4. Importance sampling for simulation of Class I and Class II OCT signals

In this section, we further improve the importance sampling technique that was described in Section 3, so we can simulate Class II OCT signals more accurately and more efficiently [19].

#### 4.1 Scattering angle of first backscattering event

In the MC simulation described in Section 3, we note that the bias function in (5) produces large values of the likelihood ratio (>>1) when photon packets are scattered in the then unlikely forward direction. These photon packets contributed to a slow decrease in the relative variation, which corresponds to relative error, with the increase in the number of photon packets launched for the calculation of the OCT signal. Referring to Figure 1, we could reduce this relative variation by choosing a distribution function for the scattering angle that limits it to the backward direction. This modified distribution is given by:

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

$$f\_B(\cos(\theta\_\mathsf{B})) = \begin{cases} \left(\mathbb{1} - \frac{1-a}{\sqrt{a^2+1}}\right)^{-1} \frac{a(\mathbbm{1}-a)}{\left(\mathbbm{1}+a^2 - 2a\cos(\theta\_\mathsf{B})\right)^{3/2}}, \cos(\theta\_\mathsf{B}) \in [\mathsf{0}, \mathbbm{1}] \\\ 0, \text{ otherwise} \end{cases},\tag{9}$$

where a is the bias coefficient that can be selected between 0 and 1. Once a biased angle θ<sup>B</sup> is randomly selected, away from the direction of the center of the OCT probe v^, where cosð Þ¼ θ<sup>B</sup> v^ � u^<sup>0</sup> , the provisional biased scattering direction u^<sup>0</sup> is rotated around v^ by an angle ϕ randomly sampled from a uniform distribution in the range from 0 to 2π. These parameters are defined in the same manner as those used in the biased distribution presented in Section 3. The only difference is that the domain of cosð Þ θ<sup>B</sup> is restricted to a maximum deviation from the biased angle by 90°. This ensures that there would not be packets with very large likelihood ratio that could reduce the efficiency of our importance sampling. The likelihood ratio of the photon packet that uses the biased probability density function in Eq. (9) is given by

$$L(\cos(\theta\_{\mathcal{B}})) = \frac{f\_{HG}(\cos(\theta\_{\mathcal{S}}))}{f\_{\mathcal{B}}(\cos(\theta\_{\mathcal{B}}))} = \frac{1-\mathbf{g}^2}{2a(1-a)} \left(\mathbf{1} - \frac{\mathbf{1}-a}{\sqrt{a^2+1}}\right) \left(\frac{\mathbf{1}+a^2-2a\cos(\theta\_{\mathcal{B}})}{\mathbf{1}+\mathbf{g}^2-2\mathbf{g}\cos(\theta\_{\mathcal{S}})}\right)^{3/2},\tag{10}$$

where cosð Þ¼ θ<sup>S</sup> u^ � u^<sup>0</sup> . We note that cosð Þ θ<sup>B</sup> is obtained using the probability density function in Eq. (9), where it is used to obtain the new propagation direction u^0 .

To sample angles according to the biased probability density function in (8), one could use any uniform pseudo-random number generator that would be typically available in scientific software libraries. For example, if ui is a random number distributed uniformly between 0 and 1, a random value for cosð Þ θ<sup>B</sup> that satisfies Eq. (9) with bias coefficient a could be generated with the following conversion formula

$$\cos \theta\_{B,i} = \frac{1}{2a} \left\{ a^2 + \mathbf{1} - \left[ u\_i \left( \frac{\mathbf{1}}{\mathbf{1} - a} - \frac{\mathbf{1}}{\sqrt{a^2 + \mathbf{1}}} \right) + \frac{\mathbf{1}}{\sqrt{a^2 + \mathbf{1}}} \right]^2 \right\}. \tag{11}$$

This conversion formula was derived using probability theory [20].

#### 4.2 Scattering angles of additional biased backscatterings

A second enhancement that could be made to the importance sampling technique, described in Section 3, is to bias the additional scatterings toward the center of the OCT probe v^ with probability 0 ≤ p ≤ 1. That contrasts with the technique presented in Section 3, in which p = 1 (all the additional scatterings were biased). An unbiased scattering is performed in case a bias scattering is not applied in a given point where scattering takes place. The likelihood ratio associated with this scattering is calculated according to the formula

$$L(\cos \theta\_B) = \frac{f\_{HG}(\cos \theta\_B)}{p \cdot f\_{HG}(\cos \theta\_B) + (1 - p) \cdot f\_{HG}(\cos \theta\_S)}. \tag{12}$$

If the biased function f <sup>B</sup>ð Þ cosð Þ θ<sup>B</sup> is selected to sample a random value of cosð Þ θ<sup>S</sup> , which is an event with probability p, cosð Þ¼ θ<sup>S</sup> u^ � u^<sup>0</sup> is a function of cosð Þ θ<sup>B</sup> that is statistically sampled from the probability density function in Eq. (9).

The second layer, extending from 2.32 to 2.42 mm from the tip of the fiber, has the same absorption and scattering coefficients as the first layer, but its refractive index is n = 1.33. The third layer, extending from 2.42 to 2.62 mm from the tip of the fiber,

Class I diffuse reflectance dependence on the distance from the center of the optical fiber for the simulation whose schematic is shown in Figure 4. The solid black line represents the result obtained with 2 <sup>10</sup><sup>5</sup> photon packets using the importance sampling technique presented in Section 3. The green long dashed line is the result obtained with 109 photon packets using standard MCML. The blue dots represent results obtained with 106 photon packets using MCML. The pink short dashed lines show the confidence interval of the importance sampling simulations with 2 105 photon packets that were estimated using a much larger ensemble of 64 105

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

From Figure 3, we note an excellent correspondence between results obtained with our new importance sampling method and results obtained using MCML, that is, standard Monte Carlo simulations. However, our results were obtained in

In this section, we further improve the importance sampling technique that was described in Section 3, so we can simulate Class II OCT signals more accurately and

In the MC simulation described in Section 3, we note that the bias function in (5)

produces large values of the likelihood ratio (>>1) when photon packets are scattered in the then unlikely forward direction. These photon packets contributed to a slow decrease in the relative variation, which corresponds to relative error, with the increase in the number of photon packets launched for the calculation of the OCT signal. Referring to Figure 1, we could reduce this relative variation by choosing a distribution function for the scattering angle that limits it to the backward

4. Importance sampling for simulation of Class I and Class II OCT

, μ<sup>s</sup> = 30 cm<sup>1</sup>

, μ<sup>s</sup> = 0 cm<sup>1</sup>

, and n = 1. After the third

, and n = 1. The

has the following parameters: μ<sup>a</sup> = 1.5 cm<sup>1</sup>

signals

10

Figure 3.

more efficiently [19].

layer, the medium was assumed to be air: μ<sup>a</sup> = 0 cm<sup>1</sup>

4.1 Scattering angle of first backscattering event

direction. This modified distribution is given by:

anisotropy factor was assumed g = 0.9 for the three diffusive layers.

simulations. Reprinted with permission from [14] © The Optical Society of America.

one-thousandth of the time required by the standard method.

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 simulated photon packet.

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

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

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

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

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

it is applied to the standard MC simulation.

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

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

Figure 5.

Figure 6.

America.

13
