Whole-body direct 4D parametric PET imaging employing nested generalized Patlak expectation–maximization reconstruction

In multi-bed dynamic PET acquisitions, the linear sPatlak graphical analysis method (Patlak et al 1983) utilizes the dynamic PET data from each bed position and the input function to estimate the kinetic macro-parameters of tracer influx rate constant ${{K}_{i}}$, in units of ml of blood per minute per gram of tissue (ml (min  ×  g)⁻¹), and total distribution volume $V$, in units of ml of blood per gram of tissue (ml g⁻¹), at each voxel (Karakatsanis et al 2013a):

$$\begin{eqnarray}&&\,\begin{array}{*{35}{l}} \frac{C\left({{t}_{n}}\right)}{{{C}_{\text{P}}}\left({{t}_{n}}\right)}={{K}_{\text{i}}}\frac{{\int}_{0}^{{{t}_{n}}}{{C}_{\text{P}}}\left({{t}^{\prime}}\right)\text{d}{{t}^{\prime}}}{{{C}_{\text{P}}}\left({{t}_{n}}\right)}+V\Rightarrow \\ \,\,\,C\left({{t}_{n}}\right)={{K}_{\text{i}}}{\int}_{0}^{{{t}_{n}}}{{C}_{\text{P}}}\left({{t}^{\prime}}\right)\text{d}{{t}^{\prime}}+V{{C}_{\text{P}}}\left({{t}_{n}}\right)={{K}_{i}}\otimes {{C}_{\text{P}}}\left({{t}_{n}}\right)+V{{C}_{\text{P}}}\left({{t}_{n}}\right),~~{{t}_{n}}>{{t}^{\ast}},~~n=1\ldots N \end{array}\end{eqnarray} \tag{ 1 }$$

where $\otimes$ denotes the convolution operation over the time variable t' and $C\left({{t}_{n}}\right)$ is the measured tissue TAC at the mid-frame time points ${{t}_{n}}$ of the $N$ dynamic PET frames, corresponding to a particular bed and voxel. Moreover, ${{C}_{\text{P}}}\left({{t}_{n}}\right)$ is the input function at the ${{t}_{n}}$ time points and ${{t}^{\ast}}$ is the p.i. time after which relative kinetic equilibrium between the blood and the tissue tracer concentration is attained. The sPatlak analysis assumes an irreversible 2-tissue-compartment tracer kinetic model, as illustrated in figure 2(a).

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Patlak and Blasberg (1985) showed that the macro-parameter ${{K}_{i}}$ can be related to the kinetic micro-parameters ${{K}_{1}}$ (ml (min  ×  g)⁻¹), ${{K}_{2}}$ (min⁻¹), ${{K}_{3}}$ (min⁻¹) and ${{K}_{4}}$ (min⁻¹) as follows:

$$\begin{eqnarray}&&{{K}_{\text{i}}}=\frac{{{k}_{1}}{{k}_{3}}}{{{k}_{2}}+{{k}_{3}}}\end{eqnarray} \tag{ 2 }$$

Standard linear Patlak analysis directly estimates K_i and V macro-parameters by assuming a 2-tissue-compartment kinetic model with an irreversible compartment, a commonly invoked model for organs and tumors exhibiting ¹⁸F-FDG uptake in PET human studies (Gunn et al 2001). However a considerable number of studies suggest uptake reversibility for a range of tracers, as presented previously (Holden et al 1997, Lodge et al 1999, Karakatsanis et al 2015a). Since the sPatlak model assumes irreversible uptake, it may underestimate K_i to compensate for lack of reversibility modeling (Messa et al 1992, Hoh et al 2011, Sayre et al 2011).

Therefore, later Patlak and Blasberg (1985) introduced a generalized graphical analysis method to account for mildly reversible uptake kinetics. A ${{k}_{\text{loss}}}$ kinetic parameter was introduced to describe the net rate constant for absorbed or metabolized tracer loss to the blood plasma. By assuming a reversible 2-tissue compartment model with $~{{k}_{\text{loss}}}\ll {{K}_{\text{i}}}$, it follows (Karakatsanis et al 2015a):

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{C\left({{t}_{n}}\right)}{{{C}_{\text{P}}}\left({{t}_{n}}\right)}={{K}_{\text{i}}}\frac{{\int}_{0}^{{{t}_{n}}}{{\text{e}}^{-{{k}_{\text{loss}}}\left({{t}_{n}}-{{t}^{\prime}}\right)}}{{C}_{\text{P}}}\left({{t}^{\prime}}\right)\text{d}{{t}^{\prime}}}{{{C}_{\text{P}}}\left({{t}_{n}}\right)}+V\Rightarrow \\ C\left({{t}_{n}}\right)={{K}_{\text{i}}}\underset{0}{\overset{{{t}_{n}}}{\int}}\,{{\text{e}}^{-{{k}_{\text{loss}}}\left({{t}_{n}}-{{t}^{\prime}}\right)}}{{C}_{\text{P}}}\left({{t}^{\prime}}\right)\text{d}{{t}^{\prime}}+V{{C}_{\text{P}}}\left({{t}_{n}}\right)\Rightarrow \\ C\left({{t}_{n}}\right)=\left({{K}_{\text{i}}}{{\text{e}}^{-{{k}_{\text{loss}}}{{t}_{n}}}}\right)\otimes {{C}_{\text{P}}}\left({{t}_{n}}\right)+V{{C}_{\text{P}}}\left({{t}_{n}}\right),~{{t}_{n}}>{{t}^{\ast}},~n=1..~.N,~{{k}_{\text{loss}}}\ll {{K}_{i}} \end{array}\end{eqnarray} \tag{ 3 }$$

The net efflux rate constant $~{{k}_{\text{loss}}}$ (min⁻¹) is related to the kinetic micro-parameters as follows:

$$\begin{eqnarray}&&{{k}_{\text{loss}}}=\frac{{{k}_{2}}{{k}_{4}}}{{{k}_{2}}+{{k}_{3}}}\end{eqnarray} \tag{ 4 }$$

Despite the presence of a non-linear term in equation (3), gPatlak analysis is characterized by a significantly lower degree of complexity and, thus higher robustness, than the standard 2-tissue compartmental kinetic modeling methods. Nevertheless, gPatlak is less robust to noise, but enhances ${{K}_{\text{i}}}$ quantification in voxels with uptake reversibility, compared to sPatlak analysis (Karakatsanis et al 2015a).

