4.3 Numerical results

We validate our importance sampling technique for simulation of OCT signals from multilayered tissue, with different refractive indices and scattering properties, by comparing its results with those obtained by the standard Monte Carlo method. We consider a tissue model that consists of multiple layers that could be imaged with an OCT system, as shown schematically in Figure 4. The modeled tissue extends from 0 to 1 mm, and consists primarily of a turbid layer with absorption coefficient μ<sup>a</sup> = 1.5 cm�<sup>1</sup> and a scattering coefficient μ<sup>s</sup> = 60 cm�<sup>1</sup> , and also contains five thin layers with absorption coefficient μ<sup>a</sup> = 3 cm�<sup>1</sup> and a scattering coefficient μ<sup>s</sup> = 120 cm�<sup>1</sup> . These five thin layers with higher scattering coefficient are located from 200 to 215 μm, from 365 to 395 μm, from 645 to 660 μm, from 760 to 775 μm, and from 900 to 915 μm. We assume that this tissue has the same refractive index n = 1 and an anisotropy factor g = 0.9. We note that our method is robust in the presence of refractive index mismatch along layer boundaries of the tissue [19]. We simulate an OCT system where the light is delivered/collected by the tip of an

#### Figure 4.

Schematic representation of a setup to simulate OCT signals similar to one in Ref. [3]. Reprinted with permission from [15] © The Optical Society of America.

Monte Carlo Methods for Simulation of Optical Coherence Tomography of Turbid Media DOI: http://dx.doi.org/10.5772/intechopen.89555

optical fiber having a radius of 10 μm and an acceptance angle of 5°. For simplicity, the light source is assumed to be a one-dimensional light beam propagating along the vertical direction as in [3, 8], since the purpose of this example is to validate and demonstrate the effectiveness of our second importance sampling technique when it is applied to the standard MC simulation.

In Figures 5 and 6, we show results obtained with 10<sup>8</sup> Monte Carlo photon packets with importance sampling, which has a computational cost of simulating about 9 <sup>10</sup><sup>8</sup> photon packets using standard Monte Carlo. The computational cost of applying this importance sampling technique depends on the target depth range, and on the average photon mean free path in the given tissue. The target depths in the shown simulations were set from 0 to 1 mm. Therefore, every single photon scattering that occurs in the depth range from 0 to 1 mm would be biased. We used a bias coefficient a = 0.925, and an additional bias probability p = 0.5, to run the Monte Carlo simulations with importance sampling. The results shown in

#### Figure 5.

Otherwise, in the case of the complementary event with probability 1 � p, the unbiased function f HGð Þ cosð Þ θ<sup>S</sup> is used to sample a random value of cosð Þ θ<sup>S</sup> and cosð Þ¼ θ<sup>B</sup> v^ � u^<sup>0</sup> depends on the value of cosð Þ θ<sup>S</sup> . Since the two random angles associated to each scattering do not depend on the random angles selected in the previous scatterings, the likelihood ratio of each collected photon packet results from the multiplication of all the likelihood ratios of all the biased scatterings in that

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

We validate our importance sampling technique for simulation of OCT signals from multilayered tissue, with different refractive indices and scattering properties, by comparing its results with those obtained by the standard Monte Carlo method. We consider a tissue model that consists of multiple layers that could be imaged with an OCT system, as shown schematically in Figure 4. The modeled tissue extends from 0 to 1 mm, and consists primarily of a turbid layer with absorption

five thin layers with absorption coefficient μ<sup>a</sup> = 3 cm�<sup>1</sup> and a scattering coefficient

from 200 to 215 μm, from 365 to 395 μm, from 645 to 660 μm, from 760 to 775 μm, and from 900 to 915 μm. We assume that this tissue has the same refractive index n = 1 and an anisotropy factor g = 0.9. We note that our method is robust in the presence of refractive index mismatch along layer boundaries of the tissue [19]. We simulate an OCT system where the light is delivered/collected by the tip of an

Schematic representation of a setup to simulate OCT signals similar to one in Ref. [3]. Reprinted with

permission from [15] © The Optical Society of America.

. These five thin layers with higher scattering coefficient are located

, and also contains

coefficient μ<sup>a</sup> = 1.5 cm�<sup>1</sup> and a scattering coefficient μ<sup>s</sup> = 60 cm�<sup>1</sup>

simulated photon packet.

4.3 Numerical results

μ<sup>s</sup> = 120 cm�<sup>1</sup>

Figure 4.

