Quantitative myocardial perfusion PET parametric imaging at the voxel-level

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Quantitative myocardial perfusion PET parametric imaging at the voxel-level

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2015 Phys. Med. Biol. 60 6013

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Physics in Medicine & Biology

Quantitative myocardial perfusion PET parametric imaging at the voxel-level

Hassan Mohy-ud-Din1,2,3, Martin A Lodge2 and Arman Rahmim1,2

1 Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA2 Department of Radiology and Radiological Science, Johns Hopkins University, Baltimore, MD 21287, USA3 Department of Diagnostic Radiology, Yale University, New Haven, CT 06520, USA

E-mail: [email protected] and [email protected]

Received 3 February 2015, revised 27 May 2015Accepted for publication 29 May 2015Published 27 July 2015

AbstractQuantitative myocardial perfusion (MP) PET has the potential to enhance detection of early stages of atherosclerosis or microvascular dysfunction, characterization of flow-limiting effects of coronary artery disease (CAD), and identification of balanced reduction of flow due to multivessel stenosis. We aim to enable quantitative MP-PET at the individual voxel level, which has the potential to allow enhanced visualization and quantification of myocardial blood flow (MBF) and flow reserve (MFR) as computed from uptake parametric images. This framework is especially challenging for the 82Rb radiotracer. The short half-life enables fast serial imaging and high patient throughput; yet, the acquired dynamic PET images suffer from high noise-levels introducing large variability in uptake parametric images and, therefore, in the estimates of MBF and MFR. Robust estimation requires substantial post-smoothing of noisy data, degrading valuable functional information of physiological and pathological importance. We present a feasible and robust approach to generate parametric images at the voxel-level that substantially reduces noise without significant loss of spatial resolution. The proposed methodology, denoted physiological clustering, makes use of the functional similarity of voxels to penalize deviation of voxel kinetics from physiological partners. The results were validated using extensive simulations (with transmural and non-transmural perfusion defects) and clinical studies. Compared to post-smoothing, physiological clustering depicted enhanced quantitative noise versus bias performance as well as superior recovery of perfusion defects (as quantified by CNR) with minimal increase in bias. Overall, parametric images obtained from the proposed methodology were robust in the presence of high-noise levels as manifested in the voxel time-activity-curves.

H Mohy-ud-Din et al

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Keywords: myocardial perfusion, coronary flow reserve, cardiovascular diseases, positron emission tomography, parametric imaging, physiological clustering, robust parameter estimation

(Some figures may appear in colour only in the online journal)

1. Introduction

Studies worldwide have attributed a high morbidity and mortality rate to cardiovascular dis-eases (Murray and Lopez 1997, Gulati et al 2003, Thom et al 2006, Mensah and Brown 2007, Schindler et al 2010). The world health organization has predicted that by 2030 approximately 23.6 million people will die due to coronary artery disease (CAD) (WHO 2011). Subsequently, there has been considerable attention and efforts devoted to instrumentation, experimentation, assessment, and analysis of the early detection, staging, and management of CAD (Dilsizian and Narula 2013).

Absolute quantification of myocardial blood flow (MBF) and flow reserve (MFR), i.e. ratios of MBF values at peak stress to rest, augments conventional myocardial perfusion imag-ing (MPI) and can potentially allow early detection of preclinical atherosclerosis, provid-ing an opportunity to modify risk factors or initiate treatment (Lodge and Bengel 2007, Pan et al 2009, Schindler et al 2010). MFR has been shown to be related to the degree of coronary artery stenosis (CAS) (Uren et al 1994). It thus allows for noninvasive assessment of the func-tional importance of CAS and may aid identification of patients with either diffuse, nonocclu-sive luminal coronary artery narrowing or a balanced reduction in coronary artery blood flow (extensive multi-vessel coronary disease) (Parkash et al 2004).

PET enables noninvasive and accurate quantification of MBF and MFR (Yoshida et al 1996, Bengel et al 2009, Dilsizian and Narula 2013, Nakazato et al 2013), and compared to SPECT, has higher sensitivity as well as superior spatio-temporal resolution, and involves more straightforward attenuation correction (Lodge et al 2005, Lodge and Bengel 2007, Rahmim and Zaidi 2008, Salerno and Beller 2009, O’Donnell et al 2010, Sciagrà 2012). Dynamic cardiac PET has also demonstrated superior diagnostic accuracy and added prog-nostic value for the detection of CAD at low levels of administered dose (Nakazato et al 2013, Small et al 2013, Dorbala and Di Carli 2014, Ohira et al 2014).

Despite all these benefits, quantification of MBF and MFR has yet to be translated into routine clinical practice and, thus far, remains primarily a research endeavor. An impediment has been the need for production and delivery of short-lived radiotracers at the clinical site. 82Rb provides great opportunities to this end (Yoshinaga et al 2010, Arumugam et al 2013, Dilsizian 2013), as it does not require an onsite cyclotron. Furthermore, its short half-life of 76 s enables fast serial imaging (~15 mins) (deKemp et al 2000) and high patient throughput. However, low count statistics produce very noisy dynamic (uptake) images adversely impact-ing accurate quantification of MBF and MFR especially at the voxel-level (parametric images) (Klein and Beanlands 2010). Figure 1 shows LV myocardial uptake images (without post-smoothing) from a clinical study and the corresponding voxel time activity curves (TACs). The TACs demonstrate an erratic behavior due to poor SNR at the voxel scale. Such a con-centration curve may be physiologically implausible resulting in highly unstable parameter estimates (Reader and Verhaeghe 2014).

