The Petri dish was placed in the middle of the field of view between the two gamma camera heads at a point 15 cm from both heads. The Symbia T6 directly measures the energy spectrum without acquiring an image. The energy spectrum of 225Ac was registered for 10 min. The same geometry and source were used to obtain a gamma camera planar sensitivity. A planar acquisition was acquired for 10 min. The planar sensitivity S was calculated in each main energy window as follows:

where CountsROI denotes the total counts measured in a region of interest (ROI) with the same area as the Petri Dish (a circle ROI of 9 cm diameter), after background subtraction, A denotes the nominal activity in the source at the time of acquisition given in MBq, and T is the acquisition duration given in s.

The standard uncertainty of S (u(S)) was calculated according to the equation and the multiplicative variant of the law of propagation of uncertainty [19]:

$$u\left(S\right)=S\cdot \sqrt_\right)}_}\right)}^+\right)}^+\right)}^}$$

(2)

The standard uncertainty in the counts u(CountsROI) within a ROI was calculated as the square root of the number of counts. The standard uncertainty on activity u(A) measurements was equal to 2% and for acquisition duration u(T) was assumed to be 1 s.

Primary and scatter energy windows for 225Ac quantitative imaging were chosen based on the acquired energy spectrum (Table 1).

The phantom was placed in the centre of the field of view of the gamma camera. The parameters of the SPECT acquisition protocol are presented in Table 2. A CT scan was acquired for attenuation correction (AC), using a standard low-dose protocol.

The acquired data with multiple energy windows were reconstructed on the Esoft workstation (Siemens Healthineers, Germany) with the following parameters: OSEM FLASH 3D algorithm, 10 iterations, and 8 subsets, with Gauss filter (FWHM = 4 mm). Attenuation correction was performed using the CT-based attenuation map with the manufacturer’s default parameters. For each main energy window the user had to indicate the appropriate correction map for AC correction. An appropriate correction map was chosen by selecting one of the radioisotopes defined in the software (which emitted radiation with energy closest to a selected energy windows). In particular, for I energy window 133Xe was chosen (scale attenuation coefficients: 80 keV), for II energy window 111In was chosen (scale attenuation coefficient: 247 keV), and for III energy window 18F was chosen (scale attenuation coefficients: 511 keV). It can be estimated that the use of selected correction maps has a small impact on attenuation correction accuracy (around 5%, taking into account the difference between linear attenuation coefficients for water in considered energy range). The projections were corrected for scatter using the corresponding scatter windows and the dual-energy window (DEW) scatter-compensation method (SC). This method for SC was used (not the triple energy window scatter-compensation method (TEW)), because during acquisition on the Symbia T6 three main energy windows were established. Since an acquisition protocol allows to define a maximum of six independent energy windows, for three main windows only three lower scatter energy windows could be defined. The reconstructed images for individual main energy windows were summed up after reconstruction before further analysis.

The phantom was scanned according to the above protocol 9 times during 2.5 months (at a medium time of interval 8 days), starting from an activity concentration of 235 kBq/mL (a total activity in the phantom equal to 18.3 MBq) and ending with an activity concentration of 1.5 kBq/mL (a total activity in the phantom equal to 0.1 MBq).

All reconstructed images were evaluated using the Esoft workstation (Siemens Healthineers, Germany) volumetric analysis application. Volumes of interests (VOIs) defined on the CT image were applied to the SPECT reconstructed data. The total numbers of counts summed over the phantom reconstructed image (a VOI defined about the CT image of the outer boundaries of the phantom) were used to determine the image calibration factor (ICF). The ICF given in cps/MBq was calculated for the entire volume of the phantom using the formula:

where CountsPhantom denotes the total number of counts in a VOI defined about the CT image of the outer boundaries of the phantom, after background subtraction, A is the activity present in the phantom at the time of acquisition given in MBq, and T denotes the acquisition duration given in s (T = 3840 s, constant for all acquisitions). The counts per mL in the background was calculated by placing eight spherical VOIs randomly in the main cylinder image (also between cylindrical sources at their height in the phantom; see Fig. 2). The standard uncertainty of the ICF (u(ICF)) was calculated as follows:

$$u\left(ICF\right)=ICF\cdot \sqrt_\right)}_}\right)}^+\right)}^+\right)}^}$$

(4)

The standard uncertainty in the counts u(CountsPhantom), as well as on measured activity u(A) and for acquisition duration u(T) were defined as the standard uncertainty of planar sensitivity calculations.

