## Systematic diffuse optical image errors resulting from uncertainty in the background optical properties

Optics Express, Vol. 4, Issue 8, pp. 299-307 (1999)

http://dx.doi.org/10.1364/OE.4.000299

Acrobat PDF (113 KB)

### Abstract

We investigated the diffuse optical image errors resulting from systematic errors in the background scattering and absorption coefficients, Gaussian noise in the measurements, and the depth at which the image is reconstructed when using a 2D linear reconstruction algorithm for a 3D object. The fourth Born perturbation approach was used to generate reflectance measurements and k-space tomography was used for the reconstruction. Our simulations using both single and dual wavelengths show large systematic errors in the absolute reconstructed absorption coefficients and corresponding hemoglobin concentrations, while the errors in the relative oxy- and deoxy- hemoglobin concentrations are acceptable. The greatest difference arises from a systematic error in the depth at which an image is reconstructed. While an absolute reconstruction of the hemoglobin concentrations can deviate by 100% for a depth error of ±1 mm, the error in the relative concentrations is less than 5%. These results demonstrate that while quantitative diffuse optical tomography is difficult, images of the relative concentrations of oxy- and deoxy-hemoglobin are accurate and robust. Other results, not presented, confirm that these findings hold for other linear reconstruction techniques (i.e. SVD and SIRT) as well as for transmission through slab geometries.

© Optical Society of America

## 1. Introduction

1. A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today **48**, 34–40 (1995). [CrossRef]

1. A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today **48**, 34–40 (1995). [CrossRef]

4. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med.Phys. **20**, 299–309 (1993). [CrossRef] [PubMed]

5. R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. **32**, 426–434 (1993). [CrossRef] [PubMed]

6. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,”Comput. Methods Programs Biomed. **47**, 131–146 (1995). [CrossRef] [PubMed]

2. 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]

8. S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and reconstruction,” Phys. Med. Biol. **42**, 841–854 (1997). [CrossRef] [PubMed]

8. S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and reconstruction,” Phys. Med. Biol. **42**, 841–854 (1997). [CrossRef] [PubMed]

9. A. J. Devaney, “Reconstruction tomography with diffractive wave-fields,” Inverse Problems **2**, 161–183 (1986). [CrossRef]

10. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A **13**, 253–266 (1996). [CrossRef]

11. A. D. Klose and A. H. Hielscher, “A transport-theory based reconstruction algorithm for optical tomography,” B. Chance, R. Alfano, and B. Tromberg ed., SPIE BiOS99, San Jose, CA, (SPIE,1999). [CrossRef]

^{th}order Born approximation. The inverse solution is based on the k-space reconstruction algorithm in reflectance mode [12

12. X. Cheng and D. A. Boas, “Diffuse optical reflectance tomography with continuous-wave illumination,” Opt. Express **3**, 118–123 (1998)http://epubs.osa.org/oearchive/source/5663.htm. [CrossRef] [PubMed]

2. 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]

## 2. Forward algorithm and K-space theory

15. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. **69**, 2658–2661 (1992). [CrossRef]

18. D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA **91**, 4887–4891 (1994). [CrossRef] [PubMed]

*ϕ*

_{sc}(

**r**

_{d}) is the scattered photon fluence measured at the detector at position

**r**

_{d}, such that what we measure at the detector is

*ϕ*(

**r**

_{s},

**r**) =

*ϕ*

_{inc}(

**r**

_{s},

**r**)+

*ϕ*

_{sc}(

**r**

_{s},

**r**).

*ϕ*

_{inc}(

**r**

_{s},

**r**) is the incident fluence at position

**r**generated in the medium by a source at position

**r**

_{s}.

*G*(

**r**,

**r**

_{d}) is the Green’s function solution of the diffusion equation for the given medium. Both

*ϕ*

_{inc}(

**r**

_{s},

**r**) and

*G*(

**r**,

**r**

_{d}) are calculated given the spatially uniform, average background optical properties of the medium

*μ*

_{s}′, the reduced scattering coefficient, and

*μ*

_{a}, the absorption coefficient.

