Open Access

Quantification of myocardial blood flow with 82Rb: Validation with 15O-water using time-of-flight and point-spread-function modeling

EJNMMI Research20166:68

DOI: 10.1186/s13550-016-0215-6

Received: 11 March 2016

Accepted: 30 June 2016

Published: 1 August 2016



We quantified myocardial blood flow with 82Rb PET using parameters of the generalized Renkin-Crone model estimated from 82Rb and 15O-water images reconstructed with time-of-flight and point spread function modeling. Previous estimates of rubidium extraction have used older-generation scanners without time-of-flight or point spread function modeling. We validated image-derived input functions with continuously collected arterial samples.


Nine healthy subjects were scanned at rest and under pharmacological stress on the Siemens Biograph mCT with 82Rb and 15O-water PET, undergoing arterial blood sampling with each scan. Image-derived input functions were estimated from the left ventricle cavity and corrected with tracer-specific population-based scale factors determined from arterial data. Kinetic parametric images were generated from the dynamic PET images by fitting the one-tissue compartment model to each voxel’s time activity curve. Mean myocardial blood flow was determined from each subject’s 15O-water k 2 images. The parameters of the generalized Renkin-Crone model were estimated from these water-based flows and mean myocardial 82Rb K 1 estimates.


Image-derived input functions showed improved agreement with arterial measurements after a scale correction. The Renkin-Crone model fit (a = 0.77, b = 0.39) was similar to those previously published, though b was lower.


We have presented parameter estimates for the generalized Renkin-Crone model of extraction for 82Rb PET using human 82Rb and 15O-water PET from high-resolution images using a state-of-the-art time-of-flight-capable scanner. These results provide a state-of-the-art methodology for myocardial blood flow measurement with 82Rb PET.


Myocardial blood flow Rubidium-82 PET Image-derived input function TOF PET


Cardiac perfusion PET with 82Rb is clinically useful for diagnosing coronary artery disease [14]. Quantification of myocardial blood flow (MBF) and coronary flow reserve (CFR) can be obtained from 82Rb PET but relies on accurately modeling the extraction fraction of rubidium by myocardial tissue, which is nonlinearly related to flow as described by the generalized Renkin-Crone model [58]. Uncertainty in the extraction model parameters causes much of the uncertainty in MBF [9].

Several groups have reported generalized Renkin-Crone model parameters for rubidium using canine or human MBF data from microspheres [10], 13N-ammonia [5, 11], and 15O-water [12, 13]. Most of these studies used older-generation PET systems with 2D or reduced-dose 3D acquisitions. To our knowledge, no extraction fraction estimations have been made using scanners with time-of-flight (TOF) capabilities; for 82Rb PET, such systems provide better signal-to-noise ratios than non-TOF systems [14, 15] and parametric images with lower standard error. Further, when point spread function (PSF) modeling is included in reconstruction, MBF estimates from 82Rb PET may be higher [16]; however, such calculations were performed using an extraction model [11] derived from non-TOF, non-PSF images. Presotto et al. [17] demonstrated the quantitative superiority of PSF + TOF for dynamic cardiac reconstructions using a thorax/heart phantom filled with either 18F alone or 18F and 13N (to simulate dynamically varying contrast), in both static and moving configurations.

Errors in the input function, another critical component of kinetic modeling, can substantially bias kinetic parameter estimates [18]. For practical reasons, image-derived input functions (IDIFs) are widely used in cardiac PET. IDIFs estimated from blood pool regions of cardiac images have been validated against the gold standard of arterial samples in dogs [5, 19] but not recently in humans with 82Rb PET.

A recent study of five extraction model fits and three IDIF estimation methods demonstrated that these choices substantially influence MBF estimates [20]. In this work, we reexamined extraction fraction estimation in humans with paired rest and stress studies with 82Rb and 15O-water acquired on a state-of-the-art system, the Siemens Biograph mCT, and reconstructed images with TOF and PSF modeling. We augmented this high-quality image data with continuously sampled arterial measurements for input function validation. From these data, we provided new parameter estimates of the generalized Renkin-Crone model for rubidium extraction.



Nine healthy subjects (five male) with no known cardiac abnormalities were studied. This study was approved by the Yale University Human Investigation Committee; all subjects signed an informed consent form. The average age was 28.4 ± 6.2 years, and average BMI was 24.7 ± 3.9 kg/m2. Subjects abstained from caffeine for 12 h pre-imaging, and from food for 4–6 h. Before scanning, an intravenous line for tracer administration and an arterial line for blood sampling were placed.