12

The Class I diffusive reflectance (thick solid black curve) and the Class II reflectance (thin solid red curve), as a function of the tissue depth using the importance sampling technique presented in Section 4 with 108 photon packets. The pink short dashed and the blue long dashed curves are results of simulating Class I reflectance and the Class II reflectance using standard Monte Carlo with 1011 photon packets, respectively. Reprinted with permission from [15] © The Optical Society of America.

#### Figure 6.

Details of reflectance results shown in Figure 5 for depths between 640 and 680 μm. The error bars shown were estimated by the same ensemble of simulations. Reprinted with permission from [15] © The Optical Society of America.

Figures 5 and 6 show that our new importance sampling technique reduces the computational cost for obtaining the Class I diffuse reflectance by approximately three orders of magnitude when compared to the standard Monte Carlo method. This algorithm is optimum when the additional bias probability is equal to p = 0.5. Since only half of the back-scatterings are biased, this choice contributes toward enabling an optimum number of Class II photons to be collected by the tip of the optical fiber.

both Class I and Class II reflectances using our importance sampling-based imple-

Monte Carlo Methods for Simulation of Optical Coherence Tomography of Turbid Media

We described two importance sampling techniques for a standard Monte Carlo (MC) method that could enable fast simulation of signals from optical coherence tomography (OCT) imaging systems. These OCT signals are generated due to diffusive reflections from either multilayered or arbitrarily shaped, turbid media, for example, tissue. Such signals typically consist of ballistic and quasi-ballistic components, of scattered photons inside the medium, in addition to photons that undergo multiple scattering. We showed that MC simulation of these OCT signals using our importance sampling reduced its computation time on a serial processor by up to three orders of magnitude compared to its corresponding standard implementation. Therefore, our importance sampling techniques enable practical simulation of OCT B-scans of turbid media, for example, tissue, using commonly

mentation.

5. Conclusions

DOI: http://dx.doi.org/10.5772/intechopen.89555

available workstations.

Conflict of interest

Author details

15

Ivan T. Lima Jr<sup>1</sup> and Sherif S. Sherif<sup>2</sup>

University, Fargo, North Dakota, USA

provided the original work is properly cited.

Winnipeg, Manitoba, Canada

\*

1 Department of Electrical and Computer Engineering, North Dakota State

\*Address all correspondence to: sherif.sherif@umanitoba.ca

2 Department of Electrical and Computer Engineering, University of Manitoba,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The authors declare no conflict of interest.

We note that the results obtained with the MCML have confidence intervals that are noticeably larger than those of the results obtained with importance sampling shown in Figure 6, even though the standard Monte Carlo simulations have a computational cost 113 times larger than those obtained with importance sampling. In Figure 6, we also note that our importance sampling technique reduces the computational cost of calculating the Class II reflectance by more than two orders of magnitude.

In Figure 7 we show the relationship between the relative error in calculating Class I and the Class II reflectances at two different depths: 400 and 670 μm and the bias coefficient a for p = 0.5. The depths at 400 and 670 μm correspond to tissue regions near the second and third regions with high local reflectance due to the higher local scattering coefficient. The relative variation in the results is the ratio between the square root of the variance, shown in Eq. (4), and the reflectance in Eq. (3).

We note that Class I reflectance has a minimum relative error at 400 μm with a = 0.925, but the minimum error at 670 μm occurs at a = 0.95 μm at 670 μm. The deeper the tissue region, the stronger the required bias because of the increase in the number of scatterings with the depth. However, as the bias coefficient is increased toward 1, larger variations in the likelihood ratio lead to an increase in the relative error. We also note that Class II reflectance has its minimum relative error at 400 μm with a = 0.91, while its minimum relative error at 670 μm increased to only about a = 0.925 μm. The optimum amount of bias required by the Classs II diffusive reflectance in both wavelengths is lower than the optimum bias coefficient observed in the Class I reflectance because the number of ballistic and quasi-ballistic photons increases with the bias, which leads to a decrease in the number of collected photon packets that undergo multiple scatterings. Figure 7 also shows that there is a range for the bias parameter a between 0.9 and 0.95 that enables fast calculation of

Figure 7.

The relative error in calculated reflectance using importance sampling as a function of bias coefficient a. Reprinted with permission from [15] © The Optical Society of America.

Monte Carlo Methods for Simulation of Optical Coherence Tomography of Turbid Media DOI: http://dx.doi.org/10.5772/intechopen.89555

both Class I and Class II reflectances using our importance sampling-based implementation.
