## Simultaneous imaging and optode calibration with diffuse optical tomography

Optics Express, Vol. 8, Issue 5, pp. 263-270 (2001)

http://dx.doi.org/10.1364/OE.8.000263

Acrobat PDF (1677 KB)

### Abstract

In order for diffuse optical tomography to realize its potential of obtaining quantitative images of spatially varying optical properties within random media, several potential experimental systematic errors must be overcome. One of these errors is the calibration of the emitter strength and detector efficiency/gain. While in principle these parameters can be determined accurately prior to an imaging experiment, slight fluctuations will cause significant image artifacts. For this reason, it is necessary to consider including their calibration as part of the inverse problem for image reconstruction. In this paper, we show that this can be done successfully in a linear reconstruction model with simulated continuous-wave data. The technique is general for frequency and time domain data.

© Optical Society of America

## 1. Introduction

1. S. B. Colak, M. B. van der Mark, G. W. Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers,“Clinical Optical Tomography and NIR Spectroscopy for Breast Cancer Detection,” IEEE Journal of Selected Topics in Quantum Electronics **5**, 1143–1158 (1999). [CrossRef]

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

*δµ*and

_{a}*δµ*’, around a known background

_{s}*µ*and

_{a,0}*µ*’. Thence the measured fluence Φ is given by a perturbation Φ

_{s,0}_{pert}on a background fluence Φ

_{o}. In this approach therefore, we first experimentally measure the photon fluence Φ and then must calculate the fluence perturbation Φ

_{pert}caused by spatial variation, (

*δµ*,

_{a}*δµ*’) in the optical properties. Accurate determination of Φ

_{s}_{pert}requires calibration of the source and detector amplitude factors (

*s*and

*d*respectively) and a reasonable estimate of Φ

_{o}. A few approaches have been discussed for obtaining a reasonable estimate of Φ

_{o}including: 1) solving for the background optical properties using all of the measurements by first assuming zero spatial perturbations, and 2) including this estimate into the non-linear inverse problem [7

7. J. C. Hebden, E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, and S. R. Arridge,“Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. **24**, 334–336 (1999). [CrossRef]

9. V. Kolehmainen, M. Vauhkonen, J. P. Kaipio, and S. R. Arridge,“Recovery of Piecewise constant coefficients in optical diffusion tomography,” Opt. Express **7**:468–480 (2000). [CrossRef] [PubMed]

*et al*. have proposed using a difference of equally distant measurements for which Φ

_{o}is equal and thus cancels [10

10. 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]

_{pert}between the two measurements is obtained accurately (assuming proper

*s*and

*d*calibration) without reference to Φ

_{o}.

*s*and

*d*are likely to be compensated in the image reconstruction through highly localized absorption and scattering perturbations appearing adjacent to the optodes. This type of image noise looks like high frequency spikes appearing preferentially near the optode locations, and has been observed by a number of groups [11]. A number of schemes for addressing this problem have been proposed, including median filtering [12

12. M. Schweiger, S. R. Arridge, and D. T. Delpy,“Application of the finite-element method for the forward and inverse models in optical tomography,” Journal of Mathematical Imaging and Vision **3**, 263–283 (1993). [CrossRef]

*et al*. used the same variable regularization method to suppress variation near the optodes to help improve image quality of objects far from the optodes [14

14. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen,“Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

*s*and

*d*can be included in the image reconstruction algorithm. When considering the log of the measured fluence as the data the perturbation to the photon fluence is linearly dependent on the logarithm of

*s*and

*d*, and nonlinearly dependent on the optical properties. When in addition we assume linear dependence on perturbation in the optical properties (the Rytov approximation) we arrive at a fully linear problem. We present simulation results for CW data that reveal significant image improvement despite uncertainties greater than 50% and biases greater than a factor of 10. This method can also be adapted for frequency and time domain data.

## 2. Theory

### 2.1 Forward Problem

17. 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.of Am.A **11**, 2727–2741 (1994). [CrossRef]

**r**,

*t*) is the photon fluence at position

**r**and time t.

*S*(

**r**,

*t*) is the source distribution of photons.

*D*=

*v*/(

*3µ*’+

_{s}*α µ*) is the photon diffusion coefficient [18

_{a}18. K. Furutsu and Y. Yamada,“Diffusion approximation for a dissipative random medium and the applications,” Phys.Rev.E **50**, 3634 (1994). [CrossRef]

19. 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*’ is the reduced scattering coefficient,

_{s}*µ*is the absorption coefficient, and

_{a}*v*is the speed of light in the medium. The dependence of D on

*µ*is expressed through the coefficient

_{a}*α*which is variously considered as 3, zero [18

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

19. 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]

20. D. J. Durian, “The diffusion coefficient depends on absorption,” Optics Letters **23**, 1502–1504 (1998). [CrossRef]

21. R. Aronson and N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A **16**, 1066–1071 (1999). [CrossRef]

**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. If the background is homogeneous then

*G*can be expressed analytically in some simple geometries[16

16. 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]

23. 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]

### 2.2 Inverse Problem

*i*is summed over measurements with each source and detector pair, and Φ

*(*

_{Theory}*x*) is given by eq. (3) and eq. (4) where

*x*is a vector giving

*δµ*at each voxel position.

