**4. Time sampling and determination of the cumulated activity**

In Section 2, the main equations of internal dosimetry were introduced, and the role played by the cumulated activity, i.e. the total number of disintegrations in the considered target, was taken in evidence.

In general, the biokinetics of a radiopharmaceutical within the human body will be influenced by the type of the carrier molecule and its physiologic and pathologic pathway, by the route of administration and by the preparation and clinical state of the patient. Clinical studies and trials give information about the average residence times of groups of patients, but, in order to plan the single treatment, only an accurate individualized dosimetry can be usefully employed.

In order to calculate the cumulated activity, the activity up-taken in each organ or region of interest must be properly sampled after administration. In principle, more measurements allow a more accurate fit of the A=A(t) curve, and, consequently, a better estimation of the total number of disintegrations.

However, we must remember that each measurement is carried out through planar scintigraphic or emission computed tomography (ECT) techniques, which are time consuming for both patient and hospital personnel.

Even in the case of non-imaging techniques, such as thyroid uptake measurements with a scintillation probe, the patient must come back to the nuclear medicine department for each measurement.

Hence the need to optimize dosimetric protocols in order to the number and timing of the acquisitions. The optimal choice will depend on the expected biokinetics of the radiopharmaceutical in the organs of interest, which can be assumed from previous clinical studies.

The simplest model applies when the uptake phase, i.e. the phase in which the radiopharmaceutical is accumulating in the organ and its radioactivity rises with time, is short enough to be considered instantaneous. Consequently, immediately after administration, the washout phase begins.

If, in the simplest assumption, the radiopharmaceutical is washed out with a monoexponential rate, the variation of *N* with time follows a law analogous to Eq. 1:

$$\frac{dN}{dt} = \lambda\_{\rm eff} N \tag{19}$$

where the effective decay constant *eff bio λ = λ + λ* is given by the sum of the physical decay constant introduced in Eq. 1 and the biological decay constant, characteristic of the biological wash-out of the radiopharmaceutical from the organ. In analogy with Eq. 3, the effective decay time τ*eff* and the effective half-life *Teff* can be defined as:

$$T\_{eff} = \frac{\text{ln2}}{\lambda\_{eff}} = r\_{eff} \text{ln2} \tag{20}$$

and the cumulated activity will be:

In Section 2, the main equations of internal dosimetry were introduced, and the role played by the cumulated activity, i.e. the total number of disintegrations in the considered target,

In general, the biokinetics of a radiopharmaceutical within the human body will be influenced by the type of the carrier molecule and its physiologic and pathologic pathway, by the route of administration and by the preparation and clinical state of the patient. Clinical studies and trials give information about the average residence times of groups of patients, but, in order to plan the single treatment, only an accurate individualized

In order to calculate the cumulated activity, the activity up-taken in each organ or region of interest must be properly sampled after administration. In principle, more measurements allow a more accurate fit of the A=A(t) curve, and, consequently, a better estimation of the

However, we must remember that each measurement is carried out through planar scintigraphic or emission computed tomography (ECT) techniques, which are time

Even in the case of non-imaging techniques, such as thyroid uptake measurements with a scintillation probe, the patient must come back to the nuclear medicine department for each

Hence the need to optimize dosimetric protocols in order to the number and timing of the acquisitions. The optimal choice will depend on the expected biokinetics of the radiopharmaceutical in the organs of interest, which can be assumed from previous clinical

The simplest model applies when the uptake phase, i.e. the phase in which the radiopharmaceutical is accumulating in the organ and its radioactivity rises with time, is short enough to be considered instantaneous. Consequently, immediately after

If, in the simplest assumption, the radiopharmaceutical is washed out with a mono-

*eff dN <sup>=</sup> <sup>λ</sup> <sup>N</sup>*

where the effective decay constant *eff bio λ = λ + λ* is given by the sum of the physical decay constant introduced in Eq. 1 and the biological decay constant, characteristic of the biological wash-out of the radiopharmaceutical from the organ. In analogy with Eq. 3, the

*eff* and the effective half-life *Teff* can be defined as:

ln2 ln2 *eff eff eff T= = τ*

*dt* (19)

*λ* (20)

exponential rate, the variation of *N* with time follows a law analogous to Eq. 1:

α*/*β=

labelled with beta-emitting radioisotopes. In such evaluations, it is usually assumed

**4. Time sampling and determination of the cumulated activity** 

2.4 Gy and for *Trep* a value of 2.8 hours.

dosimetry can be usefully employed.

consuming for both patient and hospital personnel.

administration, the washout phase begins.

τ

and the cumulated activity will be:

total number of disintegrations.

measurement.

effective decay time

studies.

was taken in evidence.

$$\tilde{A} = A \int\_0^\infty e^{-\lambda\_{eff}t} dt = \frac{A}{\lambda\_{eff}} \tag{21}$$

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 that governs the overall effective 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 tumours, was imaged at 1, 6, 24 and 48 hours post-injection.

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

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 1.8 MBq.

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 biological decay.

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.

Internal Radiation Dosimetry: Models and Applications 33

Pt. no. Umax λin λout λout(48h) Teff Teff(48h) error 1 (TNG) 23184 0.184 7.41E-3 9.72E-3 93 71 -24% 2 (TNG) 16755 0.142 5.62E-3 2.95E-3 123 235 +91% 3 (Bas.) 35920 0.740 4.42E-3 4.66E-3 157 148 -6% Table 1. Fit parameters for the complete data series, and for the first four points. Relative

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

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

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

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

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

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.

