**3. Radiobiological models of the radiation effects**

The estimation of the effect of the radiation absorbed dose in biological tissues can not neglect biological models accounting for the ability of tissues and cells to repair in some degree the radiation-induced injury. (Cremonesi, 2011; Strigari, 2011)

The radiation damage can vary due to the different tissue properties (the "five Rs" of radiobiology: *repair*, *repopulation*, *reoxigenation, redistribution* and *intrinsic radiosensitivity*) in tumours and in healthy tissues, and due to the difference in possible irradiation regimes (type and energy of the radiation, dose rate, repetition or fractionation of treatments).

In nuclear medicine therapies with radiopharmaceuticals, the radiation dose is often imparted by beta or Auger electrons, even if the role of alpha emitters as therapeutic agents is increasing again.

Internal Radiation Dosimetry: Models and Applications 29

Fig. 1. Cell surviving fraction as a function of the radiation dose, following the linear-

α:

α

In the proposed example,

relationships:

calculated as:

related to *BED* through the parameter

irradiation imparting a radiation dose *D*:

α/β= 25 Gy.

quadratic (LQ) model. Linear (L) and quadratic (Q) part of the equation are represented, too.

The Biologically Effective Dose (*BED*) is defined in a way such that the effect *E* is linearly

*E = BED* α

From Eq. 13, *BED* is related to the cell surviving fraction and to the dose with the

( ) ( ) <sup>1</sup> <sup>2</sup> log *<sup>β</sup> BED = SF = D + g T D*

For radionuclide therapies, Eq. 14 holds for the correction factor *g*, resulting, for a single

*β T BED = D + D* α

In the case of repeated irradiations, each one imparting a dose *Di*, the total *BED* can be

*β T BED = D + D* α

In the next Sections, we will see how BED is related to the impairment of kidneys after repeated cycles of peptide radio-receptor therapies (PRRT) with somatostatin analogues

 α

*rep* 2 *rep eff*

*rep* 2

*i i rep eff*

(15)

<sup>−</sup> (16)

*T +T* (17)

*T +T* ∑ ∑ (18)

where *Teff* is the effective half-life of the radionuclide in the target tissue.

Thus, the dose and the dose rate in the single treatment are governed by the biokinetics of the radiopharmaceutical and by the administered activity and route of administration.

The biological effect of such irradiation in tumours and in healthy tissues will depend firstly on the *repair* ability of the sub-lethal damage with related repair time *Trep*, which is due to the mechanisms that counteract all the natural damages to the DNA. This is the fastest mechanism influencing the response to irradiation, since its effects are detectable in external irradiations already after 30 minutes.

The cell life cycle is divided in four consecutive steps: G1, S, G2 and M. The two gaps of apparent inactivity, G1 and G2, divide the two active phases, the DNA synthesis S and the mitosis M. Radiobiology studies demonstrated that the highest cell radiosensitivity belongs to G2 and M phases. After an irradiation, the survival fraction will be higher for cells in the G1 or S phases; thus a *redistribution* of population is initiated, with a synchronization of cell life cycles.

This effect could, in principle, play a certain role in external irradiations repeated at the times of higher sensitivity, but no clear evidence of efficacy has been demonstrated yet.

Cells surviving to an exposure to radiations will continue to proliferate; such a repopulation has a detrimental effect on therapeutic results. On the other hand, tumour cell death leads to tumour tissue shrinkage and, consequently, can improve the *reoxigenation* of the residual hypoxic cells. Since hypoxic cells are more radio-resistant than the oxygenated ones, repeated cycles of irradiation are useful to improve therapeutic outcomes.

As a consequence of these mechanisms, the effect of a radiation therapy depends not only on the radiation absorbed dose, but also on the dose rate and on the fractionation regime.

