## Reliable recovery of the optical properties of multi-layer turbid media by iteratively using a layered diffusion model at multiple source-detector separations |

Biomedical Optics Express, Vol. 5, Issue 3, pp. 975-989 (2014)

http://dx.doi.org/10.1364/BOE.5.000975

Acrobat PDF (1815 KB)

### Abstract

Accurately determining the optical properties of multi-layer turbid media using a layered diffusion model is often a difficult task and could be an ill-posed problem. In this study, an iterative algorithm was proposed for solving such problems. This algorithm employed a layered diffusion model to calculate the optical properties of a layered sample at several source-detector separations (SDSs). The optical properties determined at various SDSs were mutually referenced to complete one round of iteration and the optical properties were gradually revised in further iterations until a set of stable optical properties was obtained. We evaluated the performance of the proposed method using frequency domain Monte Carlo simulations and found that the method could robustly recover the layered sample properties with various layer thickness and optical property settings. It is expected that this algorithm can work with photon transport models in frequency and time domain for various applications, such as determination of subcutaneous fat or muscle optical properties and monitoring the hemodynamics of muscle.

© 2014 Optical Society of America

## 1. Introduction

1. A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. **104**(10), 4014–4019 (2007). [CrossRef] [PubMed]

2. S. H. Tseng, P. Bargo, A. Durkin, and N. Kollias, “Chromophore concentrations, absorption and scattering properties of human skin in-vivo,” Opt. Express **17**(17), 14599–14617 (2009). [CrossRef] [PubMed]

3. A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. **44**(11), 2689–2702 (1999). [CrossRef] [PubMed]

4. V. Toronov, A. Webb, J. H. Choi, M. Wolf, L. Safonova, U. Wolf, and E. Gratton, “Study of local cerebral hemodynamics by frequency-domain near-infrared spectroscopy and correlation with simultaneously acquired functional magnetic resonance imaging,” Opt. Express **9**(8), 417–427 (2001). [CrossRef] [PubMed]

2. S. H. Tseng, P. Bargo, A. Durkin, and N. Kollias, “Chromophore concentrations, absorption and scattering properties of human skin in-vivo,” Opt. Express **17**(17), 14599–14617 (2009). [CrossRef] [PubMed]

5. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A **11**(10), 2727–2741 (1994). [CrossRef] [PubMed]

6. S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt. **14**(5), 054043 (2009). [CrossRef] [PubMed]

7. S. H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative spectroscopy of superficial turbid media,” Opt. Lett. **30**(23), 3165–3167 (2005). [CrossRef] [PubMed]

3. A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. **44**(11), 2689–2702 (1999). [CrossRef] [PubMed]

8. B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia **2**(1/2), 26–40 (2000). [CrossRef] [PubMed]

*et al.*studied the influence of the top layer on the bottom layer optical properties recovery, and they reported that the semi-infinite assumption could lead to a recovery error higher than 100% [9

9. T. J. Farrell, M. S. Patterson, and M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. **37**(10), 1958–1972 (1998). [CrossRef] [PubMed]

*et al.*used the eigenfunction method to solve the time-dependent diffusion equation for two- and three-layered samples [10

10. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(5), 056623 (2003). [CrossRef] [PubMed]

11. A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. **15**(2), 025002 (2010). [CrossRef] [PubMed]

3. A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. **44**(11), 2689–2702 (1999). [CrossRef] [PubMed]

12. A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, and H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. **37**(4), 779–791 (1998). [CrossRef] [PubMed]

13. T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. **39**(25), 4733–4745 (2000). [CrossRef] [PubMed]

*et al.*and Kienle

*et al.*used layered diffusion models with frequency-domain and time resolved DRS system, respectively, to determine five parameters (absorption and reduced scattering coefficients of the first and second layers, and the thickness of the first layer) of two-layered samples. They both reported that simultaneous recovery of five parameters was possible only when the initial values provided for the inverse problem were very close to the true values; otherwise the errors of the recovered values would be unacceptably large [12