Considering a radionuclide which emits mono-energetic and beta electrons, and gamma

*<sup>β</sup> <sup>e</sup> <sup>γ</sup>* ( ) ( ) *e,i e,i e,i <sup>γ</sup>,i <sup>γ</sup>,i <sup>γ</sup>,i*

<sup>0</sup> <sup>1</sup> *<sup>s</sup> <sup>ρ</sup> φ ρ = +*

*ρ* <sup>−</sup> ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

*i*

are the mono-energetic electron and gamma photon emission probabilities,

1

<sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> <sup>∫</sup> <sup>∑</sup> , (23)

are the electron and photon absorbed fractions.

(24)

( )

, *φe* and *φ*

( )

*D=D +D +D = E<sup>φ</sup> E dE + n E <sup>φ</sup> +n E <sup>φ</sup> m dE*

γ

*A dn E*

γ

error in Teff determination are reported.

radionuclides (Snyder et al., 1975).

humans.

2009, 2011).

where *ne* and *n*

γ

respectively, of energies *Ee* and *E*

**5. Anthropomorphic models for internal dosimetry** 

in internal contamination evaluations for radiological protection purposes.

radionuclide emission and for the geometry of the source-target couple.

a whole, neglecting inhomogeneity in the uptake within each organ or tissue.

photons, the average dose to the target volume is given by:

These absorbed fractions can be derived from the equation:

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

Measurements can proceed at 24, 48, 72 (or 96, or 120) and 168 hours, in order to detect the maximum uptake and the decay phase.

Points can be then fitted with the function:

$$
\Delta U(t) = \frac{\lambda\_{\rm in} \mathcal{U}\_{\rm max}}{\lambda\_{\rm out} - \lambda\_{\rm in}} \left( e^{-\lambda\_{\rm in} t} - e^{-\lambda\_{\rm out} t} \right) \tag{22}
$$

where the uptake and washout constants λin and λout depend upon the rapidity of the increase and decrease in uptake, respectively.

Even if the maximum uptake is expected, on average, around 24 hours, this is not a rigid rule, of course. Patient 3, for example, reached the maximum uptake already at the sixth hour, and the fit of patient 1 data show its maximum earlier than 24 hours; cases of late maxima are observed, too.

In order to evaluate properly the effective half-time of the washout phase, measurements should extend at least up to 120 hours, better up to 168 hours. In fact, since each point is affected by statistical fluctuations and by minor biases due to the practical procedure, repeated measurements help in reducing errors in fit.

As an example, in Figure 3 we represent, together with the fits of the whole data series (solid lines), the fits obtained by considering only the first four points, i.e. 3, 6, 24 and 48 hours (dashed lines).

It is apparent from figure the underestimation of the decay time for the first patient, and, on the contrary, the overestimation for the second one.

Such outcome is confirmed numerically by the results reported in Table 1, where the effective half-time of the washout, *Teff*, is calculated from the decay constants obtained from the complete fit and from the 3-48 hours data. Results from incomplete data give a 24% shorter half-time for the first patient, while for the second one an overestimation as high as 91% comes from a dosimetric protocol limited to 48 hours.

Fig. 3. 131I uptake curves for the three patients. Solid curves represent the analytical fits of complete data series with Eq. 22, dashed lines represent fits of 3-48 hours data.

Measurements can proceed at 24, 48, 72 (or 96, or 120) and 168 hours, in order to detect the

( ) ( ) in in out

where the uptake and washout constants λin and λout depend upon the rapidity of the

Even if the maximum uptake is expected, on average, around 24 hours, this is not a rigid rule, of course. Patient 3, for example, reached the maximum uptake already at the sixth hour, and the fit of patient 1 data show its maximum earlier than 24 hours; cases of late

In order to evaluate properly the effective half-time of the washout phase, measurements should extend at least up to 120 hours, better up to 168 hours. In fact, since each point is affected by statistical fluctuations and by minor biases due to the practical procedure,

As an example, in Figure 3 we represent, together with the fits of the whole data series (solid lines), the fits obtained by considering only the first four points, i.e. 3, 6, 24 and 48 hours

It is apparent from figure the underestimation of the decay time for the first patient, and, on

Such outcome is confirmed numerically by the results reported in Table 1, where the effective half-time of the washout, *Teff*, is calculated from the decay constants obtained from the complete fit and from the 3-48 hours data. Results from incomplete data give a 24% shorter half-time for the first patient, while for the second one an overestimation as high as

Fig. 3. 131I uptake curves for the three patients. Solid curves represent the analytical fits of

complete data series with Eq. 22, dashed lines represent fits of 3-48 hours data.

<sup>−</sup> <sup>−</sup> <sup>−</sup> <sup>−</sup> (22)

out in *<sup>λ</sup> Umax <sup>λ</sup> <sup>t</sup> <sup>λ</sup> <sup>t</sup> Ut= e e*

*λ λ*

maximum uptake and the decay phase. Points can be then fitted with the function:

increase and decrease in uptake, respectively.

repeated measurements help in reducing errors in fit.

the contrary, the overestimation for the second one.

91% comes from a dosimetric protocol limited to 48 hours.

maxima are observed, too.

(dashed lines).


Table 1. Fit parameters for the complete data series, and for the first four points. Relative error in Teff determination are reported.
