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| {{Context|date=December 2009}}
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| <!-- Deleted image removed: [[Image:Broad-beam-schematic.png|thumb|250px|Schematic drawing of a broad beam incident on a multi-layer scattering medium. {{deletable image-caption}}]] -->
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| Photon transport theories, such as the [[Monte Carlo methods|Monte Carlo method]], are commonly used to model [[Monte Carlo method for photon transport|light propagation in tissue]]. The responses to a [[pencil beam]] incident on a scattering medium are referred to as [[Green's function|Green’s functions]] or [[impulse response]]s. Photon transport methods can be directly used to compute broad-beam responses by distributing photons over the cross section of the beam. However, [[convolution]] can be used in certain cases to improve computational efficiency.
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| ==General convolution formulas==
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| In order for convolution to be used to calculate a broad-beam response, a system must be [[time-invariant system|time invariant]], [[linear]], and [[Translational symmetry|translation invariant]]. Time invariance implies that a photon beam delayed by a given time produces a response shifted by the same delay. Linearity indicates that a given response will increase by the same amount if the input is scaled and obeys the property of [[Superposition principle|superposition]]. Translational invariance means that if a beam is shifted to a new location on the tissue surface, its response is also shifted in the same direction by the same distance. Here, only spatial convolution is considered.
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| Responses from photon transport methods can be physical quantities such as [[Absorption (electromagnetic radiation)|absorption]], [[fluence]], [[reflectance]], or [[transmittance]]. Given a specific physical quantity, ''G(x,y,z)'', from a pencil beam in Cartesian space and a collimated light source with beam profile ''S(x,y)'', a broad-beam response can be calculated using the following 2-D convolution formula:
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| :<math>C(x,y,z)=\int_{-\infty}^\infty \int_{-\infty}^\infty \ G( x-x',y-y',z)S(x',y')\, dx'\,dy'. \qquad(1) </math>
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| Similar to 1-D convolution, 2-D convolution is commutative between ''G'' and ''S'' with a change of variables <small><math>x''=x-x'\,</math></small> and <small><math>y''=y-y'\,</math></small>:
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| :<math> C(x,y,z)=\int_{-\infty}^{\infty}\!\int_{-\infty}^{\infty}\ G( x'',y'',z)S(x-x'',y-y'')\, dx''\,dy''. \qquad(2)</math>
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| Because the broad-beam response <math>C(x,y,z)\,</math> has cylindrical symmetry, its convolution integrals can be rewritten as:
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| :<math> C(r,z) = \int_0^\infty \ S(r')r' \left [ \int_{0}^{2\pi} \ G\left (\sqrt{r^2+r'\,^2-2rr'cos\phi'},z \right )\, d\phi' \right ]dr' \qquad(3)</math>
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| :<math>C(r,z)=\int_0^\infty G(r'',z)r'' \left [ \int_{0}^{2\pi} S \left ( \sqrt{r^2+r''\,^2-2rr''cos\phi''} \right ) \, d\phi'' \right ]dr''\qquad(4)</math>
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| where <math> r'=\sqrt{x'^2+y'^2}</math>. Because the inner integration of Equation 4 is independent of ''z'', it only needs to be calculated once for all depths. Thus this form of the broad-beam response is more computationally advantageous.
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| ==Common beam profiles==
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| ===Gaussian beam===
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| For a [[Gaussian beam]], the intensity profile is given by
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| :<math> S(r') = S_0 \exp \left [ -2 \left( \frac{r'}{R} \right )^2 \right]. \qquad(5)</math>
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| Here, ''R'' denotes the <math>\tfrac{1}{e^2} \,</math> radius of the beam, and ''S''<sub>0</sub> denotes the intensity at the center of the beam. ''S''<sub>0</sub> is related to the total power ''P''<sub>0</sub> by
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| :<math> S_0 = \frac{2P_0}{\pi R^2}.\qquad(6)</math>
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| Substituting Eq. 5 into Eq. 4, we obtain
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| :<math> C(r,z) = 2\pi S(r)\int_{0}^{\infty} G(r'',z)\exp\left [-2\left (\frac{r''}{R} \right )^2 \right ]I_0\left (\frac{4rr''}{R^2} \right ) r'' \, dr'', \qquad(7)</math>
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| where ''I''<sub>0</sub> is the zeroth-order [[modified Bessel function]].
