This study was based on 33 procedures of Lutathera®-based PRRT performed in the setting of marketing authorization in 11 patients (6 women, 5 men, median age: 62 years) with metastatic grade I and II small-intestine neuroendocrine tumors progressing under cold somatostatin analogues treatment. The mean ± SD injected and residual activities, which were measured with a dose calibrator, were 7191 ± 140 MBq and 142.9 ± 121.3 MBq, respectively.

Lutathera® administration procedure

thRPM was intravenously administered over approximately 40 min using the gravity infusion method [9, 17]. In summary, RPM administration was performed by directly infusing saline (NaCl 0.9%) into the vial with a gravity drip (Fig. 1A). The increased pressure in the vial pushes 177Lu-DOTATATE into the patient’s intravenous line. The flow rate at the vial outlet is therefore imposed by the flow rate of the saline solution. In the first ten minutes, the flow rate was fixed at 0.25 drop s−1 (~ 50 mL h−1) followed by 0.5 drops s−1 from 10 to 20 min and finally at 1 drop s−1 from 20 min to the end of the infusion. Concomitant with the infusion of Lutathera®, an amino acid solution (LysaKare® 25 g/25 g) was administered by contralateral intravenous infusion.

Fig. 1
figure 1

Illustration of the gravity infusion device (A) and equivalent dose rate measurements on the patient’s arm (B)

Equivalent dose rate measurements

During each Lutathera® infusion, the EDR was monitored by an expert operator with a AT1123 (Atomtex, Minsk, Republic of Belarus) survey meter at both the injection site (Fig. 1A) and the patient’s abdomen. Measurements were performed every 2 min at a distance of 1 cm from the patient’s skin (Fig. 1B). The AT1123 survey meter was calibrated in equivalent dose rate Hp(10) (µSv h−1) by APVL (France, Saint-Cyr-sur-Loire) with a beam of radiological quality (mean energy 83 keV) and an accuracy of ± 20%.

Extravasation process modeling

Infusion model

As illustrated in Fig. 2, a simplified model of the infusion process was proposed. The radius ((R) = 2.5 mm) and length (({L}_{mathrm{T}}) = 115 cm) of the tubing line were modeled according to the manufacturer’s specifications, and the vein in the forearm was considered to be the same diameter as the tubing line and 30 cm long. The long needle that joins the vial to the tube was not modeled. The flow rates entering and leaving the vial were considered equal at all times (({Q}_{mathrm{out}}={Q}_{mathrm{in}}=Q)), and the increase in flow rate at times ({T}_{1}) = 10 min and ({T}_{2}) = 20 min was taken into account. The influence of the blood pressure on the flow rate in the patient’s arm was not modeled. The flow rates in the tube and the patient’s arm were therefore assumed to be equal at all time. The total infusion time was fixed at ({T}_{mathrm{inf}}) = 40 min, and the radioactive decay was neglected as the ratio of the infusion time to the half-life of 177Lu was 0.004.

Fig. 2
figure 2

Simplified geometry used to model the infusion process

Assuming a homogeneous mixture, the following dilution equation for the activity in the vial (({A}_{V})) can be considered:

$${A}_{V}(t)=left{begin{array}{ll}{A}_{0}{.e}^{-{alpha }_{1}.t} &quad {text{if}},, tle {T}_{1}\ {A}_{0}.{e}^{-{alpha }_{1}.{T}_{1}-{alpha }_{2}.left(t-{T}_{1}right) }&quad {text{if}},, {T}_{1}le tle {T}_{2}\ {A}_{0}.{e}^{-{alpha }_{1}.{T}_{1}-{alpha }_{2}.left({T}_{2}-{T}_{1}right)-{alpha }_{3}.left(t-{T}_{2}right)}&quad {text{if}},, {T}_{2}le tle {T}_{mathrm{inf}}end{array}right.$$

(1)

where ({A}_{0}) is the median initial activity in the vial (7232 MBq) and (left{{alpha }_{1}=frac{{Q}_{1}}{{V}_{V}}, {alpha }_{2}=frac{{Q}_{2}}{{V}_{V}}, {alpha }_{3}=frac{{Q}_{3}}{{V}_{V}}right}) are the dilution rate parameters imposed by the volume of vial (({V}_{V}) = 25 mL) and the three consecutive flow rates (({Q}_{1}) = 50 mL h−1, ({Q}_{2}) = 100 mL h−1, ({Q}_{3}) = 200 mL h−1).

