Numerical values are reported as mean ± standard deviation [minimum; maximum], unless otherwise stated.

### Radiopaque microspheres

The Eye90 microspheres (Eye90) were composed of a proprietary, radiopaque glass composition and were similar in density and size to TheraSphere microspheres (20–30 µm diameter, *ρ* = 3.4 g/cm^{3}). The ^{90}Y in Eye90 was produced through thermal neutron absorption of yttrium-89 embedded within the microsphere’s glass matrix. A nominal bolus of 40 mg (~ 981,000 microspheres) was measured for administration to each rabbit. The average microsphere activity \(A_{{{\text{MS}}}}\) at the time of administration was 156 ± 18 Bq [142; 182].

### CT calibration phantom

It has been previously demonstrated that a linear relationship, defined by Eq. 1, exists between CT Hounsfield units HU and radiopaque microsphere concentration MS_{con} over a clinically relevant range of values,

$${\text{HU}} = m_{{{\text{cal}}}} \cdot {\text{MS}}_{{{\text{con}}}} + b_{{{\text{cal}}}}$$

(1)

where \(m_{{{\text{cal}}}}\) and \(b_{{{\text{cal}}}}\) are the slope and intercept (defined below) of the calibration curve, respectively [27]. Based on this relationship, a calibration phantom was designed to contain cylindrical posts composed of a tissue-equivalent resin (\(Z_{{{\text{eff}}}}\) = 6.45, \(\rho\) = 1.03 g/cm^{3}) and infused with Eye90 in nominal concentrations of 0.5 mg/mL, 5.0 mg/mL, 25.0 mg/mL. The calibration posts were produced by mixing the microspheres with a viscous resin then immediately pouring the mixture into a cylindrical mould. At room temperature, the resin-microsphere mixture cured within two minutes, allowing the microspheres to remain in suspension within the mixture. Evaluation of each post with CT imaging (data not shown) revealed excellent uniformity across the entire post length. There were nine posts per microsphere concentration with a post length of 40 mm and varying diameters ranging from 2 to 9 mm in 1 mm increments, with an additional post having a diameter of 15 mm. The central axis of all posts was placed equidistant (100 mm) from the central longitudinal axis and was embedded in the resin background material having a radius of 150 mm, as shown in Fig. 1a.

This phantom was imaged with a clinical CT (Celesteion™ PET/CT, Canon Medical Systems, Ōtawara, Japan) using a tube potential of 100 kVp and an exposure of 270 mAs. Images were reconstructed with filtered back projection in a 16.2 cm field of view (FOV) having voxel dimensions of 0.313 mm × 0.313 mm × 2.000 mm. An axial CT slice of the phantom is shown in Fig. 1b and is displayed with a voxel intensity range of − 100 to 200 HU. Within the MIM Software platform v6.9.4 (MIM Software Inc., Cleveland, OH, USA), structures were created for all 27 posts based on the known geometry of the phantom. Segmented post structures were reduced by a 1 mm radial margin and 5 mm longitudinal margin to reduce partial volume effects between the background-post and background-air interfaces, respectively. The 2 mm-diameter post segmentations were not reduced by the 1 mm radial margin as this would eliminate the structures entirely. Instead, they were reduced to a 1 mm-diameter cylinder centred on the post’s central longitudinal axis. An additional cylindrical structure with a 30 mm diameter was segmented in the centre of the background region to quantify the intensity of a uniform volume void of microspheres. All segmented structures within the calibration phantom are shown in Fig. 1c.

The mean HU was extracted from each structure, and a calibration curve based on Eq. 1 was determined through a linear least-squares fit of the HU and MS_{con} data. The slope *m*_{cal} was extracted from the fit, while the intercept *b*_{cal} was calculated independently for each rabbit according to Eq. 2.

$$b_{{{\text{cal}}}} = \mu_{{{\text{bkg}}}} + 1.645_{{{\text{bkg}}}}$$

(2)

Here \(\mu_{{{\text{bkg}}}}\) and \(\sigma_{{{\text{bkg}}}}\) are the mean and standard deviation, respectively, of CT voxel values in a non-embolized background region \(L_{{{\text{bkg}}}}\) within each rabbit liver to account for HU variations in the liver parenchyma between rabbits. The factor 1.645 is the Z-score for a one-sided standard normal distribution with a false positive detection rate of \(\alpha = 0.05\). As the voxel values within bkg were normally distributed, voxels with \({\text{HU}} > b_{{{\text{cal}}}}\) have a 95% probability of containing Eye90.