The most widely applied radiobiological model describing cell survival after irradiations is the linear-quadratic model, in which the effect *E*, in logarithmic relation with the surviving fraction *SF*, is a function of the dose *D* and the dose squared:

$$E = -\log\left(SF\right) = \alpha D + \beta D^2\tag{12}$$

The linear component accounts for the lethal cell damage given by a single radiation producing, for example, double-strand breaks of the DNA helix, while the quadratic component accounts for the lethal damage obtained by summing up the effects of two consecutive ionizing radiations. It should be noticed that the parameter α has dimensions of Gy-1, while β of Gy-2. The dose in correspondence of which the linear contribution L equals the quadratic one Q (see Fig. 1), is given by *D*= α/β (Gy). This value expresses the *intrinsic radiosensitivity* of the tissue, and external as well as internal radiation therapies exploit the higher radiosensitivity of cancer cells with respect to normal tissue cells.

When the radiation dose is imparted in a time *T* comparable or even longer than the repair time of the sub-lethal damage, *Trep*, Eq. 12 must be corrected in order to account for the competition between radiation-induced damage and cell repair rate:

$$E \equiv -\log\left(SF\right) \equiv aD + g\left(T\right)\beta D^2\tag{13}$$

where *g*(*T*) is a properly chosen function of the irradiation time. When *T*>>*Trep*, as in the case of targeted radionuclide therapies, it was demonstrated that a good approximation is:

$$\text{kg} = \frac{T\_{rep}}{T\_{rep} + T\_{eff}} \tag{14}$$

Thus, the dose and the dose rate in the single treatment are governed by the biokinetics of the radiopharmaceutical and by the administered activity and route of administration. The biological effect of such irradiation in tumours and in healthy tissues will depend firstly on the *repair* ability of the sub-lethal damage with related repair time *Trep*, which is due to the mechanisms that counteract all the natural damages to the DNA. This is the fastest mechanism influencing the response to irradiation, since its effects are detectable in external

The cell life cycle is divided in four consecutive steps: G1, S, G2 and M. The two gaps of apparent inactivity, G1 and G2, divide the two active phases, the DNA synthesis S and the mitosis M. Radiobiology studies demonstrated that the highest cell radiosensitivity belongs to G2 and M phases. After an irradiation, the survival fraction will be higher for cells in the G1 or S phases; thus a *redistribution* of population is initiated, with a synchronization of cell

This effect could, in principle, play a certain role in external irradiations repeated at the times of higher sensitivity, but no clear evidence of efficacy has been demonstrated yet. Cells surviving to an exposure to radiations will continue to proliferate; such a repopulation has a detrimental effect on therapeutic results. On the other hand, tumour cell death leads to tumour tissue shrinkage and, consequently, can improve the *reoxigenation* of the residual hypoxic cells. Since hypoxic cells are more radio-resistant than the oxygenated ones,

As a consequence of these mechanisms, the effect of a radiation therapy depends not only on the radiation absorbed dose, but also on the dose rate and on the fractionation regime. The most widely applied radiobiological model describing cell survival after irradiations is the linear-quadratic model, in which the effect *E*, in logarithmic relation with the surviving

> log( ) *<sup>2</sup> E = SF = D +* − α

The linear component accounts for the lethal cell damage given by a single radiation producing, for example, double-strand breaks of the DNA helix, while the quadratic component accounts for the lethal damage obtained by summing up the effects of two

*radiosensitivity* of the tissue, and external as well as internal radiation therapies exploit the

When the radiation dose is imparted in a time *T* comparable or even longer than the repair time of the sub-lethal damage, *Trep*, Eq. 12 must be corrected in order to account for the

> log( ) ( ) *<sup>2</sup> E = SF = D + g T* − α

where *g*(*T*) is a properly chosen function of the irradiation time. When *T*>>*Trep*, as in the case of targeted radionuclide therapies, it was demonstrated that a good approximation is:

> *rep rep eff*

*T*

of Gy-2. The dose in correspondence of which the linear contribution L equals

α/β

*βD* (12)

α

(Gy). This value expresses the *intrinsic* 

*βD* (13)

*g = T +T* (14)

has dimensions of

repeated cycles of irradiation are useful to improve therapeutic outcomes.

consecutive ionizing radiations. It should be noticed that the parameter

higher radiosensitivity of cancer cells with respect to normal tissue cells.

competition between radiation-induced damage and cell repair rate:

fraction *SF*, is a function of the dose *D* and the dose squared:

the quadratic one Q (see Fig. 1), is given by *D*=

irradiations already after 30 minutes.

life cycles.