12. A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, and H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. **37**(4), 779–791 (1998). [CrossRef] [PubMed]

13. T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. **39**(25), 4733–4745 (2000). [CrossRef] [PubMed]

13. T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. **39**(25), 4733–4745 (2000). [CrossRef] [PubMed]

## 2. Theoretical models

### 2.1 Layered diffusion model

11. A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. **15**(2), 025002 (2010). [CrossRef] [PubMed]

*D*= 1/(3

*μ*+ 3

_{a}*μ*) and Φ are the diffusion constant and the fluence rate, respectively.

_{s}'*S*is the source term,

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

*i*is the number of layer. The pencil beam light source in frequency domain can be approximated as a point source beneath the surface and is expressed as

*S*=

_{1}*δ(x,y,z-z*, where

_{0})*z*

_{0}= 1/(

*μ*+

_{a}*μ*) is the location of the point source and

_{s}'*ω*= 2

*πf*with

*f*representing the source modulation frequency. By applying the extrapolated boundary condition and assuming the fluence and the flux are continuous at the boundary, the fluence rate of the diffusion equation system can be solved in the Fourier domain using the 2-D Fourier transform:

*R*

_{eff}_{1}representing the fraction of photons that is internally reflected at the air-sample boundary. For a two-layered sample,

*Z(z,s)*and

*N(z,s)*have the following form: For a three-layered sample,

*Z(z,s)*and

*N(z,s)*are:

*l*and

_{i}*n*are the thickness and refractive index of the layer, respectively and

_{i}*L*

_{1}at the boundary, where

*R*(

_{fres}*θ*) is the Fresnel reflection coefficient for a photon with an incident angle

*θ*relative to the normal to the boundary.

### 2.2 Layered Monte Carlo model

## 3. Results and discussion

### 3.1 Direct optical properties recovery of two-layered samples with the layered diffusion model

*μ*and

_{a2}*μ*, respectively) are smaller than those of the top layer optical properties (maximal recovery percent errors are 71% and 49% for

_{s2}'*μ*and

_{a1}*μ*, respectively). The recovery error of the top layer thickness can be as high as 177%. It is found that the bottom layer optical properties and the top layer reduced scattering coefficient could be more accurately recovered at large SDSs than at short SDSs. Among the five parameters, the top layer absorption coefficient and the top layer thickness both have higher recovery errors than other parameters in general.

_{s1}'*Y*) using the following parameters:

*μ*= 0.05 mm

_{a1}^{−1},

*μ*= 1.5 mm

_{s1}'^{−1},

*μ*= 0.002 mm

_{a2}^{−1},

*μ*= 1.0 mm

_{s2}'^{−1}, n = 1.43, top layer thickness = 1 mm, modulation frequency = 0 to 500 MHz with 1 MHz step, and the SDS = 6 mm. Next, by keeping the parameters other than

*μ*and

_{a1}*μ*the same as those used in the Monte Carlo simulation, we utilized the two-layered diffusion model to calculate the frequency domain reflectance at various

_{s1}'*μ*and

_{a1}*μ*combinations (

_{s1}'*Y*). Here the subscript

_{j}*j*is the number of the

*μ*and

_{a1}*μ*combinations, and it was set in the range from 1 to 2500 for plotting Fig. 2. Finally, we compute χ

_{s1}'^{2}= (

*Y –Y*)

_{j}^{2}at various

*μ*and

_{a1}*μ*combinations and plot Fig. 2(a). In computing χ

_{s1}'^{2}displayed in Fig. 2 and 3, the amplitude magnitudes were magnified by the ratio of average phase to average amplitude, and we summed up the results at all modulation frequencies. It can be clearly seen in Fig. 2(a) that as there are only two variables to be fit (