In addition, the Reynolds number of the “tube + vein” compartment depends on the average flow velocity (({v}_{mathrm{mean},j}, j=left{mathrm{1,2},3right})) and can be defined as follows:

$$begin{aligned} {text{Re}}_{j} & = frac{{2.rho .v_{{{text{mean}},j}} .R}}{eta } \ v_{{{text{mean}},j}} & = frac{{Q_{j} }}{{pi .R^{2} }} \ end{aligned}$$

(2)

Assuming a density and a dynamic viscosity of the diluent + thRPM mixture equal to that of saline solution (NaCl 0.9%) at room temperature (ρ = 1.0053 g cm−3, η = 1.02 × 10−3 Pa s [18]), the Reynolds number in the “tube + vein” compartment are 3.5, 7.0, 13.9 for the three consecutive flow rates. The flow regime is laminar of the Poiseuille type with a parabolic radial velocity profile (({v}_{j}left(rright), j=left{mathrm{1,2},3right})):

$${v}_{j}left(rright)={2.v}_{mathrm{mean},j}left(1-frac{{r}^{2}}{{R}^{2}}right)$$

(3)

Based on Eqs. (1), (2) and (3) activities in the tubing ((i=T)) and the arm ((i=A)) during the infusion can be expressed by the integral in cylindrical coordinates of the activity concentration (({C}_{V}=frac{{A}_{V}}{{V}_{V}})) in the corresponding curvilinear s-axis domains [({s}_{i}^{mathrm{min}} {s}_{i}^{mathrm{max}})]:

$$begin{aligned} & A_{i} left( t right) = mathop smallint limits_{{s_{i}^{{{text{min}}}} }}^{{{s_{i}{{text{max}}}} }} mathop smallint limits_{0}^{R} C_{V} left( {t – T_{{{text{delay}}}} left( {s,r,t} right)} right).2pi .r.{text{d}}r.{text{d}}s \ & begin{array}{*{20}c} {{text{With}};;C_{V} left( {t – T_{{{text{delay}}}} left( {s,r,t} right)} right) = 0} & {{text{if}};;s > Dleft( {r,t} right)} \ end{array} \ end{aligned}$$

(4)

where (D(r,t)) is the total distance traveled by the fluid at radius r and time t and ({T}_{mathrm{delay}}(s,r,t)) is the time for a fluid element to reach the coordinates (s, r) at time (t). As illustrated in Fig. 3, these two parameters depend on the three consecutive radial velocity profiles (({v}_{j}left(rright), j=left{mathrm{1,2},3right})) according to the following equations:

