### Patient characteristics

Fourteen patients with hepatic malignancies who underwent RE with ^{90}Y-resin microspheres in our institution from 2013 to 2015 were retrospectively evaluated. The inclusion criteria were availability of a contrast-enhanced CT or MRI within 4 weeks prior to treatment, lesions that could be unequivocally delineated, and similar positioning of the catheter both in the simulation with ^{99m}Tc-MAA and in the therapeutic ^{90}Y-microespheres administration. No other clinical or demographic data was taken into account for the patient selection because it is not required to achieve the principal aim of this study: to compare dosimetry methods.

### (^{99m}Tc-MAA) protocol scan and activity planning

Once ^{99m}Tc-MAA were injected trough the selected arteries during hepatic arteriography, planar and SPECT-CT images were acquired in a Symbia T2 (Siemens Medical Solutions, Erlangen, Germany) with a dual-head variable-angle gammacamera and a two-slice spiral CT scanner. A low-energy high-resolution (LEHR) collimator was used with an energy window centered at 140 keV and 15% wide.

For planar imaging, anterior and posterior images of the abdomen and the thorax (10-min acquisition) were taken in a 128 × 128 matrix. No zoom was applied.

For SPECT acquisition, 128 images (20 s per projection) were acquired over 360° using a 128 × 128 matrix with a pixel size of 4.8 × 4.8 mm^{2}. Images were reconstructed using a Flash 3D algorithm (8 iterations, 4 subsets, 8.4 mm FWHM Gaussian post-filter), an iterative algorithm considering a 3D collimator beam modeling, CT-based attenuation correction, and energy window-based scatter correction. The scan parameters for CT were 130 kV, 25 mAs, and 5-mm slices. Both SPECT and CT images were fused using an Esoft 2000 application package (Siemens Medical Solution, Erlangen, Germany).

As previously published by Gil-Alzugaray et al. [12], in our center, the administered ^{90}Y activity was planned by means of PM for lobar and segmental treatments and by body surface area model for whole liver treatments. This methods were applied according to the microspheres’ manufacturer recommended guidelines [13].

The lung shunt fraction (LSF) was calculated by Eq. (1), where *C*_{lung} and *C*_{WL} are the geometric mean of total counts (anterior and posterior images) registered within lungs and whole liver, respectively:

$$ \mathrm{LSF}\left(\%\right)=100\bullet \frac{C_{\mathrm{lung}}}{C_{\mathrm{lung}}+{C}_{\mathrm{WL}}} $$

(1)

Planar images may not be used to determine accurately the tumor to non-tumor liver activity concentration ratio (TNR) [14]; therefore, attenuation-corrected SPECT images were used instead. TNR was calculated by Eq. (2), where *C*_{TL} and *C*_{NLt} are the total counts registered within TL and NL_{t} volumes respectively:

$$ \mathrm{TNR}=\frac{C_{\mathrm{TL}}/{V}_{\mathrm{TL}}}{{\mathrm{C}}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}/{\mathrm{V}}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}} $$

(2)

### Contouring

The first step for this retrospective investigation was the anatomic VOIs segmentation. These VOIs were contoured on the CT from ^{99m}Tc-MAA SPECT-CT with the aid of a rigidly registered diagnostic scan (contrast enhanced CT or MRI) using a commercial treatment planning software (Pinnacle, Philips Medical System, Anover, MA). A process similar to the one used in external beam radiation therapy was followed. The VOIs were then exported as DICOM-RT structure sets. To avoid inter-operator bias, all VOIs were delineated by a single physician.

For each patient, individual tumors (T_{i}), the planning target volume (PTV), and the whole liver (WL) were delineated. The PTV refers to the portion of the liver in which it is intended to deliver the radiation dose: one or more segments, one lobe or the whole liver depending on whether the treatment is segmental, lobar, or total. Tumoral liver volume (TL), corresponding to the aggregated tumor volume was generated by summing all the T_{i} volumes. Target normal liver volume (NL_{t}) was defined by subtracting the TL volume from the PTV volume. Whole normal liver (NL_{w}) was also determined by subtracting the TL volume from the WL. Volumes in mL for individual tumors, aggregated tumoral liver, target normal liver, and whole normal liver (V_{Ti}, V_{TL}, \( {V}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}} \), and \( {V}_{\mathrm{N}{\mathrm{L}}_{\mathrm{w}}} \)) were calculated for the 14 patients.

