Generalized whole-body Patlak parametric imaging for enhanced quantification in clinical PET

Abstract

We recently developed a dynamic multi-bed PET data acquisition framework to translate the quantitative benefits of Patlak voxel-wise analysis to the domain of routine clinical whole-body (WB) imaging. The standard Patlak (sPatlak) linear graphical analysis assumes irreversible PET tracer uptake, ignoring the effect of FDG dephosphorylation, which has been suggested by a number of PET studies. In this work: (i) a non-linear generalized Patlak (gPatlak) model is utilized, including a net efflux rate constant k_loss, and (ii) a hybrid (s/g)Patlak (hPatlak) imaging technique is introduced to enhance contrast to noise ratios (CNRs) of uptake rate K_i images. Representative set of kinetic parameter values and the XCAT phantom were employed to generate realistic 4D simulation PET data, and the proposed methods were additionally evaluated on 11 WB dynamic PET patient studies. Quantitative analysis on the simulated K_i images over 2 groups of regions-of-interest (ROIs), with low (ROI A) or high (ROI B) true k_loss relative to K_i, suggested superior accuracy for gPatlak. Bias of sPatlak was found to be 16–18% and 20–40% poorer than gPatlak for ROIs A and B, respectively. By contrast, gPatlak exhibited, on average, 10% higher noise than sPatlak. Meanwhile, the bias and noise levels for hPatlak always ranged between the other two methods. In general, hPatlak was seen to outperform all methods in terms of target-to-background ratio (TBR) and CNR for all ROIs. Validation on patient datasets demonstrated clinical feasibility for all Patlak methods, while TBR and CNR evaluations confirmed our simulation findings, and suggested presence of non-negligible k_loss reversibility in clinical data. As such, we recommend gPatlak for highly quantitative imaging tasks, while, for tasks emphasizing lesion detectability (e.g. TBR, CNR) over quantification, or for high levels of noise, hPatlak is instead preferred. Finally, gPatlak and hPatlak CNR was systematically higher compared to routine SUV values.

1. Introduction

Dynamic PET imaging allows for acquisition of valuable spatiotemporal activity concentration measurements to enable in-vivo tracking of the physiological time course of the radiotracer. The obtained information, when utilized by tracer kinetic modeling methods, can then enable parametric PET imaging, a complementary framework of enhanced quantification (Carson et al 2005, Zaidi et al 2006, Bentourkia and Zaidi 2007). The quantitative benefits of dynamic PET have motivated an increasing number of research studies covering a wide spectrum of medical imaging applications including in oncology (Hawkins et al 1992a and b, Messa et al 1992, Fischman and Alpert et al 1993, Eary and Mankoff et al 1998, Gupta et al 1998, Delbeke 1999, Dimitrakopoulou-Strauss et al 2003, Prytz et al 2006).

Nevertheless, dynamic PET acquisitions have been primarily constrained to single-bed axial field-of-views (FOVs), due to preference for continuous temporal sampling of the FOV (Torizuka et al 1995 and 2000). Thus, dynamic PET has been mainly associated with oncology studies, involving specific tumor types, as well as cardiac or neurological PET studies, where single-bed acquisitions are adequate (Jagust et al 1991, Lortie et al 2007). As a result, dynamic PET has not yet been translated into clinical routine, where whole-body (WB) acquisitions are important. In particular, oncology would appreciate the enhanced quantification across WB FOVs, as the likelihood for metastases from the primary tumor location to other regions dictates multi-bed diagnostic evaluations (Wahl and Buchanan 2002).

On the other hand, single-frame (i.e. static) WB PET imaging, relying on the standardized uptake value (SUV) metric, has been established in routine clinical practice, mainly because of the simplicity of WB PET scan protocols, its sufficient test-retest repeatability and extensively validated clinical utility (Kubota et al 1985, Wahl and Buchanan 2002, Thie 2004). At the same time, SUV can be considered as a semi-quantitative metric, since it does not account for tracer concentration in blood (input function) and largely depends on the post-injection scan time and the current metabolic status of each patient (Keyes 1995, Huang 2000, Thie 2004, Paquet et al 2004, Durand and Besson 2015). As a result, SUV PET evaluations and treatment response assessments may suffer from poor accuracy (Hoekstra et al 2000 and 2002, Adams et al 2010, Boellaard et al 2011, de Langen et al 2012). A number of studies have attempted to partially overcome these limitations by employing either (i) delayed or dual-time point WB SUV PET imaging (Hustinx et al 1999, Nakamoto et al 2000, Zhuang et al 2001, Kubota et al 2001, Matthies et al 2002, Nishiyama et al 2006), (ii) various normalization factors for SUV, such as patient weight, lean body mass and body surface area (Zasadny and Wahl 1993, Kim et al 1994) or (iii) simplified kinetic analysis (SKA) methods over regions-of-interest (ROIs) (Hunter et al 1996, Sadato et al 1998, Graham et al 2000, Freedman et al 2003, Sundaram et al 2004). Nevertheless, all methods above heavily rely on very specific assumptions to retain simplicity for clinical protocols. Examples include assumption of the time integral of input function as being proportional to injected dose divided by lean body mass, or that the proportion of non-phosphorylated to phosphorylated FDG is minimal at time of imaging, which may not be valid, compromising the accuracy of these methods (Freedman et al 2003, Sundaram et al 2004).

Recently, we proposed a PET acquisition and imaging framework enabling clinically feasible WB dynamic PET imaging, thus combining the benefits of multi-bed FOVs with the ability to obtain images of Patlak kinetic macro-parameters (Karakatsanis et al 2013a and c). Quantitative analysis demonstrated enhanced tumor detectability over SUV in regions exhibiting high background signal, such as the liver. In this framework, the standard Patlak (sPatlak) linear graphical analysis had been selected as a robust modeling approach to directly estimate, from the reconstructed dynamic WB PET images, the tracer influx or uptake rate constant K_i and blood distribution volume V (Patlak et al 1983).

The sPatlak graphical analysis method arrives at a linear model equation only when an irreversible 2-tissue compartment tracer kinetic model is assumed, a common model for fluorodeoxyglucose (FDG) tracer in human organs and tumors (Phelps et al 1979, Patlak et al 1983, Kelloff et al 2005). However, a number of studies have reported clinical kinetic parameter values suggesting a non-negligible degree of apparent FDG uptake reversibility, or dephosphorylation, for many normal organs such as the liver, brain and lung as well as for tumors (Anchors et al 1977, Gallagher et al 1978, Phelps et al 1979, Huang et al 1980, Messa et al 1992, Torizuka et al 1995 and 2000, Nelson et al 1996, Delbeke 1999, Zhuang et al 2001, Okazumi et al 1992 and 2009, Dimitrakopoulou-Strauss et al 2006, Prytz et al 2006). As the sPatlak model inherently assumes irreversibility, it is forced to underestimate K_i in regions where a non-negligible underlying reversibility truly exists, to explain the progressive loss of the signal portion originating from the erroneously assumed “irreversible” compartment (Patlak et al 1983, Patlak and Blasberg 1985). Consequently, when reversibility is neglected from model assumptions, the quantitative accuracy of K_i images may be compromised in certain regions, with negative implications for highly quantitative imaging tasks, such as treatment response monitoring for prognostic or theranostic purposes.

On the other hand, a reversible 2-tissue compartment kinetic model can properly account for underlying tracer dephosphorylation. Thus, an extended version of the sPatlak model, accepting the possibility of relatively small uptake reversibility, could enhance its accuracy by attributing the observed progressive signal intensity loss to an additional kinetic parameter rate constant: the tracer efflux rate constant. In fact, the theoretically expected K_i underestimation, when an irreversible compartment replaces a reversible one, has been also confirmed by a number of clinical dynamic PET studies, thus further suggesting the possibility for underlying FDG uptake reversibility in certain normal organs and tumors (Lammertsma et al 1987, Messa et al 1992, Choi et al 1994, Hasselbalch et al 2001, Graham et al 2002, Wu et al 2003, Sayre et al 2011, Hoh et al 2011).