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| ===Top-hat beam===
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| For a [[tophat beam|top-hat beam]] of radius ''R'', the source function becomes
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| :<math> S(r') =
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| \begin{cases}
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| S_0, & \text{if }r'\leq R \\
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| \,0, & \text{if }r'> R
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| \end{cases}\qquad(8)</math>
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| where ''S''<sub>0</sub> denotes the intensity inside the beam. ''S''<sub>0</sub> is related to the total beam power ''P''<sub>0</sub> by
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| :<math> S_0 = \frac{P_0}{\pi R^2}.\qquad(9)</math>
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|
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| Substituting Eq. 8 into Eq. 4, we obtain
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| :<math>C(r,z) = 2\pi S_0\int_{0}^{\infty}G(r'',z)I_\phi (r,r'')r''\,dr'',\qquad(10)</math>
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| where
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| :<math> I_\phi (r,r'') =
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| \begin{cases}
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| 1, & \mbox{if }R\geq r+r'', \\
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| \tfrac{1}{\pi}\cos^{-1} \left(\tfrac{r^2+r''^2-R^2}{2rr''} \right ), & \mbox{if } \left |r-r'' \right |\leq R < r+r'', \\
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| 0, & \mbox{if }R < \left | r+r'' \right |.
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| \end{cases}\qquad(11)</math>
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| ==Errors in numerical evaluation==
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| ===First interactions===
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| First photon-tissue interactions always occur on the z axis and hence contribute to the specific absorption or related physical quantities as a [[Dirac delta function]]. Errors will result if absorption due to the first interactions is not recorded separately from absorption due to subsequent interactions. The total impulse response can be expressed in two parts:
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| :<math> C(r,z)=G_1(0,z)\frac{\delta (r)}{2\pi r}+ G_2(r,z),\qquad(12)</math>
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| where the first term results from the first interactions and the second, from subsequent interactions.
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| For a Gaussian beam, we have
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| :<math> C(r,z)=G_1(0,z)S(r)+2\pi S_0\int_{0}^{\infty} G_2(r'',z)\,exp\left [-2\left (\frac{r''-r}{R} \right )^2 \right ]I_{0e}\left(\frac{4rr''}{R^2} \right )r''\,dr''.\qquad(13)</math>
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| For a top-hat beam, we have
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| :<math> C(r,z)=G_1(0,z)S(r)+2\pi S_0\int_{0}^{\infty} G_2(r'',z)I_\phi (r,r'')r''\,dr''.\qquad(14)</math>
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| ===Truncation error===
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| For a top-hat beam, the upper integration limits may be bounded by ''r''<sub>max</sub>, such that ''r'' ≤ ''r''<sub>max</sub> − ''R''. Thus, the limited grid coverage in the ''r'' direction does not affect the convolution. To convolve reliably for physical quantities at ''r'' in response to a top-hat beam, we must ensure that ''r''<sub>max</sub> in photon transport methods is large enough that ''r'' ≤ ''r''<sub>max</sub> − ''R'' holds.
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| For a Gaussian beam, no simple upper integration limits exist because it theoretically extends to infinity. At ''r'' >> ''R'', a Gaussian beam and a top-hat beam of the same ''R'' and ''S''<sub>0</sub> have comparable convolution results. Therefore, ''r'' ≤ ''r''<sub>max</sub> − ''R'' can be used approximately for Gaussian beams as well.
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| ==Implementation of convolution==
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| There are two common methods used to implement discrete convolution: the definition of convolution and [[fast Fourier transform]]ation (FFT and IFFT) according to the [[convolution theorem]]. To calculate the optical broad-beam response, the impulse response of a pencil beam is convolved with the beam function. As shown by Equation 4, this is a 2-D convolution. To calculate the response of a light beam on a plane perpendicular to the z axis, the beam function (represented by a ''b × b'' matrix) is convolved with the impulse response on that plane (represented by an ''a'' × ''a'' matrix). Normally ''a'' is greater than ''b''. The calculation efficiency of these two methods depends largely on ''b'', the size of the light beam.