*μ*and

_{a1}*μ*), a least-squares fitting method can lead to a single solution for this two-layered sample problem since there is only one global minimum. It should be noted that the trough of the curve in Fig. 2(a) corresponds to (

_{s1}'*μ*= 0.0449 mm

_{a1}^{−1},

*μ*= 1.724 mm

_{s1}'^{−1}) which deviates from the benchmark top layer optical properties set in the Monte Carlo simulation. This suggests that there exists system differences between Monte Carlo method and the diffusion equation as indicated by Kienle

*et al*. [12

12. A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, and H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. **37**(4), 779–791 (1998). [CrossRef] [PubMed]

*Y*) for generating Fig. 2(a) and 2(b). From Fig. 2(b), we can find that the value of (

*Y*)

_{i}-Y_{i}^{2}at the global minimum is 0.4272 which is smaller than that of Fig. 2(a) (0.4558). This result implies that as the free parameters increase to 3 (

*μ*,

_{a1}*μ*, and top layer thickness), a least-squares fitting routine may find a solution that greatly deviates from the benchmark values for this two-layer problem. In addition, we found that increasing the number of free parameters to 4 or 5 would lead to multiple local minima in the χ

_{s1}'^{2}space for this two-layered sample problem (data not shown) and thus a least-squares fitting routine employed for solving this problem would find different answers for different initial value settings.

*μ*,

_{s1}'*μ*and

_{a2}*μ*have recovery errors less than 10%, while the recovery errors for

_{s2}'*μ*and top layer thickness are greater than 25.2% and 192.6%, respectively. In general, employing multiple source-detector pairs produced better recovery results than the single source-detector pair counterpart. For reference, the errors of (

_{a1}*μ*,

_{a1}*μ*,

_{s1}'*μ*,

_{a2}*μ*, L

_{s2}'_{1}) recovered at 6 mm SDS were −81.0%, 46.5%, 5.0%, −39.4%, and 33.3%, respectively. However, the benefit of increasing the source-detector pair number to more than 2 is found only for

*μ*and

_{s1}'*μ*but is not evident for other parameters especially the top layer thickness. Our results indicate that the two-layered sample problem cannot be reliably solved by simultaneously employing multiple source-detector pairs.

_{s2}'### 3.2 Optical properties recovery of a two-layered sample by iteratively using a layered diffusion model at multiple SDSs

#### 3.2.1 Iterative algorithm and selection of SDSs

5. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A **11**(10), 2727–2741 (1994). [CrossRef] [PubMed]

^{2}between the reference data and the reflectance determined from the two-layered diffusion model at various (

*μ*,

_{a1}*μ*) combinations to generate Fig. 3. The top layer thickness in the diffusion model was set at 1mm for Fig. 3(a) and 1.5 mm for Fig. 3(b). It can be seen in Fig. 3 that χ

_{s1}'^{2}is smaller at 1 mm than at 1.5 mm which is in contrary to the trend demonstrated in Fig. 2. In fact, as the bottom layer optical properties were fixed at the benchmark values and there were three floating variables, the minimum of the χ

^{2}for SDS = 13 mm was found at L

_{1}= 1.028 mm,

*μ*= 0.057 mm

_{a1}^{−1}, and

*μ*= 1.89 mm

_{s1}'^{−1}. This implied that top layer properties could be properly estimated at intermediate SDSs as the bottom layer optical properties were known.

^{2}are near

*μ*= 0.045 mm

_{a1}^{−1},

*μ*= 1.72 mm

_{s1}'^{−1}, which are closer to the benchmark top layer optical properties than those determined at SDS = 13 mm (

*μ*= 0.057 mm

_{a1}^{−1},

*μ*= 1.89 mm

_{s1}'^{−1}). This suggests that if the top layer thickness is fixed at a value that is close to the true value, 6 mm SDS measurement data could lead to a better top layer optical properties recovery than the 13 mm counterpart. It can thus be inferred that measurements conducted at short and intermediate SDSs are more sensitive to the variations of top layer optical properties and top layer thickness, respectively, than those conducted at other SDSs.