Let us first define the following:

• $\boldsymbol{y}^{{n}}=\left[\,y_{i}^{n}\right]_{i=1}^{I}$: nth dynamic frame of a PET sinogram or projection data vector comprised of a total of $I$ detector pair or line-of-response (LOR) bins,
• $\boldsymbol{Y}={{\left[\,\boldsymbol{y}^{{1}}\ldots \boldsymbol{y}^{{N}}\right]}^{T}}$: column vector of a set of $N$ dynamic frames of measured PET sinograms,
• $\boldsymbol{x}^{{n}}=\left[x_{j}^{n}\right]_{j=1}^{J}$: nth dynamic frame of a PET image vector comprised of a total of $J$ voxels,
• $\boldsymbol{X}={{\left[\boldsymbol{x}^{{1}}\ldots \boldsymbol{x}^{{N}}\right]}^{T}}$: column vector of a set of $N$ dynamic frames of reconstructed PET images,
• $\boldsymbol{K}=\left[K_{\text{i}}^{j}\right]_{j=1}^{J}$: parametric image vector of the Patlak slope or tracer influx rate constant ${{K}_{\text{i}}}$,
• $\boldsymbol{V}=\left[{{V}^{j}}\right]_{j=1}^{J}$: parametric image vector of the Patlak intercept or blood distribution volume $V$,
• $\boldsymbol{M}_{{s}}={{\left[\boldsymbol{K}\text{;}\boldsymbol{V}\right]}^{T}}$: ensemble standard Patlak parametric image vector
• ${{C}_{\text{P}}}(n)={{C}_{\text{P}}}\left({{t}_{n}}\right)$: measured blood plasma activity concentration at mid-frame time ${{t}_{n}}$,
• ${{S}_{\text{P}}}(n)=~{\int}_{0}^{{{t}_{n}}}{{C}_{\text{P}}}\left({{t}^{\prime}}\right)\text{d}{{t}^{\prime}}$: integral of ${{C}_{\text{P}}}\left({{t}^{\prime}}\right)$ along time variable ${{t}^{\prime}}$
• $\boldsymbol{P}=\left[\,{{p}_{ij}}\right]_{i=1,j=1}^{I,J}$: spatial system response matrix with ${{p}_{ij}}$ denoting the probability an annihilation event having occurred at jth image voxel to be recorded at ith detector pair or line or response (LOR) of the sinogram, thus $\boldsymbol{y}^{{n}}=\boldsymbol{P}\boldsymbol{x}^{{n}}$,
• $\boldsymbol{B}_{{s}}=\left[{{b}_{s,nk}}\right]_{n=1,k=1}^{N,2}=\left[\begin{array}{c} {{S}_{\text{P}}}(1) & {{C}_{\text{P}}}(1) \\ \vdots & \vdots \\ {{S}_{\text{P}}}(N) & {{C}_{\text{P}}}(N) \end{array}\right]$: standard Patlak model matrix and
• $\overline{\boldsymbol{P}}=\left[\begin{array}{c} {{S}_{\text{P}}}(1)\boldsymbol{P} & {{C}_{\text{P}}}(1)\boldsymbol{P} \\ \vdots & \vdots \\ {{S}_{\text{P}}}(N)\boldsymbol{P} & {{C}_{\text{P}}}(N)\boldsymbol{P} \end{array}\right]=\boldsymbol{P}\oplus \boldsymbol{B}_{{s}}$: spatio-temporal system response matrix derived by taking the Kronecker product $(\oplus )$ of P and B_s model response matrices.

According to standard Patlak graphical analysis, the expectations of dynamic sinograms $\widehat{\boldsymbol{Y}}={{\left[{{\widehat{\boldsymbol{y}}}^{1}}\ldots {{\widehat{\boldsymbol{y}}}^{N}}\right]}^{T}}$ and respective PET images $\widehat{\boldsymbol{X}}={{\left[{{\widehat{\boldsymbol{x}}}^{1}}\ldots {{\widehat{\boldsymbol{x}}}^{N}}\right]}^{T}}$ can be directly related to the expected ensemble parametric image ${{\widehat{\boldsymbol{M}}}_{s}}={{\left[\widehat{\boldsymbol{K}};\widehat{\boldsymbol{V}}\right]}^{T}}$ of tracer influx rate constant $\widehat{\boldsymbol{K}}$ and blood distribution volume $\widehat{\boldsymbol{V}}$ according to the following linear kinetic model equations:

$$\begin{eqnarray}&&\hat{X}=\boldsymbol{B}_{{s}}{{\widehat{M}}_{s}},\,\hat{Y}=\overline{\boldsymbol{P}}{{\widehat{\boldsymbol{M}}}_{s}}=\left(\boldsymbol{P}\oplus \boldsymbol{B}_{{s}}\right){{\widehat{\boldsymbol{M}}}_{s}}\end{eqnarray} \tag{ 5 }$$

or, equivalently:

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{x}}}^{n}}\left(\widehat{\boldsymbol{K}},\widehat{\boldsymbol{V}}\right)=\widehat{\boldsymbol{K}}{{S}_{\text{P}}}(n)+\widehat{\boldsymbol{V}}{{C}_{\text{P}}}(n),{{\widehat{\boldsymbol{y}}}^{n}}\left(\widehat{\boldsymbol{K}},\widehat{\boldsymbol{V}}\right)=\boldsymbol{P}{{\widehat{\boldsymbol{x}}}^{n}}\left(\widehat{\boldsymbol{K}},\widehat{\boldsymbol{V}}\right)=P\left(\widehat{\boldsymbol{K}}{{S}_{\text{P}}}(n)+\widehat{\boldsymbol{V}}{{C}_{\text{P}}}(n)\right)\end{eqnarray} \tag{ 6 }$$

