## Improved importance sampling for Monte Carlo simulation of time-domain optical coherence tomography |

Biomedical Optics Express, Vol. 2, Issue 5, pp. 1069-1081 (2011)

http://dx.doi.org/10.1364/BOE.2.001069

Acrobat PDF (1151 KB)

### Abstract

We developed an importance sampling based method that significantly speeds up the calculation of the diffusive reflectance due to ballistic and to quasi-ballistic components of photons scattered in turbid media: Class I diffusive reflectance. These components of scattered photons make up the signal in optical coherence tomography (OCT) imaging. We show that the use of this method reduces the computation time of this diffusive reflectance in time-domain OCT by up to three orders of magnitude when compared with standard Monte Carlo simulation. Our method does not produce a systematic bias in the statistical result that is typically observed in existing methods to speed up Monte Carlo simulations of light transport in tissue. This fast Monte Carlo calculation of the Class I diffusive reflectance can be used as a tool to further study the physical process governing OCT signals, e.g., obtain the statistics of the depth-scan, including the effects of multiple scattering of light, in OCT. This is an important prerequisite to future research to increase penetration depth and to improve image extraction in OCT.

© 2011 OSA

## 1. Introduction

1. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. **66**(2), 239–303 (2003). [CrossRef]

2. C. Xu, C. Vinegoni, T. S. Ralston, W. Luo, W. Tan, and S. A. Boppart, “Spectroscopic spectral-domain optical coherence microscopy,” Opt. Lett. **31**(8), 1079–1081 (2006). [CrossRef] [PubMed]

3. M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express **11**(18), 2183–2189 (2003). [CrossRef] [PubMed]

4. B. Liu and M. E. Brezinski, “Theoretical and practical considerations on detection performance of time domain, Fourier domain, and swept source optical coherence tomography,” J. Biomed. Opt. **12**(4), 044007 (2007). [CrossRef] [PubMed]

5. G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. **44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

6. S. S. Sherif, C. C. Rosa, C. Flueraru, S. Chang, Y. Mao, and A. G. Podoleanu, “Statistics of the depth-scan photocurrent in time-domain optical coherence tomography,” J. Opt. Soc. Am. A **25**(1), 16–20 (2008). [CrossRef] [PubMed]

7. M. J. Yadlowsky, J. M. Schmitt, and R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. **34**(25), 5699–5707 (1995). [CrossRef] [PubMed]

8. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. **10**(6), 824–830 (1983). [CrossRef] [PubMed]

11. N. Chen, “Controlled Monte Carlo method for light propagation in tissue of semi-infinite geometry,” Appl. Opt. **46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

5. G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. **44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

*importance sampling*to calculate the TD-OCT signal because the probability that a photon propagating in typical biological tissue will undergo single-scattered backreflection is very low. This very low probability of events of interest would require an unacceptably large computational time if standard Monte Carlo simulation were used. Importance Sampling is an advanced statistical method [12] that consists of biasing random events in such a way that the events of interest, which are often rare, appear more often in Monte Carlo simulations [13

13. G. Biondini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization-mode dispersion,” IEEE Photon. Technol. Lett. **14**(3), 310–312 (2002). [CrossRef]

13. G. Biondini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization-mode dispersion,” IEEE Photon. Technol. Lett. **14**(3), 310–312 (2002). [CrossRef]

16. I. T. Lima, A. O. Lima, G. Biondini, C. R. Menyuk, and W. L. Kath, “A comparative study of single section polarization-mode dispersion compensators,” J. Lightwave Technol. **22**(4), 1023–1032 (2004). [CrossRef]

17. J. M. Schmitt and K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A **13**(5), 952–961 (1996). [CrossRef] [PubMed]

11. N. Chen, “Controlled Monte Carlo method for light propagation in tissue of semi-infinite geometry,” Appl. Opt. **46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

5. G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. **44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

10. N. G. Chen and J. Bai, “Estimation of quasi-straightforward propagating light in tissues,” Phys. Med. Biol. **44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

17. J. M. Schmitt and K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A **13**(5), 952–961 (1996). [CrossRef] [PubMed]

## 2. Previous modeling and simulation parameters

20. “Monte Carlo simulations,” Oregon Medical Laser Center, accessed January 1, 2009, http://omlc.ogi.edu/software/mc/

*µ*and the scattering

_{a}*µ*coefficients. The scattering events, which take place at the end of the random steps, are produced by two random angles that determine the future direction of the photon packet scattering in three-dimensional space . To account for the photon packet scattering with arbitrary anisotropy factor,

_{s}*g*, we use the same Henyey-Greenstein probability density function used in the MCML software package that is defined as