To produce a voxelized CT-based ^{90}Y activity distribution *A*_{CT} with units of Bq, Eq. 1 was solved for \({\text{MS}}_{{{\text{con}}}}\) and multiplied by three scalar factors: the number of microspheres per milligram \({\text{MS}}_{{{\text{mg}}}}\), the microsphere specific activity \(A_{{{\text{MS}}}}\) measured at the time of administration, and the CT voxel volume \(V_{{{\text{CT}}}}\), as shown in Eq. 3.

$$A_{{{\text{CT}}}} = {\text{MS}}_{{{\text{con}}}} \cdot [{\text{MS}}_{{{\text{mg}}}} \cdot A_{{{\text{MS}}}} \cdot V_{{{\text{CT}}}} ]$$

(3)

The number of microspheres per milligram was determined by measuring the diameter of a group of microspheres (*n* = 28) through microscopy and calculating the average microsphere volume. Given the volume and a microsphere density of 3.4 g/cm^{3}, the microsphere mass was determined to be 4.09 ± 0.09 × 10^{−5} mg [3.96 × 10^{−5}; 4.33 × 10^{−5}]. Therefore, the average number of microspheres per milligram MS_{mg} is equal to 24,460 ± 542 [23,111; 25,227].

Theoretically, the administered ^{90}Y activity \(A_{0}\) should be recovered by summing the voxelized activity distribution *A*_{CT} over the segmented liver volume \(L\). A recovery coefficient RC_{CT} was defined as the ratio of this sum to \(A_{0}\), expressed as a percentage and shown in Eq. 4.

$${\text{RC}}_{{{\text{CT}}}} = 100 \cdot \left[ {\frac{{\mathop \sum \nolimits_{L} A_{{{\text{CT}}}} }}{{A_{0} }}} \right]$$

(4)

For comparison, Eq. 4 is applied to the PET-derived activity distribution *A*_{PET} for two structures: the liver volume \(L\) and an extended liver volume \(L_{{{\text{shell}}}}\), defined as \(L\) plus a 1 cm isotropic margin. The corresponding recovery coefficients are \({\text{RC}}_{{{\text{PET}}}}\) and \({\text{RC}}_{{{\text{PET}}}}^{{{\text{shell}}}}\), respectively.

### Rabbit liver model

The University of Missouri Animal Care and Use Committee approved the animal protocol (#9786) whose data were analysed for this study. Eight White New Zealand rabbits were included in this study (5 males, 3 females) weighing an average of 3.3 ± 0.2 kg [3.0; 3.5]. Rabbits are subsequently referred to as R01 through R08.

Prior to administration, each rabbit was induced with ketamine and dexmedetomidine then maintained on isoflurane and oxygen by mask. Eye90 was administered into either the left or proper hepatic arteries of the liver via a 2.4 Fr Progreat microcatheter. The average whole liver volume was 79 ± 11 mL [65; 97]. The ^{90}Y activity in Eye90 was measured with a Ludlum Model 3 survey meter (Ludlum Measurements Inc., Sweetwater, TX, USA) following neutron activation. Decay correction was applied to the time of microsphere administration. The average ^{90}Y activity at the time of administration was found to be 153.8 ± 18.0 MBq [140.0; 180.0]. Following administration, residual ^{90}Y activity in the microsphere vial was measured using the survey meter with the same geometry used to assay the activity post-activation. The residual ^{90}Y activity within the microsphere administration lines was also measured following their placement in a Nalgene container. To account for geometric variations, four measurements were acquired at 90° intervals as the container was rotated through 360°. Measurements were then averaged and added to the residual activity in the microsphere vial to determine a total residual activity of 8.9 ± 2.0 MBq [6.0; 12.9]. Therefore, the average administered activity \(A_{0}\) was 144.2 ± 17.4 MBq [128.1; 171.0]. The lung shunt fraction was expected to be negligible based on results from pathologic studies of a rabbit VX2 liver tumour model following the administration of iron oxide microspheres [28]. In this study, both intra-procedural fluoroscopic imaging and post-procedural PET imaging verified microsphere deposition only within the liver volume \(L\).