### Dosimetry assessment

For the purposes of this study, the mean absorbed dose delivered to each compartment (\( {D}_{\mathrm{mean}}^{{\mathrm{T}}_{\mathrm{i}}},{D}_{\mathrm{mean}}^{\mathrm{T}\mathrm{L}},{D}_{\mathrm{mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}} \), and\( {\mathrm{D}}_{\mathrm{mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{w}}} \)) was estimated according to MIRD formalism [11]. Both multi-compartment dosimetry methods were retrospectively applied to obtain *D*_{Mean}. To implement MTPM, in patients with two individual tumors or more, an Excel-based mean absorbed dose calculator was developed (available in additional file 1).

Additionally, DPK and LDM were applied to calculate a 3D dose map and DVHs. The actual ^{90}Y administered activity and volumes of the contoured VOIs used to determine the absorbed doses were the same for all dosimetry approaches (PM, MTPM, and both 3D-VDM).

For the ^{90}Y dosimetry calculation purposes, an identical ^{99m}Tc-MAA and ^{90}Y-microspheres’ biodistributions were assumed, based on previous studies [10, 15,16,17,18,19].

#### Multicompatimental methods

PM was applied to calculate \( {D}_{\mathrm{Mean}}^{\mathrm{TL}} \) and \( {D}_{\mathrm{Mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}} \) according to Eqs. (3) and (4), where A(^{90}Y) is the ^{90}Y-microspheres administered activity, and M_{TL} and \( {M}_{N{L}_t} \)are the masses in kg of the tumoral liver and the target normal liver, respectively. A 1 g/mL tissue density is assumed, and volumes in liters are straight converted in masses in kg.

\( {D}_{\mathrm{Mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{w}}} \) was determined by rescaling the \( {D}_{\mathrm{Mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}} \) to the NL_{w}volume, applying (5).

$$ {D}_{\mathrm{Mean}}^{\mathrm{TL}}(Gy)=\frac{49.67\ \left(\frac{J}{GBq}\right)\bullet A\Big({}^{90}Y\Big)(GBq)\left(1-\raisebox{1ex}{$ LSF$}\!\left/ \!\raisebox{-1ex}{$100$}\right.\right)\bullet \mathrm{TNR}}{M_{\mathrm{TL}}(kg)\bullet \mathrm{TNR}+{M}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}(kg)} $$

(3)

$$ {D}_{\mathrm{Mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}(Gy)=\frac{49.67\left(\frac{J}{GBq}\right)\bullet A\Big({}^{90}Y\Big)(GBq)\left(1-\raisebox{1ex}{$ LSF$}\!\left/ \!\raisebox{-1ex}{$100$}\right.\right)}{M_{\mathrm{TL}}(kg)\bullet \mathrm{TNR}+{M}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}(kg)} $$

(4)

$$ {D}_{\mathrm{Mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{w}}}(Gy)=\frac{D_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}(Gy)\bullet {M}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}(kg)}{M_{\mathrm{N}{\mathrm{L}}_{\mathrm{w}}}(kg)} $$

(5)

In patients with two or more tumors (*n*), the MTPM method, an (*n* + 2) compartment partition model, was applied to determine \( {D}_{\mathrm{Mean}}^{{\mathrm{T}}_{\mathrm{i}}} \)using Eq. (6), where TNR_{i} is the tumor to normal liver activity concentration ratio for individual tumors calculated by Eq. (7).

$$ {D}_{\mathrm{M}\mathrm{ean}}^{{\mathrm{T}}_{\mathrm{i}}}(Gy)=\frac{49.67\ \left(\frac{J}{GBq}\right)\bullet A\Big({}^{90}Y\Big)(GBq)\left(1-\raisebox{1ex}{$ LSF$}\!\left/ \!\raisebox{-1ex}{$100$}\right.\right)\bullet TN{R}_i}{{\mathrm{M}}_{\mathrm{T}\mathrm{L}}(Kg)\bullet \mathrm{TNR}+{M}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}(Kg)} $$

(6)

$$ \mathrm{TN}{\mathrm{R}}_i=\frac{C_{{\mathrm{T}}_{\mathrm{i}}}/{V}_{{\mathrm{T}}_{\mathrm{i}}}}{C_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}/{V}_{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}}} $$

(7)

#### 3D-voxel dosimetry

The first step to perform 3D image-based dosimetry using ^{99m}Tc-MAA SPECT is to convert, through a calibration factor, the counts registered in each voxel of the reconstructed image to ^{90}Y activity (in MBq). Since ^{99m}Tc-MAA administered activity (A(^{99m}Tc)) is totally uptaken in the liver with the exception of the fraction that shunts to the lung, the patient-specific calibration factor may be determined as it was described by Chiesa et al. [20].

