**5. Anthropomorphic models for internal dosimetry**

In the past decades, several schematic anthropomorphic models were proposed in order to standardize and simplify dose calculation both in medical applications of radionuclides and in internal contamination evaluations for radiological protection purposes.

Among these, a common approach to internal dosimetry is based on standardized phantoms and S factors pre-calculated for couples of Source-Target organs and for various radionuclides (Snyder et al., 1975).

Organ S factors were calculated through Monte Carlo simulations and are currently available in a tabular form. As a consequence, once calculated the cumulated activity in the source organ with the methods described in the previous Section, the dose to the target organ is calculated with Equation 7, using the proper S factor accounting for the radionuclide emission and for the geometry of the source-target couple.

An advantage of this method is the simplicity of measurements and the standardization of the dosimetric procedure, allowing easier comparisons between results. However, the simplified organ models, described by fixed geometrical shapes and volumes, do not account properly for the anatomy of the single patient. Furthermore, each organ is treated as a whole, neglecting inhomogeneity in the uptake within each organ or tissue.

In real cases, the shape and size of an organ can be varied by a disease, and pathologic tissues, which are the target of the treatment, are not present in standard models of healthy humans.

A sphere model is often employed in order to represent small target tissues such as tumours or pathologic lymph-nodes. Recently, we developed a model to calculate the absorbed fractions in ellipsoidal shapes uptaking beta-gamma emitting radionuclides (Amato et al. 2009, 2011).

Considering a radionuclide which emits mono-energetic and beta electrons, and gamma photons, the average dose to the target volume is given by:

$$\overline{D} = D\_{\beta} + D\_{e} + D\_{\gamma} = \frac{\tilde{A}}{m} \left[ \int \frac{dn(E)}{dE} E \rho(E) dE + \sum\_{i} \left( n\_{e,i} E\_{e,i} \rho\_{e,i} + n\_{\gamma,i} E\_{\gamma,i} \rho\_{\gamma,i} \right) \right] \tag{23}$$

where *ne* and *n*γ are the mono-energetic electron and gamma photon emission probabilities, respectively, of energies *Ee* and *E*γ , *φe* and *φ*γ are the electron and photon absorbed fractions. These absorbed fractions can be derived from the equation:

$$\varphi(\rho) = \left(1 + \frac{\rho\_0}{\rho^s}\right)^{-1} \tag{24}$$

where ρ is the "generalized radius", defined as:

Internal Radiation Dosimetry: Models and Applications 35

*m* <sup>←</sup>

where *Δi* is the mean energy emitted as radiation *i* per decay, *φi* is the absorbed fraction in

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

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

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

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

Hyperthyroidism is a consequence of an excess in free-thyroid hormone action on the tissues. The most frequent causes of hyperthyroidism are: Graves' Disease (GD), Toxic Adenoma (TA) or Toxic Multinodular Goiter (TMG). In any cases, hyperthyroidism can be

The treatment of GD, TA and TMG can be symptomatic with anti-thyroid drugs (often used as first line therapy) or definitive: radioiodine therapy (RaIT) or surgery (total or near-total

RaIT is a well established method for the treatment of hyperthyroidism; aim of the RaIT is to

Many authors evaluated the effectiveness of different dosimetric methodologies. The results were variable and often controversial, in order to the frequency of recurrence and

In order to determine the 131I activity to be administered for the treatment of hyperthyroidism, fixed activity and adjusted activity approaches are currently employed. In

the results to the actual values of cumulated activity (Furhang et al., 1997).

achieve an euthyroid or hypothyroid (such as in the Graves patients) status. Presently, the optimal 131I activity to be administered is still matter of debate.

*kh i*

*S = Δ*

*k* of the radiation *i* emitted in *h*, and *mk* is the mass of the k-th voxel.

radionuclides by means of a resampling method.

(Mainegra-Hing et al., 2005) (Maigne et al., 2011).

**7. 131I therapy of hyperthyroidism** 

caused by sub-acute thyroiditis or silent thyroiditis.

hypothyroidism (Regalbuto et al., 2009) (Giovannella, 2000).

can be employed.

thyroidectomy).

( ) *<sup>i</sup>*

<sup>←</sup> ∑ <sup>⋅</sup> (27)

*k φ k h*

$$
\rho = 3\frac{V}{S} \tag{25}
$$

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

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 calculated by means of a Monte Carlo simulation in Geant4.

Fig. 4. Electron and photon absorbed fraction in ellipsoidal volumes, as a function of the generalized radius (Amato et al. 2009, 2011).