Quantitative MP PET at the individual voxel-level has the potential to enable enhanced vis-ualization and quantification of MBF/MFR as well as identify abnormal decrease (gradient) in longitudinal myocardial blood flow from the base to the apex of the heart (e.g. diffuse luminal

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narrowing in patients with mild angiographic CAD) (Gould et al 2000, Gould et al 2013, Valenta et al 2014, Rahmim et al 2014a, 2014b). One way to address the issue of high noise-levels is to perform heavy post-smoothing on reconstructed dynamic images before estimat-ing functional parameters by kinetic modeling. This comes at a cost of substantial resolution degradation (high estimation bias). Another approach is to perform region-of-interest (ROI) quantification of MBF and MFR where the voxel intensities are averaged over each ROI. This also comes at a cost of loss of spatial information and underlying tissue heterogeneity.

We propose a novel framework of robust kinetic parameter estimation at the individual voxel level that substantially reduces noise using a method which we term ‘physiological clustering’ (Mohy-ud-Din et al 2014, 2015), and as elaborated in section 2. Physiological clustering is an approach of clustering a PET image driven by the underlying physiology. The resulting label map is a union of disjoint clusters each represented by its mean kinetics. The parameter estimation problem is then formulated as nonlinear least squares with spatial con-straints (NLLS-SC) where the spatial constraints are derived from the physiologically clus-tered image. Section 3 describes the experimental design used to validate our methodology, with results presented in section 4. Discussions and conclusions are presented in sections 5 and 6 respectively.

Figure 1. (Top) Short-axis view of LV myocardial uptake images for 82Rb radiotracer (without post-smoothing) from a clinical subject. Uptake images are summed images from 2–4 min. High noise-levels are clearly visible in the images. (Bottom) The LV myocardium TACs, at the voxel-scale, exhibit an erratic time-course due to poor SNR in the dynamic images. This results in highly unstable estimates of physiological parameters of interest at the voxel-level.

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2. Theory

2.1. Pharmacokinetic model for the 82Rb radiotracer

82Rb radiotracer kinetics can be described by a one-tissue compartment model (El Fakhri et al 2005, Lortie et al 2007, Klein and Beanlands 2010, Karpikov et al 2012). The myocardial activity concentration time-course, ( )C t ,myo is defined by a first-order ordinary differential equation (ODE)

( )= ( ) − ( )

C t

tK C t k C t

d

d   

myo1 a 2 myo (1)

where ( )C ta is the arterial input function (which is commonly approximated by ( )C t ,LV the LV blood concentration time-course, as used from here onwards), and K1 (uptake rate in units of mL min−1 g−1) and k2 (washout rate in units of 1 min−1) are the transport rate constants. Solving the ODE (1) for ( )C tmyo gives the following convolution equation:

( ) = ( ) ⊗ −C t C t K   e k tmyo LV 1

2 (2)

where we use ( ) = ( )C t C t .a LV Due to partial volume effects (PVE), caused by the limited reso-lution of PET scanners, the myocardial tissue concentration time course is contaminated by ‘spill-over’ of activity from the LV blood-pool and the RV blood-pool:

( ) = ( ) + ( ) + ( − − ) ( ) ⊗ −C t f C t f C t f f C t K        1        e k tPET LV LV RV RV LV RV LV 1

2 (3)

where ( )f f LV RV is the fractional blood volume spill-over from LV (RV) that accounts for contribution of blood activity in the measured myocardial tissue concentration time-course.

K1 estimates from a 82Rb study needs to be corrected for the extraction fraction, EF, which can be performed via the generalized Renkin–Crone model (Prior et al 2012).

( )= × = × −−

K a MBF EF MBF 1  e  b

1 MBF (4)

The extraction fraction, EF, is a nonlinear function of MBF and decreases with increase in MBF due to tracer extraction from blood via diffusion and active transport (Renkin 1959, Crone 1963). Many studies have reported the generalized Renkin–Crone model parameters (a and b) for various radiotracers and a comprehensive table can be found in Klein et al 2010 ((Klein and Beanlands 2010) table 4).

2.2. Image derived input functions

The input function can be measured invasively by arterial cannulation which is a cumbersome procedure, both for the clinician and the patient. Generation of image derived input func-tion (IDIFs) from the reconstructed dynamic images is a noninvasive and feasible alternative (Zanotti-Fregonara et al 2011). In the afore-mentioned commonly utilized framework (3), estimation of the parameter vector β ≜ [ ]K k f f      T

1 2 LV RV requires knowledge of the input func-tion, ( )C t ,a which is approximated by ( )C t ,LV and ( )C t ,RV which are conveniently estimated given the large blood pools of LV and right atrium (RA) in the PET field of view (Lodge and Bengel 2007).

( )C tLV and ( )C tRV were extracted from co-registered dynamic PET images by manually placing an elliptical ROI (~100 mm2) in the LV and RV blood pools. These concentration curves are generated from 4 mid-ventricular imaging planes and then averaged to reduce noise (Choi et al 1999).

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2.3. Physiological clustering

Clustering techniques for enhanced parameter estimation have been reported before (Kimura et al 1999, Wong et al 2002, Zhou et al 2002, Guo et al 2003, Huang et al 2007). However, in this work, clustering is driven by the underlying physiology of functional regions. The physi-ological clustering approach for noise reduction is motivated by the observation that accurate kinetic parameter estimation requires segmentation; ideal segmentation requires knowledge of the underlying physiological (kinetic) parameters (Saad et al 2007). This dilemma can be addressed by performing clustering and kinetic modeling in an iterative fashion. This approach reduces bias propagation from clustering into parameter estimation which may dominate if the two processes are performed separately.

Figure 2 describes a pseudocode for physiological clustering that uses K-means with kinetic modeling (KM–KM) (Saad et al 2007). K-means clustering uses the Euclidean norm of the TACs to gauge physiological similarity of a pair of voxels indexed by ( )i j,   :

( ) = − = ( − ) ( − )− TAC TAC TAC TAC W TAC TACS i j,          i j W i j i jKT

means

(5)

where W is the data weighting matrix. { }= ΔW diag tDF C

2

2  where Δt, DF, and C are vectors of

frame durations, decay correction factors, and frame counts respectively (Carson et al 2007). Algorithm 1 (figure 2) generates a label map where each functionally homogeneous region is represented by mean kinetics. Mean kinetics of each cluster forms the spatial constraints set β{ }sc for robust parameter estimation.

