## Light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities based on hybrid simplified spherical harmonics with radiosity model |

Biomedical Optics Express, Vol. 4, Issue 10, pp. 2209-2223 (2013)

http://dx.doi.org/10.1364/BOE.4.002209

Acrobat PDF (4234 KB)

### Abstract

Modeling light propagation in the whole body is essential and necessary for optical imaging. However, non-scattering, low-scattering and high absorption regions commonly exist in biological tissues, which lead to inaccuracy of the existing light transport models. In this paper, a novel hybrid light transport model that couples the simplified spherical harmonics approximation (SP_{N}) with the radiosity theory (HSRM) was presented, to accurately describe light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities. In the model, the radiosity theory was used to characterize the light transport in non-scattering regions and the SP_{N} was employed to handle the scattering problems, including subsets of low-scattering and high absorption. A Neumann source constructed by the light transport in the non-scattering region and formed at the interface between the non-scattering and scattering regions was superposed into the original light source, to couple the SP_{N} with the radiosity theory. The accuracy and effectiveness of the HSRM was first verified with both regular and digital mouse model based simulations and a physical phantom based experiment. The feasibility and applicability of the HSRM was then investigated by a broad range of optical properties. Lastly, the influence of depth of the light source on the model was also discussed. Primary results showed that the proposed model provided high performance for light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities.

© 2013 OSA

## 1. Introduction

1. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**(3), 313–320 (2005). [CrossRef] [PubMed]

3. J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. **7**(7), 591–607 (2008). [CrossRef] [PubMed]

4. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**(4), R1–R43 (2005). [CrossRef] [PubMed]

4. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**(4), R1–R43 (2005). [CrossRef] [PubMed]

6. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. **202**(1), 323–345 (2005). [CrossRef]

5. S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. **25**(12), 123010 (2009). [CrossRef]

6. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. **202**(1), 323–345 (2005). [CrossRef]

4. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**(4), R1–R43 (2005). [CrossRef] [PubMed]

6. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. **202**(1), 323–345 (2005). [CrossRef]

**50**(4), R1–R43 (2005). [CrossRef] [PubMed]

7. W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**(18), 6756–6771 (2005). [CrossRef] [PubMed]

9. V. A. Markel and J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **64**(3), 035601 (2001). [CrossRef] [PubMed]

_{N}) [10

10. K. Peng, X. Gao, X. Qu, N. Ren, X. Chen, X. He, X. Wang, J. Liang, and J. Tian, “Graphics processing unit parallel accelerated solution of the discrete ordinates for photon transport in biological tissues,” Appl. Opt. **50**(21), 3808–3823 (2011). [CrossRef] [PubMed]

11. Z. Yuan, X.-H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. **54**(1), 67–88 (2009). [CrossRef] [PubMed]

_{N}) [12

12. S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. **18**(1), 247–260 (2007). [CrossRef]

13. P. Surya Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. **230**(19), 7364–7383 (2011). [CrossRef]

14. W. Cong, A. Cong, H. Shen, Y. Liu, and G. Wang, “Flux vector formulation for photon propagation in the biological tissue,” Opt. Lett. **32**(19), 2837–2839 (2007). [CrossRef] [PubMed]

_{N}) [15

15. A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**(1), 441–470 (2006). [CrossRef]

17. Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. **54**(21), 6477–6493 (2009). [CrossRef] [PubMed]

_{N}, P

_{N}and PA, can accurately model light propagation in a scattering medium, but the computational burden limits their application in complex heterogeneities. The DE, however, provides a high computational efficiency and is only valid in the high scattering region [18

18. A. D. Klose, “The forward and inverse problem in tissue optics based on the radiative transfer equation: A brief review,” J. Quant. Spectrosc. Radiat. Transf. **111**(11), 1852–1853 (2010). [CrossRef] [PubMed]

_{N}has been recently presented for 3D optical imaging, which provides high accuracy for light transport in low scattering or high absorption regions [15

15. A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**(1), 441–470 (2006). [CrossRef]

17. Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. **54**(21), 6477–6493 (2009). [CrossRef] [PubMed]

19. H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. **44**(12), 2897–2906 (1999). [CrossRef] [PubMed]

22. D. Yang, X. Chen, S. Ren, X. Qu, J. Tian, and J. Liang, “Influence investigation of a void region on modeling light propagation in a heterogeneous medium,” Appl. Opt. **52**(3), 400–408 (2013). [CrossRef] [PubMed]

