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Light Transport in Tissue Convolution Potentially, the fluence rate for any irradiation profile may be obtained by launching photons distributed spatially with a probability density function equal to the irradiation profile. Since many photons must be launched at a fixed point before the random fluctuation of the Monte Carlo process becomes small, launching photons from different places increases the total number of photons which must be launched before statistical errors become negligible. Fortunately, the fluence rate which results from photons launched at a single point corresponds to the Green's function G(x,y,z) for the medium. Since the irradiation source profile is not a function of depth, the convolution is independent of z. Figure 2.6: Illustration of the difference between convolution Equations (2.18) and (2.19). Figure A shows a circle of constant Green's function and Figure B shows the circle of constant source. Cylindrical symmetry in both the Green's function and the source has been assumed. The fluence rate for an arbitrary irradiation profile may be obtained by convolving the Green's function profile with the irradiation source function S(x,y) $$\Phi(x,y,z) = \int_{-\infty}^\infty \int_{-\infty}^\infty G(x',y',z')S(x-x', y-y')\,dx' dy'$$ (2.17) In cylindrical coordinates, the convolution of a cylindrically symmetric irradiation source S(r) and Green's function G(r) will be cylindrically symmetric. This convolution may be written (referring to Figure 2.6) $$\Phi(r) =\int_0^\infty S(r') \left[\int_0^{2\pi} G(\sqrt{ {r'}^2+r^2-2r r'\cos\theta}\, d\theta\right]\, r'dr'$$ (2.18) or alternatively as, $$\Phi(r) =\int_0^\infty G(r') \left[\int_0^{2\pi} S(\sqrt{ {r'}^2+r^2-2r r'\cos\theta}\, d\theta\right]\, r'dr'$$ (2.19) Both integrals should give the same results, providing a convenient check on any convolution implementation. The advantage of Equation (2.19) is that the integral over the source needs to be done just once for a particular radius r and absorption distributions at all depths for the radius r can be calculated. A Gaussian source function with a e-2 radius of R is given by S(r)=S0 e-2(r/R)2 (2.20) where S0 is related to the total power P of the beam by $$S_0 = {2P \over \pi R^2}$$ (2.21) Substituting Equation (2.20) into Equation (2.19) yields $$\Phi(r) = S_0 e^{-2(r/R)^2} \int_0^\infty G(r') e^{-2(r'/R)^... ...[\int_0^{2\pi} e^{4rr'\cos\theta/R^2} \, d\theta\right]\,r'dr'$$ (2.22) The integral in brackets reduces to a zero order modified Bessel function, and the Equation for the fluence becomes $$\Phi(r,z) = S_0e^{-2(r'/R)^2}\int_0^\infty G(r',z)e^{-2(r/R)^2} I_0(4rr'/R^2) \, 2\pi r'\,dr',$$ (2.23) Figure 2.7: Illustration of different relations between the flat beam diameter R, the radius at which the fluence is desired r and the integration radius r' of a constant Green's function value. Integration of the source function around the gray circles yields $2\pi S_0$ in the top case, $2\Theta S_0$ in the next, and zero for the last two cases. The source function for a flat beam with radius R is $$S(r') = \cases{ S_0 & if $r'\le R$,\cr 0 & otherwise.\cr}$$ (2.24) where S0 equals the total power divided by the area of the beam. Substituting Equation (2.24) into Equation (2.19) yields $$\Phi(r,z) = S_0 \int_0^\infty G(r',z) \Theta(r,r') 2\pi r'\, dr'$$ (2.25) where the different cases for Q(r,r') are shown in Figure 2.7. Specifically, $$\Theta(r,r') = \cases{ 1 & if $0 \le r \le R-r'$,\cr {1\ove... ...& if $\vert R-r'\vert \le r \le R+r'$,\cr 0 & otherwise. \cr}$$ (2.26) These convolution equations are used to calculate the finite beam fluence rates used in Chapter 4 to evaluate the accuracy of the three-dimensional delta- Eddington fluence rate. Since the Monte Carlo method is used as the standard against which the delta-Eddington approximation is compared, the Monte Carlo method was tested extensively. The results of these tests are presented below.

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.