Fig. 3
figure 3

Illustration of the time delay (({T}_{mathrm{delay}})) between the vial concentration and the fluid element concentration as a function of time (t) and the position of the fluid element in cylindrical coordinates ((s,r)). A For (t < {T}_{1}), the fluid element traveled the distance (s) at the velocity (overrightarrow{{v}_{1}}(r)). ({T}_{mathrm{delay}}) is thus expressed as the ratio between the fluid element curvilinear abscissa (s) and the velocity (overrightarrow{{v}_{1}}(r)({T}_{mathrm{delay}}=frac{s}{overrightarrow{{v}_{1}}left(rright) , })). B For ({T}_{1}<mathrm{t }le {T}_{2}), there are two cases depending on the curvilinear abscissa s of the fluid element: Case 1. (s{le D}_{2}(r,t)), the fluid element has traveled the distance (s) at the velocity (overrightarrow{{v}_{2}}(r),({T}_{mathrm{delay}}=frac{s}{overrightarrow{{v}_{2}}left(rright) , })). Case 2. (s>{D}_{2}(r,t)), the fluid element has traveled the distance ({D}_{2}(r,t)) at the velocity (overrightarrow{{v}_{2}}(r)) and the distance (s-{D}_{2}(r,t)) at the velocity (overrightarrow{{v}_{1}}(r)) (({T}_{mathrm{delay}}=frac{{D}_{2}(r,t)}{overrightarrow{{v}_{2}}(r) , }+frac{s-{D}_{2}(r,t)}{overrightarrow{{v}_{1}}(r) , })). C For (t>{T}_{2}), there are three cases depending on the curvilinear abscissa (s) of the fluid element. The mathematical expression of ({T}_{mathrm{delay}}) is obtained by following the same method as for ({T}_{1}<t le {T}_{2})

$$Dleft(r,tright)=left{begin{array}{ll}{{D}_{1}left(r,tright) =v}_{1}left(rright).t &quad {text{if}},, tle {T}_{1}\ {{D}_{1}left(r,{T}_{1}right)+{D}_{2}left(r,tright) =v}_{1}left(rright).{T}_{1}+ {v}_{2}left(rright).left(t-{T}_{1}right)&quad {text{if}},, {T}_{1}<tle {T}_{2}\ {D}_{1}left(r,{T}_{1}right)+{D}_{2}left(r,{T}_{2}right)+{D}_{3}left(r,tright)={v}_{1}left(rright).{T}_{1}+{v}_{2}left(rright).left({T}_{2}-{T}_{1}right)+{v}_{3}left(rright).left(t-{T}_{2}right)&quad {text{if}},, {T}_{2}<tle {T}_{mathrm{inf}}end{array}right.$$

(5)

$${T}_{mathrm{delay}}(s,r,t)=left{begin{array}{ll}frac{s}{{v}_{1}left(rright)} &quad {text{if}},, tle {T}_{1} \ frac{s}{{v}_{2}left(rright)} &quad {text{if}},, {T}_{1}<tle {T}_{2} ,,{text{and}},, sle {D}_{2}left(r,tright)\ frac{{D}_{2}left(r,tright)}{{v}_{2}left(rright)}+frac{s-{D}_{2}left(r,tright)}{{v}_{1}left(rright)} &quad {text{if}},, {T}_{1}<tle {T}_{2} ,,{text{and}},, s> {D}_{2}left(r,tright)\ frac{s}{{v}_{3}left(rright)} &quad {text{if}},, t>{T}_{2} ,,{text{and}},, sle {D}_{3}left(r,tright)\ frac{{D}_{3}left(r,tright)}{{v}_{3}left(rright)}+frac{s-{D}_{3}left(r,tright)}{{v}_{2}left(rright)}&quad {text{if}},, t>{T}_{2} ,,{text{and}},, {D}_{3}left(r,tright)+{D}_{2}left(r,{T}_{2}right)ge s>{D}_{3}left(r,tright)\ frac{{D}_{3}left(r,tright)}{{v}_{3}left(rright)}+frac{{D}_{2}left(r,{T}_{2}right)}{{v}_{2}left(rright)}+frac{s-{D}_{3}left(r,tright)-{D}_{2}left(r,{T}_{2}right)}{{v}_{1}left(rright)} &quad {text{if}},, t>{T}_{2},,{text{and}},, s>{D}_{3}left(r,tright)+{D}_{2}left(r,{T}_{2}right)\ end{array}right.$$

(6)

