### Study population and image acquisition

The study population consisted of 44 patients (19 female) referred to standard stress/rest 1-day myocardial perfusion SPECT study. Their (mean ± standard deviation) characteristics were as follows: age, 68 ± 10 years; height, 169 ± 10 cm; weight, 78 ± 13 kg; body mass index, 27.4 ± 4.1 kg/m^{2}. Each patient received 300 MBq of Tc-99m tetrofosmin before the stress imaging, followed by 705 ± 12 MBq before the rest imaging 3 h later according to our institutional guidelines. To ensure clinical workflow, data were collected during the rest phase only for this study. Written informed consent was obtained from all patients and the study was approved by the Research Ethics Committee of the Northern Savo Hospital District (Dno 90/2011; March 20, 2012).

The patients were imaged in a supine position with a dual-detector SPECT/CT system (Precedence; Koninklijke Philips N.V., Amsterdam, Netherlands). The following SPECT list-mode image acquisition protocol was used: 90° detector configuration, low-energy high-resolution collimators, noncircular detector orbit, 64 projection angles from the right anterior oblique to the left posterior oblique, acquisition time of 30 s per projection angle, energy window of 140 keV ± 10%, and ECG gating acceptance window of ±20%. R triggers for ECG gating were generated using Cardiac Trigger Monitor 3000 unit (Ivy Biomedical Systems, Inc., Branford, CT, USA). For off-line list-mode data processing, we also recorded a signal-form ECG with a data acquisition system (MP150 and ECG100C; BIOPAC Systems, Inc., Goleta, CA, USA).

The list-mode data were binned into ECG-gated projection images using custom-made MATLAB scripts (MATLAB R2015b; The MathWorks, Inc., MA, USA). Matrix size of 96×96 and pixel size of 6.22 mm were used. ECG gating was realized as fixed forward gating, dividing R–R intervals into eight ECG bins; the length of each bin was calculated as one-eighth of the average duration of those R–R intervals that were accepted in the ECG gating.

### Reconstructions

Reconstructions were carried out in MATLAB environment. Rotation-based reconstruction approach was adopted and the rotation matrices were computed using Gaussian interpolation [7]. In forward and backward projections, the CDR was modeled as a distance-dependent Gaussian function [8]. For both OSEM and TOSEM reconstructions, ten iterations and eight subsets were used, as suggested in previous studies [9, 10]. Reconstructed transaxial images were rotated into short-axis images, smoothed with a three-dimensional Gaussian filter with a standard deviation of 1 voxel [11] and masked to eliminate extracardiac activity.

The general OSEM/TOSEM iterative algorithm is presented in matrix format in Algorithm 1. In Algorithm 1, *f̂*^{(n)} denotes the image vector at update number *n*, *H* denotes the transition matrix, *g* denotes the measured projection image vector, *1* denotes a vector of ones, *T* denotes matrix transpose, *S*_{n} denotes the *n*th ordered subset, *N* denotes the number of subsets and *M* denotes the number of iterations. The notation “*ℑ*∈*S*_{n}” indicates that only the matrix/vector rows that belong to subset *S*_{n} are used in the calculation. See [12].

In our case, the transition matrix is formed of matrix blocks *H*^{l}, *l* = 1,…,*L*, where *L* is the number of projection angles. That is,

$$ \boldsymbol{H}={\left[{\left[{\boldsymbol{H}}^1\right]}^T,\dots, {\left[{\boldsymbol{H}}^L\right]}^T\right]}^T. $$

In the standard OSEM algorithm, an individual matrix block is

$$ {\boldsymbol{H}}^l={\boldsymbol{P}}^l{\boldsymbol{R}}^l, $$

where *P*^{l} and *R*^{l} are the forward projection matrix (including CDR modeling) and rotation matrix at projection angle *l*, respectively. In the TOSEM algorithm, the only difference compared to OSEM is that we add to each transition matrix block a term *τ*^{l}, which is the duration of projection angle *l* [2], to obtain

$$ {\boldsymbol{H}}^l={\tau}^l{\boldsymbol{P}}^l{\boldsymbol{R}}^l. $$

Reconstructing each ECG bin individually with TOSEM results in a series of images where the total activity in the image remains constant from one ECG bin to another. This is not necessarily the case with the standard OSEM reconstruction due to the lack of duration factors *τ*^{l} in the transition matrix, as shown in Figure 1.