Furthermore, although reversible FDG uptake kinetics are not as common for tumors as for normal tissues (Fischman and Alpert 1992, Messa et al 1992, Hawkins et al 1992b, Okazumi et al 1992, Nelson et al 1996, Huang et al 2000, Graham et al 2000, Zhuang et al 2001, Lin et al 2005, Prytz et al 2006), certain tumor types with non-negligible reversibility have been reported, such as in the case of hepatocellular carcinomas (HCC) tumors (Messa et al 1992, Torizuka et al 1995). As a result, K_i estimates may be under- or even over-estimated and in various degrees at the target and background regions, depending on the presence (or not) of reversible kinetics in each region, thus affecting not only image quantification but also tumor-to-background contrast, potentially compromising tumor detectability as well. Moreover, as many normal tissues across the body reportedly exhibit some degree of reversibility, the likelihood of erroneous K_i quantification within PET FOV becomes higher with multi-bed acquisitions. Consequently, the need to include reversibility in the kinetic model analysis becomes even more evident in WB PET parametric imaging applications.

Therefore, in this study, we propose a novel generalized Patlak (gPatlak) multi-bed framework to enhance quantitative WB PET imaging including in regions exhibiting non-negligible uptake reversibility. We focus on oncology applications when the imaging task involves tumor detection and staging or treatment response assessments, but our findings could also apply to other tasks, such as quantitative PET-based differentiation of malignant vs. benign tumors and WB imaging of inflammation, infection, etc. (Hübner et al 1996, Dimitrakopoulou-Strauss et al 2002, Basu and Alavi 2007, Sanz and Fayad 2008, Oo et al 2013, Choi et al 2013). In addition, we currently evaluate our proposed method for ¹⁸F-FDG, the most common PET oncological radiotracer with well-modeled kinetics for a range of normal tissues and tumor regions across the human body (Phelps et al 1979, Hustinx et al 2002), though this approach could also apply to different radiotracers as well, such as ¹⁸F-FLT (Been et al 2004), ¹⁸F-FMISO (Thorwarth et al 2005), ¹⁸F-Fluoride (Grant et al 2008, Siddique et al 2011, Siddique et al 2012 and 2014), ¹⁸F-Choline (Husarik et al 2008) and others (Groves et al 2007).

The extended Patlak method, originally proposed as theory and for region-based analysis by Patlak and Blasberg (1985), is equipped with an additional net efflux rate constant, k_loss, to properly account for tracer net uptake reversibility from the trapping or metabolic tissue compartment to the blood plasma. In fact, this model has been previously used in few PET studies involving FDG (Lodge et al 1999), ¹⁸F-Fluorodopa brain (Holden et al 1997, Doudet et al 1998, Sossi et al 2001) as well as ¹⁸F-Fluoride bone imaging (Siddique et al 2014) but only for ROI-based analysis and limited to a single bed.

In the present study, for the first time to our knowledge, an extended gPatlak method is implemented and assessed: 1) in the context of parametric imaging, i.e. involving kinetic parameters estimation at the voxel level, and 2) for WB dynamic acquisitions. The presented framework, also includes application of the basis function method (BFM) to effectively linearize the parameter estimation process in a computationally efficient algorithm (Gunn et al 1997). In addition, a hybrid parametric imaging method (hPatlak) is proposed which selectively applies either standard or generalized Patlak method at each voxel, according to a Patlak correlation-based binary classification scheme, to enhance tumor contrast-to-noise ratio (CNR) and detectability in K_i images at the cost of quantification compared to gPatlak.

2. Materials and Methods

2.1. Generalized Patlak graphical analysis

The linear Patlak model utilizes dynamic PET image data and the time course of blood plasma tracer concentration (input function) to estimate through linear regression the kinetic macro-parameters of tracer net influx rate constant K_i and the total blood distribution volume V at each voxel employing the following model equation (Patlak et al 1983):

$$\frac{C(t_n)}{C_P(t_n)}=K_i\frac{{\int }_0^{t_k}{C}_P(\tau )d\tau }{C_P(t_k)}+V,t_n>t^{\ast },n=1\dots N$$ (1)

where C (t) is the measured time activity curve (TAC) at each voxel, C_P(t) is the blood plasma TAC or input function estimated either from an image ROI or from blood sampling, t_k with k=1…m denote the mid-time points for the N dynamic PET frames/measurements and t* is the time after which relative kinetic equilibrium between the blood and the reversible compartment is attained. The standard Patlak equation describes the linear relationship between (i) the ratio of the measured tissue C(t) to the plasma C_P(t) activity concentration and (ii) the ratio of the running time integral of the plasma TAC to the plasma TAC, the latter also denoted as “normalized” or “stretched” time (Holden et al 1997).

Later, Patlak and Blasberg (1985) introduced an extended and more general graphical analysis model to account for potential reversible uptake rate. Thus, an additional kinetic parameter was introduced, here denoted as k_loss, to describe the net rate constant for metabolized tracer loss to the blood plasma (net efflux rate constant). By assuming k_loss ≪ k_i the following non-linear Patlak equation can be obtained for the kinetic analysis of m dynamic frames:

$$\frac{C(t_n)}{C_P(t_n)}=K_i\frac{{\int }_0^{t_k}{e}^{-k_{\mathit{loss}}(t_n-\tau )}{C}_P(\tau )d\tau }{C_P(t_n)}+V,t_n>t^{\ast },n=1\dots N$$ (2)

Where K_i = K₁k₃/(k₂ + k₃) and k_loss = k₂k₄/(k₂ + k₃) as defined by Patlak and Blasberg (1985).

As the newly introduced k_loss parameter is included in an exponential term, the kinetic parameter estimation problem now becomes non-linear. As it will be demonstrated later, the accuracy of the two Patlak models depends on the ratio of k_loss/K_i. Sossi et al (2001) in particular also studied, in the case of region-based ¹⁸F-fluorodopa brain PET tracer kinetic analysis for Parkinson’s disease evaluations, the effect of this ratio, which they term as “effective dopamine turnover” (EDT).

The standard Patlak model assumes a 2-tissue compartment model involving 3 parameter rate constants (K₁, k₂ and k₃) with the second compartment considered irreversible, i.e. k₄ = 0 (Figure 1a). The measured tracer tissue concentration C(t) is defined as the sum of the extravascular C_e(t) and the metabolized C_m(t) tracer tissue concentrations. By reformulating equation 1 that:

$$C(t_n)=K_i{\int }_0^{t_n}{C}_P(\tau )d\tau +V{C}_P(t_n)=K_i\otimes C_P(t_n)+V{C}_P(t_n),t_n>t^{\ast },n=1,\dots ,N$$ (3)

where ⊗ denotes the mathematical operation of single-dimensional (1D) convolution in the time domain.

Figure 1.

(left) Two-tissue compartment kinetic models (a) without and (c) with reversibility for FDG tracer

On the other hand, the non-linear Patlak model assumes a two compartment kinetic model with a nonnegative k₄ rate constant (Figure 1, bottom left). Similarly, by restructuring equation 2 we have:

$$\begin{array}{l}C(t_n)=K_i{\int }_0^{t_n}{e}^{-k_{\mathit{loss}}(t_n-\tau )}{C}_P(\tau )d\tau +V{C}_P(t_n)\\ =(K_i{e}^{-k_{\mathit{loss}}{t}_n})\otimes C_P(t_n)+V{C}_P(t_n),t_n>t^{\ast },n=1,\dots ,N\end{array}$$ (4)

2.2. Whole-body non-linear parametric image estimation

In the case of the linear sPatlak model (equation 1 or 3), the two parameters of interest, K_i and V, can be estimated directly using the Ordinary Least Squares (OLS) linear regression method. Let us consider the following sPatlak regression model:

$${\mathit{Y}}^j={\mathit{X}}_s{\mathit{\beta }}^j+{\mathit{\epsilon }}^j$$ (5a)

where:

$${\mathit{Y}}^j={[\begin{array}{ccc}{C}^j(t_1)& \dots & C^j(t_N)\end{array}]}^T,{\mathit{X}}_s=\left[\begin{array}{cc}{\int }_0^{t_1}{C}_P(\tau )d\tau & C_P(t_1)\\ ⋮& ⋮\\ {\int }_0^{t_N}{C}_P(\tau )d\tau & C_P(t_N)\end{array}\right],{\mathit{\beta }}^j={\left[\begin{array}{cc}{\mathit{K}}_i^j& V^j\end{array}\right]}^T$$