*μ*,

_{a2}*μ*),

_{s2}'*L*, and (

_{1}*μ*,

_{a1}*μ*), respectively. New reference values are calculated by referring to previous reference values, and such iteration keeps running until the convergence condition of the five parameters is met. The configuration of the algorithm is described in detail in the following. The first step in our algorithm for determining the five parameters of a dermis-fat sample is to calculate the reference bottom layer optical properties at 20 mm SDS. (The other three parameters are also floating variables in this step.) Second, the top layer optical properties and the top layer thickness are calculated at the intermediate 13 mm SDS. It should be noted that in our program, the bottom layer optical properties are fixed at the reference values obtained in the first step here. Third, similar to the second step, the three parameters of the top layer are calculated at the 6 mm SDS with the bottom layer optical properties fixed at the values calculated in the first step and the top layer thickness and optical properties are allowed only to vary within 5% and 10%, respectively, of those calculated from the second step. Larger variation percentages for the top layer properties could be used in the third step to reduce the number of iterations but this may also lead to an instable system which produces values that diverge far from true values. Finally, in the fourth step, the bottom layer optical properties are calculated again at 20 mm SDS with the three parameters of the top layer fixed at the values calculated in the third step and then updated. This finishes the first round of iteration, and starting from the second round of iteration only step 2 to step 4 described above are carried out until stable results are obtained. A flowchart including all key steps in the iterative algorithm is illustrated in Fig. 4.

_{s1}'**39**(25), 4733–4745 (2000). [CrossRef] [PubMed]

*μ*,

_{a1}*μ*,

_{s1}'*μ*,

_{a2}*μ*, and

_{s2}'*L*were −11.5%, 15.7%, 5.0%, −4.0%, and 11.2%, respectively, which are comparable to the values listed in the second row of Table 2. This result indicates that the iterative algorithm is still robust under typical measurement noise.

_{1}*μ*= 0.05 mm

_{a1}^{−1},

*μ*= 1.5 mm

_{s1}'^{−1},

*μ*= 0.05 mm

_{a2}^{−1},

*μ*= 0.5 mm

_{s2}'^{−1}, and top layer thickness = 1 mm. We employed the (6, 13, 20) mm SDS combination in the algorithm. In Fig. 6(c), at the first iteration, the percent error of the recovered top layer thickness is 95.4%, and we can observe that the top layer thickness approaches the benchmark value as the iteration goes. This iteration process was terminated because the variation of the recovered top layer thickness met the convergence termination condition described earlier. The percent errors of the final recovered

*μ*,

_{a1}*μ*,

_{s1}'*μ*,

_{a2}*μ*, and

_{s2}'*L*were reduced to −13.2%, 13.7%, −5.4%, −2.0%, and 23.9%, respectively. Under a similar setup, we tested our algorithm on two-layered skin structures with various optical property settings (

_{1}*μ*= 0.01-0.1 mm

_{a1}^{−1},

*μ*= 1.0-3.0 mm

_{s1}'^{−1},

*μ*= 0.002-0.1 mm

_{a2}^{−1},

*μ*= 0.5-1.5 mm

_{s2}'^{−1}) and all demonstrated similar converging trends as those shown in Fig. 5 and 6.

#### 3.2.2 Influence of the layer thickness

16. A. Liemert and A. Kienle, “Bioluminescent light diffusion in a four-layered turbid medium,” Med. Laser Appl. **25**(3), 161–165 (2010). [CrossRef]

_{1}= 0.5 mm data is excluded and within 39% if included. Similarly, the recovered top layer optical properties shown in Fig. 7(a), 7(b), 8(a), and 8(b) have percent errors within 20% if the L

_{1}= 0.5 mm data is excluded and within 36% if included. We found that the deviation of the reflectance determined using the layered diffusion model from the benchmark values at the SDS of 6 mm increased as top layer thickness decreased from 1 to 0.5 mm. An alternative photon transport model that is more accurate in determining the reflectance at short SDSs could be employed to work with the iterative algorithm to improve the results when top layer thickness is smaller than 1 mm.