Then the two 4D maximum likelihood expectation–maximization (ML-EM) update equation follows:

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{K}}}_{\text{new}}}=\frac{{{\widehat{\boldsymbol{K}}}_{\text{old}}}}{\boldsymbol{P}^{{T}}\mathbf{1}\underset{n=1}{\overset{N}{\sum}}\,{{S}_{\text{P}}}(n)}\underset{n=1}{\overset{N}{\sum}}\,{{S}_{\text{P}}}(n)\boldsymbol{P}^{{T}}\left[\frac{\boldsymbol{y}^{{n}}}{{{\widehat{\boldsymbol{y}}}^{n}}\left({{\widehat{\boldsymbol{K}}}_{\text{old}}},{{\widehat{\boldsymbol{V}}}_{\text{old}}}\right)}\right]\,\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{V}}}_{\text{new}}}=\frac{{{\widehat{\boldsymbol{V}}}_{\text{old}}}}{\boldsymbol{P}^{{T}}\mathbf{1}\underset{n=1}{\overset{N}{\sum}}\,{{C}_{\text{P}}}(n)}\underset{n=1}{\overset{N}{\sum}}\,{{C}_{\text{P}}}(n)\boldsymbol{P}^{{T}}\left[\frac{\boldsymbol{y}^{{n}}}{{{\widehat{\boldsymbol{y}}}^{n}}\left({{\widehat{\boldsymbol{K}}}_{\text{old}}},{{\widehat{\boldsymbol{V}}}_{\text{old}}}\right)}\right]\,\end{eqnarray} \tag{ 7b }$$

or, equivalently:

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{M}}}_{s,\text{new}}}=\frac{{{\widehat{\boldsymbol{M}}}_{s\text{,old}}}}{{{\overline{\boldsymbol{P}}}^{T}}\mathbf{1}}{{\overline{\boldsymbol{P}}}^{T}}\left[\frac{\boldsymbol{y}}{\overline{\boldsymbol{P}}{{{\widehat{M}}}_{s\text{,old}}}}\right]\,\end{eqnarray} \tag{ 7c }$$

By letting $\boldsymbol{m}_{s}^{j}=\left[m_{s,k}^{j}\right]_{k=1}^{2}={{\left[K_{i}^{j}{{V}^{j}}\right]}^{T}}$ as the standard Patlak parameter vector at voxel $j$, we have:

$$\begin{eqnarray}&&m_{s,k}^{j,\text{new}}=\frac{m_{s,k}^{j,\text{old}}}{\underset{n}{\sum}\,\boldsymbol{B}_{{s,nk}}\underset{i}{\sum}\,\boldsymbol{P}_{{ij}}}\underset{n}{\sum}\,\boldsymbol{B}_{{s,nk}}\underset{i}{\sum}\,\boldsymbol{P}_{{ij}}\left[\frac{y_{i}^{n}}{\underset{j}{\sum}\,\boldsymbol{P}_{{ij}}\underset{k}{\sum}\,\boldsymbol{B}_{{s,nk}}m_{s,k}^{j,\text{old}}}\right]~\end{eqnarray} \tag{ 7d }$$

The nested 4D sPatlak image reconstruction algorithm breaks down the previous integrated EM process into two steps: (i) a single tomographic projection-based EM update of the dynamic image estimates, based on the measured 4D data, followed by (ii) multiple nested image-based EM updates of the kinetic parameter estimates, based on the dynamic image estimates from step 1. The nested ML-EM implementation utilizes the 'optimization transfer' principle (Lange et al 2000), which 'transfers' the optimization target from a single and more complex global objective function to simpler surrogate functions, that vary at each global ML-EM iteration step, as illustrated in figure 3 (Carson and Lange 1985, Wang and Qi 2010, 2012, 2013, Karakatsanis and Rahmim 2014a).

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In the nested sPatlak 4D ML-EM framework, both the global objective function $L$ and the surrogate objective function $~{{Q}_{w}}$, for each iteration $~w$, are defined as Poisson log-likelihood functions of the measured dynamic data Y and the dynamic image estimate x^w at iteration $~w$, respectively, given the sPatlak parameter vector $\boldsymbol{m}_{{s}}$. In fact, the nested section of the sPatlak 4D ML-EM algorithm for global iteration $w$ utilizes the latest dynamic image estimate x^w from step 1 to return, after several sub-iterations, the m_s parameter vector that maximizes the $w$th iteration surrogate log-likelihood function ${{Q}_{w}}\left(\boldsymbol{x}^{{w}}|\boldsymbol{m}_{{s}}\right)$ (figure 3(c)). Subsequently, the returned value $\boldsymbol{m}_{s}^{w}$ initializes the tomographic ML-EM update (step 1) of the next, i.e. $~(w+1)$th iteration.

We note that our scheme employs an ML-EM optimization algorithm for both the external tomographic and the nested image-based iterative update processes, while the respective Poisson log-likelihood functions satisfy the criteria described in figures 3(a) and (b). In addition, the external and nested Poisson log-likelihood maximization problems described above are equivalent to minimizing the Kullback–Leibler (KL) distance metrics (Barrett and Myers 2004) between the measured dynamic data Y and dynamic images x^w, for the tomographic estimation problem, and between the estimated dynamic images x^w and the new sPatlak parameter estimates $\boldsymbol{m}_{s}^{w}$ for the image-based parametric estimation problem. Under these conditions, the 4D ML-EM nested estimation of the sPatlak parameters is legitimately performed, as the Poisson distribution in the measured counts is fully accounted, and the EM convergence of the nested 4D ML-EM algorithm is ensured, as illustrated in figure 3(d).

