model[1

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

]. The most commonly used model is the diffusion approximation which is the lowest order non-trivial approximation to the more correct Radiative Transfer Equation (RTE)[1

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

, 2

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]

]. The reason for its success is that tissues of interest in clinical applications, such as the human breast and peripheral muscles, are of a scale such that detected photons have undergone a high enough number of scattering events that the dependence of fluence on direction is only linear.

However, one major application interest for optical tomography is its use for imaging of the brain. Within the head the regions of cerebrospinal fluid (CSF) around the brain and in the ventricles are non-scattering while still absorbing. The presence of CSF prevents the accurate modeling of photon propagation within the regions of interest when using the diffusion approximation and leads naturally to the attempt to develop a modelling scheme based on the RTE[3

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

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

, 5

5. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 261698–1707 (1999). [CrossRef] [PubMed]

, 6

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

]. The principle difficulty then is that accurate handling of the non-scattering regions implies the use of a very computationally expensive numerical method.

One proposed alternative to the full RTE is the 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

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]

]. Propagation in the void is handled via geometrical optics and the interface requires the development of 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]

]. The advantage is the greatly increased computational efficiency whilst maintaining good accuracy. This efficient model has allowed the investigation of the effect of voids on the accuracy of image reconstruction in 2D [10

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]

]. However optical tomography is in reality a 3D problem [11

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

]. In this paper we report the development of the radiosity-diffusion model in 3D and present the first assessment of optical tomography in a 3D geometry involving voids.

2 The Radiosity-Diffusion Model

Let Ω represent a domain consisting of R diffusing regions {Ω1, Ω2, … Ω R } and V void regions {Ξ1, Ξ2,… Ξ V }. Let Ω d =i=1RΩ i be the union of all diffusing regions and Ξ d =i=1V Ξ i be the union of all void regions. Thus Ω=Ω d ∪ Ξ d . Each region has an outer boundary i+ and a non-negative number of inner boundaries i,k. We define ∂Ω i =∂Ωi+ ∪(∪ kΩi,k ) for diffusing regions, with similarly ∂Ξ i =∂Ξi+ ∪(∪ kΞi,k ) for voids. The outer boundary ∂Ω1+ will also be denoted by Ω.

We consider a frequency-domain system so that the photon density Φ(r; ω) at modulation frequency ω satisfies the homogeneous equation

·κ(r)Φ(r;ω)+(µa(r)+iωc)Φ(r;ω)=0rΩd(ΞdΩd)
(1)

where 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

Φ(m;ω)+2Aκ(m)Φ(m;ω)ν=η(m;ω)mΩ1+
(2)

where 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]

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

Φ(m;ω)+2AκΦ(m;ω)ν=1πΞcosθcosθΦ(m;ω)2Ah(m,m)×
exp[(μa+iωc)mm]mm2dmm,mΞ
cosθ=ν̂(m)·mmmm,cosθ=ν̂(m)·mmmm
(3)

and 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)/2A in eq(3)by

(Φ(m;ω)2κ(m)1+R(1)1-R(0)Φ(m;ω)ν)(1R(θ)2)

where R (0), R (1) are derived from the Fresnel coefficients taken over local coordinates

R (0)=2π/2π R(ϑ)sinϑcosϑdϑ, R (1)=3π/2π R(ϑ)sin ϑcos2ϑdϑ

See [12

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

] for a detailed derivation. These two forms were compared in [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]

] and found to give comparable results. The constant for the domain boundary in eq(2) and (3) is given by A=(1-R (1))/(1-R (0)). For each Neumann source term η the measureable is

yη(m;ω)=𝓟η(μaκ)κ(m)Φη(m;ω)ν,mΩ1+
(4)

3 Inverse Problem

In the adjoint field approach [14

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

] we assume a finite number of input fluxes {ηj ; j=1,…, S}, and given data {gj ;j=1,…, S}, and look for the minimisation of the norm

C=12j=1sgj𝓟jμaκ,gj𝓟j(μaκ)L2(Ω)
(5)

A descent direction for minimising C with respect to the data from the jth source is given by

(αβ)=(Re(Φj-Ψj)Re(Φj-·Ψj))
(6)

with ψ j the solution to the adjoint diffusion equation

·κ(r)Ψj(r;ω)+(μa(r)iωc)Ψj(r;ω)=0rΩd\(ΞdΩd)
(7)

with boundary conditions

Ψj(m;ω)+2Aκ(m)Ψj(m;ω)ν=gj(m;ω)𝓟j(μaκ)mΩ1+
Ψj(m;ω)+2AκΨj(m;ω)ν=1πΞcosθcosθ'Ψj(m;ω)2Ah(m,m)x
exp[(μaiωc)mm]mm2dmm,mΞ
(8)