Extravasation model

Under normal conditions, the activity of the vial passes through the arm to reach the patient’s body. The activity inside the tube at time (t) is then provided by the infusion model (Eq. 4). In the event of extravasation, the activity is gradually stored in the arm and therefore no longer reaches the blood compartment. Assuming that extravasation does not affect the pressure in the “vial + tube” compartment, the dynamics of the vial and the flow rate in the tube remains unchanged. For extravasation occurring at injection site ({s}_{A}) and time ({t}_{E}), the extravasated activity in forearm (({A}_{E}(t,{t}_{E}))) at time (tge {t}_{E}) can simply be expressed as the difference of activity in the “vial + tube” compartment between times ({t}_{E}) and (t):

$${A}_{E}left(t,{t}_{E}right)={A}_{V}left({t}_{E}right)+{A}_{T}left({t}_{E}right)-{A}_{V}left(tright)-{A}_{T}left(tright)$$

(7)

Depending on the values of the ((t,{t}_{E})) couple with respect to the times ({T}_{1}) and ({T}_{2}), the extravasated volume in the arm at time (tge {t}_{E}) is given by:

$${V}_{E}left(t,{t}_{E}right)=left{begin{array}{ll}left(t-{t}_{E}right).{Q}_{1} &quad {text{if}},, tle {T}_{1} \ left(t-{t}_{E}right).{Q}_{2} &quad {text{if}},, {T}_{1}<tle {T}_{2},,{text{and}},, {t}_{E}>{T}_{1}\ left({T}_{1}-{t}_{E}right).{Q}_{1}+left(t-{T}_{1}right){.Q}_{2} &quad {text{if}},, {T}_{1}<tle {T}_{2},,{text{and}},, {t}_{E}le {T}_{1}\ left(t-{t}_{E}right).{Q}_{3} &quad {text{if}},, {T}_{2}<tle {T}_{text{inf}},,{text{and}},, {t}_{E}>{T}_{2}\ left({T}_{2}-{t}_{E}right).{Q}_{2}+left(t-{T}_{2}right).{Q}_{3}& quad {text{if}},, {T}_{2}<tle {T}_{text{inf}} ,,{text{and}},, {t}_{E}le {T}_{2}\ left({T}_{1}-{t}_{E}right).{Q}_{1}+left({T}_{2}-{T}_{1}right).{Q}_{2}+left(t-{T}_{2}right).{Q}_{3}& quad {text{if}},, {T}_{2}<tle {T}_{text{inf}},,{text{and}},, {t}_{E} le {T}_{1}end{array}right.$$

(8)

The concentration of extravasated activity can then be expressed as follows:

$${C}_{E}left(t,{t}_{E}right)=frac{{A}_{E}left(t,{t}_{E}right)}{{V}_{E}left(t,{t}_{E}right)}$$

(9)

Conversion to equivalent dose rates

Survey meter calibration

A proper calibration procedure of the AT1123 survey meter theoretically allows equivalent dose rates (EDRs) to be estimated from the simulated activities. This calibration was performed using a 177Lu calibrated point source (({A}_{mathrm{cal}}) = 11.01 MBq) inserted in the same tube as the one used during the administration (i.e., with an internal diameter of 2.5 mm). The activity of the point source was determined using a dose calibrator by difference between the measurements of a syringe filled with Lutathera® before and after injection of the point source into the tube. As illustrated in Fig. 4A, the AT1123 survey meter was then placed at a fixed distance of 1 cm from the tubing, and the EDR of the point source was measured according to its position along the (s) axis. Twenty-five equidistant measurements were performed for source positions ranging from 0 to 25 cm. The following function was used to fit the relationship between EDR measurements ({(H}_{T})) and positions of the point source ((s)):

$${H}_{T}left(sright)=a.{left({s}^{2}+{b}^{2}right)}^frac{c}{2}$$

(10)

where (a), (b), and (c) are the coefficients providing the best fit (Fig. 4B). These coefficients and goodness-of-fit criteria (R2 and RMSE) are given in Table 1.