The ^{90}Y-microspheres activity in a liver voxel at the image acquisition time (A_{voxel}(^{90}Y)) is directly proportional to the total counts registered within a voxel of ^{99m}Tc-MAA SPECT image (*C*_{voxel}(^{99m}Tc)). Thus, A_{voxel}(^{90}Y) may be estimated by means of Eq. (8) where *C*_{WL} is the total counts of ^{99m}Tc registered within the WL volume.

$$ {A}_{\mathrm{voxel}}\left({}^{90}Y\right)={C}_{\mathrm{voxel}}\left({}^{99m} Tc\Big)\frac{A\ \left({}^{99m} Tc\right)\left(1-\frac{LSF}{100}\right)}{C_{WL}\left({}^{99m} Tc\right)}\frac{A\Big({}^{90}Y\Big)}{A\Big({}^{99m} Tc\Big)}={C}_{\mathrm{voxel}}\right({}^{99m} Tc\Big)\frac{\left(1- LSF/100\right)A\Big({}^{90}Y\Big)}{C_{WL}\left({}^{99m} Tc\right)} $$

(8)

Unlike other internal radionuclide therapy, RE has the advantage of negligible biological clearance following the infusion. Thus, assuming the permanent trapping of microspheres, fitting of time-activity curves is not required, and the total number of disintegrations in a voxel (Ã _{voxel}(^{9o}Y)) was calculated as described by Eq. (9), where *T*_{1/2}(^{90}*Y*) is the physical ^{90}Y half-life (64.2 h).

$$ {\overset{\sim }{A}}_{\mathrm{voxel}}\left({}^{90}Y\right)=\int {A}_{\mathrm{voxel}}\left({}^{90}Y\right)\bullet {e}^{\left(\raisebox{1ex}{$- Ln(2)t$}\!\left/ \!\raisebox{-1ex}{${T}_{\frac{1}{2}}$}\right.\right)}\bullet dt=1.443\bullet {T}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left({}^{90}Y\right).\kern0.5em {A}_{\mathrm{voxel}}\left({}^{90}Y\right) $$

(9)

To convert the cumulative activity in each voxel to a tridimensional ^{90}Y absorbed dose map, two different 3D-VDM approaches were applied: LDM and DPK. For that purpose, a software tool based in MATLAB v.R2016a (The Math Works, Natick, MA) code was developed.

DPK takes into account the high-energy beta particles transport to adjacent voxels. The absorbed dose within the target voxel t \( \left({D}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}}\left({}^{90}Y\right)\right) \) was calculated by the convolution of the 3D cumulative activity matrix with a cubic dose kernel, as described in Eq. (10). Where \( {\overset{\sim }{A}}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}}\left({}^{90}Y\right) \) is the time-integrated activity within the source voxel *s,* and *S*(voxel_{t} ← voxel_{s}) is the well-known *S* value defined as the absorbed dose to the target voxel *t* per unit of cumulative activity in the voxel *s*. The dose kernels used in this work were extracted from Lanconelli database [21].

$$ {D}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}}\left({}^{90}Y\right)=\sum \limits_{s=0}^N{\overset{\sim }{A}}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}}\left({}^{90}Y\right)\otimes S\left(\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}\leftarrow \mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}\right) $$

(10)

LDM assumes that the kinetic energy from each beta particle is deposited within the voxel where the emission occurs. The source voxel *s* in this case is also the target voxel *t*. The absorbed dose in each voxel was then determined by Eq. (11), multiplying the cumulative activity within the voxel by a constant scalar factor, which is the *S* value considering an absorbed fraction equal to 1 in each voxel (\( {\left.S\left(\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}\leftarrow \mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}\right)\right|}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}=\mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}}\Big) \). *S* is calculated by means of Eq. (12), where \( {\left\langle {E}_{\beta}\left({}^{90}Y\right)\right\rangle}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}}=\left(\frac{0.9267\ MeV}{\mathrm{disintegration}}\right)\bullet \left(\frac{1.6022\bullet {10}^{-13}J}{MeV}\right)\bullet \left(\frac{Gy\bullet Kg}{J}\right)\bullet \left(\frac{10^9\mathrm{disintegrations}}{s\bullet GBq}\right) \) is the deposited β-energy per disintegration in average, and \( {M}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}} \)is the target voxel mass. For a given cubic voxel size (4.48 mm side), *S* is 1.603 Gy/GBq.s.

