## Analytical model for dual-interfering sources diffuse optical tomography

Optics Express, Vol. 10, Issue 1, pp. 2-14 (2002)

http://dx.doi.org/10.1364/OE.10.000002

Acrobat PDF (418 KB)

### Abstract

An analytical model to perform tomographic reconstructions for absorptive inclusions in highly scattering media using dual interfering sources was derived. A perturbation approach within the first order Rytov expansion was used to solve the heterogeneous diffusion equation. Analytical weight functions necessary to solve the inverse problem were obtained. The reconstructions performance was assessed using simulated data of breast-like media after contrast agent enhancement. We further investigated the reconstruction quality as a function of object depth location, modulation frequency and source separation. The ability of the algorithm to resolve multi-objects was also demonstrated.

© Optical Society of America

## 1. INTRODUCTION

^{1–31. D. Hawrys and E. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents,” Neoplasia 2, 388–417 (2000). [CrossRef] }and brain function monitoring.

^{44. A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human function,” Trends Neurosci. 20, 435–442 (1997). [CrossRef] [PubMed] }

^{55. M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon-tomography,” Opt. Lett. 20, 426–428 (1995). [CrossRef] }Such analytical expressions have already given valuable results in the clinical setting.

^{66. V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000). [CrossRef] [PubMed] }The key of the success of the Rytov approach is that it has been shown to be more accurate than the Born-type perturbative solution

^{77. A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press , N-Y (1987).,88. M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996).}and allows performing differential DOT.

^{99. V. Ntziachristos, B. Chance, and A. Yodh, “Differential diffuse optical tomography,” Opt. Express 5, 230–242 (1999). http://www.opticsexpress.org/opticsexpress/tocv5n10.htm [CrossRef] [PubMed] }

^{1010. A. Knuttel, J.M. Schmitt, and J.R. Knutson, “Spatial localization of absorbing bodies by interfering diffuse photon-density waves,” Appl. Opt. 32, 381–389 (1993). [CrossRef] [PubMed] }This method has been suggested to be more sensitive to the presence of absorption abnormalities than single-source schemes.

^{1111. M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997). [CrossRef] }

^{1212. Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source”, Opt. Express 9, 212–224 (2001). http://www.opticsexpress.org/opticsexpress/tocv9n4.htm [CrossRef] [PubMed] }The principle of operation is based on a pair of 180°-phase-shifted sources that yield two diffuse photon density waves (DPDW), which propagate as scalar waves. The superposition of these waves in a homogeneous medium creates an ac amplitude null plane located at the mid distance from the two sources, and a 180° sharp transition in the phase at each side of this plane. The presence of heterogeneity modifies this balance pattern, which is expressed by a perturbation of the null-plane and a corresponding phase shift. These characteristics have been shown to lead to a very sensitive detection

^{1313. B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA 90, 3423–3427 (1993). [CrossRef] [PubMed] }of abnormalities with important clinical potential.

^{1414. B. Chance and E. Conant, “A novel tumor imager using NIR light,” in preparation.,1515. Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000). [CrossRef] }

^{1515. Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics 5, 206–213 (2000). [CrossRef] }Despite the sensitive detection, such an approach does not allow for resolving depth. A more elaborate approach that uses single sources and performs the superposition of the detected DPDW by post-processing has been proposed by O’Leary.

^{88. M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996).}This scheme has been shown to enhance the quality of the reconstruction especially when the bulk optical properties of the medium are miss-estimated. On the other hand, this data processing technique is more sensitive to random and biological noise since the “interfering” measurements are not acquired simultaneously. Furthermore this scheme can work only with detectors that are on the null plane; therefore its true tomographic application is limited.

## 2. METHODS

### 2.1 Theory

^{1919. A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Physics Today 48, 34–40 (1995). [CrossRef] }It is well known that the P

_{1}approximation

^{2020. P. Morse and H. Feshbach, “Methods of theoretical physics,” Mc Graw Hill, N-Y (1953).,2121. A. Ishimaru, “Wave propagation and scattering in random media,” Vol.1, Academic Press, N-Y (1980).}leads to an accurate model of near infrared (NIR) light propagation in diffuse media for source-detector distances greater than 10 mean free paths.