Note in equation 5a that the input function measurements in standard Patlak model matrix X_s and the TACs column vector Y are linearly related with the unknown sPatlak kinetic macro-parameters β (slope K_i and intercept V). This relationship can be expanded into the following set of sPatlak N bilinear equations:

$$C(t_n)=K_i{\int }_0^{t_n}{C}_P(\tau )d\tau +V{C}_P(t_n)+{\epsilon }_n,t_n>t^{\ast },n=1,\dots ,N$$ (5b)

However, for the non-linear gPatlak model (equation 2 or 4), we propose an efficient implementation of the Basis Function Method (BFM) (Gunn et al 1997) as we have presented in a preliminary work (Karakatsanis et al 2013d). Initially, a pool of discrete candidate k_loss values has to be determined. The range [10⁻⁵, 10⁻¹] of M = 10⁴ candidate k_loss values, in units of sec⁻¹, is recommended, according to equation 7 and a wide collection of k-values, as reported in the literature (Torizuka et al 1995 and 2000, Dimitrakopoulou-Strauss et al 2006, Okazumi et al 1992 and 2009), a subset of which was later utilized to conduct simulations (Table I). Then a set of M basis functions, each representing one of the M candidate k_loss values, is constructed to effectively linearize the non-linear parameters estimation problem:

$${\omega }_m(t_n)=C_P(t_n)\otimes e^{-k_{\mathit{loss}}^m{t}_n},m=1\dots M,n=1\dots N$$ (6)

Table I.

K Rate Parameters.

Regions	K1	k2	k3	k4	VB
Normal Liver	0.864	0.981	0.005	0.016	-
Liver Tumor	0.243	0.78	0.1	0	-
Normal Lung	0.108	0.735	0.016	0.013	0.017
Lung tumor	0.301	0.864	0.097	0.001	0.168
HCC Tumor	0.283	0.371	0.057	0.012
Myocardium	0.6	1.2	0.1	0.001	-

After replacing the exponential term in equation 4 with each of the M basis functions (equation 6), the following set of M bilinear Patlak equations is constructed:

$${\mathit{Y}}^j={\mathit{X}}_m^j{\mathit{\beta }}_m^j+{\mathit{\epsilon }}_m^j,m=1\dots M$$ (7a)

The definitions of the generalized Patlak model matrix ${\mathit{X}}_m^j$ and the unknown kinetic macro-parameters vector ${\mathit{\beta }}_m^j$ of Patlak slope $K_i^{j,m}$ and intercept V^j,m depend, each time, on the currently selected m^th basis function ω_m as follows:

$${\mathit{X}}_m^j=\left[\begin{array}{cc}{\int }_0^{t_1}{e}^{-k_{\mathit{loss}}^{j,m}(t_1-\tau )}{C}_P(\tau )d\tau & C_P(t_1)\\ ⋮& ⋮\\ {\int }_0^{t_N}{e}^{-k_{\mathit{loss}}^{j,m}(t_N-\tau )}{C}_P(\tau )d\tau & C_P(t_N)\end{array}\right],{\mathit{\beta }}_m^j={\left[\begin{array}{cc}{K}_i^{j,m}& V^{j,m}\end{array}\right]}^T$$ (7b)

Or, equivalently, we have M different sets of equations, where each set is composed of N bilinear Patlak formulas:

$$C(t_n)=K_i^{j,m}{\omega }_m(t_n)+V^{j,m}{C}_P(t_n)+{\epsilon }_n,t_n>t^{\ast },m=1\dots M,n=1,\dots ,N$$ (7c)

Thus the original non-linear parameter estimation problem (equation 4) has now been translated to a set of n linear estimation problems (equation 7c), as many as the initially selected number of Basis functions or candidate k_loss values. Therefore, the standard linear OLS regression can then be applied to each of the n linearized Patlak equations in order to estimate the respective set of ${\mathit{K}}_i^j$ and V^j parameters for the selected $k_{\mathit{loss}}^j$ value. Subsequently, the corresponding residual sum of squared error RSS_j between our measurements Y and our estimates Ŷ_j = Xβ̂_j is calculated:

$${\mathit{RSS}}_{j,m}={(\mathit{Y}-\mathit{X}{\widehat{\mathit{\beta }}}_m^j)}^T(\mathit{Y}-\mathit{X}{\widehat{\mathit{\beta }}}_m^j)$$ (8)

The paired set of estimated parameters ${\widehat{\mathit{\beta }}}_m^j$ and $k_{\mathit{loss}}^{j,m}$, for which the minimum RSS_j,m, for a fixed j^th voxel over all M RSS scores, was observed, is finally selected as the final BFM parametric estimate:

$${\widehat{\mathit{\beta }}}_{\mathit{BFM}}^j=\left\{{\widehat{\mathit{\beta }}}_{m_{\mathit{opt}}}^j,k_{\mathit{loss}}^{{j,m}_{\mathit{opt}}}\right\},m_{\mathit{opt}}=\mathit{arg}\underset{m}{min}\{{\mathit{RSS}}_{j,m}\},{\widehat{\mathit{\beta }}}_{m_{\mathit{opt}}}^j={\left[\begin{array}{cc}{K}_i^{{j,m}_{\mathit{opt}}}& V^{{j,m}_{\mathit{opt}}}\end{array}\right]}^T$$ (9)

By repeating the BFM estimation algorithm for each voxel, a set of K_i, k_loss and V parametric images can be produced in the end.

2.3 Hybrid generalized Patlak imaging

Multi-bed dynamic PET acquisition involves short frames (e.g. 45sec/bed in this study) non-continuously acquired over time at each bed position and with relatively large time gaps between the frames, thus tending to enhance noise levels compared to single-bed dynamic protocols. In addition, gPatlak method is nonlinear involving three parameters as opposed to two for the linear sPatlak. As a result, noise may be considerable for gPatlak, though it is still expected to be lower than for full compartmental kinetic analysis (Carson et al 2005). For that reason, although gPatlak might enhance K_i estimates at the tumor ROIs, the respective (tumor-to-background) contrast to (background) noise ratio (CNR) metric, an important detectability index, may not be similarly enhanced, due to noise elevation at the background.

Therefore, in this work we also propose a novel hybrid parametric imaging method, namely hybrid Patlak (hPatlak), involving the selective application of either sPatlak or gPatlak analysis at every voxel, aiming for higher K_i CNR scores than those achieved by gPatlak or sPatlak alone, by exploiting the expected higher tumor contrast of the former and the low background noise of the latter within the same analysis framework. Consequently, hPatlak is not expected to be as quantitative as gPatlak, although it is more accurate than sPatlak. Our main motivation with hPatlak is to provide a complimentary method that would improve tumor detectability by targeting enhancement of tumor CNR scores.

The selection of the applied Patlak method at each voxel is based on the quantitative criterion of weighted Patlak correlation-coefficient WR, an index of the degree of linear correlation between the dynamic PET measurements and the sPatlak assumptions. By considering the sPatlak analysis, equation 1 can be rewritten, for a given dynamic frame n, as follows:

$$y_n=K_i{x}_n+V,y_n=\frac{C(t_n)}{C_P(t_n)},x_n=\frac{{\int }_0^{t_n}{C}_P(\tau )d\tau }{C_P(t_n)},n=1\dots N$$ (10)

Then the weighted voxel-wise Patlak correlation coefficient WR can be calculated as follows [16]:

$$WR=\frac{{\sum }_n{w}_n{\sum }_k{w}_n{x}_n{y}_n-{\sum }_n{w}_n{x}_n{\sum }_n{w}_n{y}_n}{\sqrt{[{\sum }_n{w}_n{\sum }_n{w}_n{x}_n^2-{({\sum }_n{w}_n{x}_n)}^2][{\sum }_n{w}_n{\sum }_n{w}_n{y}_n^2-{({\sum }_n{w}_n{y}_n)}^2]}}$$ (11)

The weights are defined as a function of the time duration Δt_n and sinogram total counts c_n in frame n:

$$w_k=\frac{{(\mathit{\Delta }{t}_n)}^2}{c_n},\mathit{\Delta }{t}_n\stackrel{\text{def}}{=}{t}_n^{\mathit{end}}-t_n^{\mathit{start}},n=1\dots N$$

After repeating the previous Patlak correlation calculations for every voxel TAC, a weighted Patlak correlation image (WR-image) can be generated from the acquired set of dynamic PET images.