#### 3.2.3 Hemodynamics

**44**(11), 2689–2702 (1999). [CrossRef] [PubMed]

15. G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. **40**(22), 3810–3821 (2001). [CrossRef] [PubMed]

^{−1}to simulate muscle hemodynamics and the other parameters were kept fixed. The layer thickness of dermis was 1 mm. The recovery results are shown in Fig. 9. In Fig. 9(a), the recovered muscle absorption coefficients are systematically lower than the benchmark values by 0.0039 mm

^{−1}in average (6% error in average) and the slope of the regression line of the two sets of data is 1.069. Our results suggest that muscle hemodynamics could be tracked using our method with a modest error. The other four parameters plotted in Fig. 9(a) to 9(c) do not show clear dependence with the bottom layer absorption coefficient.

^{−1}.On the other hand, the reduced scattering coefficients recovered using the semi-infinite model do not show strong dependence on the variation of bottom layer absorption, but they are less accurate than those recovered using the iterative algorithm.

### 3.3 Optical properties recovery of a three-layered sample by iteratively using a layered diffusion model at several SDSs

#### 3.3.1 Iterative algorithm and selection of SDSs

17. S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol. **38**(1), 28–34 (2012). [CrossRef] [PubMed]

*μ*,

_{a1}*μ*,

_{s1}'*L*) for the top layer, (

_{1}*μ*,

_{a2}*μ*,

_{s2}'*L*) for the middle layer, and (

_{2}*μ*,

_{a3}*μ*) for the bottom layer. We utilize a long SDS to calculate the reference values of (

_{s3}'*μ*,

_{a3}*μ*), two intermediate SDSs to calculate the reference values of (

_{s3}'*μ*,

_{a2}*μ*,

_{s2}'*L*), and two short SDSs to calculate the reference values of (

_{2}*μ*,

_{a1}*μ*,

_{s1}'*L*). For instance, our algorithm consists of following four major steps when the SDS combination is (6, 10, 13, 17, 20) mm: 1. The reference bottom layer optical properties as well as the other six parameters are first calculated at 20 mm SDS. 2. The top layer optical properties and the top layer thickness are calculated at 6 and 10 mm SDS with the middle and bottom layer optical properties fixed at the values calculated in the previous step. In this step, 10 mm data is used to calculate the top layer thickness and optical properties are allowed to vary within 10% and 5% of those calculated from the first step. The updated results are then passed to the 6 mm data processing where the top layer thickness and optical properties are allowed to vary within 5% and 10% of the updated reference values. 3. Similar to the second step, the three parameters of the middle layer are calculated at the 13 and 17 mm with the top and bottom layer properties fixed. 4. The bottom layer optical properties are calculated again at 20 mm SDS with the other six parameters fixed at the values calculated in the second and the third steps. After the first round of iteration is completed, the second and further iterations only need to carry out steps from 2 to 4. The iteration of algorithm stops when the two termination conditions described in the previous section are satisfied. We tested the algorithm on a Monte Carlo simulated dermis-fat-muscle data. The thicknesses of dermis and fat layers were 1 and 5 mm, respectively. The recovery results and their percent errors are listed in Table 3. It can be seen that the bottom layer optical properties were less accurately recovered than the other parameters. By looking up the recorded photon trajectories, we found that this phenomenon could be caused by the fact that some photons arriving at the 20 mm SDS detector did not ever enter the bottom layer and thus did not carry the bottom layer information. This situation became worse when the middle layer thickness increased and could be alleviated by increasing the distance of the longest SDS pair. However, measurement noise increases as the SDS increases. The attainable longest SDS is system dependent and is typically less than 30 mm [1