^{2222. K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. 24, 2647–2650 (1990). [CrossRef] ,2323. X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media 9, 489–499 (1999). [CrossRef] }This approximation, which is valid for the frequencies of interest (<1GHz) and the optical values encountered in biomedical applications (μ

_{a}<< μ′

_{s}), yields a Helmoltz like equation that describes the propagation of the fluence in an infinite homogeneous medium,

*i.e*.:

_{s}) is the fluence [W.cm

^{-2}] (isotropic term), D the diffusion coefficient given by D = (3μ′

_{s})

^{-1}[cm], μ

_{a}the absorption coefficient [cm

^{-1}], μ′

_{s}the corrected transport scattering coefficient [cm

^{-1}], AS(r⃗

_{s}) the source term and k the wave vector is given by

^{2424. R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994). [CrossRef] ,2525. S. Arridge, “Photon-measurement density function.I Analytical forms,” Appl. Opt. 34, 7395–7409 (1995). [CrossRef] [PubMed] }In the case of heterogeneities, this equation is implemented with a heterogeneous operator. In this study, we considered only purely absorptive objects. This choice is justified since the main imaging applications pertaining to cancer detection and functional monitoring are realized through mapping of the absorption coefficient with or without enhancement from the contrast agent.

^{66. V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA 97, 2767–2772 (2000). [CrossRef] [PubMed] ,2626. S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270.}The heterogeneous diffusion equation can be written as:

^{88. M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996).}

*i.e*.:

_{a}(r⃗) is the spatially varying part of the absorption coefficient μ

_{a}(r⃗) =

_{a}(r⃗).

^{2727. S. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999). [CrossRef] }Herein we will use the Rytov expansion as it has shown to be more effective with experimental measurements.

^{77. A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press , N-Y (1987).,88. M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996).}Under the Rytov expansion the diffuse field can be written as:

_{0}(r⃗,r⃗

_{s}) is the solution of the homogeneous equation (1) and Φ

_{sc}(r⃗, r⃗

_{s}) the diffuse Rytov phase. We are going to solve the heterogeneous equation for the specific case of a dual–interfering-source system. In this case the source term is:

_{s1}and r⃗

_{s2}respectively. The method to obtain analytical solution of the heterogeneous equation is derived from Kak.

^{77. A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press , N-Y (1987).}If we use the Rytov expression (4) in equation (2), we obtain:

_{0}(r⃗,r⃗

_{s}) = e

^{Φ0(r⃗,r⃗s)}and as the scattered signal is negligible compared to the delta sources functions, the homogeneous equation could be written as:

_{0}(r⃗,r⃗

_{s}),

*i.e*. the solution of the homogeneous equation (1)). In equation (10), the source term is a sum of two delta functions. To obtain a solution in this case we use the superposition theorem, which enable us to use the Green function theorem to solve the Helmoltz equation. So, we solve independently two equations for each delta function and the total field is the sum of the two separated fields obtained:

_{0}(r⃗,r⃗

_{s}) =

_{s1}) +

_{s2}).

### 2.2 Simulations

^{1616. V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998). [CrossRef] }The dimensions of the medium simulated were 20×5 cm

^{2}which correspond to the typical thickness of the compressed breast. This surface was sampled by 401×101 nodes (Δx=0.5mm; Δy =0.5mm). The optical properties of the background were chosen after the typical average values of breast

^{2828. T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000).},

*i.e*.

^{-1},

^{-1}We further assumed objects, representing tumors after contrast agent enhancement with

^{-1}and

^{-1}.

### 2.3 Reconstructions

_{S}× N

_{D}set of measurements that contain the modulated amplitude and the phase for a set of dual-interfering source system at each spatial node sampling the media, where N

_{s}is the number of source pairs and N

_{d}the number of detectors. In each case, 64 detectors and 17 source-pairs positions were considered. The detectors and the source pairs were positioned in the 8cm-wide middle part of the forward mesh to avoid lateral boundary interference. Simulations were performed for both the heterogeneous medium considered and for a homogeneous medium with the same background optical properties as the heterogeneous medium. The diffuse Rytov phase “measured” is then given by:

_{X}× N

_{Y}voxels. The inverse problem in its matrix form can be written as:

_{sc}(r⃗

_{si},r⃗

_{di}) is the diffuse Rytov perturbative phase for the

*i*

^{th}source-detector pair, W

_{ij}is the weight function for the

*j*

^{th}voxel and the

*i*

^{th}source-detector pair, and δμ

_{a}(r⃗

_{j}) is the differential absorption coefficient of the

*j*

^{th}voxel.