$$ {\left.{D}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}}\left({}^{90}Y\right)={\overset{\sim }{A}}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}}\left({}^{90}Y\right)\times S\left(\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}\leftarrow \mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}\right)\right|}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}=\mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}} $$

(11)

$$ {\left.S\left(\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}\leftarrow \mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}\right)\right|}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}=\mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}}={\left.\frac{{\left\langle {\mathrm{E}}_{\upbeta}\left({}^{90}\mathrm{Y}\right)\right\rangle}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}}}{M_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}}}\ \right|}_{\mathrm{voxe}{\mathrm{l}}_{\mathrm{t}}=\mathrm{voxe}{\mathrm{l}}_{\mathrm{s}}} $$

(12)

### Dosimetry comparisons and statistical analysis

\( {D}_{\mathrm{mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{t}}} \) and \( {D}_{\mathrm{mean}}^{\mathrm{N}{\mathrm{L}}_{\mathrm{w}}} \)calculated by PM, LDM, and DPK methods were compared using a paired Student’s *t* test or Wilcoxon test in case differences between methods do not meet normal criteria.

\( {D}_{\mathrm{mean}}^{{\mathrm{T}}_{\mathrm{i}}} \) and \( {D}_{\mathrm{mean}}^{\mathrm{TL}} \)calculated by PM, MTPM, and both 3D-VDM were also compared using a paired Student’s *t* test or Wilcoxon test, as corresponds. For MTPM, LDM, and DPK, \( {D}_{\mathrm{mean}}^{\mathrm{TL}} \) was calculated for each patient as the average of all \( {D}_{\mathrm{mean}}^{\mathrm{Ti}} \). The standard deviation (SD) was also determined. \( {D}_{\mathrm{mean}}^{\mathrm{Ti}} \) calculated by PM was the same for all individual tumors of the same patient, and equal to \( {D}_{\mathrm{mean}}^{\mathrm{TL}} \), as tumoral liver compartment in PM approach is defined as an aggregated tumor including all T_{i}.

The heterogeneity of ^{90}Y-microspheres distribution among the tumors for each patient was evaluated through the TNR_{i} coefficient of variation (COV(TNR_{i})).

A comparison among the studied dosimetry methods for all VOIs was performed in terms of mean absorbed dose differences (\( \Delta {D}_{\mathrm{mean}}^{\mathrm{VOI}} \)) in Gy.

The correlation between differences in *D*_{mean} between PM and the other studied dosimetry methods (MTPM, LDM, and DPK) and TNR-TNR_{i} differences was evaluated by means of the Spearman’s correlation coefficient (rho).

Dosimetry comparison between DPK and LDM methods was also managed in terms of DVHs. Some metrics were extracted from the DVHs: the minimum dose to 5%, 25%, 50%, 70%, and 95% in the corresponding VOI (*D*_{5}, *D*_{25}, *D*_{50}, *D*_{70}, and *D*_{95}, respectively), the percentage of the tumor volume receiving at least 100 Gy (V_{100}) and the percentage of the NL_{w} and NL_{t} volumes receiving at least 20 Gy (V_{20}). A paired Student’s *t* test or Wilcoxon test, as appropriate, was applied. Absorbed dose differences in Gy were also determined for each VOI.

Bland-Altman analysis was used to evaluate the agreement among the studied dosimetry methods (PM, MTPM, LDM, and DPK), in terms of *D*_{mean}, for both tumoral and non-tumoral volumes (NL_{t}, NL_{w}, TL, and T_{i}). The agreement of DVH between both 3D-VDM methods was also evaluated by means of a Bland-Altman analysis. Pearson correlation (*ρ*) and Lin concordance (ρ_{c}) coefficients were reported.

All analyses were performed with statistical STATA v.15 software (StataCorp, TX, USA). A *p* value of 0.05 or less was considered statistically significant.

Differences between LDM and DPK methods were also assessed by a voxel by voxel analysis. A voxel based subtraction of the parametric images (in Gy) calculated by both methods was performed, and the calculation of the normalized mean square error (NMSE) between dose absorbed maps obtained applying Eq. (13), as described previously by Pacilio et al. [22], where *x*_{i} is the *i*th voxel of the DPK image and *p*_{i} the *i*th voxel of the LDM image (used as a reference).

$$ NMSE=100\bullet \frac{\sum \limits_i{\left({x}_i-{p}_i\right)}^2}{\sum \limits_i{p}_i^2} $$

(13)