Gy-1, while

β

where *Teff* is the effective half-life of the radionuclide in the target tissue.

Fig. 1. Cell surviving fraction as a function of the radiation dose, following the linearquadratic (LQ) model. Linear (L) and quadratic (Q) part of the equation are represented, too. In the proposed example, α/β= 25 Gy.

The Biologically Effective Dose (*BED*) is defined in a way such that the effect *E* is linearly related to *BED* through the parameter α:

$$E \equiv \alpha BED \tag{15}$$

From Eq. 13, *BED* is related to the cell surviving fraction and to the dose with the relationships:

$$BED = -\frac{1}{a}\log\left(SF\right) = D + \frac{\beta}{a}g\left(T\right)D^2\tag{16}$$

For radionuclide therapies, Eq. 14 holds for the correction factor *g*, resulting, for a single irradiation imparting a radiation dose *D*:

$$BED = D + \frac{\beta}{\alpha} \frac{T\_{rep}}{T\_{rep} + T\_{\text{eff}}} D^2 \tag{17}$$

In the case of repeated irradiations, each one imparting a dose *Di*, the total *BED* can be calculated as:

$$BED = \sum D\_i + \frac{\beta}{\alpha} \frac{T\_{rep}}{T\_{rep} + T\_{eff}} \sum D\_i^2 \tag{18}$$

In the next Sections, we will see how BED is related to the impairment of kidneys after repeated cycles of peptide radio-receptor therapies (PRRT) with somatostatin analogues

Internal Radiation Dosimetry: Models and Applications 31

*<sup>λ</sup>eff*

In more complex cases, the uptake phase can require a certain amount of time and, consequently, the assumption of instantaneous uptake must be released and the uptake phase can be usually described by an exponential growth. Furthermore, the washout phase can be not accurately described by a simple mono-exponential decay. For example, a biexponential curve can fit better to a biokinetical behaviour composed by a first phase of rapid clearance in which the biologic half-life is much smaller than the physical half-life, followed by a slower retention phase in which, on the contrary, it is the physical half-life

In Figure 2, an example of near-instantaneous uptake, followed by a washout phase described by a bi-exponential decay, is shown. The renal uptake of a diagnostic dose of 111In-DTPA-Octreotide, a somatostatin analogue used for the diagnosis of neuroendocrine

Fig. 2. Uptake curves for kidneys in four patients after 111In-DTPA-Octreotide intravenous

In Figure 3 we present three examples of 131I uptake curves for hyperthyroid patients (pt. 1 and 2 affected by toxic nodular goitre, TNG, and the third by Basedow disease), acquired by means of a scintillation probe at six times after oral administration of a diagnostic activity of

In these cases, the uptake phase is expected to last up to one day, followed by a decay phase with a characteristic half-life of 100-200 hours, deriving from both physical (eight days) and

Thanks to the simplicity and rapidity of thyroid uptake measurements with a gamma probe, it is possible to sample properly in time these patients. Usually, two measurements during the first day, 3 and 6 hours after oral administration, properly characterize the uptake phase.

*<sup>t</sup> <sup>A</sup> A = A e dt =*

*eff*

∫ (21)

*λ*

0

that governs the overall effective half-life.

administration.

biological decay.

1.8 MBq.

tumours, was imaged at 1, 6, 24 and 48 hours post-injection.

<sup>∞</sup> <sup>−</sup>

labelled with beta-emitting radioisotopes. In such evaluations, it is usually assumed α*/*β = 2.4 Gy and for *Trep* a value of 2.8 hours.