Below, we present the theoretical framework of the nested sPatlak 4D ML-EM algorithm (Wang and Qi 2010). In this work, we extended its application to raw PET data from multiple beds. Initially, for every global ML-EM iterative cycle, an updated dynamic image set $~{{\widehat{\boldsymbol{x}}}_{\text{new}}}$ is estimated for each bed utilizing the large tomographic system matrix P and the respective bed dynamic data Y (step 1):

$$\begin{eqnarray}&&\widehat{\boldsymbol{x}}_{\text{new}}^{n}=\frac{{{\widehat{\boldsymbol{x}}}^{n}}\left({{\widehat{\boldsymbol{K}}}_{\text{old}}},{{\widehat{\boldsymbol{V}}}_{\text{old}}}\right)}{\boldsymbol{P}^{{T}}\mathbf{1}}\boldsymbol{P}^{{T}}\left[\frac{\boldsymbol{y}^{{n}}}{{{\widehat{\boldsymbol{y}}}^{n}}\left({{\widehat{\boldsymbol{K}}}_{\text{old}}},{{\widehat{\boldsymbol{V}}}_{\text{old}}}\right)}\right]~\end{eqnarray} \tag{ 8a }$$

or, equivalently:

$$\begin{eqnarray}&&\boldsymbol{x}_{j,\text{new}}^{n}=\frac{\boldsymbol{x}_{j,\text{old}}^{n}}{\underset{i}{\sum}\,\boldsymbol{P}_{{ij}}}\underset{i}{\sum}\,\boldsymbol{P}_{{ij}}\left[\frac{\boldsymbol{y}_{{i,n}}}{\underset{j}{\sum}\,\boldsymbol{P}_{{ij}}\underset{k}{\sum}\,\boldsymbol{B}_{{s,nk}}m_{s,jk}^{\text{old}}}\right]\,\end{eqnarray} \tag{ 8b }$$

Subsequently, the algorithm performs a series of nested ML-EM updates of the kinetic parameter images K and V, corresponding to each bed, by employing the considerably smaller in size sPatlak model matrix $\boldsymbol{B}_{{s}}$ and the respective PET image estimates ${{\widehat{\boldsymbol{x}}}_{\text{new}}}$ from step 1 as a reference (step 2):

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{K}}}_{\text{new}}}=\frac{{{\widehat{\boldsymbol{K}}}_{\text{old}}}}{\underset{n=1}{\overset{N}{\sum}}\,{{S}_{\text{P}}}(n)}\underset{n=1}{\overset{N}{\sum}}\,{{S}_{\text{P}}}(n)\left[\frac{\widehat{\boldsymbol{x}}_{\text{new}}^{n}}{{{\widehat{\boldsymbol{x}}}^{n}}\left({{\widehat{\boldsymbol{K}}}_{\text{old}}},{{\widehat{\boldsymbol{V}}}_{\text{old}}}\right)}\right]\,\end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{V}}}_{\text{new}}}=\frac{{{\widehat{\boldsymbol{V}}}_{\text{old}}}}{\underset{n=1}{\overset{N}{\sum}}\,{{C}_{\text{P}}}(n)}\underset{n=1}{\overset{N}{\sum}}\,{{C}_{\text{P}}}(n)\left[\frac{\widehat{\boldsymbol{x}}_{\text{new}}^{n}}{{{\widehat{\boldsymbol{x}}}^{n}}\left({{\widehat{\boldsymbol{K}}}_{\text{old}}},{{\widehat{\boldsymbol{V}}}_{\text{old}}}\right)}\right]\,\end{eqnarray} \tag{ 9b }$$

or, equivalently:

$$\begin{eqnarray}&&m_{s,jk}^{\text{new}}=\frac{m_{s,jk}^{\text{old}}}{\underset{n}{\sum}\,\boldsymbol{B}_{{s,nk}}}\underset{n}{\sum}\,\boldsymbol{B}_{{s,nk}}\left[\frac{\boldsymbol{x}_{j,\text{new}}^{n}}{\underset{k}{\sum}\,\boldsymbol{B}_{{s,nk}}\boldsymbol{m}_{s,jk}^{\text{old}}}\right]\end{eqnarray} \tag{ 9c }$$

Then, the nested sPatlak 4D ML-EM steps above are repeated for the data of the remaining beds to produce the respective sPatlak images. Finally, all images of the same parameter type are combined, after accounting for any axial overlapping slices between beds, to create multi-bed or WB sPatlak images.

The first step of each global EM iteration cycle involves forward- and back-projection 3D tomographic operations, which are often computationally expensive due to the large size of P. On the contrary, the nested EM loop of the second step employs the much smaller model matrix B_s, thus allowing for much faster forward- and back-projection operations to transform between parametric and dynamic image space. Thus, by nesting multiple faster update steps of the kinetic parameter estimates (equations (9a) and (9b)) within every tomographic update step of the dynamic images (equation (8a) or (8b)), the global convergence rate of the 4D reconstruction algorithm is effectively accelerated, in terms of total computation time (Wang et al 2010, Wang and Qi 2013, Karakatsanis and Rahmim 2014a).

For the non-linear gPatlak model let us denote:

• K, k_loss and V: column vectors denoting respective ${{K}_{i}}$, ${{k}_{\text{loss}}}$ and $V$ parametric images,
• $\boldsymbol{M}_{{g}}={{\left[\boldsymbol{K};\boldsymbol{k}_{{\text{loss}}};\boldsymbol{V}\right]}^{T}}$: the overall gPatlak parametric image matrix,
• $\boldsymbol{m}_{g}^{j}=\left(K_{i}^{j},k_{\text{loss}}^{j},{{V}^{j}}\right)$: vector of the three gPatlak parameters at voxel $j$,
• $t_{d}^{\prime},~d=1\ldots D$: variable denoting each of the $D$ convolution time points (different from ${{t}_{n}},~n=1\ldots N$, variable for the $N$ mid-frame time points)
• $h_{d}^{j}\left(K_{i}^{j},k_{\text{loss}}^{j},t_{d}^{\prime}\right)=K_{i}^{j}{{\text{e}}^{-k_{\text{loss}}^{j}t_{d}^{\prime}}}$: gPatlak impulse response element at time point $t_{d}^{\prime}$ for voxel $j$,
• $\boldsymbol{h}^{{j}}=\left[h_{d}^{j}\right]_{d=1}^{D}$: impulse response column vector at voxel $j$ and
• $\boldsymbol{x}_{{j}}=\left[x_{j}^{n}\right]_{n=1}^{N}$: TAC at voxel $j$.