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| In direct convolution, the solution matrix is of the size (''a'' + ''b'' − 1) × (''a'' + ''b'' − 1). The calculation of each of these elements (except those near boundaries) includes ''b'' × ''b'' multiplications and ''b'' × ''b'' − 1 additions, so the [[analysis of algorithms|time complexity]] is [[big O notation|O]][(''a'' + ''b'')<sup>2</sup>''b''<sup>2</sup>]. Using the FFT method, the major steps are the FFT and IFFT of (''a'' + ''b'' − 1) × (''a'' + ''b'' − 1) matrices, so the time complexity is O[(''a'' + ''b'')<sup>2</sup> log(''a'' + ''b'')]. Comparing O[(''a'' + ''b'')<sup>2</sup>''b''<sup>2</sup>] and O[(''a'' + ''b'')<sup>2</sup> log(''a'' + ''b'')], it is apparent that direct convolution will be faster if ''b'' is much smaller than ''a'', but the FFT method will be faster if ''b'' is relatively large.
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| ==Computational examples==
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| The fate of photons can be modeled using a Matlab implementation of the Monte Carlo method (''n''<sub>rel</sub> = 1, ''μ''<sub>a</sub> = 0.1, ''μ''<sub>s</sub>=100, ''g'' = 0.9, 100,000 photons). Using this Matlab model, the fluence of a 3 × 3 × 3 cm<sup>3</sup> region is recorded and the fluence distribution of a broad-beam response is plotted. Figure 1 and Figure 2 show the responses to a pencil beam and a 1-cm top-hat broad-beam, respectively. Direct convolution was used to calculate the broad-beam response in Figure 2. Figure 3 shows the broad-beam response calculated using the FFT method. When the diameter of the light beam is 0.2 cm, direct convolution costs 1.93 seconds, and the FFT method costs 7.35 seconds. When the diameter of the light beam is 2 cm, direct convolution costs 90.1 seconds, and FFT method costs 16.8 seconds. Of course, the absolute computation time depends on the processing speed of the computer being used. These two comparisons were made on the same computer. Although the computation times differ, the plots in Figures 2 and 3 are indistinguishable.
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| {|
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| |<!-- Deleted image removed: [[Image:PencilBeam-plot.png|thumb|300px|Figure 1. Model of photon transport through a scattering medium of a pencil beam calculated using the Monte Carlo simulation.]] -->
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| | <!-- Commented out: [[Image:TopHat-response.png|thumb|300px|Figure 2. Model of photon transport through a scattering medium of a broad beam calculated using the direct convolution method.]] -->
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| |<!-- Deleted image removed: [[Image:TopHat-response-FT.png|thumb|300px|Figure 3. Model of photon transport through a scattering medium of a broad beam calculated using the fast Fourier transform (FFT) method.]] -->
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| |}
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| ==See also==
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| *[[Radiative transfer equation and diffusion theory for photon transport in biological tissue]]
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| *[[Monte Carlo method]]
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| *[[Monte Carlo method for photon transport]]
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| ==Links to other Monte Carlo resources==
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| *[http://labs.seas.wustl.edu/bme/Wang/mc.html Optical Imaging Laboratory at Washington University in St. Louis (MCML)]
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| *[http://omlc.ogi.edu/software/mc/ Oregon Medical Laser Center]
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| ==References==
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| *L.-H. Wang and H.-I. Wu. <u>Biomedical Optics: Principles and Imaging.</u> Wiley 2007.
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| *L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "Monte Carlo modeling of photon transport in multi-layered tissues," Computer Methods and Programs in Biomedicine 47, 131–146 (1995).
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| *L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "Convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Computer Methods and Programs in Biomedicine 54, 141–150 (1997). [http://labs.seas.wustl.edu/bme/Wang/epub/1997LWCMPBConv.pdf Download article].
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| [[Category:Scattering theory]]
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