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Simulation speed

In [2], we compared the latest versions of several optical Monte Carlo (MC) simulation packages with our recently developed TIM-OS [1]. Particularly, MMCM was downloaded on September 29, 2010 from its website (http://mcx.sourceforgo.net/mmc) and compiled with the best setting in the package. As shown in Dr. Fang’s comment [5], he recently updated the MMCM package that now takes advantage of the SSE instructions and the Intel compiler, yielding a substantial performance gain. However, the latest MMCM still does not take the thread racing condition into account. As pointed out by Alerstam [4], thread racing may compromise data integrity. We also observed this problem in the MMCM results.

It is underlined that TIM-OS photon-tetrahedron intersection style has a less computational complexity than the Plücker-coordinate scheme used in MMCM [2,5]. When we do photon-tetrahedron intersection tests, a photon is actually inside a tetrahedron. Such a tight restriction on the position of the photon greatly reduces the computational complexity. As a result, while the Plücker-coordinate algorithm utilizes all the equations in [3], the original TIM-OS algorithm only uses the popular ray-plane intersection equation.

Simulation accuracy

Figure 1 illustrates the problem in [5]. While the solid curve shows the true value $y_{truth}$, $y_{mmc}(i)$ and $y_{timos}(i)$ are the values used in [5] to compare MMCM and TIM-OS. However, each $y_{timos}(i)$ datum he used had two parts: $y_{timos}(i)=({\displaystyle {\int }_{(i-1)\Delta x}^{i\Delta x}f(x)dx}+{\displaystyle {\int }_{i\Delta x}^{(i+1)\Delta x}f(x)dx})/2$, where ${\displaystyle {\int }_{(i-1)\Delta x}^{i\Delta x}f(x)dx}$ and ${\displaystyle {\int }_{i\Delta x}^{(i+1)\Delta x}f(x)dx}$ were the values TIM-OS estimated at the positions $(i-1/2)\Delta x$ and $(i+1/2)\Delta x$, respectively. Hence,$y_{timos}(i)$ actually was a linear interpolation of two TIM-OS results. It is not fair to compare a linearly interpolated TIM-OS result to a directly computed MMCM result.

 

Fig. 1 Illustration of the problem in Dr. Fang’s Comment.

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To address this discrepancy for the problem shown in Fig. 1, we compared the results of MMCM and TIM-OS to the true value $1/(i\Delta x)$ at an arbitrarily selected point $i\Delta x$. In this case, by the meshing requirements of the two simulators, the integral range for MMCM was from $(i-1)\Delta x$ to $(i+1)\Delta x$ and the range for TIM-OS was from $(i-1/2)\Delta x$ to $(i+1/2)\Delta x$. We have

Then, the relative errors for MMCM and TIM-OS were derived as

Therefore $\underset{i->\infty }{\lim }erro{r}_{mmc}/erro{r}_{timos}=2$. Figure 2 plots $erro{r}_{mmc}/erro{r}_{timos}$ for $2\le i\le 20$.

 

Fig. 2 Comparison of MMCM and TIM-OS in terms of the relative error.

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Furthermore, we considered a more realistic example in which a pencil beam passed through an absorbing-only media, and the intensity of the light beam would obey Beer’s law along the light path. We got similar result: $\underset{\Delta x->0}{\lim }erro{r}_{mmc}/erro{r}_{timos}=2$ and $erro{r}_{mmc}/erro{r}_{timos}>1$ for $\Delta x>0$. We also set up a mesh to test MMCM and TIM-OS under the above condition. Our experimental results are in an excellent agreement with the analytical prediction. We prepared a package containing all the files for the reader to repeat the experiments, which can be downloaded from http://imaging.sbes.vt.edu/software/tim-os.