19. H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. **44**(12), 2897–2906 (1999). [CrossRef] [PubMed]

26. X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. **17**(6), 066015 (2012). [CrossRef] [PubMed]

27. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. **50**(20), 4913–4930 (2005). [CrossRef] [PubMed]

29. D. Gorpas and S. Andersson-Engels, “Evaluation of a radiative transfer equation and diffusion approximation hybrid forward solver for fluorescence molecular imaging,” J. Biomed. Opt. **17**(12), 126010 (2012). [CrossRef] [PubMed]

30. T. Hayashi, Y. Kashio, and E. Okada, “Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region,” Appl. Opt. **42**(16), 2888–2896 (2003). [CrossRef] [PubMed]

19. H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. **44**(12), 2897–2906 (1999). [CrossRef] [PubMed]

24. 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 nonscattering regions,” Med. Phys. **27**(1), 252–264 (2000). [CrossRef] [PubMed]

26. X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. **17**(6), 066015 (2012). [CrossRef] [PubMed]

*nm*) [31

31. V. Y. Soloviev, G. Zacharakis, G. Spiliopoulos, R. Favicchio, T. Correia, S. R. Arridge, and J. Ripoll, “Tomographic imaging with polarized light,” J. Opt. Soc. Am. A **29**(6), 980–988 (2012). [CrossRef] [PubMed]

32. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. **50**(17), 4225–4241 (2005). [CrossRef] [PubMed]

_{N}with the radiosity theory was presented in this paper, called HSRM for short, which is the abbreviation of the hybrid SP

_{N}-radiosity method. In the presented model, the radiosity theory was used to characterize the light transport in non-scattering regions and SP

_{N}was selected to handle the scattering problems. The hybrid model was coupled by a Neumann source which was constructed by the light transport in the non-scattering region and formed at an interface between the non-scattering and scattering regions. The Neumann source was superposed onto the original light source to construct a new synthetic light source. The accuracy and effectiveness of the HSRM were first verified with both regular and digital mouse model based simulations and a physical phantom based experiment. The feasibility and applicability of the HSRM was then investigated by the simulations performed over a broad range of optical properties. Lastly, the influence of depth of the light source on the model was also discussed.

## 2. Methods

_{N}was used to depict the light transport in the scattering region. Based on RTE, the three dimensional SP

_{N}was obtained after a series of inferential reasonings in the planar geometry with the spherical harmonics approximation [15

15. A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. **220**(1), 441–470 (2006). [CrossRef]

_{N}can be described as follows [15

**220**(1), 441–470 (2006). [CrossRef]

33. Y. Lu, H. B. Machado, A. Douraghy, D. Stout, H. Herschman, and A. F. Chatziioannou, “Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework,” Opt. Express **17**(19), 16681–16695 (2009). [CrossRef] [PubMed]

**220**(1), 441–470 (2006). [CrossRef]

33. Y. Lu, H. B. Machado, A. Douraghy, D. Stout, H. Herschman, and A. F. Chatziioannou, “Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework,” Opt. Express **17**(19), 16681–16695 (2009). [CrossRef] [PubMed]

**220**(1), 441–470 (2006). [CrossRef]

33. Y. Lu, H. B. Machado, A. Douraghy, D. Stout, H. Herschman, and A. F. Chatziioannou, “Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework,” Opt. Express **17**(19), 16681–16695 (2009). [CrossRef] [PubMed]

**220**(1), 441–470 (2006). [CrossRef]

**17**(19), 16681–16695 (2009). [CrossRef] [PubMed]

34. J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. **91**(10), 103901 (2003). [CrossRef] [PubMed]

*N*and could be deduced according to Eq. (2) and Eq. (3). The specific expression for the third order is presented as follows:

7. W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**(18), 6756–6771 (2005). [CrossRef] [PubMed]

16. K. Liu, Y. Lu, J. Tian, C. Qin, X. Yang, S. Zhu, X. Yang, Q. Gao, and D. Han, “Evaluation of the simplified spherical harmonics approximation in bioluminescence tomography through heterogeneous mouse models,” Opt. Express **18**(20), 20988–21002 (2010). [CrossRef] [PubMed]

17. Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. **54**(21), 6477–6493 (2009). [CrossRef] [PubMed]

20. 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**(9), 1659–1670 (2000). [CrossRef] [PubMed]

22. D. Yang, X. Chen, S. Ren, X. Qu, J. Tian, and J. Liang, “Influence investigation of a void region on modeling light propagation in a heterogeneous medium,” Appl. Opt. **52**(3), 400–408 (2013). [CrossRef] [PubMed]

24. 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 nonscattering regions,” Med. Phys. **27**(1), 252–264 (2000). [CrossRef] [PubMed]

29. D. Gorpas and S. Andersson-Engels, “Evaluation of a radiative transfer equation and diffusion approximation hybrid forward solver for fluorescence molecular imaging,” J. Biomed. Opt. **17**(12), 126010 (2012). [CrossRef] [PubMed]

*k*th node, and

## 3. Experiments and results

_{N}can obtain an ideal balance between accuracy and computation time [16

16. K. Liu, Y. Lu, J. Tian, C. Qin, X. Yang, S. Zhu, X. Yang, Q. Gao, and D. Han, “Evaluation of the simplified spherical harmonics approximation in bioluminescence tomography through heterogeneous mouse models,” Opt. Express **18**(20), 20988–21002 (2010). [CrossRef] [PubMed]

**54**(21), 6477–6493 (2009). [CrossRef] [PubMed]

35. H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. **11**(9), 1029–1038 (2004). [CrossRef] [PubMed]

36. N. Ren, J. Liang, X. Qu, J. Li, B. Lu, and J. Tian, “GPU-based Monte Carlo simulation for light propagation in complex heterogeneous tissues,” Opt. Express **18**(7), 6811–6823 (2010). [CrossRef] [PubMed]

*ARE*) was introduced as:where

*f*is the value of the intensity at the

_{i}*i*th sample point;

*N*is the total number of sample points; the superscript

*std*denotes that the intensity was obtained from MOSE or measurements and

*cal*represents that the intensity was calculated by the HSRM or HRDM.

*ARE*was used to reflect the discrepancy between the calculation of HSRM or HRDM and the standard of MOSE or measurements. It means that the closer

*ARE*gets to zero, the better the performance of the calculated method.

### 3.1 Accuracy verification experiments

#### 3.1.1 3D regular geometry based simulations

*ARE*less than 2%, some other interesting conclusions could also be addressed. Firstly, the HSRM performed much better than the HRDM when the high absorption or low scattering regions existed in the solving subject. Secondly, the HSRM exhibited good performance in the non-scattering regions, such as the detection points at about 0-15 and 45-60 in Fig. 2(a)-2(c) and points at about 0-10 and 45-50 in Fig. 2(d)-2(e) where the results were affected greatly by the non-scattering regions. Last but not least, although the HSRM had good agreement with MOSE, both of the results of the HSRM and MOSE seemed a little fluctuant, as shown in Fig. 2. This may have been induced by the stochastic nature of the Monte Carlo method and insufficient mesh discretization of the HSRM. The comparison results demonstrated the accuracy of the HSRM model and illustrated its superiority over the HRDM.

#### 3.1.2 3D digital mouse based simulation

*in vivo*studies, a physical model should be first constructed as follows. Firstly, the tissue regions should be properly segmented from the anatomical structure of small animals which can be scanned by micro-CT or MRI. Secondly, the tissue regions were divided into high scattering, low scattering, and non-scattering regions according to their optical properties. Accurately, the optical properties should be ideally measured

*in vivo*. Considering the difficulty in obtaining

*in vivo*measurements of the optical properties, the optical properties commonly adopted the calculated values according to an empirical formula presented in the literature [32

32. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. **50**(17), 4225–4241 (2005). [CrossRef] [PubMed]

*in vivo*at the highest extent, a digital mouse model was employed to verify the accuracy of HSRM in an irregular model in this subsection. The torso of a digital mouse atlas extracted from the CT and cryosection data was used to construct the digital mouse model, and the main organs were selected to mimic the heterogeneities, as shown in Fig. 1(c) [37

37. B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. **52**(3), 577–587 (2007). [CrossRef] [PubMed]

*nm*[32

32. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. **50**(17), 4225–4241 (2005). [CrossRef] [PubMed]

*mm*was located beside the stomach with the coordinates of (30.4, 15.2, 26.4)

*mm*. To obtain relatively smooth results, the torso of the digital mouse was discretized into 94738 tetrahedral elements and 16998 nodes.