Data acquisition

PET scans were acquired on the Biograph mCT 1104 (Siemens Healthcare, Knoxville, TN) at rest and under pharmacological stress with 82Rb and 15O-water for each subject. For seven subjects, the scan sequence was: 82Rb rest; 15O-water rest; 15O-water stress; and 82Rb stress. For the remaining subjects, the 82Rb stress acquisition was performed directly after the 82Rb rest acquisition. A 1-h interval separated each stress scan from the following acquisition, with confirmation that heart rate and blood pressure had returned to baseline. Low-dose CT attenuation correction scans were acquired before each rest scan and after each stress scan. Pharmacological stress was induced with 0.4 mg of regadenoson, injected over 30 s, 1 min before tracer injection. The 82Rb injections were performed with the CardioGen-82 (Bracco Diagnostics, Princeton, NJ) system with an infusion rate of 50 mL/min, duration of 18 ± 4 s, and mean ± SD dose of 663 ± 82 MBq. 15O-water infusions with mean dose 690 ± 136 MBq were delivered over 20 s. Dosing was independent of body weight.

Arterial blood sampling and data analysis

Arterial blood was drawn from the radial artery for 7 min per scan at 4 mL per minute for seven of nine subjects and radioactivity measured with a cross-calibrated radioactivity monitor (PBS-101, Veenstra Instruments, Joure, The Netherlands). One subject chose not to have the arterial line. In another subject, the arterial line was not successfully placed. Because IDIFs were corrected with population-based scale factors, these subjects’ image data were not excluded. Because the 1.25-mL infusion line for 82Rb was not flushed, residual activity remained at end-of-elution. This unshielded activity contributed to the background signal detected by the radioactivity monitor, visible in the initial portion of the 82Rb arterial readings before the input function peak (when measurements should be 0). To model this background signal, a decaying exponential with the 82Rb decay constant was fit to the raw count data for each acquisition, between end-of-elution and the rise of the input function peak. This fitted curve was subtracted from the arterial measurements (see Additional file 1, page 1). Apart from removal of the background signal, 82Rb and 15O-water data were processed analogously. Corrections were applied for sensitivity, decay, and external dispersion. Sensitivity was measured by cross-calibration with phantom measurements per isotope. To correct for time delay between the left ventricle (LV) and the arterial sampling site, each acquisition’s time shift was estimated by maximizing the correlation between the corrected arterial samples and the LV time activity curve (TAC).

Image reconstruction

For each injection, list-mode data were acquired for 4 min post-injection and reconstructed into 32 frames (20x3s,6x10s,6x20s) with mCT software using TOF and PSF modeling, and, for 82Rb, prompt-gamma correction (OSEM with 2 iterations of 21 subsets, voxel size 2.036x2.036x2.0 mm3). Images were post-smoothed with a 3 mm-FWHM Gaussian kernel. Summed dynamic PET images were inspected for alignment with the corresponding attenuation correction CT images, and manually realigned and re-reconstructed as necessary. Images were transformed to short-axis orientation.

Kinetic modeling

The one-tissue compartment model
$$ {C}_{\mathrm{T}}(t)={K}_1{e}^{-{k}_2t}\otimes {C}_{\mathrm{A}}(t) $$
was used to describe the kinetics in the myocardium tissue, where K 1 is the myocardial influx rate constant, k 2 is the efflux rate constant, and C T(t) is the tissue TAC [11, 21, 22]. Tissue TACs were converted from Bq/mL to Bq/g with an assumed tissue density of 1.05 g/mL. The arterial input function C A(t) was estimated from either the arterial samples or the images (described below).
Partial volume, motion effects, and arterial blood volume were accounted for with one or two additional parameters: V A, for the LV cavity spillover and arterial blood volume, and, optionally, V RV, for the right ventricle (RV) cavity spillover:
$$ {C}_{\mathrm{PET},\mathrm{m}\mathrm{y}\mathrm{o}}(t)=\left(1-{V}_{\mathrm{A}}-{V}_{\mathrm{RV}}\right){C}_{\mathrm{T}}(t)+{V}_{\mathrm{A}}{C}_{\mathrm{A}}(t)+{V}_{\mathrm{RV}}{C}_{\mathrm{RV}}(t) $$
Equation 2 was fit to each voxel TAC using the basis function method [23] and a weighted least squares (WLS) criterion, with weights based on noise equivalent counts. 15O-water K 1 images were registered to the corresponding 82Rb K 1 images for each subject/condition using rigid 3D versor transforms optimized by the Mattes mutual information metric. The k 2, V A, and V RV images were realigned using the same transforms. Left ventricular myocardium volumes-of-interest (VOIs) with approximate thickness of 4 mm were automatically determined for each subject’s rest and stress scans from K 1 images, using in-house software (Fig. 1a). Model fits were performed both as three-parameter (without V RV) and four-parameter fits.
Fig. 1

a Example myocardium volume of interest (VOI), overlaid on a K 1 image in horizontal long axis (HLA) and short axis (SA). b Example VOI for estimating image-derived right ventricle (blue) and left ventricle (red) TACs, overlaid on a composite image of three early 82Rb frames