*D*=

*v*/(3

*μ*

_{s}′) is the photon diffusion coefficient [19, 20

20. 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]

*v*is the speed of light in the medium, and

*δμ*

_{a}(

**r**) is the spatial variation of the absorption coefficient from the average background value ma at position

**r**. The second and n

^{th}order approximation are given by

*δμ*

_{a}(

**r**) from measurements of

*ϕ*

_{sc}(

**r**

_{d}) at multiple source,

**r**

_{s}, and detector,

**r**

_{d}, positions. Recognizing that eq. (1) resembles the Sommerfeld - Kirchoff scalar diffraction integral, we can use “diffraction tomography” to reconstruct an image of

*δμ*

_{a}(

**r**) from measurements of

*ϕ*

_{sc}(

**r**

_{d}) in a plane [21–23

21. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express **1**, 6–11 (1997).http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]

_{d}, and y

_{d}, and the object function

*A*(

*ω*

_{x},

*ω*

_{y},

*z*), which we wish to reconstruct an image of, is given by

*h*at position

*z*. Thus, the analytic solution for the spatially varying absorption coefficient becomes

21. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express **1**, 6–11 (1997).http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]

12. X. Cheng and D. A. Boas, “Diffuse optical reflectance tomography with continuous-wave illumination,” Opt. Express **3**, 118–123 (1998)http://epubs.osa.org/oearchive/source/5663.htm. [CrossRef] [PubMed]

## 3. Spectroscopy for Oxy- and Deoxy-hemoglobin

*ε*

_{HbO}and

*ε*

_{Hb}are the extinction coefficients of oxy- and deoxy-hemoglobin respectively and [HbO] and [Hb] are the corresponding concentrations. Below we represent the total hemoglobin concentration as [HbT] = [HbO] + [Hb] and the oxygen saturation as SO2 = [HbO] / [HbT]. The extinction spectra of hemoglobin are published in the literature [24–26] and are shown in figure 2. Measurements of the absorption coefficient at two or more wavelengths are necessary in order to solve for [HbO] and [Hb] as shown in eq. (8a) and eq. (8b).

*HbO*] and Δ[

*Hb*] refer to the change in oxy- and deoxy- hemoglobin concentration corresponding to the reconstructed absorption coefficient,

## 4. Simulations

^{-1}and an absorption coefficient of 0.05 cm

^{-1}. The object has dimensions of 0.5 by 0.5 by 0.2 cm located at a depth of 2 cm. It has the same scattering coefficient as the background but a different absorption coefficient. For our single wavelength simulation, the object has an absorption coefficient of 0.1cm

^{-1}. For the two-wavelength simulation, the background has an [HbT] of 100 μmole/liter and SO2 = 80%. For the object, [HbT] = 120 μmol/liter and SO2 = 60%. We used a wavelength of 780 nm and 830 nm in the simulation. From eq. (7), the background absorption coefficient is found to be 0.078 cm

^{-1}and 0.092 cm

^{-1}at 780 nm and 830 nm respectively and the absorption perturbation at the two wavelengths is 0.024 cm

^{-1}and 0.012 cm

^{-1}respectively. The measurement geometry consists of an array of 12 by 12 detectors uniformly distributed over an area of 6 by 6 cm. The light source is located at the center of the probe.

## 5. Results and discussions

**cannot**be greater than -100%. These large systematic errors originate from the exponentially decaying nature of the diffuse photon density wave inside the turbid media. This suggests that the systematic image errors will have a nonlinear relationship with the errors in the background absorption and scattering coefficient.