The WR correlation metric essentially quantifies how well correlated each measured voxel TAC is with the sPatlak assumptions. In a hypothetical noise-free scenario, all measured voxel TACs would highly correlate with sPatlak model assumptions, regardless of the presence of uptake reversibility. Also, WR would approach unity in all voxel regions where underlying k_loss is nearly zero. However, in real dynamic PET scans, particularly when extended to the WB, noise can be considerably high, due to the short frames and the sparse temporal sampling at each bed. As the noise increases, WR is expected to drop in values. Thus, it is reasonable to expect that regions of lower uptake and, therefore, higher noise, such as most of the background normal tissue regions, will exhibit lower Patlak correlation coefficients, while regions of higher uptake and therefore better count statistics and lower noise, such as myocardium and most of suspected tumors, will be associated with relatively higher correlation coefficients. In fact, we have systematically observed that high Patlak correlation clusters are likely associated with voxel TACs of low noise, usually corresponding to tumors or high uptake regions, while low Patlak correlation cluster often characterizes voxel TACs in regions of low uptake, such as in tumor background (Zasadny and Wahl 1996, Karakatsanis et al 2013a and d). In addition, WR correlation may be also diminished for voxel TACs affected by bulk body motion across the different WB dynamic frames.

As a result, the proposed WR correlation-coefficient metric may act as an effective binary classification method between regions of low and high levels of noise or motion contamination. This feature can be utilized to selectively apply the less precise but more accurate non-linear BFM gPatlak parameter estimation method only to the highly correlated and statistically more reliable voxel TACs. By contrast, the more robust but potentially not as accurate sPatlak method can be selected for the remaining less correlated voxel TACs. An extra degree of freedom is introduced with the proposed hPatlak regression method thanks to the role of WR threshold, a free parameter quantitatively determining the level of Patlak correlation above which a voxel TAC can be characterized as of high correlation. Based on our clinical findings, we propose an initial range of possible WR threshold values between 0.75 and 0.98.

Initially, a user-defined correlation threshold is picked to classify the voxels of the WR-image into 2 clusters of relatively high (hWR cluster) and low (lWR cluster) Patlak correlation. Subsequently, gPatlak is assumed for the hWR voxel TACs, while sPatlak is used for the lWR voxel TACs. Non-linear BFM is employed for the estimation of the K_i, k_loss and V parameters in the hWR cluster, while OLS is applied for K_i and V parameter estimation in the lWR cluster. Thus, the k_loss parameter is assumed to be zero for the lWR voxels. In the end, a set of hybrid K_i, k_loss and V parametric images is created. A flow chart illustrating the principal steps of the proposed hybrid parametric imaging method is presented in figure 2.

Figure 2.

Flow chart illustrates the WR-based hPatlak imaging method. In the WR image, all voxels have been assigned either to the hWR (white) or lWR (black) cluster, according to their calculated WR coefficient value and a user-selected WR threshold (0.95 in this case). The depicted WR image classification may considerably change for different WR thresholds and patients but hWR cluster usually tends to include the high focal uptake regions.

Obviously, hPatlak images will depend on the selection of the WR threshold. Therefore, for the clinical routine application of hPatlak through an image analysis software, we suggest performing this method for a predetermined number of WR thresholds, within our recommended range of [0.75 0.98]. Then, a sliding bar interface can be provided to enable online inspection of the corresponding hPatlak images while sliding over the predetermined WR threshold values. Thus, users may instantly visualize the effect of WR threshold on the hPatlak K_i images over a range of correlation-coefficients and select the most appropriate threshold for the targeted imaging task. The extra computational cost of such type of analysis is negligible, as hPatlak images can be readily synthesized from the generated gPatlak and sPatlak images.

2.4. Simulation and Clinical Studies

2.4.1. Dynamic Multi-bed Acquisition

In both simulations and clinical studies, a previously validated dynamic multi-bed PET acquisition protocol was employed, after being optimized for clinically feasible acquisitions (Karakatsanis et al 2011) and according to the system specifications of current commercial clinical PET/CT scanners. The protocol consists of:

1. an initial single-bed dynamic scan over the heart to acquire the initial phase (first 6-min post-injection) of the input function, followed by

2. a sequence of 6 WB passes, corresponding to a total time window of 10–45min post-injection.

All bed positions were scanned in step-and-shoot mode for 45sec/bed to ensure enough counts per dynamic frame while allowing for a sufficient number of 6 passes in the clinically available time window (10–45min post-injection) before the SUV static PET WB scan (figure 3).

Figure 3.

Flowchart of the two WB dynamic acquisition protocols involving an initial 6min single-bed dynamic scan over the heart to acquire the initial part of the input function followed by m dynamic WB passes scanned in the same direction (head-to-feet in this case), each consisting of 6 beds, each scanned for 45sec.

In addition, two commercially available clinical PET/CT scanners were used for performance evaluation of the proposed method on clinical WB dynamic data. Initially, the GE Discovery RX scanner was utilized where the optimization of the acquisition protocol had been performed (Karakatsanis et al 2011), while later the more recent Siemens Biograph mCT scanner was employed to assess the performance of the proposed methods on a state-of-the-art clinical PET/CT scanner with time-of-flight (TOF) and point-spread-function (PSF) resolution modeling capabilities (Jacoby et al 2011). A preliminary evaluation study of the benefits of TOF and PSF features in WB parametric PET imaging are presented here (Karakatsanis et al 2014b). The same acquisition protocol was applied for both scanners, to ensure a common ground when comparing the results.

2.4.2. Generation of simulated data

An extensive literature review was conducted to collect a characteristic set of FDG k-values (Table I) (Torizuka et al 1995 and 2000, Dimitrakopoulou-Strauss et al 2006, Okazumi et al 1992 and 2009). The kinetic data were used together with a 2-compartment 4-parameter kinetic model (figure 1c) and the Feng input function model (Feng et al 1993) to generate realistic noise-free TACs (figure 4a), that were later assigned to the XCAT phantom (figure 4b). Then, lung and liver spherical tumors of 15mm and 10mm in diameter were placed within the normal XCAT organs of lung and liver respectively. Table 1 includes a wide range of k-micro parameter values corresponding to a k_loss range from zero or negligible up to very high values, i.e. comparable to K_i. We chose to evaluate such a wide range of FDG kinetics in order to validate the proposed gPatlak method under all potential clinical scenarios. We should also note that the very high k₄ and respective k_loss values were considerably less frequent in the literature.

Figure 4.

(a) Noise-free time activity curves for different regions, as generated using a 2 compartment kinetic model and the k-values of Table I. (b) the XCAT phantom where the TACs were assigned to produce the dynamic data.

The resulting dynamic XCAT bed frames were used as input to perform analytic 4D simulations, using STIR open-source package, with realistic levels of Poisson noise determined by the number of counts per frame, as calculated based on the reported sensitivity of the mCT scanner, the 3D acquisition properties (e.g. max. ring difference and span factor), the injected activity, the FDG tracer decay and the time duration of each frame (Thielemans et al 2012). A total of n=15 Poisson noise realizations were produced. Subsequently, the generated 4D simulated data were reconstructed with the ML-EM iterative reconstruction algorithm as implemented within the OS-MAP-OSL application of STIR reconstruction suite (Thielemans et al 2012). Finally the dynamic reconstructed PET images from various iteration steps and all noise realizations were analyzed according to sPatlak, gPatlak and hPatlak methods to produce respective WB K_i images. The generated images were then quantitatively analyzed in terms of noise vs. bias, tumor-to-background (TBR) contrast and CNR performance at selected lung and liver tumor ROIs.

