Biomedical Imaging

$D(z)=\widehat{D}(z)*\frac{1}{\sqrt{2\pi }\sigma }\exp \left(-\frac{z^2}{2{\sigma }^2}\right)=\frac{1}{\sqrt{2\pi }\sigma }{\displaystyle \underset{-\infty }{\overset{R_{\text{p}}}{\int }}\widehat{D}(z^{\prime })}\exp \left(-\frac{{(z-z^{\prime })}^2}{2{\sigma }^2}\right)d{z}^{\prime }.(5.58)$

This leads to a rather complex, but closed analytic expression for the ‘real’ proton Bragg curve D(z), including some tabulated special functions. The Bragg peak now has a finite maximum and is in very good agreement with experimental data [12]. Here, we have carried out the convolution numerically (σ = 0.16 cm). The resulting Bragg curve is plotted in Figure 5.12 (solid line). A finite spectrum of initial energies further contributes to the broadening of realistic Bragg peaks. In radiotherapy with protons, the superposition of several quasi monoenergetic beams is employed to create a so called spread out Bragg peak with an almost homogeneous dose plateau in a certain ...