Input function estimation

IDIFs were estimated from fixed-volume (6.5 mL) cylindrical VOIs manually placed towards the base of the LV and atrium blood pools of each image (Fig. 1b). The resulting TACs were compared to the measured arterial input functions (AIFs) with regard to peak concentration, tail concentration, and area under the curve (AUC). For comparisons, AIFs were resampled to the image times by averaging values within each frame. Peaks were computed as the maximal activity of each TAC. Tail activity was computed as the average concentration over 1 min starting at 2 min, 40 s post-injection. Percent difference in each metric was computed for corresponding pairs of IDIFs and AIFs and averaged across subjects.

With a sufficiently small LV VOI, activity is often assumed to be fully recovered [1012]; alternatively, IDIFs are sometimes corrected for partial-volume effects. For instance, [13] assumed the LV cavity TAC is a partial-volume mixture of 85 % arterial blood and 15 % myocardial tissue. We investigated partial-volume correction (PVC) methods using the AIF as the gold standard.

To assess IDIF PVC, the “true” tissue TAC C T was estimated from a mean global myocardium TAC using the AIF in Eq. 1 and 2 (omitting the V RV term). First, a one-parameter model was investigated, where the LV recovery coefficient β was estimated from the LV TAC C PET,LV and AIF:
$$ {C}_{\mathrm{PET},\mathrm{L}\mathrm{V}}(t)=\beta {C}_{\mathrm{A}}(t)+\left(1-\beta \right){C}_{\mathrm{T}}(t). $$
An alternative two-parameter model did not constrain the sum of the coefficients to 1:
$$ {C}_{\mathrm{PET},\mathrm{L}\mathrm{V}}(t)={\beta}_1{C}_{\mathrm{A}}(t)+{\beta}_2{C}_{\mathrm{T}}(t). $$

The sum β1 + β2 might be <1 if partial-volume mixing occurs with signal outside the heart (e.g., the lung).

To perform scale correction, the parameter β s is estimated from:
$$ {C}_{\mathrm{PET},\mathrm{L}\mathrm{V}}(t)={\beta}_{\mathrm{S}}{C}_{\mathrm{A}}(t). $$

This correction was used by [5], with tracer-independent β s ≈ 0.90 estimated from canine 82Rb and 13N-ammonia PET and well counter measurements of arterial samples.

An alternative scale correction β AUC was estimated as the ratio of the IDIF AUC to the AIF AUC:
$$ {\displaystyle {\int}_0^T}{C}_{\mathrm{PET},\mathrm{L}\mathrm{V}}(t)\mathrm{d}\mathrm{t}={\beta}_{\mathrm{AUC}}{\displaystyle {\int}_0^T}{C}_A(t)\mathrm{d}\mathrm{t} $$

where T is the duration of the dynamic acquisition. While individually estimated β AUC cannot outperform β s in terms of weighted sum-of-squared residuals (WSS) (β s minimizes WSS by design), a population-based β AUC might give better kinetic parameter concordance.

Parameters for the one-parameter PVC, two-parameter PVC, and scaling models (Eq. 3, 4, and 5, respectively) were estimated via WLS for each acquisition. Model fits were compared by F tests. The 82Rb K 1 estimates and 15O-water k 2 estimates from AIFs were compared to those from β AUC scale-corrected IDIFs by linear Deming regression, which models error in both variables, and by the Lin concordance coefficient [24], which provides a measure of absolute agreement between two estimates.

MBF estimation

Using myocardial VOIs (Fig. 1a), MBF was estimated from the mean myocardial 15O-water k 2 values, corrected with a partition coefficient of p = 0.91 mL/g (MBF = k 2 p) [25]. Finally, the parameters a and b of the generalized Renkin-Crone model [5, 7, 8]
$$ {K}_1=\mathrm{M}\mathrm{B}\mathrm{F}\cdot \left(1-a{e}^{-b/\mathrm{M}\mathrm{B}\mathrm{F}}\right) $$
were estimated from the mean myocardial 82Rb K 1 values and 15O-water MBF. In this model, b reflects the basal permeability-surface area (PS) product and a accounts for MBF-dependent PS changes. Fits used weighted orthogonal distance regression [26] to account for errors in both variables, with weights set to the reciprocals of the variance of voxel values in the myocardium VOIs. There were two datapoints per subject (rest and stress). Renkin-Crone parameters were independently estimated from mean kinetic parameters using (1) uncorrected IDIFs, (2) scale-corrected IDIFs, and (3) AIFs. Analyses were also performed without the V RV term and separately for the lateral and septal walls.