_{1}1. A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. **104**(10), 4014–4019 (2007). [CrossRef] [PubMed]

15. G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. **40**(22), 3810–3821 (2001). [CrossRef] [PubMed]

*μ*,

_{a3}*μ*) had percent errors of −62.7% and 93.9%, respectively. When the SDS combination was adjusted to (6, 10, 20, 25, 35) mm, the errors of recovered (

_{s3}'*μ*,

_{a3}*μ*) were reduced to −36.0% and −34.1%, respectively. Therefore, picking a proper combination of SDS based on the structure of the site under investigation is the prerequisite for obtaining accurate properties of three-layer samples.

_{s3}'#### 3.3.2 Hemodynamics

^{−1}. The recovered muscle layer absorption and reduced scattering coefficients using the SDS combination of (6, 10, 13, 17, 20) mm are shown in Fig. 10 as filled triangles. In Fig. 10(a), the recovered muscle absorption coefficients increase with the benchmark values and the linear regression slope is 0.538. Also depicted in Fig. 10 as empty circles are the recovered muscle optical properties using the semi-infinite diffusion model at 20 mm SDS. It can be seen that the muscle optical properties determined using the iterative algorithm are closer to the benchmark values than those determined using the semi-infinite model. The absorption coefficients calculated using the semi-infinite model barely follow the variation trend of the benchmark values and have a linear regression slope of 0.145. Our simulation results suggest that the semi-infinite model is not suitable for tracking muscle hemodynamics in a three-layered tissue model.

_{1}= 1.0448 mm, L

_{2}= 4.9716 mm) and then performing the data recovery for the rest of experiments (with 6 floating parameters), we obtained bottom layer optical properties as those illustrated in crosses in Fig. 10. The linear regression slope of the crosses in Fig. 10(a) is 1.262, which means the sensitivity to the bottom layer absorption variation is ameliorated by locking the top and the middle layer thicknesses. For most clinical applications, it is reasonable to keep the top layer and middle layer thicknesses fixed at the values determined at a certain time point and then track the time evolution of optical properties. Our algorithm with a three-layered diffusion model provides a robust and sensitive way for tracking the hemodynamics of muscle tissue.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. |

2. | S. H. Tseng, P. Bargo, A. Durkin, and N. Kollias, “Chromophore concentrations, absorption and scattering properties of human skin in-vivo,” Opt. Express |

3. | A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. |

4. | V. Toronov, A. Webb, J. H. Choi, M. Wolf, L. Safonova, U. Wolf, and E. Gratton, “Study of local cerebral hemodynamics by frequency-domain near-infrared spectroscopy and correlation with simultaneously acquired functional magnetic resonance imaging,” Opt. Express |

5. | R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A |

6. | S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt. |

7. | S. H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative spectroscopy of superficial turbid media,” Opt. Lett. |

8. | B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia |

9. | T. J. Farrell, M. S. Patterson, and M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. |

10. | F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

11. | A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. |

12. | A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, and H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. |

13. | T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. |

14. | L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput. Meth. Prog. Bio. |

15. | G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. |

16. | A. Liemert and A. Kienle, “Bioluminescent light diffusion in a four-layered turbid medium,” Med. Laser Appl. |

17. | S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol. |

**OCIS Codes**

(170.5270) Medical optics and biotechnology : Photon density waves

(170.5280) Medical optics and biotechnology : Photon migration

(290.1990) Scattering : Diffusion

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: January 1, 2014

Revised Manuscript: February 9, 2014

Manuscript Accepted: February 21, 2014

Published: February 27, 2014

**Citation**

Yu-Kai Liao and Sheng-Hao Tseng, "Reliable recovery of the optical properties of multi-layer turbid media by iteratively using a layered diffusion model at multiple source-detector separations," Biomed. Opt. Express **5**, 975-989 (2014)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-5-3-975

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### References

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