^{2424. R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994). [CrossRef] }For four couples of positive and negative image source pairs the analytical form of the weight function for the case of dual interfering sources is:

^{+}denote positive image and

^{-}negative image source pairs. Generally, four image source pairs yield satisfactory accuracy.

^{2929. M. Patterson, B. Chance, and B. Wilson, “Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989). [CrossRef] [PubMed] }The exact expression of the position of the image source pairs can be obtained from Arridge

^{2525. S. Arridge, “Photon-measurement density function.I Analytical forms,” Appl. Opt. 34, 7395–7409 (1995). [CrossRef] [PubMed] }or Wang

^{3030. L. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J.Opt.Soc.Am.A 15, 936–944 (1998). [CrossRef] }for example. Similar to the simulations performed, we considered an air-tissue boundary (n

_{tissue}=1.33), and used the Fresnel reflection coefficient for unpolarized light

^{2424. R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A 11, 2727–2741 (1994). [CrossRef] }to estimate the position of the extrapolated boundary condition, at z

_{boundary}= 1.69/(μ′

_{s}+ μ

_{a}). It has been shown that the position of the extrapolated boundary is a crucial theoretical parameter especially for the transmission-slab geometry.

^{3131. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997). [CrossRef] [PubMed] }

^{77. A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press , N-Y (1987).}(ART) with a positive constraint. More precisely, we used the Simultaneous Iterative Reconstruction Technique (SIRT) in this study,

*i.e*.:

^{th}estimate of j

^{th}element of the object function, b

_{i}the i

^{th}measurement, W

_{ij}the i-j

^{th}element of the weight matrix and

*λ*the relaxation parameter. This last parameter was set to 0.1 for all the reconstructions herein. This technique is known to produce smoother reconstructions but attains slower convergence than ART. To speed-up the SIRT procedure and to improve the reconstructions quality, we used a random projection access order.

^{3232. X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002). [CrossRef] }

*i.e*. iteration number), we employed numerical estimators, which compare the reconstructions to the model used. The best reconstructions were selected in accordance to these evaluators. Hence, the optimum regularization parameter is selected using prior knowledge.

^{3232. X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. 47, N1–N10 (2002). [CrossRef] }The first evaluator that we used was the correlation coefficient between the reconstruction and the model. The mathematical expression of this coefficient is:

_{i}(t¯) and

^{(k)}) each represent the pixel value (average value) in the original and k

^{th}reconstructed images, respectively. The value of this coefficient ranges between 0 (no correlation) and 1 (total correlation).

*i.e*.:

## 3. RESULTS

### 3.1 Synthetic measurements

**Fig. 1**. This example shows the perturbation at detectors in transmission geometry when three heterogeneities are embedded in the media (

*cf*.

**Fig. 5**(a)).

**Fig. 5**(a) is on the null-line and the perturbations due to the other objects are somehow compensating themselves.

^{10–1310. A. Knuttel, J.M. Schmitt, and J.R. Knutson, “Spatial localization of absorbing bodies by interfering diffuse photon-density waves,” Appl. Opt. 32, 381–389 (1993). [CrossRef] [PubMed] }

### 3.2 Sensitivity profiles

**Fig. 2**(

*ν*: modulation frequency of the sources, d: separation between the two out of phases sources constituting a pair.).

*2.2*. The sensitivity is displayed in terms of natural logarithm of the weight function for a specific source detector configuration.

**Fig. 2**, the sensitivity maps exhibit some common features such as the null-sensitivity line corresponding to the plane separating the two out of phase sources. This plane is characteristic to the phased array configuration. It corresponds to the plane of 180° transition of the phase.

^{1111. M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A 14, 3083–3092 (1997). [CrossRef] ,3333. X. Intes, B. Chance, M. Holboke, and A. Yodh, “Interfering diffusive photon-density waves with an absorbing-fluorescent inhomogeneity,” Opt. Express 8, 223–231 (2001). http://www.opticsexpress.org/opticsexpress/tocv8n3.htm [CrossRef] [PubMed] }No perturbation can be detected along this plane since the excitation pattern is cancelled.