Figure 2 compares rate-pressure products (RPPs) for each pair of 82Rb and 15O-water scans. Mean (±standard deviation) absolute percent difference between 82Rb and 15O-water RPP was 12 ± 9 % at rest and 11 ± 9 % under pharmacological stress. Two subjects had RPPs with greater than 20 % difference between 82Rb and 15O-water scans at rest, and two subjects had RPPs with greater than 20 % difference between 82Rb and 15O-water scans under pharmacological stress. No significant group differences between 82Rb and 15O-water RPPs were found for rest (p = 0.40) or stress (p = 0.44).
Fig. 2

Rate-pressure products (RPP) for 82Rb scans versus 15O-water scans. The dashed line is the identity line. Rest RPP was measured immediately before scan start. Stress RPP is averaged over 4 min post-injection

IDIF validation

Table 1 gives means and standard deviations of the percent differences (across seven subjects) in peak activity, tail activity, and AUC of uncorrected IDIFs with respect to AIFs. The peaks of the 82Rb IDIFs were underestimated by 13 and 19 % (rest, stress) compared to the AIFs, with smaller bias in AUC. An ideal IDIF might be expected to have a higher peak value, since no correction for internal-body dispersion was applied to the AIFs. Mean percent differences for all 15O-water metrics were within ±10 %. These results suggest that IDIF correction could be beneficial for 82Rb kinetic modeling, due to poorer resolution from larger positron range and higher myocardium-to-blood-pool contrast.
Table 1

Comparison of uncorrected IDIFs to AIFs





% difference mean ± SD

% difference mean ± SD

% difference mean ± SD



−11 ± 12

−13 ± 9.2

−7.1 ± 18


−6.0 ± 13

−19 ± 18

2.6 ± 23



−5.1 ± 10

−8.4 ± 14

−5.4 ± 9.0


−1.2 ± 9.9

2.4 ± 18

−3.5 ± 7.8

SD standard deviation, AIF arterial sample-based input function, IDIF image-derived input function, AUC area under curve, % difference 100 × (IDIF − AIF)/AIF

The mean and standard deviation of the parameter β from the one-parameter PVC model (Eq. 3) was 0.87 ± 0.09 (0.86 ± 0.12) for 82Rb (15O-water), which are similar to the 0.85 assumed by [13]. However, the tails of 82Rb IDIFs corrected by this method were consistently underestimated (−54 ± 29 %) compared to the AIFs; neither the two-parameter PVC model (Eq. 4) nor scaling model (Eq. 5) demonstrated this deficiency (Fig. 3a). For 15O-water IDIFs, there was no apparent qualitative difference among correction methods (Fig. 3b).
Fig. 3

Arterial input functions (AIFs) and image-derived input functions (IDIFs) corrected using parameters estimated for these scans from a typical a 82Rb scan and b 15O-water scan. In both cases, weighted-least-squares (WLS)-based scaling (green) and area-under-the-curve (AUC)-based scaling corrections are nearly identical. The two-parameter (partial-volume-corrected) PVC IDIFs were omitted from these plots, as they were virtually identical to the scale-corrected IDIFs (green, orange curves)

Using the two-parameter PVC method, the mean and standard deviation of β 1 + β 2 (Eq. 4) were 0.85 ± 0.10 (0.98 ± 0.10) for 82Rb (15O-water). This indicates that while a partial-volume mixture model of arterial blood and myocardium tissue may be sufficient for water, rubidium could require a different model of recovery-diminishing effects.

The mean and standard deviation of the scaling parameter β s (Eq. 5) was 0.83 ± 0.09 (0.94 ± 0.10) for 82Rb (15O-water). The scaling parameter β AUC (Eq. 6) was 0.92 ± 0.12 (0.97 ± 0.10) for 82Rb (15O-water). Additional file 1: Table S1 gives mean estimated correction parameters by tracer and condition.

Additional file 1: Figure S3 gives results of F-tests comparing the two-parameter PVC model to either the one-parameter PVC or scaling model (Eq. 5) for each acquisition. For 11 of 14 82Rb scans, the two-parameter PVC model outperformed the one-parameter PVC model. For only four 82Rb scans, the two-parameter model outperformed scaling with β s. For most 15O-water acquisitions, the two-parameter model was not superior to either one-parameter model (Eqs. 3 and 5).