According to gPatlak model assumptions in (3) the TAC at every voxel $j$ can be modeled as follows:

$$\begin{eqnarray}&&\boldsymbol{x}_{{j}}=\boldsymbol{h}^{{j}}\left(K_{i}^{j},k_{\text{loss}}^{j},{{t}^{\prime}}\right)\otimes \boldsymbol{C}_{{\text{P}}}\left({{t}^{\prime}}\right)+{{V}^{j}}\boldsymbol{C}_{{\text{P}}}\left({{t}^{\prime}}\right)\end{eqnarray} \tag{ 10a }$$

By approximating the above time convolution operation with a summation over $D$ finely sampled time convolution points $~t_{d}^{'}$, we can also model every voxel TAC ${{x}_{j}}$ as a vector-matrix product:

$$\begin{eqnarray}&&\boldsymbol{x}_{{j}}=\Theta\boldsymbol{r}^{{j}}\end{eqnarray} \tag{ 10b }$$

where $\boldsymbol{r}^{{j}}={{\left[\boldsymbol{h}^{{j}};{{V}^{j}}\right]}^{T}}$ is the Patlak response vector at voxel $j$, constructed by appending the ${{V}^{j}}$ unknown parameter at the end of the impulse response vector $\boldsymbol{h}^{{j}}$, and $\mathbf{\Theta}$ is the $N\times (D+1)$ matrix derived from the Toeplitz matrix (Heinig and Ross 1984) of $\boldsymbol{C}_{{\text{P}}}\left({{t}_{n}}\right)$ for $D$ temporal convolution points:

$$\begin{eqnarray}\mathbf{\Theta}=\left[\begin{array}{c} {{C}_{\text{P}}}\left({{t}_{1}}-t_{1}^{\prime}\right) & \cdots \\ \vdots & \ddots \\ {{C}_{\text{P}}}\left({{t}_{N}}-t_{1}^{\prime}\right) & \cdots \end{array}~\begin{array}{c} {{C}_{\text{P}}}\left({{t}_{1}}-t_{D}^{\prime}\right) & {{C}_{\text{P}}}\left({{t}_{1}}\right) \\ \vdots & \vdots \\ {{C}_{\text{P}}}\left({{t}_{N}}-t_{D}^{\prime}\right) & {{C}_{\text{P}}}\left({{t}_{N}}\right) \end{array}\right]~\end{eqnarray} \tag{ 11 }$$

For the proposed nested gPatlak 4D ML-EM WB reconstruction algorithm, each global iteration step is now decomposed into three distinct respective steps, unlike the two steps previously described for nested sPatlak 4D algorithm.

The first step, which is identical with step 1 of the nested sPatlak 4D method, involves a single update of the estimated TAC $\boldsymbol{x}_{{j}}=\left[x_{j}^{n}\right]_{n=1}^{N}$ at voxel $j$, through a tomographic EM estimation process, and is often the most computationally expensive, as it is applied consecutively to all $N$ dynamic frames and employs the large tomographic matrix P. Thus, for $d=1\ldots D~$ and $~n=1\ldots N$, we have for step 1:

$$\begin{eqnarray}&&x_{j,\text{new}}^{n}=\frac{x_{j,\text{old}}^{n}}{\underset{i}{\sum}\,\boldsymbol{P}_{{ij}}}\underset{i}{\sum}\,\boldsymbol{P}_{{ij}}\left[\frac{\boldsymbol{y}_{{i,n}}}{\underset{j}{\sum}\,\boldsymbol{P}_{{ij}}\underset{l}{\sum}\,{{\mathbf{\Theta}}_{nl}}r_{l,\text{old}}^{j}}\right]\end{eqnarray} \tag{ 12 }$$

Subsequently, the previously estimated TAC $\boldsymbol{x}_{{j,\text{new}}}$ of voxel j from step 1 and the measured data in $\mathbf{\Theta}$ are employed to estimate the Patlak response vector $\boldsymbol{r}^{{j}}$ of size $D+1$, through the following nested iterative ML-EM process (step 2):

$$\begin{eqnarray}&&\boldsymbol{r}_{\text{new}}^{j}=\frac{\boldsymbol{r}_{\text{old}}^{j}}{\underset{n}{\sum}\,{{\mathbf{\Theta}}_{nl}}}\underset{n}{\sum}\,{{\mathbf{\Theta}}_{nl}}\left[\frac{x_{j,\text{new}}^{n}}{\underset{l}{\sum}\,{{\mathbf{\Theta}}_{nl}}r_{l,\text{old}}^{j}}\right]~\end{eqnarray} \tag{ 13 }$$

Thus, after all nested sub-iterations in step 2 of current global iteration have been completed, $\boldsymbol{r}_{\text{new}}^{j}$ is estimated, which includes the impulse response vector h^j and gPatlak parameter $~{{V}^{j}}$. Subsequently, in step 3 of current global iteration, the gPatlak parameters $~K_{l}^{j}$ and $~k_{\text{loss}}^{j}$ are analytically derived, as it will be described later. By repeating the previous 3 steps for a number of ML-EM iterations in all voxels of a particular bed position, the gPatlak images are reconstructed for that bed. Finally, this process is repeated for the dynamic data of the rest of the beds, to ultimately produce WB gPatlak images.

Similarly with sPatlak, the presented nested gPatlak 4D algorithm targets at Poisson log-likelihood types of global and surrogate functions and employs the ML-EM algorithm for the external and the nested optimization problems. The main difference lies in the type of nested estimates targeted by the gPatlak algorithm. Due to the non-linear relationship between the gPatlak parameters and dynamic image space, the latter could not be estimated directly from the nested ML-EM approach employed in the previous section for nested sPatlak 4D case. Instead, the Patlak response vector r^j at each voxel j is now estimated through the same nested ML-EM update process, as it is linearly related with the dynamic image estimates, according to equation (10b). Therefore, the same conditions apply to gPatlak case, as those illustrated in figure 3, if sPatlak parameter vector m_s is replaced by the Patlak impulse response vector r. In fact, if k_loss is set to zero, the gPatlak 4D formulation in equation (10b) reduces to the sPatlak framework and the direct linear relationship between parametric and dynamic image space is restored.