Fig. 4
figure 4

A Illustration of the calibration procedure of the AT1123 survey meter according to the position of an 11.01 MBq point source along the axis of the tubing(s) and for a fixed tubing-survey meter distance of 1 cm. B Result of the calibration that provides the relationship between EDR measurements and positions of the point source (circular symbols) with the corresponding regression (dashed line)

Table 1 Parameters of the function used to model the relationship between equivalent dose rates measurements ({(H}_{T})) and positions of the point source ((s))

Infusion model

EDR during the infusion may be estimated at 1 cm from the injection site (({s}_{a})) by integrating the product of the theoretical activity concentration by the function ({H}_{T}) over a domain ranging from (s={s}_{a}-m) to (s={s}_{a}+m):

$${mathrm{EDR}}_{{s}_{a}}left(tright)=frac{1}{{A}_{mathrm{Cal}}}underset{{s}_{a}-m}{overset{{s}_{a}+m}{int }}left(underset{0}{overset{R}{int }}{C}_{V}left({t-T}_{mathrm{delay}}(s,r,t)right).2pi .r.mathrm{d}rright).{H}_{T}left({s}_{a}-sright)mathrm{d}s$$

(11)

m is set at 20 cm because only activity present within ± 20 cm of the injection site is considered to contribute to the EDR measurements.

Extravasation model

Although the EDR during nonextravasated infusion could be estimated by a linear integration of weighted activity (Eq. 11), the EDR during extravasation requires integration over a spatial extent. In our study, we considered the extravasation regions to be elliptical in shape, centered on the injection site (({s}_{a})) and homogeneous in activity concentration.

In addition, the thickness of the extravasation region is fixed to the diameter of the tubing (2.R) to be in line with the conversion function ({H}_{T}left(rright)). Under these conditions, EDR may be estimated 1 cm above the ({s}_{a}) point by integrating the function ({H}_{T}) over an elliptical disc:

$${dot{D}}_{{s}_{a}}left(tright)=frac{{C}_{E}(t,{t}_{E})}{{A}_{mathrm{Cal}}}underset{0}{overset{2pi }{int }}underset{0}{overset{r(theta )}{int }}{H}_{T}left(rright).r.mathrm{d}r.mathrm{d}theta$$

(12)

where r(θ) is the polar equation of an ellipse whose semi major axis β and semi minor axis α are parallel and perpendicular to the arm direction, respectively:

$$rleft(theta right)=frac{beta }{sqrt{1-{mathrm{cos}}^{2}left(theta right)+{left(frac{beta }{alpha }right)}^{2}.{mathrm{cos}}^{2}left(theta right)}}$$

(13)

For a given extravasation region, the elliptical aspect ratio ((mathrm{AR}=frac{beta }{alpha })) is fixed, and the (beta .alpha) product is conditioned by the extravasated volume (({V}_{E})) according to the following equation:

$$beta .alpha = frac{{V}_{E}(t,{t}_{E})}{2Rpi }$$

(14)

Implementation

The tubing and vein were sampled with 1 mm bins along the curvilinear abscissa (s) and 0.005 mm along the radial direction (r). The infusion model (Eq. 4) and the corresponding EDR (Eq. 11) were computed for times ((t)) ranging from 0 to ({T}_{mathrm{inf}}) = 40 min with a 1-min sampling. Elliptical extravasation regions were sampled with 0.5° bins along the angular direction (theta) and 0.1 mm along the radial direction (r). The extravasation model (Eqs. 7, 8 and 9) and the corresponding EDR (Eqs. 12, 13 and 14) were computed for both times ((t)) and extravasation times (({t}_{E})) ranging from 0 to ({T}_{mathrm{inf}}) with a sampling time of 1 min. As EDR measurements are strongly dependent on the shape of the elliptical extravasation region, EDRs were simulated for four different extravasation aspect ratios ((mathrm{AR}) = 2, 3, 4, and 5).

These calculations were performed using Python 3.6. All the processes, including the extravasation model and the conversion from MBq into µSv h−1, take approximately 1 h on a single core i7 8700 K @ 3.7 GHz processor.

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