The AUC scale correction (Eq. 6) cannot be compared to scaling with β s by F-test, since the WSS of the AUC scale correction will always be at least that of the β s correction. With the β s correction, the difference in IDIF peak compared to AIF peak was 1 ± 17 % (3 ± 17 %) for 82Rb (15O-water); the difference in AUC becomes 10 ± 15 % (3 ± 10 %) for 82Rb (15O-water). With the β AUC correction, the difference in peaks becomes −9 ± 16 % (0 ± 17 %) for 82Rb (15O-water); the difference in AUC becomes 0 ± 13 % (0 ± 10 %) for 82Rb (15O-water). Though β s correction provides better peak agreement, β AUC correction improves the peak agreement while also improving AUC agreement.

Based on these results, AUC-based scaling correction was adopted. All IDIFs were corrected by multiplication with the reciprocal of the average β AUC per tracer (1.09 for 82Rb, 1.03 for 15O-water). Eleven of 14 82Rb and 10 of 14 15O-water IDIFs had better agreement (lower WSS) with the AIF after scaling, with an average decrease in WSS of 18 ± 23 and 3 ± 16 %, respectively.

Parametric images

Example parametric images for one subject from the three-parameter fit are shown in Fig. 4, using each of: AIF, scaled IDIF, and uncorrected IDIF (Additional file 1: Figure S4 shows parametric images with and without the V RV term). K 1 is an estimate of MBF for 15O-water and nonlinearly related to MBF for 82Rb due to incomplete extraction, so myocardial K 1 (Fig. 4a) is lower for 82Rb than 15O-water (note different display scales). Naturally, stress values exceed rest values for both tracers. Because we studied healthy subjects, K 1 is relatively uniform in the myocardium. The LV cavity is more blurred in the K 1 images at stress than at rest for both tracers, presumably due to greater motion during stress. K 1 images are noisiest for 15O-water stress, where higher k 2 results in greater correlation between K 1 and V A, posing a more difficult estimation problem. The noisy 82Rb k 2 images (Fig. 4b) show poor delineation of the myocardium due to minimal tracer washout, as rubidium is trapped by viable myocardial tissue. In contrast, 15O-water freely diffuses in and out of tissue, so k 2 is proportional to MBF, and the k 2 images mirror the K 1 images. In the blood pool, k 2 values are very noisy since k 2 has minimal effect on model fits with K 1 ≈ 0. Since the V RV term was not included here, both the RV and LV blood pools are distinctly visible in the V A images (Fig. 4c). V A is overestimated from images using the uncorrected IDIF as compared to the AIF or scaled IDIF. The 15O-water V A images are slightly sharper than the 82Rb V A images, particularly near the septum, which may be explained by the poorer resolution of 82Rb.
Fig. 4

Short-axis parametric images for one subject’s 15O-water and 82Rb rest and stress scans, generated with different input functions. a K 1 parametric images, b k 2 parametric images, and c V A parametric images. Background outside the heart has been omitted for display. AIF arterial sample-based input function, IDIF image-derived input function. Right ventricle spillover correction term (V RV) not included

For this subject, the 82Rb K 1 images from corrected IDIFs showed qualitatively better agreement with those from AIFs than those using uncorrected IDIFs (Fig. 4a). Because corrected IDIFs were generated using a population-based correction factor, not all scans had comparable improvement in agreement between corrected IDIF- and AIF-based images. The IDIF correction factor was closer to unity for 15O-water than 82Rb, so 15O-water K 1 images were less affected by IDIF correction. The IDIF correction had virtually no impact on k 2 parametric images (Fig. 4b), as K 1 and V A compensate changes in input function scale.

Table 2 lists mean myocardial parameter estimates across subjects (using subject-specific regions) for the uncorrected IDIFs, scaled IDIFs, and AIFs (three-parameter fit; see Additional file 1: Table S2 for four-parameter results). Standard deviations as a percent of the mean are similar between tracers for K 1. The V A estimates, which account for both fractional blood volume and partial-volume effects, are lower for 15O-water than 82Rb, presumably due to the better resolution of 15O. The K 1 data agreed better between the scaled IDIF and AIF estimates than between the uncorrected IDIF and AIF estimates; k 2 estimates were similar for all IFs.
Table 2

Mean kinetic parameter estimates from three-parameter fit (without V RV)