2.4. Robust parameter estimation

Robust estimation of physiologically meaningful parameters (like the uptake rate K1) is achieved by solving the following nonlinear least squares with spatial constraints (NLLS-SC) optimization problem for each voxel:

β β β βγ= ( ) − ( ) + ‖ − ‖β

t C t;ˆ  arg min  C  PET Wmeas2

sc 22

(6)

Figure 2. Physiological Clustering: a pseudocode that outlines the steps for functional clustering of dynamic PET images where clustering is driven by kinetic modeling. The output of Algorithm 1 is a label map, where each voxel belongs to a functionally homogenous region (cluster), and a representative set of kinetic parameter vectors for each cluster i.e. cluster centroids.

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where β ≜ [ ]K k f fˆ       T1 2 LV RV is the estimated parameter vector, β( )C t;PET models the measured

PET signal (3), ( )C tmeas is the measured voxel TAC, βsc is the spatial constraint derived from physiological clustering of dynamic images (Algorithm 1; as concluded with step 11), and W is the data weighting matrix. Each voxel belongs to one of the N clusters and the representative kinetic parameter vector for that cluster forms the spatial constraint vector. γ penalizes large deviations from mean kinetics. Figure 3 depicts a flowchart of the overall proposed approach. (6) was solved using the basis function method (Gunn et al 1997) for weighted least squares. 200 basis functions ( ( ) = ⋯ )B tk i; ;   1, , 200i

i2 were assumed with the nonlinear parameter, k ,i2

logarithmically sampled in the range [ ]0 1 . MBF is estimated from K1 by solving (4) using a fixed-point iteration approach (Leader 2004).

3. Methods

3.1. Simulations

3.1.1. Perfusion defects. Transmural and non-transmural perfusion defects were simulated at different locations along the LV wall using the NCAT phantom (Segars et al 1999). Figure 4 illustrates short-axis views of the perfusion defects and the corresponding parameters used

Figure 3. Flowchart of the proposed approach: Kinetic parameter estimation is formulated as a nonlinear least squares with spatial constraints problem (NLLS-SC) where spatial constraints are computed from a physiologically driven clustering of dynamic images. Physiological clustering of dynamic images is performed using K-means clustering with kinetic modeling in an iterative fashion. This gives a map of labels where each functionally homogeneous cluster is represented by mean kinetics (cluster centroid). Parametric images are acquired by solving the NLLS-SC problem for each voxel by penalizing spatial variations from mean kinetics. This substantially reduces noise in the estimation process for each voxel by utilizing kinetic information from physiologically similar voxels (cluster members). Resolution degradation may be substantially minimized as regression is performed on non-smoothed (raw) dynamic images.

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to generate the phantoms. Defects were simulated in stress images and were placed over the anterior-lateral and lateral-inferior regions.

3.1.2. Simulation and reconstruction. Original TACs were generated for five tissues (liver, lung, muscle, normally perfused myocardium, and myocardium defect) using realistic kinetic parameters and the one tissue-compartment model. Table 1 lists the kinetic parameters used in the simulations. Arterial blood fraction for the myocardial tissue was simulated as 25%.

The dynamic data set consisted of 32 time frames (24 × 5 s, 4 × 20 s, and 4 × 40 s) for a total duration of 6 min. Analytic simulations were performed for the dynamic images recon-structed using 315 angular samples covering 180°, 323 radial bins with 2.26 mm spacing, and 47 axial bins with 3.27 mm spacing (Fessler 2012). Reconstructions were performed utilizing

× ×128 128 47 voxels in the transaxial (x, y) and axial (z) directions, respectively, with voxel dimensions × ×3.27 3.27 3.27 mm3. Attenuation, normalization, randoms, scatter, and decay were also incorporated. For randoms and scatter, a uniform distribution was assumed in the projection space. Poisson noise was then added to generate 30 realizations of noisy dynamic

Figure 4. (Top) Short-axis and long-axis views of the LV wall with the region of perfusion defect shaded. Perfusion defects are parameterized by (θ θΔ Δz z,   ,   ,   ,center center and wall fraction). (Below) Table shows the short-axis views of the simulated perfusion defects and the corresponding parameters used to generate the phantoms (Tang et al 2009).

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sinograms which were reconstructed using the ordinary-subset expectation-maximization (OSEM) (3 iterations, 21 subsets) (Hudson and Larkin 1994, Fessler 2012)).

3.2. Clinical studies

3.2.1. Patient dataset. We studied 3 patients (2 males and 1 female) that underwent 82Rb PET/CT scans for a rest-stress study. The average age of these patients was ±65 16 years, and the average weight of these patients was ±182 61 lbs. All patients had a history of CAD and showed perfusion defects along the inferior-lateral wall. Table 2 summarizes the characteris-tics of the clinical data.

3.2.2. PET/CT protocol. Sinograms were acquired using a GE LYSO-crystal, 64-slice Discov-ery Rx VCT scanner. Subjects were positioned supine using a scout scan. This was followed by a low-dose CT scan (120 kV, 80 mA) with shallow breathing for attenuation correction of PET data. Using a large intravenous line, 35–50 mCi of 82Rb was infused at 50 ml min−1 over 30 s, and a list-mode 2D PET scan was acquired for 8 min. After rest acquisition, subjects received 0.4 mg of A2A adenosine receptor agonist Regadenoson, infused intravenously over 10 s. Within 20 s of pharmacologic induced stress, a second dose of 35–50 mCi of 82Rb was injected followed by another 8 min acquisition of list-mode 2D PET scan. Dynamic sinograms were reconstructed using the OSEM algorithm (2 iterations, 21 subsets) into 32 dynamics frames (framing sequence: 20 × 6 s, 5 × 12 s, 4 × 30 s, and 3 × 60 s) where each frame is a 128 × 128 × 47 matrix with a voxel volume of 3.27 × 3.27 × 3.27 mm3.