The updated Patlak response vector r^j maximizes now a surrogate Poisson log-likelihood given the current TAC estimate x_j from step 1, as illustrated below:

$$\begin{eqnarray}&&\boldsymbol{r}_{\text{new}}^{j}=\underset{\boldsymbol{r}^{{j}}}{{\text{argmax}}}\,\underset{n=1}{\overset{N}{\sum}}\,x_{j,\text{new}}^{n}\log x_{j}^{n}\left(\boldsymbol{r}^{{j}}\right)-x_{j}^{n}\left(\boldsymbol{r}^{{j}}\right)\end{eqnarray} \tag{ 14 }$$

The first parameter to be updated at every nested ML-EM iteration of equation(13) is $~{{V}^{j}}$, as the last element of the updated vector r^j. Then, inspired by a similar analytical derivation for a reversible 1-tissue compartment kinetic model (Yan et al 2012, Wang and Qi 2013), the analytical solutions for K and k_loss gPatlak parameters at every voxel j can be calculated as follows (Karakatsanis and Rahmim 2014a):

$$\begin{eqnarray}&&k_{\text{loss},\text{new}}^{j}={{S}^{-1}}\left\{\frac{\underset{d=1}{\overset{D}{\sum}}\,t_{d}^{\prime}h_{d,\text{new}}^{j}}{\underset{d=1}{\overset{D}{\sum}}\,h_{d,\text{new}}^{j}}\right\}~\end{eqnarray} \tag{ 15 }$$

where ${{S}^{-1}}$ is the inverse of the function below:

$$\begin{eqnarray}&&S\left({{k}_{\text{loss}}}\right)=\frac{\underset{d=1}{\overset{D}{\sum}}\,t_{d}^{\prime}{{\text{e}}^{-k_{\text{loss}}^{j}t_{d}^{\prime}}}}{\underset{d=1}{\overset{D}{\sum}}\,{{\text{e}}^{-k_{\text{loss}}^{j}t_{d}^{\prime}}}}\end{eqnarray} \tag{ 16 }$$

In order to enhance computational efficiency, a look-up table for $S\left({{k}_{\text{loss}}}\right)$ can be pre-calculated and loaded to computer memory at the start of the reconstruction algorithm for a range of possible ${{k}_{\text{loss}}}$ initial values. Then, during reconstruction, this look-up table can be utilized to invert $S\left({{k}_{\text{loss}}}\right)$ function and efficiently determine the updated estimate $k_{\text{loss},\text{new}}^{j}$ with equation (15).

Finally, the tracer influx rate constant parameter $K_{\text{i},\text{new}}^{j}$ can be also analytically calculated from the current estimates $\boldsymbol{r}_{\text{new}}^{j}$ and $k_{\text{loss},\text{new}}^{j}$ as follows:

$$\begin{eqnarray}&&K_{\text{i},\text{new}}^{j}=\frac{\underset{d=1}{\overset{D}{\sum}}\,h_{d,\text{new}}^{j}}{\underset{d=1}{\overset{D}{\sum}}\,{{\text{e}}^{-k_{\text{loss},\text{new}}^{j}t_{d}^{\prime}}}}\end{eqnarray} \tag{ 17 }$$

Although the $S\left({{k}_{\text{loss}}}\right)$ look-up table is pre-loaded, the estimation of gPatlak parameters $K_{\text{i}}^{j}$ and $k_{\text{loss}}^{j}$ from the current r^j estimate may be computationally inefficient, if repeated for each nested sub-iteration. Besides, only the EM update of r^j is strictly required to maximize the surrogate EM log-likelihood as in equation (14). Therefore, here we propose updating only the r^j vector at every nested sub-iteration, except for the last one wherein the gPatlak parameters $K_{\text{i}}^{j}$ and $k_{\text{loss}}^{j}$ are estimated as well.

Furthermore, we recommend not using the newly estimated $K_{\text{i}}^{j}$ and $k_{\text{loss}}^{j}$ parameters to update ${{h}^{j}}$ estimates of the new global iteration cycle. Aside from the observation that such an update would be redundant and only add computational cost, as ${{h}^{j}}$ is already updated before, it can also be 'risk-prone' for the proper global EM convergence of the algorithm. The risk lies in the estimation of $K_{\text{i}}^{j}$ and $k_{\text{loss}}^{j}$ parameters, which is not exclusively driven by the nested ML-EM process (steps 1 and 2), as was the case with sPatlak 4D method. Now, an analytical derivation is additionally employed in the end (step 3), which forces the new estimates $K_{\text{i}}^{j}$ and $k_{\text{loss}}^{j}$ to be related with $\boldsymbol{h}^{{j}}=\left[h_{d}^{j}\right]$ according to the following equation: $h_{d}^{j}=K_{\text{i}}^{j}{{\text{e}}^{-k_{\text{loss}}^{j}t_{d}^{\prime}}}$, $d=1\ldots D$, provided $S\left({{k}_{\text{loss}}}\right)$ inversion in equation (15) is accurate. Therefore, depending on the sampling density of the $S\left({{k}_{\text{loss}}}\right)$ discrete look-up table and its range (equation (16)), which can both be freely determined by the user, the linear interpolation accuracy of $S\left({{k}_{\text{loss}}}\right)$ inversion in equation (16) may be degraded, therefore affecting the bias in the parametric ${{K}_{i}}$ and ${{k}_{\text{loss}}}$ estimates. As a result, the optimization transfer requirements may not be strictly fulfilled, if the inversion of the $S\left({{k}_{\text{loss}}}\right)$ look-up table is interfering with ML-EM estimation at every global iteration step. Although we have observed a negligible error associated with the analytic calculations even when moderate sampling rates are selected (1000 samples uniformly drawn from a (10⁻⁵,1) k_loss range), the overall convergence of the ML-EM algorithm may nevertheless be affected after several global ML-EM iterations. Therefore, to ensure the proper EM convergence properties of the gPatlak 4D algorithm and save computational time, the r^j vector of the next global tomographic iteration in equation (12) should be updated directly from the r^j estimate of the last nested sub-iteration, denoted as $\boldsymbol{r}_{\text{old}}^{j}$ in equation(13).