Scaled IDIF

Uncorrected IDIF

K 1 (mL/min/g) mean ± SD



0.43 ± 0.09

0.45 ± 0.05

0.53 ± 0.06

H2 15O

0.87 ± 0.21

0.86 ± 0.15

0.91 ± 0.16



0.99 ± 0.19

1.11 ± 0.13

1.30 ± 0.17

H2 15O

3.43 ± 1.62

3.53 ± 0.85

3.68 ± 0.89

k 2 (1/min) mean ± SD



0.11 ± 0.05

0.13 ± 0.04

0.13 ± 0.04

H2 15O

1.10 ± 0.31

1.05 ± 0.22

1.05 ± 0.22



0.21 ± 0.06

0.23 ± 0.08

0.23 ± 0.08

H2 15O

3.76 ± 1.24

4.10 ± 1.06

4.10 ± 1.06

V A mean ± SD



0.32 ± 0.05

0.37 ± 0.04

0.40 ± 0.05

H2 15O

0.29 ± 0.08

0.33 ± 0.05

0.34 ± 0.06



0.31 ± 0.06

0.40 ± 0.06

0.44 ± 0.06

H2 15O

0.27 ± 0.06

0.27 ± 0.06

0.28 ± 0.06

SD standard deviation, AIF arterial sample-based input function, IDIF image-derived input function

This study’s mean 82Rb rest and stress K 1 and k 2 estimates using scaled IDIFs were comparable to those reported by [11]. The V A estimates were approximately 20 % lower than those in [11], likely attributable to IDIF correction and the improved resolution of this study’s images.

Figure 5a, b shows the good concordance between corrected IDIF- and AIF-based 82Rb K 1 and 15O-water k 2 mean myocardial estimates, respectively. Regression slopes (±standard error) for both 82Rb K 1 and 15O-water k 2 were close to 1.0 (1.06 ± 0.23 and 1.05 ± 0.36, respectively), and both intercepts were nearly 0 (0.025 ± 0.12 and -0.027 ± 0.51, respectively). The concordance correlation coefficient was 0.84 for both 82Rb K 1 and 15O-water k 2. Datapoints with poorer concordance may indicate subject motion, particularly during stress acquisitions, which affects the accuracy of IDIFs.
Fig. 5

a Comparison of mean myocardial 82Rb K 1 values using scale-corrected IDIFs vs. AIF. b Comparison of mean myocardial 15O-water k 2 values. Parameters were estimated with the three-parameter model (omitting right ventricle spillover term). The solid lines and equations represent fits from Deming regression. The dashed line is the identity line. AIF arterial sample-based input function, IDIF image-derived input function

Extraction fraction parameter estimates

Figure 6 shows the Renkin-Crone model fits based on three-parameter kinetic model fits using scaled IDIFs; Table 3 gives corresponding parameter estimates. The V RV term did not greatly impact Renkin-Crone fits estimated from global myocardial parameters, though omitting it resulted in greater differences between separate septal and lateral regional fits (Additional file 1: Figure S8). Figure 7 and Table 3 compare this study’s extraction model parameter estimates (with and without IDIF correction) to several previously published fits. This study’s a estimates are in reasonable agreement with that of the six comparison studies. Though the parameter b estimated using uncorrected IDIFs agrees with previous estimates, IDIF correction results in reduced b. The standard error of the estimates of a and b of this study were similar to or lower than those previously published.
Fig. 6

Renkin-Crone model fit of K 1 and myocardial blood flow from three-parameter kinetic model using scale-corrected image-derived input function. The dashed lines represent 95 % confidence interval of the regression line

Table 3

Renkin-Crone parameter estimates from this and published studies


Renkin-Crone parameter estimates


Flow measurement

Kinetic model

Input function correction

a ± SE

b ± SE

This study





0.74 ± 0.03

0.51 ± 0.09



0.77 ± 0.03

0.39 ± 0.06

Yoshida 1996 [5]





0.85 ± 0.03

0.45 ± 0.08

Lortie 2007 [11]





0.77 ± 0.05

0.63 ± 0.17

Lautamaki 2009 [10]







Prior 2012 [12]





0.80 ± 0.04

0.59 ± 0.14

Katoh 2012 [13]




2-parameter partial-volume correction



Renaud 2013 [30]







IDIF image-derived input function, SE standard error

Fig. 7

Comparison of this study’s Renkin-Crone model fit to published fits

MBF estimated from 15O-water k 2 (82Rb K 1) was 0.96 ± 0.20 (0.91 ± 0.19) at rest and 3.73 ± 0.96 (3.59 ± 0.55) under pharmacological stress (Table 4). These flows are consistent with previously published measurements in healthy controls [27]. A Bland-Altman analysis shows no systematic bias between 82Rb and 15O-water flows (p = 0.51 for paired t test; Fig. 8). The reproducibility coefficient, defined as 1.96 times the standard deviation of the differences between the water and rubidium-based MBFs, normalized to the mean of each water-rubidium pair, was 39 %. While this study used 15O-water MBF estimated from k 2, K 1-based flows were similar (Additional file 1: Figures S7 and S8).
Table 4