*θ*is the angle between the photon packet propagation direction

_{s}*ϕ*that is randomly picked from a uniform probability density function from 0 to 2π. At each scattering event, where the light-matter interaction is modeled, the weight

*W*of the photon packet is decreased by an amount determined by the absorption coefficient

*µ*at the scattering location. The weight

_{a}*W*, which is initialized at 1, is an estimate of the residual number of photons left in the packet. When the packet weight reaches

*W*

_{th}= 10

^{−4}, the photon packet is either eliminated with probability 1/

*m*or is left to continue propagating with probability 1 – 1/

*m*and weight equal to

*mW*. In this work we use m = 10. This elimination process, called a Russian roulette technique, is an unbiased way to remove from the simulation the photon packets that have a negligible contribution to the scattering and absorption in the tissue, so that a new photon packet can be simulated.

*z*is obtained by calculating the mean value of the indicator function

*I*

_{1}of the spatial and temporal filter of the Class I diffuse reflectance for all the photon packets (samples) in the ensemble. The indicator function

*I*

_{1}of the spatial and temporal filter for the

*i*th photon packet is defined as

*l*is the coherence length of the source,

_{c}*r*is the distance of the

_{i}*i*th reflected photon packet to the origin along the plane

*z*= 0, where the collecting optical system is located,

*d*

_{max}and

*θ*

_{max}are the maximum collecting diameter and angle, respectively,

*θ*is the angle with the

_{z,i}*z*-axis, which is the axis normal to the tissue interface, Δ

*s*is the optical path, and

_{i}*z*is the maximum depth reached by the photon packet. The diffuse reflectance

*R*

_{1}at any depth, which is the expected value of

*I*

_{1}at that corresponding depth, and its corresponding standard deviation

*σ*

_{R}_{,1}can be estimated by the expressions

*N*is the initial number of photon packets in the Monte Carlo simulations [21].

## 3. Novel Importance sampling to simulate Class I diffuse reflectance in TD-OCT

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

### 3.1 Calculation of the apparent position of the collecting optics

*z*-coordinate of the detector in the

*j*th layer,

*j*th layer and

*j*th layer. We obtained Eq. (5) from the law of refraction using the paraxial approximation:

*j*would arrive at the origin (0,0,0) in the absence of additional scattering if it is directed towards the coordinate (0,0,

**R**located at the

*j*th layer is defined as

### 3.2 Calculation of scattering angle of the first backscattering event

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

*g*. Our probability density function of the biased angle is defined as

*a*is a bias coefficient. After randomly picking a biased angle

*ϕ*that is randomly picked from a uniform probability density function from 0 to 2π. This last procedure is equivalent to the one used in MCML software package to enable a full three-dimensional scattering, except that the scattered angle

13. G. Biondini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization-mode dispersion,” IEEE Photon. Technol. Lett. **14**(3), 310–312 (2002). [CrossRef]

16. I. T. Lima, A. O. Lima, G. Biondini, C. R. Menyuk, and W. L. Kath, “A comparative study of single section polarization-mode dispersion compensators,” J. Lightwave Technol. **22**(4), 1023–1032 (2004). [CrossRef]

*ϕ*. The likelihood ratio in Eq. (8) is the ratio of the probability that this biased scattering angle would have been observed in an unbiased simulation divided by the probability of the biased scattering angle specified by the biased distribution. While

*a*, enabled a rapid convergence of the Importance Sampling method for the cases that we reported in Sec. 4 as a result of the strong bias used.

### 3.3 Calculation of scattering angles of further backscattering events

*R*

_{1}and its corresponding standard deviation

*σ*

_{R,1}can be estimated by the following expressions

### 3.4 Comparison with a previously proposed bias method

10. N. G. Chen and J. Bai, “Estimation of quasi-straightforward propagating light in tissues,” Phys. Med. Biol. **44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

11. N. Chen, “Controlled Monte Carlo method for light propagation in tissue of semi-infinite geometry,” Appl. Opt. **46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

10. N. G. Chen and J. Bai, “Estimation of quasi-straightforward propagating light in tissues,” Phys. Med. Biol. **44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

**46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

**44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

**46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

**44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

**46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

### 3.5 Importance sampling statistics as a function of the depth

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

**46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

*N*in Eqs. (9) and (10), since the use of the likelihood ratio associated to each photon packet in these equations will produce the correct statistical result. Once a photon packet exceeds the region within the depth target layer, it will no longer be biased and will likely be terminated after exceeding the boundary of the last layer while propagating in the forward direction. Then a new photon packet will be created at the origin as in the standard MCML method, and a new Monte Carlo sample will be simulated. Despite the higher computational cost per photon packet, the computation of the Class I diffuse reflectance in the case that we studied using Monte Carlo simulations with importance sampling required as little as one-thousandth of the computational time required to achieve the same accuracy in the diffuse reflectance calculation using the standard Monte Carlo method.