3.3. Parametric imaging at the voxel-level

3.3.1. Parametric imaging using physiological clustering. Physiological clustering was per-formed on post-smoothed reconstructed dynamic images to generate a set of spatial constraint vectors, one for each functional cluster. Three clusters were assumed a priori (background, normal myocardium, and perfusion defect). Parametric images were generated using non-smoothed reconstructed dynamic images and varying penalty parameter, γ ∈ [ × ]−  0 1 10 1 (section 2.4). γ = 0 corresponds to no spatial constraint (unconstrained minimization).

3.3.2. Parametric imaging following post-smoothing. To evaluate the efficacy of our approach we computed parametric images by first post-smoothing the reconstructed dynamic images to reduce noise followed by solving the following weighted nonlinear least squares problem using basis function method:

Table 1. Kinetic parameters used in simulation of cardiac PET images for 82Rb radiotracer.

Tissue K1 (mL min−1 g−1) k2 (1 min−1)

Liver 0.57 0.97Lung 0.18 0.98Muscle 0.06 0.21Myocardium (Rest) 0.70 0.16Myocardium (Stress) 1.48 0.32Perfusion Defect (Stress) 0.74 0.25

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β β= ( ) − ( )β

C t C;ˆ  arg min  t  WPET measpsps 2

(7)

where ( )C tmeasps denotes the voxel TAC from post-smoothed dynamic images. Post-smoothing

was performed using Butterworth filter of order 4 and cutoff frequencies ranging from 0.2 cycle cm−1 to 1 cycle cm−1 (0.1 cycle cm−1 increments).

3.4. Figures of merit

We quantitatively analyzed parametric images (from physiological clustering, sec-tion 3.3.1, and post-smoothing, section 3.3.2) by computing the following figures of merit (Su et al 2013):

⎛

⎝⎜⎜

⎞

⎠⎟⎟

∑∑ θ θ

θ

−

∈

=

NBias:

1 

j

Ri

R

ij j

jp ROI

1

1

est true

true

(8)

∑∑∑

∑

θ θ

θ( )

−

∈

−= =

=

⎛

⎝⎜

⎞

⎠⎟

N

RNSD Normalized Standard Deviation :

1 1

j ROI

Ri

R

iji

R

ij

Ri

R

ijp

1

11

est

1

est2

1

1

est

(9)

∑ ∑ θ θ

( )

( − )

×= =

N RRMSE Root Mean Squared Error :

 j

N

i

R

ij j1 1

est true 2

p

p

(10)

θθ

−Contrast: 

 1N

D (11)

Table 2. Characteristics of the clinical dataset.

No Sex Age (y) BP (mm Hg) HR (bpm) Weight (lb) LVEF (%) Defect Type

1 M 64 116/69–103/55 80–146 172 30%–42% Large fixed inferior wall defect

2 M 49 119/65–133/60 75–96 247 45%–50% Fixed perfusion defect in lateral wall

3 F 81 98/61–120/62 59–162 126 29%–32% Fixed large inferior-lateral wall defect

BP (mm Hg): rest systolic/diastolic blood pressure—stress systolic/diastolic blood pressure.HR (bpm): rest heart rate—peak heart rate.LVEF (%): Left ventricle ejection fraction (rest—stress).

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where θjtrue is the jth pixel value of the true parametric image, θij

est is the jth pixel value of the estimated parametric image from the ith noise realization, R is the total number of noise real-izations, and Np is the total number of pixels in an ROI. Two ROIs were drawn on the normally perfused myocardium and the perfusion defect. Overall Bias, NSD, and RMSE were calcu-lated by computing a weighted average of the ROI values (weighted by the size of the ROI). Contrast was calculated from the mean uptake values from the normal and defect regions, θN and θ ,D respectively.

For clinical studies we computed contrast-to-noise ratio, ( )=CNR   ,contrast

NSD where contrast

and NSD are defined as follows:

⎛

⎝⎜⎜

⎞

⎠⎟⎟∑ ∑

∑

θ θ

θ

−−

∈ ∈

∈

NNSD:

 1

Ni

ii

i

Ni

i

1

1   B

est

B    B

est

2

1

   B

est

B

B

(12)

∑

∑

θ

θ∈

∈

Contrast:  – 1N

ii

Ni

i

1

   B

est

1

   D

est

B

D

(13)

θiest is the ith pixel value of the estimated parametric image, NB is the number of pixels in the

background ROI, B, and ND is the number of pixels in the perfusion defect ROI, D.

4. Results

4.1. Simulations

We compared parametric images of uptake rate ( )K1 obtained using the proposed approach of physiological clustering and the conventional approach of post-smoothing.

4.1.1. Transmural perfusion defect. Figure 5 shows that with decreasing values of w ,  cutoff and increasing values of γ, for the conventional post-smoothing and proposed physiological clus-tering methods, respectively, noise is considerably reduced in K1 parametric images. In the post-smoothing approach, at cutoff frequencies typically encountered in clinical settings ( ⩽w 0.5cutoff cycles cm−1), the images are blurred especially around the septal, anterior-lateral and lateral-inferior regions of the myocardium. By comparison, visual assessment reveals that the proposed approach reduces noise while maintaining distinct physiological regions.