2.4.3. Clinical studies

In addition, the clinical performance of the proposed gPatlak and hPatlak WB K_i imaging methods was evaluated for a set of n=11 WB patient dynamic PET studies, of which the first six were obtained on the GE Discovery RX PET/CT scanner while the last five were acquired on the state-of-the-art Siemens Biograph mCT TOF PET/CT scanner, both installed at the Johns Hopkins PET center. In all clinical cases, the optimized WB dynamic acquisition protocol of figure 3 has been applied. The utilization of two widely applicable modern clinical PET/CT scanners and the acquisition of oncology patient data allowed us to validate the proposed methods under a broad range of acquisition conditions often present in PET imaging centers nowadays.

3. Results

3.1. Tracer kinetic simulation studies

Figure 5a shows, on the left, the true K_i image from a cardiac bed FOV, as constructed by directly assigning the true K_i values, calculated from the k-micro-parameters of Table I and equation 6, to the respective ROIs of normal organs (including liver, lung and myocardium) and tumor types A and B of 15mm (A1, B1) and 10mm (A2, B2) diameter respectively.

Figure 5.

(a) Simulated Patlak K_i images of a cardiac XCAT bed, where 2 sets of spherical homogeneous tumor ROIs are identified: A1 (large), A2 (small) and B1 (large), B2 (small). Same underlying (true) kinetics were originally assigned between A1 and A2 ROIs (type A kinetics) and between B1 and B2 ROIs (type B kinetics) of the true PET frames. True K_i image was derived from direct assignment of true K_i values, while sPatlak and gPatlak noise-free K_i images were generated with sPatlak-OLS and gPatlak-BFM analysis, respectively, on 6 noise-free PET frames.

(b) Noise-free simulated input function and tumor ROI B1 or B2 TACs (same kinetics assigned to B1 and B2), as generated using the 2 compartment model (figure 1c), demonstrating decrease of activity at later times, due to underlying k_loss reversibility.

(c) Noise-free plots for sPatlak and gPatlak models after analyzing simulated tumor ROI B2 TAC, either directly modeled from k-values or extracted from B2 ROI on reconstructed dynamic PET images. The reduced slope of sPatlak and gPatlak plots for the B2 ROI extracted data, with respect to modeled B2 TAC case, is attributed to the partial volume error for the small B2 tumor.

The central and right images of figure 5a represent the estimated sPatlak and gPatlak K_i cardiac images, as derived by application of the respective Patlak model on 6 noise-free dynamic PET cardiac frames, acquired at exactly the time frames corresponding to our proposed WB dynamic PET scan protocol (figure 3). By visual comparison of sPatlak and gPatlak with respect to true K_i image, significant underestimation (bias) in the normal liver and tumor ROIs B1 and B2 is evident for sPatlak, while gPatlak nearly eliminates bias. As the data are noise-free, underestimation of K_i can be attributed to model-related factors and, in particular, the erroneous sPatlak assumption for zero k_loss while, in fact it had been simulated as positive for those two particular ROIs. On the contrary, gPatlak method is able to estimate K_i with a considerably smaller bias after accounting for the underlying presence of non-zero k_loss. Moreover, we plot in figure 5b the Feng input function model (Feng et al 1993) and the simulated noise-free FDG TACs for B2 tumor. The latter was produced after convolving the impulse response function of the 2-tissue compartment 4-parameters model (figure 1c) with the modeled input function (Carson et al 2005). Due to a non-negligible simulated k_loss value, with respect to K_i, B2 tumor TAC is reduced at later times.

Then, we present in figure 5c two groups of noise-free Patlak plots, as derived from simulated tumor B2 kinetics employing either sPatlak or gPatlak graphical analysis. The true k-values from Table I are used as simulations input. For each method, the modeled (true) and ROI-based Patlak curves are plotted. The former is calculated from Patlak analysis on TACs directly modeled from the true k-values after convolving with the Feng input function model, while the latter is derived from ROI-based Patlak analysis on noise-free reconstructed images (ML-EM, 5 iterations), thus suffering from partial volume error (PVE) effect. All 4 Patlak curves are plotted in the same diagram, as gPatlak definition for stretched time variable in horizontal axis applies for sPatlak model too, assuming zero k_loss in that case. By comparing the true and ROI-based plots for each model, an underestimation of the curves slope (K_i) is observed for the latter, which is attributed to PVE, an effect inherent in the tomographic reconstruction process, particularly for smaller tumors (such as B2).

In addition, the modeled (true) data, when plotted in the Patlak diagram assuming zero k_loss, are not forming straight lines, as would have been expected by sPatlak, but instead exhibit a curvature (negative second derivative), resulting in OLS-fitted straight lines of reduced slope (first derivative), thus underestimating K_i. On the other hand, after accounting for true k_loss, the same data are aligned to form a nearly straight line. As a result, the OLS-fitted straight line is then associated with a relatively increased slope or K_i. The same effect between the two models was also observed for the ROI-based analysis.

Since sPatlak plot assumes k_loss=0 (equation 1) while in fact simulation input involves positive k_loss (B2 tumor true k_loss=0.0086), sPatlak is consequently overestimating the stretched time (horizontal axis variable) for a given ratio of tissue to plasma signal (vertical axis variable). On the other hand, gPatlak recovers the expected linear relationship in the Patlak plot by correcting for the loss of metabolized tracer after multiplying the stretched time with the term exp(−k_losst) (equation 2). In figure 5c, the negative curvature of Patlak plot for standard model, as well as its linearization when gPatlak assumptions are utilized, provide intuitive graphical illustrations for the actual causes of sPatlak K_i underestimation and the correcting effect of gPatlak.

The % error (bias) quantitative analysis on the noise-free dynamic data for K_i and k_loss estimates is presented in figures 5 and 6, respectively. The plots suggest a considerable reduction of % bias for both parameters and for a large range of k_loss/K_i ratios when gPatlak replaces sPatlak method. However, our results also confirm the limitations of gPatlak accuracy when true k_loss becomes too high, i.e. comparable to K_i, as also expected from the theory (Patlak and Blasberg 1985). In particular, in figure 6a, the % bias is equally zero for both models when underlying k_loss=0, irrespective of k₃ or K_i, as expected from equations 1 and 3 too. However, as true k_loss increases (assuming only k₄ increases, with the rest of the parameters constant) sPatlak K_i % bias increases dramatically, while for gPatlak it does not exceed 5%, for small k_loss values. In addition, a further decrease of k₃, such that 0<k_loss/K_i <2, assuming K₁ and k₂ constant, causes an additional but relatively milder increase of K_i % bias which however never exceeds 10% for gPatlak. It should be noted that, based on our collection of FDG tracer k-values from literature, the range of k_loss and K_i is always such that 0<k_loss/K_i<1.5 with ratio values higher than 1 rarely observed in real data. In this study, we initialized our investigation with a set of k-values from literature and then carefully expanded our search space to more extreme cases such that 0<k_loss/K_i<2. This allowed us to validate gPatlak K_i bias in extreme cases, i.e. when k_loss is comparable to K_i, in the absence of noise.

Figure 6.

Noise-free K_i estimation error (% bias) curves vs. true k_loss/K_i (by effectively increasing k₄ for each curve) for (b) 10–45m and (c) 30–90m acquisition protocol. The different color curves in both (a) and (b) correspond to a range of k₃ published values such that true 0<k_loss/K_i<2. Also K₁ and k₂ are constant and correspond to the lung tumor case (Table I).

The previous K_i % bias analysis for the same tumor region was repeated for a 30–90min post-injection (p.i.) acquisition time window (figure 6b) confirming our conclusions above. However, the observed sPatlak K_i % bias for the 30–90min protocol is relatively larger than for 0–45min window. On the other hand, gPatlak K_i % bias is consistently low, irrespective of the acquisition time window, demonstrating the quantitative significance of gPatlak when a) dynamic acquisitions last longer, as is inherently the case with WB dynamic protocols, or b) they are delayed to include the time window of static SUV PET scans (60–80min p.i.), as we recently proposed for combined SUV/Patlak imaging (Karakatsanis et al 2015).

A similar % bias analysis for noise-free simulated data was repeated for the k_loss parameter as well. Figure 7a shows a very good correlation between the estimated noise-free k_loss and the true k_loss value, as the latter increases, for a range of true k_loss values commonly found in the literature (0<k_loss<0.04). The % k_loss bias was evaluated with respect to k_loss/K_i ratio, assuming K₁, k₂ and k₃ constant for each error plot and only decreasing k₃ between different plots. As figure 7b illustrates, k_loss % bias never exceeds 10% for 0<k_loss/K_i<1. The same k-values search space as for K_i bias analysis has been used here too.