## 4. Validation of our new importance sampling method

*µ*= 1.5 cm

_{a}^{−1}and a scattering coefficient

*µ*= 60 cm

_{s}^{−1}. The second layer extends from 330 µm to 360 µm and has an absorption coefficient

*µ*= 3 cm

_{a}^{−1}and a scattering coefficient

*µ*= 120 cm

_{s}^{−1}. We assume the three layers to have the same refractive index

*n*= 1 and an anisotropy factor

*g*= 0.9, as in [5

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

**46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

^{6}Monte Carlo simulations with importance sampling, which has a computational cost slightly smaller than 10

^{7}standard Monte Carlo simulations. In Figs. 3 and 4, we also show results obtained with the angle biasing method described in [5

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

*a*= 0.9. The choice of

*a*=

*g*is the value for the bias coefficient that enabled the fastest conversion of the statistical results in (9) and (10) with respect to the number of simulated photon packets because it produces the best combination of strong bias with a limited range of variation of the likelihood ratio in Eq. (8). Therefore, the results shown in Figs. 3 and 4 indicate that our new importance sampling procedure reduces the computational cost to obtain the Class I diffuse reflectance by about three orders of magnitude.

^{7}standard Monte Carlo simulations produces a very small number of Class I diffusely reflected photon packets, an ensemble with 2 × 10

^{6}Monte Carlo simulations with importance sampling produces a number of Class I diffusely reflected photon packets that exceeds the number of Class I diffusely reflected photon packets produced by as many as 10

^{10}standard Monte Carlo simulations at depths greater than 132 µm. The agreement between the Class I diffuse reflectance, obtained with our importance sampling and with the standard Monte Carlo method with a much larger number of samples, indicates that the biased samples with importance sampling are weighted in a way that eliminates any residual statistical bias. Therefore, not only does our importance sampling method converge to the true statistical result, but it does so at a much faster rate than the standard Monte Carlo method.

**44**(9), 2307–2320 (1999). [CrossRef] [PubMed]

**44**(7), 1669–1676 (1999). [CrossRef] [PubMed]

**46**(10), 1597–1603 (2007). [CrossRef] [PubMed]

*n*= 1, absorption coefficient

*µ*= 1.5 cm

_{a}^{−1}, and the scattering coefficient

*µ*= 60 cm

_{s}^{−1}. The second diffusive layer, which is extends from 2.32 mm to 2.42 mm from the center of the fiber, has the same absorption and scattering coefficient as the first layer, but its refractive index is

*n*= 1.33. The third layer, which extends from 2.42 mm to 2.62 mm from the center of the fiber, has the following specifications: refractive index

*n*= 1, absorption coefficient

*µ*= 1.5 cm

_{a}^{−1}, and the scattering coefficient

*µ*= 30 cm

_{s}^{−1}. We assume there is air at the end of the third layer. We consider the three diffusive layers have anisotropy factor

*g*= 0.9. In Fig. 7 we show the simulation results of this TD-OCT setup probing a tissue with multiple layers with different refractive indices. We observed an excellent convergence between the results obtained with our new importance sampling method and the results obtained using standard Monte Carlo simulations using MCML. Our results, however, were obtained in one-thousandth of the time required by standard method.

## 5. Conclusion

## References and links

1. | A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. |

2. | C. Xu, C. Vinegoni, T. S. Ralston, W. Luo, W. Tan, and S. A. Boppart, “Spectroscopic spectral-domain optical coherence microscopy,” Opt. Lett. |

3. | M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express |

4. | B. Liu and M. E. Brezinski, “Theoretical and practical considerations on detection performance of time domain, Fourier domain, and swept source optical coherence tomography,” J. Biomed. Opt. |

5. | G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. |

6. | S. S. Sherif, C. C. Rosa, C. Flueraru, S. Chang, Y. Mao, and A. G. Podoleanu, “Statistics of the depth-scan photocurrent in time-domain optical coherence tomography,” J. Opt. Soc. Am. A |

7. | M. J. Yadlowsky, J. M. Schmitt, and R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. |

8. | B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. |

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

10. | N. G. Chen and J. Bai, “Estimation of quasi-straightforward propagating light in tissues,” Phys. Med. Biol. |

11. | N. Chen, “Controlled Monte Carlo method for light propagation in tissue of semi-infinite geometry,” Appl. Opt. |

12. | R. Y. Rubinstein, |

13. | G. Biondini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization-mode dispersion,” IEEE Photon. Technol. Lett. |

14. | S. L. Fogal, G. Biondini, and W. L. Kath, “Multiple importance sampling for first- and second-order polarization-mode dispersion,” IEEE Photon. Technol. Lett. |