Extraction-corrected population estimates of myocardial blood flow, mean (±standard deviation)


82Rb MBF (mL/min/g)

15O-water MBF (mL/min/g)

Uncorrected IDIF


0.92 ± 0.19

0.96 ± 0.20


3.65 ± 0.64

3.73 ± 0.96

Scaled IDIF


0.91 ± 0.19

0.96 ± 0.20


3.59 ± 0.55

3.73 ± 0.96

IDIF image-derived input function, MBF myocardial blood flow

Fig. 8

Bland-Altman plot comparing 15O-water myocardial blood flow (MBF) to 82Rb MBF, using scaled image-derived input functions. Mean percent difference is −3.2 %. The dashed lines show 95 % confidence intervals


This study estimated Renkin-Crone extraction model parameters for rubidium using state-of-the-art 82Rb and 15O-water TOF PET images with PSF reconstruction. IDIF estimation was validated with continuously sampled arterial blood measurements (AIFs).

Armstrong et al. [16] provided a comparison of reconstructions with TOF and PSF to standard reconstructions; they reported an average increase in MBF of 10–14 % in advanced reconstructions compared to standard OSEM, which is consistent with improved recovery of signal in the myocardium. Because that study used only 82Rb, extraction could not be estimated from the advanced reconstructions.

The PSF modeling used here was not isotope-specific. Because 15O and 82Rb have poorer resolution than 18F, the PSF modeling will provide only partial resolution recovery. Though the employed reconstruction should provide better resolution than reconstruction without PSF modeling, there is further room for improvement.

Resolution modeling in PET reconstruction can produce ringing artifacts that significantly impact quantification [28]. These effects are most often detectible in simulation and phantom studies with well-defined object borders. Here, because of the additional blurring incurred by uncompensated cardiac and respiratory motion and modest Gaussian filtering applied pre-modeling, Gibbs-like artifacts were not observed.

While PSF modeling and TOF can provide improved resolution, the primary benefit in this application is reduced noise in the parametric images. For representative maps of the standard error of 82Rb K 1 and 15O-water k 2, see Additional file 1: Figure S9.

In 3D PET with high injected activities, inaccurate corrections for detector deadtime can impact the accuracy of reconstructed activity in early frames. A previous patient study on the Biograph mCT suggests that doses of 82Rb <1110 MBq can avoid significant detector saturation [29]. The doses used in the present study were on average ~60 % of this, and none exceeded it. Based on the peak singles rates, the average peak deadtime was 35 ± 8 % (32 ± 7 %) for 82Rb (15O-water).

PET image resolution is affected by positron range, detector resolution, smoothing in the reconstruction, and motion. Poor resolution affects quantification of myocardial activity and IDIFs. To minimize the impact of these effects on IDIFs, VOIs are typically small and central in the blood cavity where spill-in and spill-out are presumed insignificant. IDIF accuracy is important to kinetic modeling results; simulations show that a 10 % error in the IF peak can bias 82Rb K 1 estimates by up to 25 % [18]. Thus, verification of IDIF accuracy is highly important.

IDIFs had lower correspondence with AIFs for rubidium than water, particularly in terms of peak activity. Unlike water, rubidium is retained in myocardial tissue, causing the tail of the blood pool TAC to fall below that of the myocardium TAC (water IDIFs will have matched activity in the tails of the LV cavity and myocardium TACs). When 82Rb images are degraded by motion and resolution effects, expected consequences for IDIFs are reduced peak activity (spill-out from LV cavity) and increased tail activity (spill-in from myocardium). Though prior publications’ images [11, 12] were likely of poorer resolution than this study’s, their IDIFs were uncorrected. In the current study, we primarily observed reduced peak activity in 82Rb IDIFs, with differences in tail activity inconsistent with a strict model of geometric PVC (Eq. 3). In Katoh et al. [13], LV TACs were corrected using a PVC model; that model overcorrected the tails of our 82Rb IDIFs. A mismatch between 82Rb IDIFs and AIFs was better described using a two-parameter model (Eq. 4), which provides for signal mixing with background regions. The simpler scale factor correction (Eq. 5), which recovers activity from an unspecified combination of resolution degradation effects, gave comparable results. For half the 82Rb acquisitions, the AUC-based scale factor β AUC (Eq. 6) was approximately equal to the scan-specific β s (Eq. 5). For the rest, β AUC was markedly higher than β s; in these cases, the IDIF more greatly underestimated the AIF peak, with lower error in the tail. When β s is used to correct these cases, though the average peak error is reduced to ~0, the average AUC is overestimated, because tail activity is overestimated.