Normally, conventional 3D ML-EM iterative reconstruction algorithms are associated with objective functions that do not require any special initialization scheme. In this study, all 3D ML-EM methods have been initialized with unity values. The sPatlak 4D algorithms are also characterized by a sufficiently stable EM convergence when initialized with unity K_i and V images, due to their linearity and robustness, and thus no special initialization was applied to this class of methods.

Nevertheless, non-linear 4D reconstruction methods involve more complex objective functions, and a more advanced initialization scheme may be helpful. In particular, gPatlak 4D algorithms involve non-linear parameters, and thus, their EM convergence is sensitive to initialization. Therefore, for gPatlak 4D nested algorithm we evaluated (i) a conventional scheme involving initialization of K_i and V estimates with unity values, and (ii) a novel sPatlak-based scheme, where K_i and V parameters were initialized with respective sPatlak 4D estimates. In both cases, k_loss initial value was set to zero, which is equivalent to the sPatlak method. Although initialization with zero values is not recommended in ML-EM algorithms to avoid trapping of estimates to zeroes in subsequent iterations due to the multiplicative update mechanism, k_loss belongs to an exponential term in the gPatlak model and thus zero is effectively translated as the unity value. The number of sPatlak ML-EM iterations employed to produce the parameter values for gPatlak initialization were determined based on noise-bias trade-off performance in simulated data.

The noise-free dynamic PET SUV cardiac images in figure 5(a) (1st row) illustrate the variability introduced to each simulated lesion uptake and contrast during the first 45 min p.i. due to the modeled kinetics (table 1). The simulated dynamic PET images were produced from 3D ML-EM reconstruction (3 cycles of 21 iterations each) of dynamic cardiac data which were sampled according to our validated WB dynamic PET protocol (figure 1). Moreover, the reconstructed noise-free indirect and direct (s/g)Patlak K_i images in the 2nd row of figure 5(a) converged to higher lesion TBR contrast scores than any of the dynamic noise-free PET images for both ROI groups A and B. Therefore, in the absence of noise, Patlak may theoretically offer information beyond SUV and thus the complementary application of the two may enhance lesion detectability performance.

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In noise-free conditions, indirect and direct methods are expected to match in performance, after convergence is attained. Indeed, no visually distinct difference was observed in the noise-free K_i images between the two method classes (figure 5(a)). In the presence of noise, however, the benefit in noise and resolution of properly initialized direct 4D versus indirect Patlak analysis is illustrated when visually comparing the noisy K_i simulated images (figures 4 and 5(b)), especially for the tumor lesions of B group in the right lung. Moreover, the noise-bias trade-off curves (figure 5(c)) clearly demonstrated, for all evaluated ROIs, the superiority of 4D sPatlak and properly initialized 4D gPatlak algorithms, relative to the respective indirect methods. In particular, we observed significantly reduced noise at matched bias (resolution) and vice versa for the direct 4D versus the indirect methods in all ROIs. Finally, the 4D methods converged to distinctly smaller bias values than indirect methods, thus suggesting reduced noise-induced bias compared to indirect Patlak.

These observations were further supported by the TBR and CNR quantitative analysis on the same four K_i image ROIs in figure 6. Nevertheless, it should be noted that 4D imaging methods were associated with a relatively higher gain in CNR rather than TBR scores, as the main benefit of direct over indirect parametric reconstruction is the reduction of the noise in the K_i images. The TBR relative enhancements of 4D over indirect algorithms can be attributed to the reduction of noise-induced bias for the former, as also indicated by the noise-bias trade-off analysis in figure 5(c). Nevertheless, the ground true TBR K_i contrast, as calculated from the true input values of our simulation study, was underestimated in all cases. In all cases, the observed bias in the lesion K_i estimates and respective underestimated TBR scores becomes higher for smaller diameters (A2 and B2 ROIs), which we attribute to the partial volume effects.

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A visual inspection of the ground truth K_i and k_loss images and their comparison with the noise-free reconstructed K_i images in figure 5(a) (2nd row) suggests that, in general, the gPatlak indirect 3D and direct 4D methods were associated with more accurate K_i estimates than respective sPatlak methods. Furthermore, both noise-free and noisy gPatlak 4D reconstructions yielded relatively higher K_i TBR contrast scores, than respective sPatlak reconstruction, for tumor ROIs of group B, where a relatively higher degree of uptake reversibility (k₄  =  0.012) was introduced in our simulations (table 1). Thus, our observations demonstrated the theoretical advantage of gPatlak over sPatlak algorithms, when evaluating regions exhibiting non-negligible uptake reversibility. However, the same results indicated lower K_i image noise for sPatlak versus the gPatlak 4 D methods. The respective noise-bias curves (figure 5(c)) confirmed the previous findings, as they revealed smaller bias at matched noise levels and higher noise at matched resolution (bias) for gPatlak 4D reconstruction methods.

Furthermore, in terms of lesion detectability performance, the results in figure 6 suggest that the main differences between sPatlak and gPatlak 4D methods were observed in TBR and CNR scores, with TBR being affected more profoundly. We attribute this finding to the relatively higher noise levels for gPatlak imaging, even within the 4D framework. Although bias and TBR contrast are enhanced with gPatlak 4D methods, the increased noise associated with gPatlak non-linear model eventually limits gPatlak 4D CNR scores. As a result, gPatlak 4D is not increasing the CNR scores as much as it enhances the TBR scores.