15. | I. T. Lima Jr, A. O. Lima, J. Zweck, and C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. |

16. | I. T. Lima, A. O. Lima, G. Biondini, C. R. Menyuk, and W. L. Kath, “A comparative study of single section polarization-mode dispersion compensators,” J. Lightwave Technol. |

17. | J. M. Schmitt and K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A |

18. | H. Iwabuchi, ““Efficient Monte Carlo method for radiative transfer modeling,” J. of the Atmosph,” Science |

19. | I. T. Lima, Jr., “Advanced Monte Carlo methods applied to Optical Coherence Tomography” (invited), presented at the 2009 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference, Belém, Brazil, 3–6 Nov. 2009. |

20. | “Monte Carlo simulations,” Oregon Medical Laser Center, accessed January 1, 2009, http://omlc.ogi.edu/software/mc/ |

21. | S. Kay, |

22. | E. Hecht, Optics, 4th ed. (Pearson Addison Wesley, 2003). |

**OCIS Codes**

(110.4500) Imaging systems : Optical coherence tomography

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

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: February 3, 2011

Revised Manuscript: March 31, 2011

Manuscript Accepted: March 31, 2011

Published: April 4, 2011

**Citation**

Ivan T. Lima, Anshul Kalra, and Sherif S. Sherif, "Improved importance sampling for Monte Carlo simulation of time-domain optical coherence tomography," Biomed. Opt. Express **2**, 1069-1081 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-5-1069

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### References

- A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003). [CrossRef]
- C. Xu, C. Vinegoni, T. S. Ralston, W. Luo, W. Tan, and S. A. Boppart, “Spectroscopic spectral-domain optical coherence microscopy,” Opt. Lett. 31(8), 1079–1081 (2006). [CrossRef] [PubMed]
- M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003). [CrossRef] [PubMed]
- B. Liu and M. E. Brezinski, “Theoretical and practical considerations on detection performance of time domain, Fourier domain, and swept source optical coherence tomography,” J. Biomed. Opt. 12(4), 044007 (2007). [CrossRef] [PubMed]
- G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44(9), 2307–2320 (1999). [CrossRef] [PubMed]
- S. S. Sherif, C. C. Rosa, C. Flueraru, S. Chang, Y. Mao, and A. G. Podoleanu, “Statistics of the depth-scan photocurrent in time-domain optical coherence tomography,” J. Opt. Soc. Am. A 25(1), 16–20 (2008). [CrossRef] [PubMed]
- M. J. Yadlowsky, J. M. Schmitt, and R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34(25), 5699–5707 (1995). [CrossRef] [PubMed]
- B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983). [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(2), 131–146 (1995). [CrossRef] [PubMed]
- N. G. Chen and J. Bai, “Estimation of quasi-straightforward propagating light in tissues,” Phys. Med. Biol. 44(7), 1669–1676 (1999). [CrossRef] [PubMed]
- N. Chen, “Controlled Monte Carlo method for light propagation in tissue of semi-infinite geometry,” Appl. Opt. 46(10), 1597–1603 (2007). [CrossRef] [PubMed]
- R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, 1981).
- G. Biondini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization-mode dispersion,” IEEE Photon. Technol. Lett. 14(3), 310–312 (2002). [CrossRef]
- S. L. Fogal, G. Biondini, and W. L. Kath, “Multiple importance sampling for first- and second-order polarization-mode dispersion,” IEEE Photon. Technol. Lett. 14(9), 1273–1275 (2002). [CrossRef]
- I. T. Lima, A. O. Lima, J. Zweck, and C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15(1), 45–47 (2003). [CrossRef]
- I. T. Lima, A. O. Lima, G. Biondini, C. R. Menyuk, and W. L. Kath, “A comparative study of single section polarization-mode dispersion compensators,” J. Lightwave Technol. 22(4), 1023–1032 (2004). [CrossRef]
- J. M. Schmitt and K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A 13(5), 952–961 (1996). [CrossRef] [PubMed]
- H. Iwabuchi, ““Efficient Monte Carlo method for radiative transfer modeling,” J. of the Atmosph,” Science 63, 2324–2339 (2006).
- I. T. Lima, Jr., “Advanced Monte Carlo methods applied to Optical Coherence Tomography” (invited), presented at the 2009 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference, Belém, Brazil, 3–6 Nov. 2009.
- “Monte Carlo simulations,” Oregon Medical Laser Center, accessed January 1, 2009, http://omlc.ogi.edu/software/mc/
- S. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory (Prentice-Hall, 1993).
- E. Hecht, Optics, 4th ed. (Pearson Addison Wesley, 2003).

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