In particular, VOIs were manually defined about the physical boundary of each cylindrical source on the CT image. These CT-based VOIs were copied to the SPECT image. RCs were calculated according to the formula:

$$RC=\frac_}_\cdot T}$$

(5)

where CountsCylinder denotes the total reconstructed counts within the target VOI, after background subtraction, ACylinder is the activity present in each cylinder at the time of acquisition given in MBq, and T denotes the acquisition duration given in s. To summarize, RCs were calculated by dividing the SPECT-based activity in each of the cylindrical sources of nominal volume by the activity dispensed to the cylindrical source known from the phantom experiment preparation. The recovery coefficients were fitted to a three-parameter model using the following equation:

$$RC\left(V\right)=\frac\right)}^}$$

(6)

where V is each cylindrical source volume given in mL and α, β, and γ are the fit parameters. To fit RC vs. V, as given in Eq. (6), a curve-fit function from scipy python library was used. This function returns both fitted parameters: α, β and γ of the model, and their standard errors (SE). Based on residual values mean squared error was calculated from which the confidence interval for the fitted curve was obtained.

The uncertainty of RC(V) (u(RC(V)) was calculated following Eq. (7), under the assumption of spherical sources and the definition of target VOI on SPECT imaging [20]:

$$u\left(RC\left(V\right)\right)=RC\sqrt\right)}^^+\right)}^^+\right)}^^+\right)}^^}$$

(7)

An experiment was performed to validate the accuracy of the defined protocol and quantitative imaging procedure. A three-dimensional printed model of glioblastoma was prepared. The model was designed based on real CT images of the patient with a brain tumour. In collaboration with the Faculty of Physics, University of Warsaw, the stl format file was created in Slicer 3D and the 3D phantom was printed on Prusa MK3 3D printer with polylactic acid (PLA). The volume of the 3D model was 25 mL. The printed model was checked for liquate before filling with a 225Ac solution. Finally, it was filled with 250 kBq/mL of 225Ac. The total activity in the 3D model was (6.1 ± 0.1) MBq. The filled model was placed in the middle of the mail cylinder, while other inserts were removed from it (Fig. 3).

The 3D printed model and its positions in the main cylinder (I and IV). The 3D printed model was created based on CT scan: clinical target volume (II). 3D printed model located in the patient’s brain (III). The line in (III) represents the CT scan shown in (II)

The phantom was scanned just after preparation. A SPECT/CT acquisition of the phantom was acquired with the same acquisition parameters described above on the Symbia T6 gamma camera with HE collimators. The same reconstruction parameters were then applied to obtain the final image. A VOI was contoured in the CT image of the 3D model, and the total counts inside the boundaries of that VOI in the SPECT image were determined. Additionally, eight background VOIs were defined. The mean counts per mL in the background was calculated. Taking into account the 3D model volume, the background counts were subtracted from the total counts in the 3D model VOI. The activity in the model (A3DPhantom) was estimated using the imaging calibration factor and the recovery coefficient obtained from experiments with the phantom and compared with the actual activity. The activity in the 3D model was calculated with the equation:

where Counts3DPhantom denotes the total reconstructed counts within the 3D model VOI, after background subtraction, ICF is the calculated imaging calibration factor, RC(V) denoted the recovery coefficient calculated according to Eq. 6, and T is the acquisition duration given in s.

The standard uncertainty of A3DPhantom (u(A3DPhantom) was calculated as follows:

$$u\left(_\right)=_\cdot \sqrt_\right)}_}\right)}^+\right)}^+\right)}^+\right)}^}$$

(9)

The standard uncertainty in the counts u(Counts3DPhantom), as well as for acquisition duration u(T), were defined as above. The standard uncertainty of ICF (u(ICF)) was assumed to be 1,8 cps/MBq (see Sect. 4). The uncertainty in recovery at a given volume was calculated following the law of propagation of uncertainties, under the assumption of spherical sources and the definition of target VOIs in SPECT imaging [20].

The percentage difference (%D) between the SPECT-measured activity A3DPhantom, and the activity measured using the radionuclide dose calibrator (the actual activity A equal to (6.1 ± 0.1) MBq), was used to evaluate the quantitative accuracy:

$$\%D=100\cdot \frac_-A}$$

(10)