Figure 7.

(a) Correlation between estimated (noise-free) and true k_loss and (b) plot of k_loss estimation error (% bias) as a function of true k_loss/K_i for 30–90m acquisition protocol. The different color curves in both (a) and (b) correspond to a range of k₃ published values such that true 0<k_loss/K_i<2. Also K₁ and k₂ are constant and correspond to the lung tumor case (Table I).

3.2. Realistic 4D tomographic simulations

The resulting K_i images from the sPatlak, gPatlak and hPatlak analysis of the reconstructed noisy simulated data are presented in figure 8a along with the quantitative noise vs. bias analysis over 4 tumor ROIs: A1, A2, B1 and B2 in figures 8b, 8c, 8d and 8e respectively. For the hPatlak method, 4 different WR thresholds were evaluated starting from 0.8 up to 0.95 with a step of 0.5. In this simulation study, we considered the cardiac bed sufficient as it includes in its FOV both the left-ventricle blood pool region, from which the input function is extracted, as well as all major tissue regions referred in Table 1. Besides, all particularities of WB dynamic acquisitions were retained in our simulation study, as the data were acquired at exactly the times corresponding to our multi-bed dynamic PET acquisition protocol (figure 3).

Figure 8.

Simulation results: (a) parametric K_i images as generated by gPatlak (1^st column), hPatlak of increasing WR thresholds (2^nd – 5^th column) and sPatlak (6^th column) methods. Indirect Patlak imaging has been conducted on reconstructed dynamic PET images after 21 (1^st row) and 105 (2^nd row) MLEM iterations. Normalized standard deviation (NSD) vs. normalized bias (NBias) quantitative analysis for 15 noise realizations and all 6 Patlak methods above on tumor ROIs: (b) A1, (c) A2, (d) B1 and (e) B2.

By visual inspection of K_i images in figure 8, the simulated tumor regions B1 and B2 with non-negligible underlying k_loss, are becoming less visible as we move from left (gPatlak) to right (sPatlak) through the intermediate (hPatlak) images. Meanwhile, visual detectability of tumors A1 and A2 is not significantly affected, since true k_loss is very small in their case. Moreover, the spatial image noise in the background regions is also gradually decreased along the same direction.

The above qualitative evaluations are also confirmed from the subsequent noise vs. bias quantitative analysis. The % bias is gradually increasing with WR thresholds from gPatlak to sPatlak. The most apparent bias enhancement is observed in the case of highly k_loss-reversible B1 and B2 tumors with a total increase of 20–40% from sPatlak to gPatlak, while the corresponding bias enhancement was only 16–18% in the case of A1 and A2 less k_loss-reversible tumors. Furthermore, the stronger PVE in smaller A2 and B2 regions results in relatively larger K_i underestimation and thus % K_i bias enhancement relative to A1 and B1 larger tumor regions. Finally the % noise, as evaluated across the 15 realizations, decreased gradually for hPatlak for higher WR threshold levels, with the total reduction being 10% from gPatlak to sPatlak.

In addition, in terms of TBR (figure 9) and CNR (figure 10) detectability metrics, hPatlak with a WR threshold of 0.95 consistently achieved the best scores, along all MLEM iteration steps evaluated. However, between gPatlak and sPatlak methods, the former method produced higher scores for tumors B1 and B2 while the latter performed better for tumors A1 and A2.

Figure 9.

Simulation results: Target-to-background ratio (TBR) as a function of MLEM iterations between sPatlak, gPatlak and hPatlak (various WR thresholds) K_i images for tumor ROIs: (a) A1, (b) A2, (c) B1 and (d) B2.

Figure 10.

Simulation results: Contrast-to-noise ratio (CNR) as a function of MLEM iterations between sPatlak, gPatlak and hPatlak (various WR thresholds) K_i images for tumor ROIs: (a) A1, (b) A2, (c) B1 and (d) B2.

3.3. Clinical studies

The quantitative performance of the proposed Patlak imaging methods has also been demonstrated on n=11 WB dynamic patient datasets. From these collection, a total of 6 characteristic high uptake (foci) target and respective background ROIs were selected for quantitative analysis, of which the first 4 were extracted from GE RX cases (figures 11–13) while the last two from state-of-the-art Siemens Biograph mCT scans (figure 14) in order to evaluate performance for a range of clinical study set-ups. Except for the mCT patient case, where 2 ROIs were identified (foci 1 and 2 in the thorax, figure 14), for the GE RX cases 1 characteristic ROI has been evaluated for each patient.

Figure 11.

Clinical results: RX patient case #3, scanned on the GE Discovery RX, designated focal uptake ROI in the lung (a) 1^st row: 6 WB dynamic PET passes (45sec/bed) timely-ordered from left to right and a WB static SUV PET image (5min/bed) beginning 60min post FDG injection, all reconstructed after 2 full OSEM iterations with 21 subsets, 2^nd row (left-to-right): WB K_i images derived with gPatlak, hPatlak of increasing WR thresholds and sPatlak methods, as well as a Patlak correlation-coefficient image thresholded at WR=0.95 (white color assigned only to voxels with values >=0.95). Quantitative evaluation on the designated ROI of: (b) K_i ROI mean value and (c) TBR and CNR over all 6 Patlak methods and the SUV image.

Figure 13.

Clinical results: RX patient case #6, scanned on the GE Discovery RX, designated focal uptake ROI in the thyroid (a), (b) and (c) same as figure 11.

Figure 14.

Clinical results: mCT patient case #1, scanned on the Siemens Biograph mCT, two designated focal uptake ROIs in the thorax (a) same as figure 11, (b) same as figure 11 but for two target ROIs: ROI 1 and (c) ROI 2.

Figures 11–14 illustrate, for four patient cases, the reconstructed dynamic PET frames along with the SUV image, as well as the corresponding K_i images of six Patlak methods (gPatlak, hPatlak of 4 different WR thresholds and sPatlak) and the WR image. Moreover, the K_i ROI mean values between all Patlak methods as well as the TBR and CNR scores between the Patlak and SUV images are also presented.

Overall the clinical data analysis agreed with our simulation findings regarding the TBR and CNR performance of the proposed methods. Thus, hPatlak with a 0.95 WR threshold was associated with the best TBR and CNR performance among all other methods, except from GE RX patient case #5 (figure 13c) where gPatlak achieved the best scores, possibly because of potentially high k_loss presence at the particular patient ROI. Furthermore, SUV consistently provided lower TBR and CNR scores, except from mCT patient case #1, where it outperformed gPatlak CNR score, due to gPatlak higher noise levels.

In addition, gPatlak always provided the highest K_i mean value among the presented Patlak methods, suggesting reduced bias after accounting for underlying k_loss reversibility. However, since the ground truth is not known in clinical cases, a relatively higher K_i mean value may not be necessarily associated with less bias. For instance, K_i may be overestimated too in some cases, due to underlying kinetic heterogeneity within each voxel, as discussed later. Overall, any potential K_i overestimation effects are expected to be at least partially counteracted by PVE-induced K_i underestimation.

4. Discussion

4.1. Patlak graphical analysis as the method of choice for WB parametric PET imaging

The proposed gPatlak method retains, unlike full kinetic modeling, much of sPatlak robustness to high levels of noise (Graham et al 2000, Hoekstra et al 2002, Krak et al 2003, Tomasi et al 2012). Furthermore, in terms of quantitative accuracy performance, despite the gPatlak assumption for a small efflux (k_loss) relative to influx (K_i) rate constant (Patlak and Blasberg 1985), our results demonstrated that the relative noise-free gPatlak K_i bias is limited to less than 10% in all cases for a wide set of true k_loss values, ranging from zero up to extremely rare and high values (figure 6). In addition, its WB dynamic PET application remains clinically feasible because, unlike Logan graphical analysis method (Logan et al 1990), it only requires the time integral of the input function from time of injection and not that of every voxel TAC across the entire WB FOV. By contrast, Logan analysis additionally requires TACs for all voxels across the entire WB FOV from injection time, which can depict very rapid dynamics at early times. Application of Logan analysis is thus significantly more challenging in WB dynamic PET studies, because of the need to perform considerably faster passes which may result in significantly higher noise levels for the given sensitivity of clinical PET scanners. For instance, 10 seconds or less per frame at the beginning of conventional single-bed dynamic scans must now be allocated to 6 or more beds in WB mode, thus substantially limiting the scan time per bed),. Moreover, the observed underestimation of the total tissue distribution volume (DV) after applying simple linear regression with Logan technique needs to be evaluated against alternative graphical analysis methods (Zhou et al 2009).