### Post-treatment imaging

Following microsphere administration, each rabbit was imaged with a time-of-flight (TOF) PET/CT scanner (Celesteion™ PET/CT, Canon Medical Systems, Ōtawara, Japan) while under anaesthesia. The radioisotope ^{90}Y was selected for the PET acquisition. Data were acquired using four overlapping bed positions with seven minutes/position. The lower and upper energy level discriminators were set to 435 keV and 650 keV, respectively. Prior to reconstruction, sinograms were corrected for scatter using a model-based scatter correction method [29] and for attenuation using CT image data [30]. Random coincidences were corrected for using a delayed coincidence approach [31]. Images were reconstructed in a 128 × 128 × 240 matrix using a 26.0 cm transaxial FOV with isotropic voxel sizes of 2.039 mm × 2.039 mm × 2.039 mm. The reconstruction algorithm employed in this study was the ordered subset expectation maximization algorithm with three iterations and ten subsets [32]. Post-filtering of reconstructed images was performed with a 4 mm FWHM Gaussian filter to reduce image noise.

Following the PET/CT acquisition, an additional four-phase CT was acquired with acquisition parameters set to match the parameters used during CT imaging of the calibration phantom. CT scans included a baseline unenhanced, arterial, portal, and delayed venous phase. Within MIM, the liver volume \(L\) was contoured using portal phase CT to provide maximum contrast between liver parenchyma and surrounding soft tissue. The extended liver volume \(L_{{{\text{shell}}}}\) was generated by isotropically extending the liver volume \(L\) by a 1.0 cm margin. This margin was chosen to account for the reduced PET spatial resolution relative to CT as well as perceived ^{90}Y activity outside of the liver volume \(L\) resulting from respiration, which was shown during intra-procedural angiographic imaging to displace the rabbit livers by a maximum of 1.0 cm in the cranial-caudal direction. A third structure \(B\) was generated around the exterior of the rabbit body. In the unenhanced CT, a planar structure *L*_{bkg} was generated in a non-embolized, homogeneous background region of the liver to account for HU variations in the non-embolized liver parenchyma between rabbits. In Fig. 2, all structures in R03 are visible in a baseline, unenhanced axial CT slice with a voxel intensity range of − 100 to 200 HU.

### Dosimetry

#### MIRD

Treatment planning for commercially available TheraSphere microspheres is based on a MIRD model that assumes a uniform ^{90}Y activity distribution within a target volume [33]. In this model, the mean dose \(D_{{{\text{MIRD}}}}\) to a target volume is defined in Eq. 5.

$$D_{{{\text{MIRD}}}} = \frac{{A_{0} \cdot 50 \cdot \left( {1 - R} \right)}}{M}$$

(5)

Here \(A_{0}\) is the administered ^{90}Y activity in GBq, \(M\) is the mass of the target in kg, and \(R\) is the fractional residual activity. To serve as a reference for PET- and CT-based dosimetry, \(D_{{{\text{MIRD}}}}\) was calculated for each rabbit liver given \(R\), \(A_{0}\), the liver volume \(L\), and an assumed liver density of 1.03 g/mL.