^{-5}% noise will cause ~5% uncertainty in the image value. In figure 5b, the Gaussian noise is fixed at 10

^{-5}% and the object depth is varied. We see that the image uncertainty rises significantly as the depth of the object increases. This means that imaging a deep absorbing object is more prone to uncertainty than imaging a shallow object. Although this result is obvious, it does give us a method for designing the measurement geometry to optimize the depth sensitivity for a given application. Note that for noise levels as small as 10

^{-5}% we obtain image uncertainties greater than 10%. Noise levels in practice are expected to be on the order of 0.1%. This extreme sensitivity to noise in our example arises for two reasons: 1) the object contrast is small (<25%) and 2) the measurement geometry is not optimal.

## 6. Summary

*Hb*]/Δ[

*HbO*] is much less sensitive to the depth of the object than the absolute measures of Δ[

*Hb*] and Δ[

*HbO*]. The ratio is also less sensitive to errors in the background optical properties. This distinction between absolute and relative imaging will play an important role in the application of diffuse optical tomography to studying hemodynamics. Ultimately, these results motivate the development of 3D reconstruction algorithms to better localize object position and to better determine background optical properties in order to minimize the systematic errors addressed here.

## 7. Acknowledgements

## References and links

1. | A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today |

2. | 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. |

3. | “Trends in Optics and Photonics Series,” R. Alfano ed., Advances in Optical Imaging and Photon Migration,Orlando, FLA, (OSA,1996). |

4. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med.Phys. |

5. | R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. |

6. | L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,”Comput. Methods Programs Biomed. |

7. | S. R. Arridge, J. P. Kaltenbach, R. L. Barbour, and G. Muller ed., Bellingham, Wa, (Proc. SPIE,1993). |

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

9. | A. J. Devaney, “Reconstruction tomography with diffractive wave-fields,” Inverse Problems |

10. | H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A |

11. | A. D. Klose and A. H. Hielscher, “A transport-theory based reconstruction algorithm for optical tomography,” B. Chance, R. Alfano, and B. Tromberg ed., SPIE BiOS99, San Jose, CA, (SPIE,1999). [CrossRef] |

12. | X. Cheng and D. A. Boas, “Diffuse optical reflectance tomography with continuous-wave illumination,” Opt. Express |

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

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

15. | M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. |

16. | J. M. Schmitt, A. Knuttel, and J. R. Knutson, “Interference of diffusive light waves,” J.Opt.Soc.Am.A |

17. | J. B. Fishkin and E. Gratton, “Propagation of photon density waves in strongly scattering media containing an absorbing ‘semi-infinite’ plane bounded by a straight edge,” J. Opt. Soc. Am. A |

18. | D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA |

19. | K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J.Math.Phys. |

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

21. | C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express |

22. | X. D. Li, T. Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. |

23. | X. D. Li, T. Durduran, A. G. Yodh, B. Chance, and D. N. Pattanayak, “Diffraction tomography for biomedical imaging with diffuse photon density waves: errata,” Opt. Lett. |

24. | M. Cope, “The Development of a Near-Infrared Spectroscopy System and Its Application for Noninvasive Monitoring of Cerebral Blood and Tissue Oxygenation in the Newborn Infant,” University College London (1991). |

25. | S. Wray, M. Cope, and D. T. Delpy, “Characteristics of the near infrared absorption spectra of cytochrome aa3 and hemoglobin for the noninvasive monitoring of cerebral oxygenation.,” Biochim Biophys Acta |

26. | S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope, and D. T. Delpy, “Performance comparison of several published tissue near-infrared spectroscopy algorithms,” Anal. Biochem. |

**OCIS Codes**

(170.5280) Medical optics and biotechnology : Photon migration

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Focus Issue: Biomedical diffuse optical tomography

**History**

Original Manuscript: February 25, 1999

Published: April 12, 1999

**Citation**

Xuefeng Cheng and David Boas, "Systematic diffuse optical image errors resulting
from uncertainty in the background optical properties," Opt. Express **4**, 299-307 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-8-299