For quantitative analysis, figure 6 shows plots of overall noise (NSD) versus overall bias for the K1 parametric images for both rest (left) and stress (stress) MP simulations. Clearly, enhanced quantitative noise versus bias performance is observed for the proposed framework. Table 3 also highlights key numerical performance of the two approaches in comparison to baseline parametric imaging ( =w 0cutoff and γ = 0). Thus, by comparison, physiological clus-tering resulted in superior quantitative accuracy by substantially reducing the overall bias and RMSE without degrading bias. The computed CNR in stress images (where perfusion defect was present) showed that physiological clustering, compared to post-smoothing approach, resulted in a higher CNR at optimal performance.

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We also studied the quantitative performance of the two approaches at the ROI level. Figure 7 shows the NSD versus bias curves for each of the normal myocardium and the perfu-sion defect. Numerical comparisons between the two approaches are summarized in table 4 for both ROIs. Interestingly, physiological clustering approach substantially reduces both noise and bias in the perfusion defect.

To quantify the ability to resolve perfusion defect, we also studied CNR as varied via wcutoff and γ for the post-smoothing and physiological clustering approaches, respectively. Figure 8 depicts plots for RMSE versus CNR as well as bias versus CNR. At higher CNR (⩾ )3 , physi-ological clustering, compared to post-smoothing, resulted in K1 parametric images with sig-nificantly lower RMSE and estimation bias. With decreasing wcutoff (especially ⩽w 0.5cutoff cycles cm−1 as commonly encountered in clinical practice), the estimation bias (and RMSE) were severely degraded resulting in unstable K1 parametric estimates. On the contrary, physi-ological clustering resulted in improved CNR without aggravating RMSE and estimation bias. Doubling the CNR (from ~5  to 10) was at the cost of %11 increase in estimation bias and %3.6 increase in RMSE for post-smoothing, while this was at a cost of <1% increase in estimation bias and an actual improvement of %8 in RMSE in the proposed framework.

4.1.2. Non-transmural perfusion defect. We evaluated qualitative and quantitative perfor-mance for non-transmural perfusion defect placed along the lateral myocardium wall (wall fraction = 0.75, figure 4). The defect was only simulated in stress images and, hence, we focus on stress uptake parametric images (figure 9) obtained using post-smoothing and physiologi-cal clustering approaches.

Figure 5. K1 Parametric images: we show K1 images from rest-stress simulation. Perfusion defect was placed along the lateral myocardium wall in stress images. For each set of images we show: (row 1) true K1 image, (row 2) K1 images obtained from post-smoothed dynamic images with varying cutoff frequencies: (L-to-R)

=w 0,  1,  0.9,  0.8,  0.7,  0.6,  0.5,  0.4,  0.3,  and 0.2cutoff cycles cm−1 where =w 0cutoff corresponds to no post-smoothing, and (Row 3) K1 images obtained

from physiological clustering approach by varying the penalty parameter: (L-to-R) γ = × × × × × × × ×− − − − − − − −  0,  1 10 ,  5 10 , 8 10 , 1 10 , 3 10 , 6 10 , 9 10 , 1 10 , and5 5 5 4 4 4 4 3 1 × 10−2 where γ = 0 corresponds to no spatial constraint. =w 0cutoff and γ = 0 are identical scenarios which involve weighted nonlinear regression on raw dynamic images (as seen from equations (6) and (7)).

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Figure 6. Transmural perfusion defect: plots of overall NSD versus overall bias comparing the K1 parametric images obtained from post-smoothing approach and physiological clustering approach. Points on the post-smoothing curve correspond to different cutoff frequencies of Butterworth filter ( )wcutoff and points on the physiological clustering curve correspond to different values of penalty parameter (γ). ⩽w 0.5cutoff cycles cm−1 are typically found in clinical settings (Cullom et al 2013, Tahari et al 2014).

Table 3. Transmural perfusion defect: summary of the quantitative performance of physiological clustering and post-smoothing approaches, compared to the baseline ( =wno processing;  0cutoff and γ = 0. We include results for =w 0.5cutoff cycles cm−1 and =w 0.3cutoff cycles cm−1 which are typically found in clinical settings and may not be the optimal value (Cullom et al 2013, Tahari et al 2014).

Simulations Methodology Parameter

Quantitative Metrics

Bias increase (%)

NSD reduction (%)

RMSE reduction (%) CNR

Rest Post-smoothing (wcutoff in cycles cm−1)

0.3 cycles cm−1 10.5 40.0 40.3 —0.5 cycles cm−1 3.3 34.0 49.0 —0.6 cycles cm−1 (optimal value)a

1.0 31.0 50.0 —

Physiological clustering −1 × 10 2 (optimal value)

−5.1b 34.8 61.2 —

Stress Post-smoothing (wcutoff in cycles cm−1)

0.3 cycles cm−1 15.6 37.8 22.2 9.90.5 cycles cm−1 8.4 33.0 34.3 9.00.7 cycles cm−1 (optimal value)

3.9 27.5 38.2 7.2

Physiological clustering −4 × 10 3 (optimal value)

1.2 32.6 46.0 13.7

a Optimal parameter value corresponds to minimum overall RMSE.b Negative values indicate a decrease in overall bias.

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Figure 9 shows that with increasing values of γ and decreasing values of wcutoff noise is considerably reduced in K1 parametric images. However, in the post-smoothing approach, it becomes increasingly difficult to distinguish between transmural and non-transmural perfu-sion defects due to mixing of kinetic information from heterogeneous voxels on the boundary of the defect. At lower cutoff frequencies typically encountered in clinical settings ( ⩽w 0.5cutoff cycles cm−1), the images are blurred, especially around the septal, the anterior-lateral, and the lateral-inferior regions of the myocardium. On the contrary, the physiological clustering approach is able to resolve the non-transmural perfusion defect by only utilizing kinetic infor-mation from homogenous voxels to reduce noise.