_{a}*I*is the identity matrix,

*λ*is the Tikhonov regularization parameter, and

*y*is the measured data. Each element of the matrix A is

**r**

_{s,i}and

**r**

_{d,i}is the position of the i

^{th}source and detector and

**r**

_{j}is the position of the j

^{th}voxel. Within the Rytov approximation

*λ*=10

^{-3}of the maximum value within

*AA*[24

^{T}24. H. Dehghani, D. C. Barber, and I. Basarab-Horwath, “Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography,” Physiol Meas **20**, 87–102 (1999). [CrossRef] [PubMed]

25. V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen, and J. P. Kaipio,“Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns,” Physiol Meas **18**, 289–303 (1997). [CrossRef] [PubMed]

### 2.3 Image and Coupling Coefficient Reconstruction — Rytov Approximation

**A**and

**y**requires accurate knowledge of the background optical properties

*µ*and

_{ao}*µ*’ and of the

_{so}*s*and

*d*amplitudes. Here we consider

*µ*and

_{ao}*µ*’ known, and the

_{so}*s*and

*d*amplitude for each source and detector as unknowns in the inverse problem.

*y*written in terms of the unknown source coupling amplitudes

_{i}*s*, detector coupling amplitudes

_{k}*d*, and the absorption perturbations

_{l}*δµ*becomes

_{a,j}*k*(

*i*) and

*l*(

*i*) identifies the source and detector respectively used for the

*i*measurement. Note that the

^{th}*s*and

*d*appearing in eq. (8) represent the difference from the assumed values as used in Φ

*and the real values for the measurement of Φ. We can write this in matrix form as*

_{o}**y**=

**Bξ**where

**B**=[

**Ã S D**] and

*N*is the number of voxels,

_{v}*N*is the number of sources, and

_{s}*N*is the number of detectors. Scaling

_{d}*δµ*by

_{a,j}*µ*makes the elements dimensionless and of the same order as ln

_{ao}*s*and ln

*d*.

**Ã**=

*µ*

_{ao}**A**is the rescaling of the matrix given in eq. (4), and

**S**and

**D**have block diagonal form.

**S**has a one in the

*k*column for each measurement corresponding to source k, and

^{th}**D**has a one in the

*l*column for each measurement corresponding to detector

^{th}*l*. For instance, with 4 measurements between 2 sources and 2 detectors,

## 3. Simulation Results

*µm*’=10 cm

_{so}^{-1}and

*µ*=0.05 cm

_{ao}^{-1}(see fig. 1). A 1.6 cm diameter absorbing object with

*µ*’=10 cm

_{so}^{-1}and

*µ*=0.15 cm

_{a}^{-1}was centered at (x,y,z)=(1, -1, 3) cm. Sixteen sources were placed in a 4×4 grid at z=0, spanning x=-3 to 3 cm and y=-3 to 3 cm in 2 cm steps. Sixteen detectors were placed in a 4×4 grid at z=6 cm spanning x=-3 to 3 cm and y=-3 to 3 cm in 2 cm steps. All simulated measurements were made with continuous-wave sources.

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

*s*and

*d*) is clearly degrading image quality by increasing artefact.

*µ*≤0.05(red) cm

_{a}^{-1}. When there is no variance in the coupling coefficients, we see that the location of the object is determined properly in X and Y and is within 0.5 cm in Z. The resolution (blurring or point spread function) is poorer relative to the case without simultaneous reconstruction of the coupling coefficients. This is because we now have more unknowns in the reconstruction being considered. Furthermore, notice that the artefact observed previously in fig. 2a due to error between the exact and Rytov approximation to the forward problem, has been reduced. We therefore conclude that reconstruction for the coupling coefficients has slightly compensated for model mismatch in the Rytov approximation. This is a qualitative comparison between the reconstruction without and with simultaneous determination of the coupling coefficients. A more quantitative comparison would closely examine the singular value spectrum and chose the regularization parameter based on the signal-to-noise ratio in the measurement.

*s*) for each measurement. For the examples shown here we see that the products are reconstructed with an accuracy of a few percent despite uncertainties exceeding 80%. Finally, when we include additive shot or electronic measurement noise in our simulations, we then observe a greater sensitivity to the measurement noise as the source and detector variance increases (results not shown).