*z*= 26.5 and 34.5

*mm*, as shown in Fig. 3(d) and 3(e), and their relative errors are listed in Table 3. In Table 3,

*ARE*is the average relative error between the calculated results of the HSRM or HRDM and the simulated one of MOSE on the whole detection points;

*LARE*was defined to depict the average relative error between the calculated results and the one of MOSE on the partial detection points where the photon density was highly affected by the low-scattering regions, such as the detection points 15 to 25 in Fig. 3(d) and 15 to 20 in Fig. 3(e).

*ARE*listed in Table 3, it is obvious that both results of HSRM and HRDM were close to those of MOSE. However, the profiles at the detection positions of 15 to 25 in Fig. 3(d) and 15 to 20 in Fig. 3(e) show that the HSRM performed better than the HRDM as shown by

*LARE*in Table 3. Because the influence of the low-scattering region mainly focuses on the detection points of 15 to 25 in Fig. 3(d) and 15 to 20 in Fig. 3(e), the HSRM and HRDM both performed well on all of the detection points. However, the HSRM performed better on the selected partial influenced detection points. On the other hand, there are some fluctuations in all of the results of the HSRM, HRDM and MOSE, which might be caused by the fact that the mesh was derived from the irregularity of the body surface of the digital mouse. The digital mouse based simulation demonstrated that the HSRM performed better than the HRDM when the low-scattering region existed simultaneously with the non-scattering regions and was more suitable for depicting light propagation in whole-body small animal imaging.

#### 3.1.3 Physical cube phantom based experiment

*mm*in width and 30

*mm*in height. Two holes of different sizes were drilled in the phantom as shown in Fig. 1(d). The first hole with a radius of 1

*mm*and a depth of 16

*mm*was injected with the fluorescent solution of 2

*mm*in height to simulate the light source. The center of the light source was at the coordinate of (12, 12, 15)

*mm*. The second hole was employed to simulate the non-scattering region, with a radius of 3

*mm*and a depth of 18

*mm*. The cavity region of 13

*mm*in height and 3

*mm*in radius was constructed by blocking up the top of the second hole with a nylon rod of 5

*mm*in height. The peak wavelength of the light source was about 660

*nm*, so the measured optical properties of the nylon phantom were listed as: the absorption coefficient was 0.023

*mm*

^{−1}and the scattering coefficient was 20

*mm*

^{−1}[7

7. W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**(18), 6756–6771 (2005). [CrossRef] [PubMed]

*z*= 15 and 11

*mm*were extracted and shown in Fig. 4(d) and 4(e). From Fig. 4, we found that the similar tendency and good agreement were obtained between the measurements and the calculated results of both the HSRM and HRDM. The average relative errors between the measurements and the calculated results were about 1.824% and 1.818% respectively. Because of the high scattering of the nylon phantom, the performance of the HSRM and HRDM were both approved. Thus, the comparison results indicated that the proposed HSRM performed as well as the HRDM for simulating light transport in the medium with high scattering and non-scattering regions simultaneously, and showed good agreement with the measurement of the physical experiment.

### 3.2 Investigation of the optical proprieties

**44**(12), 2897–2906 (1999). [CrossRef] [PubMed]

23. 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**(4), 767–783 (1996). [CrossRef] [PubMed]

25. J. H. Lee, S. Kim, and Y. T. Kim, “Modeling of diffuse-diffuse photon coupling via a nonscattering region: a comparative study,” Appl. Opt. **43**(18), 3640–3655 (2004). [CrossRef] [PubMed]

27. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. **50**(20), 4913–4930 (2005). [CrossRef] [PubMed]

_{N}used in the evaluation. The performance may be improved with an increase in the order of SP

_{N}. Overall, the HSRM would achieve a much larger scope of application than the HRDM in whole-body small animal imaging.

### 3.3 Investigation of the depth of the source in light transport

*mm*, in which the distance from the source center to the phantom surface was 7, 5.5, 4

*mm*respectively, as shown in Fig. 5(b). Comparison of the results of the HSRM, HRDM and MOSE for all three experiments are shown in Fig. 7 and the quantitative comparisons are listed in Table 6.