Figure 7. Transmural Perfusion Defect: regional NSD versus bias curves for K1 parametric images obtained from post-smoothing approach and physiological clustering approach. Points on the post-smoothing curve correspond to different cutoff frequencies of Butterworth filter ( )wcutoff and points on the physiological clustering curve correspond to different values of penalty parameter (γ). ⩽w 0.5cutoff cycles cm−1 are typically found in clinical settings.

Table 4. Transmural Perfusion Defect: summary of the quantitative performance of physiological clustering and post-smoothing approaches in the two ROIs: normal myocardium and perfusion defect. In the normal myocardium the optimal parameter values were =w 0.7cutoff

opt cycles cm−1 and γ = × −4 10  opt 3 and in the perfusion defect the optimal parameter values were =w 0.3cutoff

opt cycles cm−1 and γ = × −1 10 .opt 2 Optimal parameters correspond to minimum regional RMSE.

Methodology

Quantitative metrics

Normal myocardium Perfusion defect

Bias increase (%)

NSD reduction (%)

RMSE reduction (%)

Bias increase (%)

NSD reduction (%)

RMSE reduction (%)

Post-smoothing 5.6 27.0 36.1 2.1 42.0 56.2Physiological clustering

4.6 30.7 42.8 −11a 40.6 68.3

a Negative value indicates a decrease in bias.

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Figure 10 shows plots of overall NSD versus overall bias for the K1 parametric images. Table 5 summarizes the quantitative performance of the two approaches for non-transmural perfusion defect.

Quantitative performance of the two approaches at the ROI level is also depicted in figure 11 for the normal myocardium and the perfusion defect. As before, table 6 also shows numerical comparison between the two approaches. Overall, the results demonstrate the effi-cacy of the physiological clustering approach in enabling substantially enhanced noise versus bias quantitative performance in the non-transmural perfusion defect.

Figure 8. (Left) RMSE versus CNR and (right) bias versus CNR curves for post-smoothing approach and physiological clustering approach. Points on the post-smoothing curve correspond to different cutoff frequencies of Butterworth filter ( )wcutoff and points on the physiological clustering curve correspond to different values of penalty parameter (γ). In both plots, the left-most point corresponds to =w 0cutoff and γ = 0.

Figure 9. K1 Parametric Images: we show uptake images from stress simulation. Non-transmural perfusion defect was placed along the lateral myocardium wall  (wall  fraction = 0.75, figure  4). We show: (row 1) true K1 image, (row 2) K1 images obtained from post-smoothed dynamic images with varying cutoff frequen-cies:  (L-to-R)  =w 0,  1,  0.9,  0.8,  0.7,  0.6,  0.5,  0.4,  0.3,   and 0.2cutoff   wcycles cm−1 where =w 0cutoff corresponds to no post-smoothing, and (row 3) K1 images obtained from physiological clustering approach by varying the penalty parameter: (L-to-R) γ = × × × × × × × ×− − − − − − − −  0,  1 10 ,  5 10 , 8 10 , 1 10 , 3 10 , 6 10 , 9 10 , 1 10 ,  and5 5 5 4 4 4 4 3 1 × 10−2 where γ = 0 corresponds to no spatial constraint. =w 0cutoff and γ = 0 are identical scenarios which involve weighted nonlinear regression on raw dynamic images (as seen from equations (6) and (7)).

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Figure 12 shows plots for RMSE versus CNR as well as bias versus CNR. In physiologi-cal clustering approach, as the CNR doubled from 5 to 10, RMSE was improved from %44 to 46% and the estimation bias increased from %2.3 to %3.4 . However, with post-smoothing, the achievable CNR was less than six (CNR < 6), and the RMSE and bias performance was clearly degraded in comparison to the physiological clustering approach.

4.1.3. Number of clusters. For simulations and clinical studies, we chose the true number of clusters ( = )N 3 in Algorithm 1 (figure 3) after carefully visualizing the summed dynamic images with the help of a resident. We quantitatively analyzed simulated stress uptake images obtained using the physiological clustering approach with varying number of clusters,

=N 3,  5,  8, and 11. Figure  13 shows the percentage reductions achieved in overall NSD, bias, and RMSE in the estimated K1 parametric images. For =N 3 and 5, the quantitative per-formance is almost similar (<2% difference) but, for =N 8 and 11, the performance degrades further by > %5 .

4.2. Patient dataset

Figure 14 shows that with increasing values of γ and decreasing values of wcutoff noise is con-siderably reduced in K1 parametric images. However, compared to post-smoothing approach, physiological clustering is able to resolve the perfusion defects without blurring the region of normal uptake at the interface of the defect.

For quantitative comparison, figure 15 depicts uptake rate (K1) versus CNR for the two approaches in normally perfused region and perfusion defect. Physiological clustering outper-formed post-smoothing approach in the sense that for a given uptake rate, higher CNR was obtained. In the post-smoothing approach, with decreasing w ,cutoff the uptake rate deteriorates even for normal myocardium which is attributed to the heterogeneous mixing of voxels at the

Figure 10. Non-Transmural Perfusion Defect: plots of overall NSD versus overall bias comparing the K1 parametric images obtained from post-smoothing approach and physiological clustering approach. Points on the post-smoothing curve correspond to different cutoff frequencies of Butterworth filter ( )wcutoff and points on the physiological clustering curve correspond to different values of penalty parameter (γ).

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interface of the perfusion defect. On the contrary, the physiological clustering approach is able to resolve the perfusion defect with substantial noise reduction resulting in a higher CNR.

5. Discussion

In the present work we have proposed the approach of physiological clustering for robust voxel-wise estimation of K1 parametric images in quantitative MPI. The framework was extensively evaluated using simulations and clinical data and compared with conventional post-smoothing. In the physiological clustering framework, we varied the penalty parameter, is γ, from 0 to 1 × 10−1 and in the post-smoothing framework, we varied the cutoff frequency

Table 5. Non-Transmural Perfusion Defect: summary of the quantitative performance of physiological clustering and post-smoothing approaches. We include results for

=w 0.5cutoff cycles cm−1 and =w 0.3cutoff cycles cm−1 which are typically found in clinical settings and may not be the optimal value.