The expected gain in ML-EM convergence rate for the nested relative to the conventional, i.e. non-nested, 4D sPatlak implementations was illustrated qualitatively and quantitatively in figures 5(b) and (c) respectively. In particular, visual inspection of B1 and B2 lesions contrast as a function of the iteration cycles in simulated K_i images of figure 5(b) suggested a faster contrast recovery, and thus convergence rate, for the nested sPatlak K_i images. In addition, the respective noise-bias curves in figure 5(c) indicated smaller bias values at matched noise levels for the nested sPatlak 4D implementation.

Moreover, a mildly faster 4D ML-EM convergence was recorded as the number of nested sub-iterations increased per global iteration step. This is conjectured from all three plots of the 2nd column of figure 7. However, the gain in bias and TBR contrast became progressively negligible when more than 20 sub-iterations were involved, as convergence had already been established at earlier iterations in these cases. Meanwhile, the noise was being steadily deteriorated in the same cases, due to the higher number of nested updates involved per global iteration step. As a result, for higher than 20 nested sub-iterations, image noise kept increasing relatively faster than TBR lesion contrast and, consequently, CNR started dropping at later iterations. Although not included in the results, it should be noted that a very small number of sub-iterations (<10) resulted in consistently slower convergence in all nested 4D algorithms.

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The noise-free images in figure 5(a) demonstrate that the (s/g)Patlak 4D ML-EM algorithms converge in theory to the global optimal solution regardless of the initialization method. Thus, our findings indicated proper theoretical EM convergence properties for the implemented algorithms. In the presence of noise, however, the conventional method of initializing 4D ML-EM with K_i  =  1, k_loss  =  0 and V  =  1 parameter values, yielded correct EM convergence only in the case of 4D sPatlak method, as it can be conjectured by comparing 3rd and 4th row in figure 5(b). Nevertheless, as the K_i images of the last 2 rows in figure 5(b) illustrate, higher K_i lesion contrasts were attained with 4D gPatlak, compared to sPatlak (3rd row), after initializing the gPatlak 4D method with K_i and V estimates from the first 21 (5th row) or 3  ×  21  =  63 (6th row) sPatlak iterations.

The importance of sPatlak-based initialization for gPatlak 4D algorithms was further demonstrated by the noted bias reduction as well as TBR and CNR score enhancements in figure 7 (1st column plots), when more sPatlak 4D global iterations were involved in the initialization of gPatlak 4D algorithm. However, after 3 cycles of 21 sPatlak ML-EM initial iterations, no additional benefit was observed for gPatlak 4D EM convergence rate. Thus, under noisy conditions, gPatlak 4D reconstruction may require a minimum number of sPatlak 4D iterations for its initialization, to ensure proper convergence and thus high quantification accuracy in K_i reconstructed images.

The spatial noise levels visually observed in the background regions of the selected liver and chest target ROIs of WB 4D (s/g)Patlak K_i clinical images of figure 8 were comparable to the respective static SUV image noise. This is also evident by comparing the TBR and CNR scores of respective clinical K_i and SUV images for both evaluated ROIs in the same figure. In particular, the superiority of K_i imaging, relative to SUV, in terms of TBR contrast is also retained to nearly the same or higher degree in terms of CNR score. As CNR is derived from TBR after normalizing the latter with spatial noise in the target background, the previous observation suggest similar or lower quantitative levels of spatial noise between direct 4D K_i and SUV clinical images, at least for the two evaluated ROIs. Thus, our results demonstrate the clinical feasibility of 4D WB Patlak K_i methods, when applied on a streamlined 6-pass WB PET protocol. In addition, the superior TBR and CNR 4D K_i scores, relative to SUV, on the two identified ROIs indicate potential enhancement of tumor detectability performance, when complementing the currently established in clinic WB SUV imaging protocols with the proposed direct 4 D WB (s/g)Patlak methods.

The images in figure 8 illustrated the lower noise of direct 4D relative to indirect Patlak methods. Moreover, the quantitative plots in figure 8 demonstrated the superior TBR and CNR performance for all 4D Patlak methods, compared to the respective indirect methods, particularly for the chest ROI. The improvement was more evident in terms of the CNR metric, owing to the lower noise levels observed in the 4D reconstructions versus indirect Patlak analysis. Our clinical findings confirmed the simulation results and can be explained by the more efficient utilization of the acquired data with 4D Patlak algorithms. Finally, the quantitative TBR and CNR analysis suggested that the gain observed when switching from indirect to direct 4D Patlak methods is relatively larger than the respective gain between standard and generalized Patlak models.

Our clinical validation results in figure 8 demonstrated the superior TBR lesion contrast scores for nested gPatlak 4D K_i images, both via the qualitative inspection of the respective patient WB K_i images as well as through the quantitative TBR analysis in both evaluated ROIs. Moreover, despite the higher noise levels observed in gPatlak K_i images, relative to sPatlak, the highest clinical ROI CNR scores were systematically observed for the former. Besides, the clinical TBR and CNR score differences between the two 4D Patlak methods were not as significant as the respective differences between (i) indirect sPatlak and gPatlak or (ii) direct versus indirect Patlak methods. In other words, the differences between the two Patlak models were less significant in the 4D framework.

The series of WB K_i images in figure 9 illustrate the convergence of sPatlak and gPatlak 4D methods, when applied to the same patient dataset and after being initialized with the proposed schemes. The two 4D algorithms converged to different but similar sPatlak and gPatlak solutions in the last two cycles of 21 iterations.

Moreover, the TBR and CNR plots describe the quantitative effect on 2 chest ROIs of the number of nested ML-EM sub-iterations as well as that of sPatlak-based initialization for gPatlak 4D algorithm. In particular, the plots of the 2nd row suggested superior TBR and CNR performance for both ROIs in clinical gPatlak-4D K_i images, when at least 3  ×  21 ML-EM sPatlak iterations are employed for its initialization. Any higher number of iterations only resulted in negligible convergence acceleration. Furthermore, the TBR and CNR scores of the 3rd row suggested a minimum number of 20 nested ML-EM sub-iterations to sufficiently accelerate convergence without increasing noise in the K_i images.