#### Convolution

Pathohistological studies performed on explanted human livers following ^{90}Y RE have demonstrated highly heterogeneous in vivo microsphere distributions [12, 13, 34]. Currently, no clinical imaging modality can resolve individual microspheres, so all relevant imaging methods present a reduced resolution approximation of the true ^{90}Y activity distribution. Within the constraints of this limitation, dose-voxel kernel (DVK) convolutional dosimetry can be used to calculate the dose distribution based on a heterogeneous ^{90}Y activity distribution. In this study, the dose distribution *D* was determined through the convolution of a cumulated activity distribution \(\tilde{A}\) with a spatially invariant DVK, as described in Eq. 6.

$$D = \tilde{A} \otimes {\text{DVK}} = \mathop \sum \limits_{{x^{\prime}}} \mathop \sum \limits_{{y^{\prime}}} \mathop \sum \limits_{{z^{\prime}}} \tilde{A}\left( {x^{\prime } ,y^{\prime } ,z^{\prime } } \right) \cdot {\text{DVK}}\left( {x - x^{\prime } ,y - y^{\prime } ,z - z^{\prime } } \right)$$

(6)

As microspheres are permanent implants, it is unnecessary to image at multiple time points post-administration to determine the cumulated activity. Therefore, \(\tilde{A}\) was calculated using Eq. 7,

$$\tilde{A}\left( {x^{\prime } ,y^{\prime } ,z^{\prime } } \right) = \mathop \smallint \limits_{0}^{\infty } A\left( {x^{\prime } ,y^{\prime } ,z^{\prime } ,t} \right) e^{ - \lambda t} {\text{d}}t = \frac{{A\left( {x^{\prime } ,y^{\prime } ,z^{\prime } } \right)}}{\lambda } = \tau A\left( {x^{\prime } ,y^{\prime } ,z^{\prime } } \right)$$

(7)

where \(\lambda\) is the decay constant, \(\tau\) is the mean lifetime of ^{90}Y, and \(A\left( {x^{\prime } ,y^{\prime } ,z^{\prime } ,t} \right)\) is the initial ^{90}Y activity in a voxel at coordinate \((x^{\prime } ,y^{\prime } ,z^{\prime } )\) at time \(t\) = 0. The convolution of \(\tilde{A}\) and DVK was performed in the frequency domain using the fast Fourier transform. The resulting PET- and CT-based dose distributions are subsequently referred to as \({\text{DD}}_{{{\text{PET}}}}\) and \({\text{DD}}_{{{\text{CT}}}}\), respectively. Dosimetry calculations were performed in MATLAB R2020b (MathWorks Inc., Natick, MA, USA).

#### Dose-voxel kernels

The DVKs in this study were calculated through simulations of ^{90}Y radiation transport in a voxelized sphere of water with the GATE v9.0 Monte Carlo toolkit encapsulating Geant4 10.06.p01 [35, 36]. Physics processes were enabled according to option 4 of the standard electromagnetic physics list, and electron transport was performed with an energy cut-off of 1 keV. The DVKs were calculated specific to voxel sizes in the activity distributions *A*_{CT} and *A*_{PET}, and are referred to as \({\text{DVK}}_{{{\text{CT}}}}\) and \({\text{DVK}}_{{{\text{PET}}}}\), respectively. Prior to each simulation, a ^{90}Y source was uniformly distributed within the origin voxel of a spherical water phantom where 40 million histories were set to decay. Voxels whose centre of mass was ≤ 25 mm from the origin were assigned to water, and the remaining voxels were set to air. From the simulation output, the mean absorbed dose per history was calculated in each voxel. Voxels whose centre of mass was > 25 mm were masked to zero to ensure convolution with spherically symmetric DVKs.

#### Statistical analysis

Dosimetric evaluations were carried out through a comparison of standard dose metrics including the median dose \(D_{{{\text{med}}}}\), maximum dose \(D_{{{\text{max}}}}\), mean dose \(D_{\mu }\), standard deviation \(\sigma\), and coefficient of variation (COV) defined as \(\sigma {/}D_{\mu }\). Cumulative dose-volume histograms (cDVHs) were calculated to determine \(D_{70}\). The mean dose \(D_{\mu }\) across all rabbits was compared between dose distributions DD_{PET} and DD_{CT} through linear regression, ANOVA, and Bland–Altman analysis. For DD_{CT}, \(D_{\mu }\) was calculated for the liver volume \(L\). For DD_{PET}, \(D_{\mu }\) was calculated for the structure *L*_{shell} to account for reduced spatial resolution and respiratory motion during PET image acquisition.