**6. Three-dimensional dosimetry**

Traditional approaches to internal dosimetry employ anatomic models to obtain mean organ doses. With the diffusion of SPECT-CT and PET-CT tomographs, dosimetric methods which can utilize non-uniform activity distributions to derive dose distributions within organs become increasingly important.

This result can be obtained either with dose point-kernel convolution methods, or by means of a direct Monte Carlo simulation, or exploiting the voxel S factor approach.

The voxel S factor approach, introduced in (Bolch et al., 1999), has been used more widely than dose point-kernel and direct Monte Carlo computation approaches, due to its recognized simplicity.

Following this approach, the average dose to the k-th voxel can be calculated as:

$$
\overline{D}\_k = \sum \tilde{A}\_h \cdot S\_{k \leftarrow h} \tag{26}
$$

where *Ah* is the cumulated activity in the generic h-th voxel and *Sk h* <sup>←</sup> is the voxel S factor, defined as:

3 *V*

where *V* and *S* are the volume and surface of the ellipsoid. The parameters ρ0 and *s* in Equation 24 depend on the nature (electron or photon) and energy of the radiation, and can

In Figure 4, the absorbed fraction as a function of the generalized radius is reported for photons and electrons uniformly emitted in ellipsoids of various shapes. These data are

Fig. 4. Electron and photon absorbed fraction in ellipsoidal volumes, as a function of the

Traditional approaches to internal dosimetry employ anatomic models to obtain mean organ doses. With the diffusion of SPECT-CT and PET-CT tomographs, dosimetric methods which can utilize non-uniform activity distributions to derive dose distributions within

This result can be obtained either with dose point-kernel convolution methods, or by means

The voxel S factor approach, introduced in (Bolch et al., 1999), has been used more widely than dose point-kernel and direct Monte Carlo computation approaches, due to its

is the cumulated activity in the generic h-th voxel and *Sk h* <sup>←</sup> is the voxel S factor,

*D= A S k h kh* ∑ <sup>⋅</sup> <sup>←</sup> (26)

of a direct Monte Carlo simulation, or exploiting the voxel S factor approach.

Following this approach, the average dose to the k-th voxel can be calculated as:

be derived from proper parametric functions described in (Amato et al. 2009, 2011).

calculated by means of a Monte Carlo simulation in Geant4.

generalized radius (Amato et al. 2009, 2011).

**6. Three-dimensional dosimetry** 

organs become increasingly important.

recognized simplicity.

where *Ah*

defined as:

*<sup>ρ</sup> <sup>=</sup> S* (25)

$$S\_{k \leftarrow h} = \sum \Delta\_i \cdot \frac{\varphi\_i(k \leftarrow h)}{m\_k} \tag{27}$$

where *Δi* is the mean energy emitted as radiation *i* per decay, *φi* is the absorbed fraction in *k* of the radiation *i* emitted in *h*, and *mk* is the mass of the k-th voxel.

In (Bolch et al., 1999), voxel S factors were tabulated for five radionuclides and two cubic voxel sizes. Recently, Dieudonnè et al. (2010) presented a generalization of this approach to variable voxel sizes, which allows to obtain S factors for a generic voxel size for nine radionuclides by means of a resampling method.

The dose point-kernel gives the dose per decay event at a given distance from a point source located inside an infinite and homogeneous absorbing medium. Dose point-kernels for electrons and photons were calculated by means of Monte Carlo simulations, and are available either in tabular form or as analytical functions obtained fitting the simulative data (Mainegra-Hing et al., 2005) (Maigne et al., 2011).

By convolving the dose point-kernels evaluated for the actual radionuclide spectrum in the three-dimensional distribution obtained from the volume distribution of cumulated activity, obtained from SPECT-CT or PET-CT images, one can retrieve the volume distribution of the absorbed dose. In order to speed up calculations, fast Fourier or fast Hartley transformations can be employed.

Finally, the Monte Carlo approach is the most demanding in terms of computational power, but it is the only one which can account accurately for tissue inhomogeneity. Known the three-dimensional map of tissue density and composition from CT scan and the volume distribution of the cumulated activity through SPECT or PET emission tomography, it is possible to generate a fixed number of decay events statistically distributed in space according to emission tomography data within the patient's body reconstructed from CT and track all the emitted secondaries in order to calculate the geometrical distribution of energy deposition, i.e. the three-dimensional map of absorbed dose, and finally to rescale the results to the actual values of cumulated activity (Furhang et al., 1997).
