**2. Definitions**

If we consider a radioactive volume containing, at the time *t*, *N* radioactive nuclei, we know that the activity *A*(*t*), representing the number of decays per second, is proportional to *N* through the decay constant λ:

$$A\left(t\right) = \frac{dN}{dt} = -\lambda N\tag{1}$$

Equation 1 can be integrated, leading to the exponential decay law for the number of radioatoms present at the time *t* in a radioactive sample:

$$N = N\_0 e^{-\lambda t} \tag{2}$$

The decay constant λ is related to the decay time τ and to the half-life *T1/2* by the relationships:

$$T\_{1/2} = \frac{1\text{n}\mathbb{Z}}{\lambda} = \text{r}1\text{n}\mathbb{Z} \tag{3}$$

The SI unit of activity is the *Becquerel* (Bq), defined as one disintegration per second:

$$1Bq = 1dis \cdot s^{-1} \tag{4}$$

while the old unit of activity was the *Curie* (Ci):

$$1\,\mathrm{Ci} = 3.7 \times 10^{10} Bq\tag{5}$$

The radiation absorbed dose, commonly intended as dose, is defined as the average energy imparted by the radiation per unit mass of the irradiated volume:

$$D = \frac{d\overline{E}}{dm} \tag{6}$$

In the SI system, the dose is expressed in joules per kilogram or Gray (Gy); the older unit, no longer employed but often encountered in aged texts, was the Rad (1 erg/g). The conversion is such that 1 Gy = 100 Rad.

In internal dosimetry, the dose to an organ or tissue accumulating a radiopharmaceutical can be evaluated following the MIRD approach (Snyder et al., 1975).

The dose imparted to a target volume *k* from a single source volume *h*, can be calculated as:

$$D\left(r\_k \leftarrow r\_h\right) = \tilde{A}\_h S\left(r\_k \leftarrow r\_h\right) \tag{7}$$

We describe the application of such dosimetric approaches in the main nuclear medicine therapies such as the 131I therapy of thyroid diseases, the therapy of neuroendocrine tumours (NET) with somatostatin analogues labelled with beta- or Auger-emitters, and the palliation of painful bone metastases, focusing on dose-efficacy relationships and on the

If we consider a radioactive volume containing, at the time *t*, *N* radioactive nuclei, we know that the activity *A*(*t*), representing the number of decays per second, is proportional to *N*

( ) *dN At = = <sup>λ</sup><sup>N</sup>*

Equation 1 can be integrated, leading to the exponential decay law for the number of radio-

*<sup>λ</sup><sup>t</sup> N=N e0*

ln2 *T= = <sup>τ</sup>*ln2

The radiation absorbed dose, commonly intended as dose, is defined as the average energy

*dE D =*

In the SI system, the dose is expressed in joules per kilogram or Gray (Gy); the older unit, no longer employed but often encountered in aged texts, was the Rad (1 erg/g). The conversion

In internal dosimetry, the dose to an organ or tissue accumulating a radiopharmaceutical

The dose imparted to a target volume *k* from a single source volume *h*, can be calculated as:

1/2

The SI unit of activity is the *Becquerel* (Bq), defined as one disintegration per second:

is related to the decay time τ and to the half-life *T1/2* by the

*dt* <sup>−</sup> (1)

<sup>−</sup> (2)

*λ* (3)

<sup>1</sup> 1 1 *Bq = dis s*<sup>−</sup> ⋅ , (4)

<sup>10</sup> 1 3.7 10 *Ci = B* × *q* (5)

*D r r =AS r r* ( *k h hk h* ← ← ) ( ) (7)

*dm* (6)

limiting of side effects to other potentially critical organs.

λ:

atoms present at the time *t* in a radioactive sample:

λ

while the old unit of activity was the *Curie* (Ci):

imparted by the radiation per unit mass of the irradiated volume:

can be evaluated following the MIRD approach (Snyder et al., 1975).

**2. Definitions** 

through the decay constant

The decay constant

is such that 1 Gy = 100 Rad.

relationships:

where *Ah* is the cumulated activity in the source organ and *S* is the average dose absorbed by the target per unit cumulated activity in the source. The cumulated activity in *h* is defined as the total number of disintegrations in that organ, i.e. the integral of the activity *A* over the time:

( ) 0 *A = A t dt h h* ∞ ∫ , (8)

The definition of the *S* factor appearing in Equation 7 is:

$$S\left(r\_k \leftarrow r\_h\right) = \frac{\sum \Delta\_i \wp\_i\left(r\_k \leftarrow r\_h\right)}{m\_k} \tag{9}$$

where Δ*i* is the average energy emitted per transition as *i*-th radiation, φ*<sup>i</sup>* is the "absorbed fraction", i.e. the fraction of the energy emitted in the source volume *rh* which was absorbed in the target volume *rk*, and *mk* is the mass of the target.

In general, if several organs accumulate the radiopharmaceutical, the overall dose to the target volume (organ or tissue) *k* is obtained by summing up all the contributions coming from the various regions *h*:

$$D\left(r\_k\right) = \sum\_h \tilde{A}\_h \sum\_i \Delta\_i \mathfrak{O}\_i\left(r\_k \leftarrow r\_h\right) / \ m\_h \tag{10}$$

Another usually employed quantity is the residence time, defined as the ratio between the cumulated activity in *h* and the administered activity *A*0:

$$
\pi\_{h} = \frac{\tilde{A}\_{h}}{A\_{0}} \tag{11}
$$

Even if the residence time has the physical dimensions of a time and it is often indicated with the same Greek letter, it must not be confused with the decay time of a radionuclide. In fact, while the decay time is the time necessary to reduce by 1/e = 0.37 the activity of an isolated sample, the residence time is the length of time an activity *A0* would have to reside in the volume to give that cumulated activity.