### Phase analysis

Phase analysis was performed with Quantitative Gated SPECT (QGS) 2012 program (Cedars-Sinai Medical Center, Los Angeles, CA, USA). In phase analysis, QGS samples the LV myocardial activity from up to 1008 sampling points [13] from all ECG bins to form time–activity curves (TACs) for each sampling point. These TACs are further interpolated by fitting first Fourier harmonic (FFH) function to the time–activity data. The amplitude and phase angle of each FFH are recorded, and 5% of the data that correspond to the lowest amplitudes are discarded [14]. The remaining phase angles are used to build a 360-bin histogram, which is characterized by computing the histogram bandwidth (BW), phase angle standard deviation (StD) and entropy (ENT) [14].

### Specific studies

In the first part of the study, we assessed the association between the HRV and the data shortage in the last ECG bin. To do this, we computed the standard deviation SD_{R-R} of those R–R intervals that were accepted in the ECG gating. This parameter was further normalized by dividing with the average of the accepted R–R intervals. The parameter describing the data shortage in the last ECG bin, the activity ratio (AR), was computed by summing the voxel values of OSEM-reconstructed images at each ECG bin to provide a value for total activity and calculating the ratio of the total activity in the last ECG bin and the average total activity in the first five ECG bins [6]. The association between SD_{R-R} and AR was evaluated by computing their Pearson correlation coefficient (*r*).

In the second part of the study, we assessed the robustness of phase analysis results when AR was artificially reduced. Having computed the AR for all 44 patients, we selected those 14 patients whose AR was larger than 90% [6]. For these patients, we rebinned the list-mode data such that a part of the last ECG bin data was discarded uniformly from all projection angles to yield AR values of 90%, 80%, 70%, 60% and 50%. These data were reconstructed with both OSEM and TOSEM algorithms. Phase analysis was performed on all reconstructed images.

Statistical analysis was performed in SPSS Statistics v.23 (IBM Corporation, NY, USA). Sphericity of phase analysis data was assessed with Mauchly’s test. Repeated measures analysis of variance (ANOVA) with post hoc pairwise multiple comparison tests (Bonferroni corrected) was performed to assess whether there were significant (*p* < 0.05) differences between images of different AR values. Reliability was assessed by computing Lin’s concordance correlation coefficient (CCC) between the 90% data and the lower AR values. The CCC values were computed in MATLAB environment as [15]

$$ CCC=r\bullet {C}_b=\frac{\sigma_{xy}}{\sigma_x{\sigma}_y}\bullet \frac{2{\sigma}_x{\sigma}_y}{\sigma_x^2+{\sigma}_y^2+{\left({\mu}_x-{\mu}_y\right)}^2}, $$

where *r* is the Pearson’s correlation coefficient, *C*_{b} is the bias correction factor, *σ*_{xy} is the covariance of data vectors *x* and *y*, *σ*_{x} and *σ*_{y} are the standard deviations of data vectors *x* and *y*, respectively, and *μ*_{x} and *μ*_{y} are the means of data vectors *x* and *y*, respectively.

In the third and final part of the study, we assessed how much phase analysis results change with respect to AR in the whole study population when the images are reconstructed with TOSEM instead of OSEM. Data from all 44 patients were reconstructed with both OSEM and TOSEM algorithms and subjected to phase analysis. Pearson correlation coefficients were computed between AR and the changes of phase analysis parameters (∆BW, ∆StD and ∆ENT).