^{3434. D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996). [CrossRef] [PubMed] }This is also seen in the profiles of

**Fig. 1**.

*ν*) and source separation (d), as shown in

**Fig. 2**(a) and (d). Generally, the higher the frequency and the lower the separation, the more spatially confined is the sensitivity. However, the sensitivity retains a dissymmetric spatial distribution in contrast to the banana shape of the single source technique.

### 3.3 Single objects

^{2}object was embedded in the center of the medium. Its optical values were set to have a 10:1 contrast versus the background. As described above, 17 positions of the phased sources were used with 64 detectors. The sources were separated by 2cm and modulated at 50MHz. Reconstruction for the object embedded in the center of the media is presented in

**Fig. 3**.

*2.3*(equation (21) and (22)). After this iteration, the reconstructions are getting worse as long as SIRT is a semi-convergent technique.

^{88. M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996).}

**Fig. 3**, we see that the inhomogeneity was well recovered in terms of spatial and quantitative information. The reconstruction is artifact free demonstrating that the analytical model is able to recover the information.

**Fig. 4**.

### 3.4 Multiple objects

^{2}were embedded at 1–2.5–4 cm respectively as shown in

**Fig. 5**(a). They were separated by 2cm in the longitudinal dimension and 1.5cm in the transversal dimension.

**Fig. 5**display reconstructions obtained with 17 sources and 64 detectors for different modulation frequencies and separation between the sources pair.

**Fig. 6**, reconstructions for the same configuration but with noise added to the synthetic measurements (

*c.f*.

**Fig. 6**). This uniformly distributed noise was added independently on each source of a phased pair. The reconstructions presented are for a modulation frequency of 200MHz and a separation of 2cm between the sources pair.

## 4. DISCUSSION and CONCLUSIONS

**Fig. 5**(a), the central object and the one closest to the detector plane were underestimated. For higher frequencies (

*cf*.

**Fig. 5**(d)), this non-uniform estimation is less apparent. For this case, the three objects were well quantified.

**Fig. 5**and

**Fig. 6**, the object close to the boundaries suffer from this heart-shape sensitivity profiles.

*i.e*., the modification of the relative strength

^{3636. B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. 67, 4324–4331 (1996). [CrossRef] }or relative phase difference

^{1616. V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998). [CrossRef] }of the sources in order to bend the null-line. Experimental validation of this method to produce a scanning-null line has already been performed.

^{3434. D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics 1, 305–310 (1996). [CrossRef] [PubMed] }With this approach, one could produce an apparatus that will always optimally work with a detector on the null line, enabling more uniform sensitivity across the medium of investigation

## Acknowledgements

## References and links

1. | D. Hawrys and E. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents,” Neoplasia |

2. | T. McBride, B. Pogue, S. Jiang, U. Osterberg, and K. Paulsen, “Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Let. |

3. | V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Research |

4. | A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human function,” Trends Neurosci. |

5. | M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon-tomography,” Opt. Lett. |

6. | V. Ntziachristos, A. Yodh, M. Schnall, and B. Chance, “Concurent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc.Nat.Acad.Sci. USA |

7. | A. Kak and M. Slaney, “Computerized tomographic Imaging,” IEEE Press , N-Y (1987). |

8. | M. O’Leary, “Imaging with diffuse photon density waves,” PhD University of Pennsylvania (1996). |

9. | V. Ntziachristos, B. Chance, and A. Yodh, “Differential diffuse optical tomography,” Opt. Express |

10. | A. Knuttel, J.M. Schmitt, and J.R. Knutson, “Spatial localization of absorbing bodies by interfering diffuse photon-density waves,” Appl. Opt. |

11. | M. Erickson, J. Reynolds, and K. Webb, “Comparison of sensitivity for single-source and dual-interfering-source configurations in optical diffusion imaging,” J.Opt.Soc.Am.A |

12. | Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source”, Opt. Express |

13. | B. Chance, K. Kang, L. He, J. Weng, and E. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Nat. Acad. Sci. USA |

14. | B. Chance and E. Conant, “A novel tumor imager using NIR light,” in preparation. |