## 4. Discussion and conclusion

_{N}) and radiosity theory model (HSRM) was proposed in this paper.

_{N}used in the HSRM, which will further increase the burden of the computation. Lastly, the result of the HSRM was also affected by the depth of the light source. The lower order HSRM model may not work with the increase in depth.

## Acknowledgments

## References and links

1. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. |

2. | J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. |

3. | J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. |

4. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

5. | S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. |

6. | A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. |

7. | W. X. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express |

8. | Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express |

9. | V. A. Markel and J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

10. | K. Peng, X. Gao, X. Qu, N. Ren, X. Chen, X. He, X. Wang, J. Liang, and J. Tian, “Graphics processing unit parallel accelerated solution of the discrete ordinates for photon transport in biological tissues,” Appl. Opt. |

11. | Z. Yuan, X.-H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. |

12. | S. Wright, M. Schweiger, and S. R. Arridge, “Reconstruction in optical tomography using the PN approximations,” Meas. Sci. Technol. |

13. | P. Surya Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. |

14. | W. Cong, A. Cong, H. Shen, Y. Liu, and G. Wang, “Flux vector formulation for photon propagation in the biological tissue,” Opt. Lett. |

15. | A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. |

16. | K. Liu, Y. Lu, J. Tian, C. Qin, X. Yang, S. Zhu, X. Yang, Q. Gao, and D. Han, “Evaluation of the simplified spherical harmonics approximation in bioluminescence tomography through heterogeneous mouse models,” Opt. Express |

17. | Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. |

18. | A. D. Klose, “The forward and inverse problem in tissue optics based on the radiative transfer equation: A brief review,” J. Quant. Spectrosc. Radiat. Transf. |

19. | H. Dehghani, D. T. Delpy, and S. R. Arridge, “Photon migration in non-scattering tissue and the effects on image reconstruction,” Phys. Med. Biol. |

20. | H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A |

21. | J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express |

22. | D. Yang, X. Chen, S. Ren, X. Qu, J. Tian, and J. Liang, “Influence investigation of a void region on modeling light propagation in a heterogeneous medium,” Appl. Opt. |

23. | 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. |

24. | 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 nonscattering regions,” Med. Phys. |

25. | J. H. Lee, S. Kim, and Y. T. Kim, “Modeling of diffuse-diffuse photon coupling via a nonscattering region: a comparative study,” Appl. Opt. |

26. | X. Chen, D. Yang, X. Qu, H. Hu, J. Liang, X. Gao, and J. Tian, “Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection,” J. Biomed. Opt. |

27. | T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. |

28. | T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. |

29. | D. Gorpas and S. Andersson-Engels, “Evaluation of a radiative transfer equation and diffusion approximation hybrid forward solver for fluorescence molecular imaging,” J. Biomed. Opt. |

30. | T. Hayashi, Y. Kashio, and E. Okada, “Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region,” Appl. Opt. |

31. | V. Y. Soloviev, G. Zacharakis, G. Spiliopoulos, R. Favicchio, T. Correia, S. R. Arridge, and J. Ripoll, “Tomographic imaging with polarized light,” J. Opt. Soc. Am. A |

32. | G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. |

33. | Y. Lu, H. B. Machado, A. Douraghy, D. Stout, H. Herschman, and A. F. Chatziioannou, “Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework,” Opt. Express |

34. | J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. |

35. | H. Li, J. Tian, F. P. Zhu, W. X. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Acad. Radiol. |

36. | N. Ren, J. Liang, X. Qu, J. Li, B. Lu, and J. Tian, “GPU-based Monte Carlo simulation for light propagation in complex heterogeneous tissues,” Opt. Express |

37. | B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. |

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.7050) Medical optics and biotechnology : Turbid media

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: July 18, 2013

Revised Manuscript: September 13, 2013

Manuscript Accepted: September 17, 2013

Published: September 23, 2013

**Citation**

Defu Yang, Xueli Chen, Zhen Peng, Xiaorui Wang, Jorge Ripoll, Jing Wang, and Jimin Liang, "Light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities based on hybrid simplified spherical harmonics with radiosity model," Biomed. Opt. Express **4**, 2209-2223 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-10-2209

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