In addition, Patlak was also preferred in this study as it directly estimates K_i, k_loss and V estimates, a set of macro-parameters highly relevant with the metabolic state of normal and tumor tissues and therefore widely applicable for our target oncology application of dynamic whole-body FDG imaging (Castell and Cook 2008). At the same time, we acknowledge the importance of other physiologically relevant macro-parameters, such as the total tissue distribution volume quantifying the capacity of a tracer binding to tissue and directly estimated by Logan method. Thus, we currently examine the future prospect of frameworks supporting additional WB graphical and multi-graphical analysis (Zhou et al 2009 and 2010).

On the contrary, full kinetic modeling may offer a potentially more detailed, though less robust, description to the above set of macro-parameters through the non-linear estimation of individual kinetic micro-parameters (K₁, k₂, k₃ and k₄) with minimum assumptions for the underlying 2-tissue compartment model (Zaidi et al 2006). Thus, K_i could also be indirectly provided by these estimates but with significantly less robustness (Castell and Cook 2008). Finally, WB full kinetic modeling would have required finer temporal sampling for all voxel TACs across all beds from time of injection resulting in very high noise levels with current clinical PET scanner technology. Thus, the feasibility and future prospects of the less robust compartment model-based WB parametric PET imaging need to be studied separately.

4.2. Likelihood of tracer uptake reversibility between target and background regions

It has been reported that FDG k_loss effects are more likely observed in normal tissues (background), mainly in the liver and kidney, and less often in suspected tumor or high uptake (target) regions (Messa et al 1992, Hawkins et al 1992b, Okazumi et al 1992, Huang et al 2000, Graham et al 2000, Zhuang et al 2001, Lin et al 2005, Qiao et al 2007). This finding may, at first thought, render sPatlak more desirable over gPatlak, because a potential K_i underestimation only in the background and not the target regions, as expected in such a scenario, would actually further enhance TBR contrast and even potentially favor target CNR and detectability (Messa et al 1992, Okazumi et al 1992, Lin et al 2005). Therefore depending on the kinetic properties of the targeted ROI and its background, sPatlak may be preferred over gPatlak, e.g. when targeting regions with a priori negligible k_loss expectations or when the imaging task promotes detectability over quantification.

On the other hand, such potential TBR contrast enhancement may not be quantitative for tumor assessment in oncology, as it is triggered by the incidental underestimation of K_i only in background normal tissues and not the respective target regions. In addition, the probability of k_loss effect not being expressed in tumors versus in background can vary among different regions or different tumor types of the same region (Lin et al 2005). In fact, it has been shown that the probability for non-negligible k_loss in targeted tumor regions cannot be excluded, as in the case of hepatocellular carcinoma (HCC) tumor types (Okazumi et al 1992, Lin et al 2005).

4.3 Tumor FDG dephosphorylation and kinetics heterogeneity

As discussed in sections 1 and 6.2, FDG dephosphorylation has been suggested by numerous PET studies mainly in normal healthy tissues, such as the liver, and less often, in tumor regions, such as the case of HCC liver tumors. By contrast, a limited number of dynamic brain PET studies have reported or referred to simulation results that point to considerable overestimation (up to 20%) of regional K_i from kinetically heterogeneous regions, e.g. mixing of white and grey matter values in the same ROI, due to limited PET resolution and PVE (Schmidt et al 1991 and 1992, Lucignani et al 1993, Vriens et al 2011).

That bias was observed when a 2-tissue compartment 4-parameter (4K) kinetic model was applied, assuming a non-zero FDG dephosphorylation rate constant (k₄). However, the reported bias was reduced when a 2-tissue 3-parameter (3K) kinetic model was utilized instead, i.e. assuming zero k₄. Schmidt et al (1991 and 1992) attributed this effect to the tendency of the 4K model to compensate for the reduced true FDG phosphorylation (k₃) rate constant, caused by the mixture of heterogeneous kinetics in the same region, with k₄ overestimation, thus resulting in subsequent overestimation of k₃, K_i and k_loss parameters too. In the case of the 3K model, this would not have been possible, as k₄ is forced to be zero. Thus, they concluded that the application of certain compartmental kinetic models, designed for physiologically homogeneous regions, on dynamic PET measurements extracted from heterogeneous regions may lead to erroneously overestimated K_i and k_loss regional estimates. Moreover, they conjectured that region-based kinetic analysis of dynamic brain FDG PET measurements cannot provide solid evidence of presence of true underlying FDG dephosphorylation, as non-zero k₄ estimates in specific regions may simply be the result of heterogeneity rather than actual dephosphorylation.

Although in this study we have only simulated physiologically homogeneous regions, there still remains the possibility for limited kinetic heterogeneity at the boundaries of the evaluated regions, especially in small tumors, due to PVE effects. In addition, the clinical data may also include suspected tumor regions with some heterogeneity. However, in all cases, our voxel-wise parametric estimation is expected to drastically limit this effect. When the input dynamic PET measurements are heterogeneous, the parameter estimation process may introduce bias in an attempt to explain heterogeneous input data, which are not expected by the model. An example of this effect could be the overestimation of k₄, as an attempt of the 4K model to explain a heterogeneity-induced decrease of k₃ (Schmidt et al 1992). In this study, we employed voxel-wise parametric estimation to minimize inter-voxel kinetic cross-contaminations, and constrain any bias propagation between K_i and k_loss. Nevertheless, there still remains a yet small but non-negligible probability for intra-voxel tissue kinetics heterogeneity, i.e. for each voxel TAC to be a weighted average, or mixture, of different kinetics in highly heterogeneous regions or at regional boundaries. However, our simulation results suggest that K_i bias was consistently reduced in all regions, regardless of true underlying k_loss effect, when gPatlak was applied. Thus, it is suggested that the positive effect in quantitative accuracy (bias) when accounting for non-zero k_loss, outperforms any potential negative effect of PVE-induced intra-voxel kinetic heterogeneities.

In addition, the findings of previous studies by Schmidt et al (1991 and 1992) regarding heterogeneity-induced artificially non-zero k_loss estimates, though valid and quite probable, do not necessarily exclude the probability for true underlying k_loss effects as well, as many other studies have suggested (Messa et al 1992, Hawkins et al 1992b, Okazumi et al 1992, Huang et al 2000, Graham et al 2000, Zhuang et al 2001, Lin et al 2005). Moreover, we did not observe K_i bias enhancement for any of the evaluated regions when k_loss was assumed non-zero. In clinical results, K_i estimates were systematically higher with gPatlak, an observation consistent with both theories. However, since the ground truth is not known in clinical data, it is not possible to conjecture if gPatlak relatively higher K_i estimates suggest i) bias enhancement, i.e. K_i overestimation due to heterogeneity effects, or ii) bias reduction, due to properly accounting for non-zero k_loss, or actually iii) a combination of both effects.

Finally, as the previous studies have also reported, the utilization of the Patlak framework to robustly estimate K_i and k_loss macro-parameters, as opposed to application of fully compartmental kinetic analysis (3K or 4K models) for the individual estimation of k₃ and k₄ micro-parameters, may have also further reduced any potential negative tissue kinetics heterogeneity effects, as Patlak methods are considered less sensitive to kinetically heterogeneous data (Mori et al 1990, Schmidt et al 1992, Lucignani et al 1993).

4.4 Task-based recommendations for different WB Patlak methods

Having demonstrated the pros and cons of each method, in this section we provide our recommendations regarding the choice of the most appropriate Patlak method for the imaging task and underlying type of kinetics (e.g. different tracers and targets) expected in a clinical study (figure 15).

Figure 15.