Simulations Methodology Parameter

Quantitative metrics

Bias increase (%)

NSD reduction (%)

RMSE reduction (%) CNR

Stress Post-smoothing (wcutoff in cycles cm−1)

0.3 cycles cm−1 18.1 37.4 17.8 2.30.5 cycles cm−1 10.1 33.1 31.3 5.30.7 cycles cm−1 (optimal value)

4.7 27.8 36.9 5.7

Physiological clustering −4 × 10 3 (optimal value)

3.0 33.4 47.1 11.3

Figure 11. Non-Transmural Perfusion Defect: regional NSD versus bias curves for K1 parametric images obtained from post-smoothing approach and physiological clustering approach. Points on the post-smoothing curve correspond to different cutoff frequencies of Butterworth filter ( )wcutoff and points on the physiological clustering curve correspond to different values of penalty parameter (γ).

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of the Butterworth filter, w ,cutoff from 0.2 cycles cm−1 to 1 cycles cm−1. Qualitative analysis showed that the proposed approach produced sharper images with increased noise reduction (figures 5, 9 and 14). Increasing γ values strongly penalize deviations of voxel estimates from spatial constraints which utilize kinetic information from physiologically similar voxels. This considerably reduces noise with limited mixing of information from functionally heteroge-neous regions. Quantitative analysis showed that at optimal performance (minimum overall RMSE), the physiological clustering approach, compared to post-smoothing, considerably reduced overall NSD with minimum resolution loss (overall bias) and provided a higher CNR (tables 3 and 5). We observed the efficacy of physiological clustering in substantially reducing both noise and bias in the perfusion defect (tables 4 and 6). The latter is related to the presence

Table 6. Non-Transmural Perfusion Defect: summary of the quantitative performance of physiological clustering and post-smoothing approaches in the two ROIs: normal myocardium and perfusion defect. In the normal myocardium the optimal parameter values were =w 0.7cutoff

opt cycles cm−1 and γ = × −1 10  opt 2 and in the perfusion defect the optimal parameter values were =w 0.3cutoff

opt cycles cm−1 and γ = × −4 10 .opt 3 Optimal parameters correspond to minimum regional RMSE.

Methodology

Quantitative metrics

Normal myocardium Perfusion defect

Bias increase (%)

NSD reduction (%)

RMSE reduction (%)

Bias increase (%)

NSD reduction (%)

RMSE reduction (%)

Post-smoothing 5.9 27.6 35.8 4.5 50 51.7Physiological clustering

5.0 33.7 44.9 −3.4 33.1 57.7

Figure 12. (Left) RMSE versus CNR and (right) bias versus CNR curves for post-smoothing approach and physiological clustering approach. Points on the post-smoothing curve correspond to different cutoff frequencies of Butterworth filter ( )wcutoff and points on the physiological clustering curve correspond to different values of penalty parameter (γ). In both plots, the left-most point corresponds to =w 0cutoff and γ = 0.

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of noise-induced bias in non-linear estimation tasks (Slifstein and Laruelle 2000), which can result in bias reduction when noise is handled more appropriately. We also studied the ability to resolve perfusion defects (quantified by CNR) as a function of overall RMSE and overall bias (figures 8, 12 and 15). This analysis demonstrates that physiological clustering can better resolve perfusion defects without compromising the robustness of the estimated K1 parametric images, both of which are important in clinical decision making (diagnosis, prognosis and treatment response assessment).

In our simulations, we randomly sampled the myocardium to select representative TACs to initialize the clustering process (Step 2, Algorithm 1, figure 3). Hence, parametric images obtained from the proposed approach of physiological clustering are not only robust to high noise-levels in the voxel TACs but also robust to the inherent variability in the clustering across multiple noise realizations. We also observed that in MP PET imaging, the performance is less sensitive to the number of clusters than in whole brain imaging (Lu et al 2012), which is

Figure 13. A bar graph showing percentage reduction achieved in overall NSD, bias, and RMSE by varying the number of clusters in the physiological clustering process.

Figure 14. K1 Parametric Images: we show short-axis views of stress uptake images from three patient studies. For the post-smoothing approach, K1 images were obtained by varying cutoff frequencies: (L-to-R) =w 0,  1,  0.9,  0.8,  0.7,  0.6,  0.5,  0.4,  0.3, 0.2,  and 0.1cutoff cycles cm−1 where =w 0cutoff corresponds to no post-smoothing, and, for the physiological clustering approach K1 images were obtained by varying the penalty parameter: (L-to-R) γ = × × × × × × × ×− − − − − − − −  0,   2 10 ,   5 10 , 1 10 , 5 10 , 8 10 , 1 10 , 3 10 , 6 10 ,6 6 5 5 5 4 4 4 1 × 10−4, and 1 × 10−3 where γ = 0 corresponds to no spatial constraint. Perfusion defects are shown by yellow arrows.

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Figure 15. CNR versus uptake rate curves for post-smoothing approach and physiological clustering approach. Points on the post-smoothing curve correspond to different cutoff frequencies of Butterworth filter ( )wcutoff and points on the physiological clustering curve correspond to different values of penalty parameter (γ). In both plots, the right-most point corresponds to =w 0cutoff and γ = 0. Uptake rates were computed in a small ROI (~20 voxels) in the vicinity of the perfusion defect.

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understandable given the greater number of distinct regions in the latter. Some variability in the quantitative performance is attributed to random initializations of the clustering algorithm as discussed before. Apart from the initialization of the clustering algorithm, choosing a very large number of clusters may lead to sub-optimal quantitative performance as shown in figure 13.