We chose scaling IDIFs based on AUC matching as more appropriate than WLS-based scaling. For 15O-water, the differences between the two methods were small. 82Rb, however, is more sensitive to the input function AUC, as its uptake is approximately irreversible. For such tracers, tissue activity is directly proportional to the input function AUC, so errors in the AUC propagate into the parameter estimates.

Given our AIF measurements and anticipating that population-based IDIF correction is most practical for scans without arterial sampling, we used mean tracer-dependent scale factors to correct IDIFs. However, the optimal correction factor is scan-dependent, conditional on variations in VOI size and placement, heart size, breathing pattern, and subject motion. Further investigation is required to assess generalizability to other scanners and reconstruction algorithms. Using AIFs for modeling does not guarantee accurate parameter estimates, as error in myocardial TACs not addressed by the partial-volume fractions V A and V RV (Eq. 2) could induce bias. Body and respiratory motion are likely principal sources of error. One limitation this study shares with previous publications is that no motion compensation was incorporated.

The Renkin-Crone model fits obtained in this study are similar to previously published fits, though our b estimate is lower, primarily a reflection of lower 82Rb K 1 estimates in this study compared to others, from the IDIF correction that was applied. With uncorrected IDIFs, extraction parameters obtained in this study closely match those previously published by Lortie [11]. Because scaling correction impacts 82Rb K 1 but not 15O-water k 2, scaled IDIFs resulted in a decrease in the b parameter of the Renkin-Crone model. For most previous studies, there was no gold standard measurement of the input function, and therefore, no basis for IDIF correction. Katoh et al. [13] used partial-volume corrected IDIFs, which explains the better agreement between their extraction model and that from corrected IDIFs in this study, compared to the Lortie model. The a parameter of the Renkin-Crone model is less sensitive to IDIF correction and reported values vary less across the literature.

Some differences in Renkin-Crone fits can also be explained by the flow estimation method. For instance, [11] used 13N-ammonia to measure MBF, which has a lower extraction fraction than 15O-water and will hence underestimate MBF. Using unweighted ODR to fit the data instead of weighted ODR resulted in higher parameter estimates (Additional file 1: Figure S10). Average MBFs were similar regardless of IDIF correction because water-based flows were unaffected by IDIF scaling and extraction parameters were estimated separately for each case. Therefore, accurate flows can be determined for 82Rb with or without IDIF correction, if extraction parameters have been estimated from data processed similarly. This suggests that modeling choices could have greater impact on extraction fraction estimates than TOF and PSF modeling, though TOF/PSF-based kinetic parameters have lower standard error.


We have presented parameter estimates for the generalized Renkin-Crone model of extraction for 82Rb PET using human 82Rb and 15O-water PET from high-resolution images from a state-of-the-art TOF-capable scanner with PSF reconstruction. The image-derived input functions were validated against direct arterial measurements, and a scale correction improved the accuracy of IDIFs. With this IDIF correction, MBF should be estimated from 82Rb K 1 using the Renkin-Crone parameters reported here. These results provide a state-of-the-art methodology for MBF measurement with 82Rb PET, though further validation will be necessary in patients with coronary artery disease with infarcts and ischemia.


AIF, arterial input function; IDIF, image-derived input function; LV, left ventricle; MBF, myocardial blood flow; PSF, point spread function; PVC, partial-volume correction; RPP, rate-pressure product; RV, right ventricle; TAC, time activity curve; TOF, time-of-flight; VOI, volume of interest; WLS, weighted least squares



This work was supported by Siemens Medical Solutions and NIH grants 1S10OD010322-01 and 1F31EB018720-01. This publication was made possible by CTSA Grant Number UL1 TR000142 from the National Center for Advancing Translational Science (NCATS), a component of the National Institutes of Health (NIH). Its contents are solely the responsibility of the authors and do not necessarily represent the official view of NIH.

Availability of data and materials

The data supporting the conclusions of this article are included within the article (and its additional files).

Authors’ contributions

JR and NN performed synthesis of 15O-water. CL, AS, RC, EA, and HF designed the imaging protocol and assisted in image acquisition. MG, TM, and KF performed data processing and analysis. MG, RC, AS, KF, CL, and TM prepared the manuscript. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Ethics approval and consent to participate

All procedures performed in studies involving human participants were in accordance with the ethical standards of the Yale University Human Investigation Committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. All participants signed a consent form.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

Biomedical Engineering, Yale University
PET Center, Diagnostic Radiology, School of Medicine, Yale University
Yale Cardiovascular Research Center, Section of Cardiovascular Medicine, Department of Internal Medicine, Yale University
PET Center, Yale School of Medicine


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