Flowchart illustrating our recommendations regarding the choice of WB Patlak imaging method (orange blocks) based on the imaging task (green blocks) and the a priori k_loss expectations for the target ROIs (grey blocks). According to our simulation and clinical findings, unless negligible target k_loss is expected, in which case sPatlak is most appropriated, either gPatlak or hPatlak is recommended depending on the imaging task performed.

When the imaging task involves quantitative assessments and comparisons between different scans, such as in longitudinal studies or treatment response monitoring and follow-up tasks, the gPatlak method is recommended for enhanced quantification, unless there exist sufficient indications that the targeted regions exhibit negligible k_loss, in which case sPatlak is sufficient for robust parametric imaging. On the other hand, for imaging task prioritizing detectability hPatlak method is recommended, when no underlying kinetic information is known a priori, as it can potentially trigger higher TBR and CNR scores, provided the appropriate WR correlation threshold being selected. However, for target regions with expected negligible k_loss, we suggest the more robust sPatlak analysis for detectability tasks as well.

4.5. Generalized Patlak as a complementary framework to SUV and standard Patlak

Despite its quantitative limitations, WB static SUV imaging remains the most established PET imaging technique in the clinic. Thanks to its simplicity and routine clinical implementation, SUV imaging has allowed for the standardization of diagnostic and treatment response criteria in the clinic (Wahl et al 2009, Boellaard et al 2010). On the other hand, WB Patlak imaging methods, although potentially more accurate than SUV, are associated with higher complexity and noise levels. Yet, they are nowadays steadily gaining more attention, as emerging commercial PET technologies, such as continuous bed motion, time-of-flight and resolution modeling, help to overcome these challenges, paving the way towards their clinical adoption (Karakatsanis et al 2014b). Moreover, following sPatlak, the proposed gPatlak framework can be easily applied as an additional kinetic analysis method, utilizing the same set of reconstructed images at a small computational cost.

Our primary aim here is the design of a clinically feasible multi-bed parametric PET imaging framework to enhance quantitative information content with respect to single-pass SUV imaging alone. At the same time, we recognize the important clinical value of the SUV metric as well, particularly in relation to the currently established SUV-based treatment response criteria. In this context, SUV and parametric PET imaging frameworks could efficiently complement each other. Currently, we are investigating clinically adoptable novel PET scan protocols that simultaneously enable WB SUV and Patlak imaging from a single acquisition performed over the conventional 60min post-injection SUV time window (Karakatsanis et al 2015). This study aims to introduce additional WB Patlak imaging frameworks to enhance quantification or detectability with respect to more conventional Patlak schemes. In future work, a comparative clinical study between WB Patlak and SUV imaging method would be important so as to also assess the clinical potential of combined SUV/Patlak WB imaging.

4.6. Hybrid WB Patlak regression: pearls and pitfalls

The performance of hPatlak in enhancing suspected tumor ROI CNR and, thus, detectability, depends on the user-defined WR threshold level. A very large threshold, i.e. a value very close to unity, may substantially reduce background noise, but it may also exclude some tumor voxels from being accurately estimated with the gPatlak method, especially those situated at tumor boundaries, susceptible to PVE or at kinetically heterogeneous tumor sections. As a result, a high WR threshold is expected to reduce noise at the cost of quantification and contrast, with respect to pure gPatlak, while a low WR threshold may enhance quantification and contrast at the cost of additional noise, compared to sPatlak.

As hPatlak is targeting the enhancement of tumor detectability, a potential criterion for the choice of an appropriate WR threshold could have been the tumor CNR score either at a particular target tumor ROI or weighted averaged over a set of target ROIs, with the weights defined as the relative size, in number of voxels, of each ROI. On the other hand, CNR can be substantially different among patients or even between target ROIs of the same patient and, as such, we cannot recommend a data-independent globally optimal WR threshold. Alternatively, a data-driven algorithm could automatically select for every patient a WR threshold and respective K_i clinical image based on the highest evaluated CNR score for the target ROI or set of ROIs. In fact, other nuclear imaging studies have in the past followed similar CNR-based data-driven optimization schemes when designing reconstruction algorithms (Qi and Leahy 1999), imaging systems and scan protocols (Asma and Manjeshwar 2010, da Rocha Vaz Pato et al 2012).

However, since CNR scores may depend on the tumor and background ROI delineation techniques, CNR-based optimization should be exercised with care. Therefore, in this study we suggest not to necessarily optimize WR threshold based on CNR and, instead, to provide the users in the future with the flexibility to alter this parameter (analogously to sliding the color bar) for an enhanced perspective onto the suspected tumor of interest.

4.7. Future prospects

The present work has scope for significantly enhanced performance. Kinetic parameter estimation commonly involves two main steps: (i) image reconstruction of individual dynamic frames, followed by (ii) application of kinetic modeling to the resulting dynamic images. This common process poses limitations due to the poor characterization of the complex noise distribution in the reconstructed images. On the contrary, direct 4D reconstruction schemes enable kinetic modeling within a comprehensive reconstruction framework, allowing accurate noise characterization directly in the projection-space (Tsoumpas et al 2008, Rahmim et al 2009 and 2012, Tang et al 2010, Wang et al 2010, 2012 and 2013). As such, 4D parametric reconstruction methods could be particularly important for gPatlak, where noise is relatively higher than sPatlak. By integrating the gPatlak model within 4D ML-EM reconstruction we expect Patlak images of both superior precision (low noise) and less noise-induced bias.

Previously, we confirmed the benefit of direct 4D reconstruction in limiting noise propagation in WB sPatlak imaging (Karakatsanis et al 2013b). However the convergence of the iterative algorithm can be very slow, due to the correlation between K_i and V estimates (Wang et al 2010). In addition, the nonlinearity of gPatlak 4D reconstruction, which we are presently exploring, can further impact the convergence rates (Wang et al 2012 and 2013). Therefore, we are investigating the utilization of optimization transfer methods to enable accelerated gPatlak 4D reconstruction (Karakatsanis et al 2014a).

5. Conclusion

In this work, we implemented and quantitatively assessed a family of WB parametric imaging methods, utilizing both simulated and clinical PET studies, as acquired with a clinically feasible multi-bed dynamic PET scan protocol. Specifically, we extended our standard WB Patlak (sPatlak) graphical analysis method to sufficiently account for potential underlying tracer uptake reversibility. By properly incorporating the net efflux rate constant parameter of k_loss in a generalized Patlak (gPatlak) imaging framework, it became possible to generate parametric WB tracer influx rate constant K_i images of enhanced quantitative accuracy, compared to SUV, including in regions with non-negligible k_loss, where sPatlak introduced bias. At the same time, our results also indicated reduced gPatlak robustness to high noise levels and, thus relatively lower tumor CNR scores than sPatlak, except if k_loss effect is strong enough to provide sufficiently high contrast to counteract any CNR losses due to background noise amplification.

Furthermore, in order to efficiently enhance tumor CNR and detectability and, at the same time, to retain sufficient quantitative accuracy, we proposed a hybrid Patlak (hPatlak) method to selectively apply gPatlak analysis only to voxel TACs exhibiting high Patlak correlation coefficients and sPatlak method elsewhere. According to both simulations and clinical results, hPatlak was always associated with the highest TBR and CNR scores, while its accuracy and precision performance was placed between that of gPatlak and sPatlak methods, thus demonstrating primary clinical usefulness mainly for detectability tasks. Finally, for the clinical study, the measured CNR in SUV images was consistently lower than that of sPatlak and gPatlak in most regions and that of hPatlak in all regions, demonstrating enhanced detectability performance for Patlak imaging over conventional SUV PET.

Although gPatlak and hPatlak schemes are presented here for WB dynamic PET clinical acquisitions, they are also applicable to more common dynamic PET protocols, such as single-bed dynamic cardiac or oncologic PET studies, where temporal sampling is continuous and noise levels are lower. Moreover, although the current study has focused on FDG scans, the proposed methods may also be applied to other PET tracers which are used in WB imaging and which exhibit considerable net efflux rate constants, such as ¹⁸F-FLT and ¹⁸F -fluoride bone parametric imaging (Kim et al 2008, Siddique et al 2014).

Figure 12.

Clinical results: RX patient case #5, scanned on the GE Discovery RX, designated focal uptake ROI in the bowel (a), (b) and (c) same as figure 11.