Functional clustering of non-transmural perfusion defects presents a bigger challenge com-pared to the case of transmural defects. It becomes increasingly difficult to segment out the defect for wall fractions between 0.5 and 1. Although our simulations showed superior recov-ery of non-transmural defects in K1 parametric images (figure 9), we did not simulate blurring artifacts due to motion which would add another layer of complexity. Since our proposed approach performs nonlinear regression on non-smoothed dynamic images, it is advised to perform motion correction before parametric estimation to generate more accurate dynamic images. In any case, we did not perform motion correction or resolution modeling on the clini-cal dataset and still observed superior qualitative and quantitative performance compared to post-smoothing approach (figures 14 and 15).

6. Future work

6.1. K-means clustering algorithm

We used the K-means clustering algorithm to segment dynamic PET images into function-ally homogeneous clusters. K-means clustering, though extremely simple and straightforward, is sensitive to the initialization of representative TACs due to a non-convex objective func-tion, requires a priori specification of the number of clusters, and the similarity metric does not incorporate spatial proximity of voxels. Non-convexity can be addressed by a convex relaxation of the K-means objective function (Hocking et al 2011, Lindsten et al 2011, Chi and Lange 2013). Estimating the true number of clusters is a common problem in clustering algorithms and there exist a number of efforts to address this issue, including Akaike informa-tion criterion (Akaike 1974), Bayesian information criterion (Pelleg and Moore 2000), and cross-validation (Wang 2010) etc. The similarity metric in (5) can be modified to account for spatial proximity of voxels. Such metrics are found in normalized cuts and spectral clustering algorithms (Tatiraju and Mehta 2008, Mouysset et al 2013, Zbib et al 2015). Incorporating spatial coordinates of voxels in the similarity metric encourages contiguous clusters, which is physiologically more plausible; however, this benefit comes at the cost of optimizing addi-tional parameters, which we intend to study in the future.

6.2. Image derived input function

In our experiments, we extracted the input functions by manually placing an elliptical ROI in the LV and RV blood pools. There are several approaches in the scientific literature which automatically extract the input functions from dynamic PET images e.g. utilizing factor analy-sis (FA) (Wu et al 1995, El Fakhri et al 2005), non-negative matrix factorization (NMF) (Lee et al 2001, Ahn et al 2004), and independent component analysis (ICA) (Karpikov et al 2012). Automatically extracted input functions using matrix factorization techniques may further enhance performance of the proposed approach.

6.3. Resolution degrading factors

Spatial resolution of 82Rb PET images is impacted by the high positron range, photon non-collinearity and inter-crystal scattering (Rahmim et al 2008, Le Meunier et al 2010, 2011).

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Resolution is further degraded due to patient motion (cardiac motion, respiratory motion, and bulk motion) (Rahmim et al 2013). It has been shown before (Rahmim et al 2010) that these degrading factors resulted in ~48% underestimation of myocardial activity, leading to inaccurate estimates of parametric images; however, when correcting for all degrading effects (including motion) except for 82Rb positron range, ~23% underestimation continued to remain. In our proposed approach of physiological clustering, kinetic parameters are estimated from pre-smoothed dynamic images which influence the accuracy of the estimated parameters. This reinforces the need of to incorporate resolution modeling and appropriate motion correction schemes in the reconstruction process to obtain high quality perfusion images which may lead to more accurate parametric images.

6.4. Direct 4D parametric imaging

Direct 4D parametric imaging reduces noise in the parametric images by incorporating kinetic modeling within the image reconstruction framework (Tsoumpas et al 2008, Rahmim et al 2009). There have been a few approaches to this in the specific context of MP PET imaging. Rakvonghai et al (Rakvongthai et al 2013) proposed a direct 4D reconstruction method solved using preconditioned conjugate gradient to estimate uptake rate images for 18F-Flurpiridaz radiotracer. Rahmim et al (Rahmim et al 2014c) used numerical methods with an optimiza-tion transfer approach to estimate K1 parametric images for 82Rb radiotracer. Su et al (Su et al 2013) used a modified EM method to estimate K1 parametric images for 13N-Ammonia radiotracer. A quadratic penalty function is commonly employed to minimize intensity differ-ences between neighboring voxels in the image space (Rakvongthai et al 2013). Future work involves incorporating cluster-based priors, such as utilized in the present work, within the direct 4D reconstruction to obtain substantial noise reduction with minimal increase in bias.

7. Conclusion

Parametric imaging based on physiological clustering was shown to clearly outperform con-ventional parameter estimation techniques, generating images with enhanced quantitative noise versus bias trade-off performance, as well as enhanced CNR with minimal reduction of bias. It achieved this by utilizing kinetic information from physiologically similar voxels. In this framework, functionally similar voxels are binned in the same cluster with each cluster represented by mean kinetics. The mean kinetic information for each cluster is enforced as a spatial constraint in the voxel-wise parameter estimation process thereby forcing (by tun-ing the penalty parameter γ) the estimated parameters to be close to the mean kinetics of its representative cluster. The proposed framework minimizes resolution degradation as no spatial smoothing of heterogeneous functional regions is performed. Overall, the proposed approach showed enhanced qualitative and quantitative performance as verified in simulations and patient data.

Acknowledgments

This work was in part supported by the 2014 Bradley-Alavi fellowship (Hassan Mohy-ud-Din, PhD) from SNMMI, and the NIH grant 1S10RR023623. The authors wish to thank Dr Thomas Schindler and Dr Nicolas Karakatsanis for helpful discussions, Corina Voicu for help with extraction of clinical data, as well as Dr Elcin Zan, Dr Esther Mena Gonzales, and Dr Riham Elkhouli for help with analysis of clinical data, and the reviewers for their valuable comments.

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