15. | Y. Chen, S. Zhou, C. Xie, S. Nioka, M. Delivoria-Papadopoulos, E. Anday, and B. Chance, “Preliminary evaluation of dual-wavelength phased array imaging on neonatal brain function,” Journal of Biomedical Optics |

16. | V. Ntziachristos, XuHui Ma, and B. Chance, “Time-correlated single photon counting imager for simultaneaous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. |

17. | S. Morgan, M. Somekh, and K. Hopcraqft, “Probabilistic method for phased array detection in scattering media,” Opt. Eng. |

18. | S. Morgan and K. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express |

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

20. | P. Morse and H. Feshbach, “Methods of theoretical physics,” Mc Graw Hill, N-Y (1953). |

21. | A. Ishimaru, “Wave propagation and scattering in random media,” Vol. |

22. | K. Yoo, F. Liu, and R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?,” Phys. Rev. Lett. |

23. | X. Intes, B. Le Jeune, F. Pellen, Y. Guern, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source”, Waves Random Media |

24. | R. Haskell, L. Svaasand, TT. Tsay, Tc. Feng, M. McAdams, and B. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J.Opt.Soc.Am.A |

25. | S. Arridge, “Photon-measurement density function.I Analytical forms,” Appl. Opt. |

26. | S. Nioka, S. Colak, X. Li, Y. Yang, and B. Chance, “Breast tumor images of hemodynamics information using a contrast agent with backprojection and FFT enhancement”, OSA Trends in Optics and Photonics vol. 21, Advances in Optical imaging and Photon Migration, JamesG. Fujimoto and Michael S. Patterson, eds. (Optical Society of America, Washington, DC 1998), 266–270. |

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

28. | T. Durduran, M. Holboke, J. Culver, L. Zubkov, R. Choe, D. Pattanayak, B. Chance, and A. Yodh, “Tissue bulk optical properties of breast and phantoms obtained with clinical optical imager,” in Biomedical Topical Meetings, OSA Technical Digest (Optical Society of America, Washington DC, 2000), 386–388 (2000). |

29. | M. Patterson, B. Chance, and B. Wilson, “Time-resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. |

30. | L. Wang, “Rapid modeling of diffuse reflectance of light in turbid slabs,” J.Opt.Soc.Am.A |

31. | D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. |

32. | X. Intes, V. Ntziachristos, J. Culver, A. Yodh, and B. Chance, “Projection access order in Algebraic Reconstruction Technique for Diffuse Optical Tomography,” Phys. Med. Biol. |

33. | X. Intes, B. Chance, M. Holboke, and A. Yodh, “Interfering diffusive photon-density waves with an absorbing-fluorescent inhomogeneity,” Opt. Express |

34. | D. Papaioannou, G.’ tHoof, S. Colak, and J. Oostveen, “Detection limit in localizing objects hidden in turbid medium using an optically scanned phased array,” Journal of Biomedical Optics |

35. | B. Pogue, T. Mc. Bride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

36. | B. Chance, K. Kang, L. He, H. Liu, and S. Zhou, “Precision localization of hidden absorbers in body tissues with phased-array optical systems,” Rev. Sci. Instrum. |

**OCIS Codes**

(110.5100) Imaging systems : Phased-array imaging systems

(170.0110) Medical optics and biotechnology : Imaging systems

(170.5270) Medical optics and biotechnology : Photon density waves

(170.5280) Medical optics and biotechnology : Photon migration

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 19, 2001

Published: January 14, 2002

**Citation**

Xavier Intes, Vasilis Ntziachristos, and Britton Chance, "Analytical model for dual-interfering sources diffuse optical tomography," Opt. Express **10**, 2-14 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-2

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### References

- D. Hawrys and E. Sevick-Muraca, "Developments toward diagnostic breast cancer imaging using Near-Infrared optical measurements and fluorescent contrast agents," Neoplasia 2, 388-417 (2000). [CrossRef]
- T. McBride, B. Pogue, S. Jiang, U. Osterberg and K. Paulsen, "Initial studies of in-vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging," Opt. Let. 26, 822-824 (2001). [CrossRef]
- V. Ntziachristos and B. Chance, "Probing physiology and molecular function using optical imaging: applications to breast cancer," Breast Cancer Research 3, 41-47 (2001). [CrossRef] [PubMed]
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