_{k}d_{l}## 4. Conclusions

## Acknowledgements

## References and links

1. | S. B. Colak, M. B. van der Mark, G. W. Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers,“Clinical Optical Tomography and NIR Spectroscopy for Breast Cancer Detection,” IEEE Journal of Selected Topics in Quantum Electronics |

2. | S. Fantini, M. A. Franceschini, E. Gratton, D. Hueber, W. Rosenfeld, D. Maulik, P. G. Stubblefield, and M. R. Stankoivic,“Non-invasive optical imaging of the piglet brain in real time,” Opt. Express |

3. | 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 |

4. | 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 |

5. | B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, and U. L. Osterberg,“Hemoglobin imaging of breast tumors with near-infrared tomography,” Radiology214, (in press). |

6. | S. R. Arridge,“Optical Tomography in medical imaging,” Inverse Problems |

7. | J. C. Hebden, E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, and S. R. Arridge,“Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. |

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

9. | V. Kolehmainen, M. Vauhkonen, J. P. Kaipio, and S. R. Arridge,“Recovery of Piecewise constant coefficients in optical diffusion tomography,” Opt. Express |

10. | 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. |

11. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in |

12. | M. Schweiger, S. R. Arridge, and D. T. Delpy,“Application of the finite-element method for the forward and inverse models in optical tomography,” Journal of Mathematical Imaging and Vision |

13. | S. R. Arridge and M. Schweiger, “Inverse Methods for Optical Tomography” in |

14. | B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen,“Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

15. | A. Ishimaru, |

16. | 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. |

17. | 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.of Am.A |

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

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

20. | D. J. Durian, “The diffusion coefficient depends on absorption,” Optics Letters |

21. | R. Aronson and N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A |

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

23. | 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 |

24. | H. Dehghani, D. C. Barber, and I. Basarab-Horwath, “Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography,” Physiol Meas |

25. | V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen, and J. P. Kaipio,“Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns,” Physiol Meas |

**OCIS Codes**

(170.5280) Medical optics and biotechnology : Photon migration

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 17, 2001

Published: February 26, 2001

**Citation**

David Boas, Thomas Gaudette, and Simon Arridge, "Simultaneous imaging and optode calibration with diffuse optical tomography," Opt. Express **8**, 263-270 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-5-263

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### References

- S. B. Colak, M. B. van der Mark, G. W. Hooft, J. H. Hoogenraad, E. S. van der Linden and F. A. Kuijpers,"Clinical Optical Tomography and NIR Spectroscopy for Breast Cancer Detection," IEEE Journal of Selected Topics in Quantum Electronics 5, 1143-1158 (1999). [CrossRef]
- S. Fantini, M. A. Franceschini, E. Gratton, D. Hueber, W. Rosenfeld, D. Maulik, P. G. Stubblefield and M. R. Stankoivic,"Non-invasive optical imaging of the piglet brain in real time," Opt. Express 4, 308-314 (1999). [CrossRef] [PubMed]
- 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 97, 2767-72 (2000). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman and U. L. Osterberg, "Hemoglobin imaging of breast tumors with near-infrared tomography," Radiology 214, (in press).
- S. R. Arridge,"Optical Tomography in medical imaging," Inverse Problems 15, R41-R93 (1999). [CrossRef]
- J. C. Hebden, E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy and S. R. Arridge,"Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data," Opt. Lett. 24, 334-336 (1999). [CrossRef]
- 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]
- V. Kolehmainen, M. Vauhkonen, J. P. Kaipio and S. R. Arridge,"Recovery of Piecewise constant coefficients in optical diffusion tomography," Opt. Express 7, 468-480 (2000). [CrossRef] [PubMed]
- 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]
- S. R. Arridge, M. Schweiger, M. Hiraoka and D. T. Delpy, "Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging," in Photon Migration and Imaging in Random Media and Tissues, B. Chance and R. R. Alfano, SPIE 1888, 360-371 (1993).
- M. Schweiger, S. R. Arridge and D. T. Delpy,"Application of the finite-element method for the forward and inverse models in optical tomography," J. Mathematical Imaging and Vision 3, 263-283 (1993). [CrossRef]
- S. R. Arridge and M. Schweiger, "Inverse Methods for Optical Tomography," in Information Processing in Medical Imaging (IPMI'93 Proceedings), Lecture Notes in Computer Science, (Springer-Verlag, 1993).
- B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg and K. D. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, 2950-2961 (1999). [CrossRef]
- A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, Inc., San Diego 1978).
- 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]
- 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]
- K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634 (1994). [CrossRef]
- 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]
- D. J. Durian, "The diffusion coefficient depends on absorption," Opt. Lett. 23, 1502-1504 (1998). [CrossRef]
- R. Aronson and N. Corngold, "Photon diffusion coefficient in an absorbing medium," J. Opt. Soc. Am. A 16, 1066-1071 (1999). [CrossRef]
- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York 1988).
- 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]
- H. Dehghani, D. C. Barber and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Meas. 20, 87-102 (1999). [CrossRef] [PubMed]
- V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J. P. Kaipio,"Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns," Physiol Meas 18, 289-303 (1997). [CrossRef] [PubMed]

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