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### References

- A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995). [CrossRef]
- M. A. OLeary, 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]
- "Trends in Optics and Photonics Series," R. Alfano ed., Advances in Optical Imaging and Photon Migration, Orlando, FLA, (OSA, 1996).
- S. R. Arridge, M. Schweiger, M. Hiraoka and D. T. Delpy, "A finite element approach for modeling photon transport in tissue," Med.Phys. 20, 299-309 (1993). [CrossRef] [PubMed]
- R. Graaff, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel and J. G. Aarnoudse, "Condensed Monte Carlo simulations for the description of light transport," Appl. Opt. 32, 426-434 (1993). [CrossRef] [PubMed]
- L. Wang, S. L. Jacques and L. Zheng, "MCML-Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995). [CrossRef] [PubMed]
- S. R. Arridge, J. P. Kaltenbach and R. L. Barbour, G. Muller ed., Bellingham, Wa, (Proc. SPIE, 1993).
- S. R. Arridge and J. C. Hebden, "Optical Imaging in Medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-854 (1997). [CrossRef] [PubMed]
- A. J. Devaney, "Reconstruction tomography with diffractive wave-fields," Inverse Problems 2, 161-183 (1986). [CrossRef]
- H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue and M. S. Patterson, "Optical image reconstruction using frequency-domain data: Simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996). [CrossRef]
- A. D. Klose and A. H. Hielscher, "A transport-theory based reconstruction algorithm for optical tomography," B. Chance, R. Alfano and B. Tromberg ed., SPIE BiOS99, San Jose, CA, (SPIE, 1999). [CrossRef]
- X. Cheng and D. A. Boas, "Diffuse optical reflectance tomography with continuous-wave illumination," Opt. Express 3, 118-123 (1998). http://epubs.osa.org/oearchive/source/5663.htm [CrossRef] [PubMed]
- W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambirdge Univ. Press, New York, 1988) Ch2 p52.
- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
- M. A. OLeary, D. A. Boas, B. Chance and A. G. Yodh, "Refraction of diffuse photon density waves," Phys. Rev. Lett. 69, 2658-2661 (1992). [CrossRef]
- J. M. Schmitt, A. Knuttel and J. R. Knutson, "Interference of diffusive light waves," J. Opt. Soc. Am. A 9, 1832 (1992). [CrossRef] [PubMed]
- J. B. Fishkin and E. Gratton, "Propagation of photon density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," J. Opt. Soc. Am. A 10, 127-140 (1993). [CrossRef] [PubMed]
- D. A. Boas, M. A. OLeary, B. Chance and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: analytic solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887- 4891 (1994). [CrossRef] [PubMed]
- K. Furutsu, "Diffusion equation derived from the space-time transport equation in anisotropic random media," J.Math.Phys. 24, 765-777 (1997).
- 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]
- C. L. Matson, "A diffraction tomographic model of the forward problem using diffuse photon density waves," Opt. Express 1, 6-11 (1997). http://epubs.osa.org/oearchive/source/1884.htm [CrossRef] [PubMed]
- X. D. Li, T. Durduran, A. G. Yodh, B. Chance and D. N. Pattanayak, "Diffraction tomography for biochemical imaging with diffuse-photon density waves," Opt. Lett. 22, 573-575 (1997). [CrossRef] [PubMed]
- X. D. Li, T. Durduran, A. G. Yodh, B. Chance and D. N. Pattanayak, "Diffraction tomography for biomedical imaging with diffuse photon density waves: errata," Opt. Lett. 22, 1198 (1997). [CrossRef] [PubMed]
- M. Cope, "The Development of a Near-Infrared Spectroscopy System and Its Application for Noninvasive Monitoring of Cerebral Blood and Tissue Oxygenation in the Newborn Infant," University College London (1991).
- S. Wray, M. Cope and D. T. Delpy, "Characteristics of the near infrared absorption spectra of cytochrome aa3 and hemoglobin for the noninvasive monitoring of cerebral oxygenation.," Biochim Biophys Acta 933, 184-192 (1988). [CrossRef] [PubMed]
- S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope and D. T. Delpy, "Performance comparison of several published tissue near-infrared spectroscopy algorithms," Anal. Biochem. 227, 54-68 (1995). [CrossRef] [PubMed]

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