4 Implementation in 3D

f(m,m)=cosθ(m)cosθ(m)πmm2
(9)

extracted from eq(3), is called the form factor corresponding to its use in Computer Graphics [16

16. M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).

]. In our current implementation we approximate this function by a bilinear expansion over the FEM shape functions on the surface of the void [17

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

]. Between two element faces τα , τα′ ; ∊ δΞ containing N, N′ nodes respectively the form factor can be approximated by

fτα,τα(m,m)=k=1Nk=1Nun(k)(m)un(k)(m)cosθn(k)cosθn(k)πNn(k)Nn(k)2
(10)

where n(k) maps the local node to the global nodes in the complete mesh.

5 Results

In fig.1 we show the geometry being considered. We take a sphere radius 25mm with concentric sphere radius rinner and a concentric void gap with outer radius router . 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.

Fig. 1. Left : cutaway of spherical mesh. Right : location of sources and detectors on the sphere surface
Fig. 2. distribution of photon density, photons/mm3 (top row) and mean time of photon flight, picoseconds (bottom row) over sphere surface. Left to right, solid sphere (no gap), gap widths 3mm, 4mm, 5mm.

In fig.2 we show the surface data generated by a single source for the case of a solid sphere and for three gaps defined by rinner =17,16,15mm, and in each case router =20mm. The difference between each void case and the solid sphere emphasises the increased light intensity and decreased mean time in the gap region.

Fig. 3. Sensitivity functions for the 3mm gap case. Left intensity (photons/mm2), right mean time(picoseconds mm). The functions plotted are cross-sections through the equatorial plane of the sphere. Also available as a QuickTime movie, pmdf.mov. (3.8MB)

Fig. 4. Target images (top row) and reconstructions (bottom row) for the 3mm gap case. The images are transverse, sagittal and coronal slices through the true centre of the blob, orientated according to the diagram in the top right panel. Bottom right shows a profile along the equatorial diameter through the blob centre. A movie showing a rotating orthographic view is attached rotating3d.mov (3.8MB).

In fig.4 we show the reconstruction of a blob in the 3mm gap. The blob had a radius of 3mm and an absorption of 0.02mm -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

The correct treatment of a non-scattering region in diffusing media is an important requirement for application of Optical Tomography to the brain. In this paper we showed initial results of a 3D extension to the Radiosity-Diffusion model that provides one approach to this problem. Even though the cases considered here are simple, they indicate that, if the location of the void is known, the signal detected on the domain surface still carries enough information to allow a reconstruction. A more complete treatment should consider data types in addition to just the mean time as used here, as well as simultaneous reconstruction of absorption and scattering. Furthermore, one of the interesting results found in [10

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]

] was the improvement in reconstruction quality when the boundary of the void gap is not smooth. Investigation of this case requires a more sophisticated determination of form factors which is under development.

Acknowledgments

We thank the Wellcome Trust and the EPRSC. J.Riley thanks UCL Graduate School for his PhD scholarship. J.Ripoll acknowledges a grant from Ministerio de Educación y Cultura. N.Nieto-Vesperinas thanks DGICYT and Fundación Ramón Areces. We thank F. Natterer and O. Dorn for discussions.

References and links

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. 261698–1707 (1999). [CrossRef] [PubMed]

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. 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]

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]

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. 22, 1779–1792, (1995). [CrossRef] [PubMed]

14.

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

15.

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

16.

M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).

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

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

  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, A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26 1698-1707 (1999). [CrossRef] [PubMed]
  6. 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).
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. M. Schweiger and S. R. Arridge, "Comparison of 2D and 3D reconstruction algorithms in Optical Tomography," Appl. Opt. 37, 7419-7428 (1998).
  13. J. Ripoll, Ph.D. thesis, University Autonoma of Madrid, 2000. [CrossRef] [PubMed]
  14. 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]
  15. F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
  16. J. Schoberl, "NetGen", http://www.sfb013.uni-linz.ac.at/ joachim/netgen/
  17. M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).
  18. H. R. Zatz, Master's thesis, Cornell University, 1993. [CrossRef] [PubMed]
  19. S. R. Arridge and M. Schweiger, "Photon Measurement Density Functions. Part 2: Finite Element Calculations," Appl. Opt. 34, 8026-8037 (1995).
  20. 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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