A local-neighborhood Lassen plot filter for creating occupancy and non-displaceable binding images

Abstract

For radioligands without a reference region, the Lassen plot can be used to estimate receptor occupancy by an exogenous drug ($O^{\mathit{Drug}}$). However, the Lassen plot is not well-suited for spatial variation in $O^{\mathit{Drug}}$. To overcome this limitation, we introduce a Lassen plot filter, i.e. a Lassen plot applied to local neighborhoods in PET images. Image data were simulated with regional variation in $V_{ND}$, $O^{\mathit{Drug}}$, both, or neither and analyzed using the change in binding potential (${\Delta BP}_{ND}$), the conventional Lassen plot, and the Lassen plot filter at the region of interest (ROI) and voxel level. All methods were also applied to a human [¹¹C]flumazenil occupancy study using PF-06372865. This combination of a non-selective radioligand and selective drug should lead to varying $O^{\mathit{Drug}}$provided the distribution of subtypes varies spatially. In contrast with ${\Delta BP}_{ND}$ and the conventional Lassen plot, ROI-level and voxel-level Lassen plot filter estimates remained unbiased in the presence of regional variation in $V_{ND}$ or $O^{\mathit{Drug}}$. In the [¹¹C]flumazenil data-set, $O^{\mathit{Drug}}$ was shown to vary regionally in accordance with the distribution of binding sites for [¹¹C]flumazenil and PF-06372865. We demonstrate that a local-neighborhood Lassen plot filter provides robust and unbiased estimates of $O^{\mathit{Drug}}$ and $V_{ND}$ without the need for any user intervention.

Introduction

Neuroreceptor occupancy by an exogenous drug ($O^{\mathit{Drug}}$) can be estimated in vivo using positron emission tomography (PET).¹ Estimation of $O^{\mathit{Drug}}$ is based on the principle that the exogenous drug decreases the fraction of receptors that are available for the radioligand to bind. This decrease in binding can be quantified by comparing a baseline with a post-drug scan. Volume of distribution ($V_T$) of the radiotracer can be used to estimate occupancy, but $V_T$ contains specifically ($V_S$) and nonspecifically ($V_{ND}$) bound components. Since drug occupancy will only influence $V_S$, an estimate of $V_{ND}$ is required. $V_{ND}$ can be derived from a reference region, provided a region with negligible specific binding exists. For many radioligands, an adequate reference region is not available. In such cases, $O^{\mathit{Drug}}$ can be estimated using an elegant solution first proposed by Lassen et al. in 1995 and subsequently refined by Cunningham et al. in 2010.^2,3

The Lassen plot is often burdened by certain assumptions which we seek to alleviate. The first assumption is that the $V_{ND}$ is uniform throughout the brain. The second assumption is that $O^{\mathit{Drug}}$ is uniform throughout the brain. In practice, regional variation in $O^{\mathit{Drug}}$ has been observed^4,5 and the assumption of uniform $V_{ND}$ has been challenged.^6–8 Spatially varying $O^{\mathit{Drug}}$ will bias whole brain estimates of $O^{\mathit{Drug}}$ based on the Lassen plot. Similarly, spatially varying $V_{ND}$ will bias Lassen plot estimates of $O^{\mathit{Drug}}$. This paper employs a Lassen plot “filter” that operates at the local voxel neighborhood level to create parametric images of $O^{\mathit{Drug}}$ and $V_{ND}$ thus eliminating the assumption of uniformity across the whole brain. We applied our Lassen plot filter approach, at both the region and voxel levels, to simulated images and to human [¹¹C]flumazenil data to evaluate the ability of the method to recover regional variation in $O^{\mathit{Drug}}$ and $V_{ND}$ without bias.

Materials and methods

Definitions and dimensions of all variables and conventions for use of different typefaces are listed in Supplemental Table S1.

Theory

$V_T$ values obtained at baseline and post-drug can be written as³

$${{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}={\mathit{V}}_{\mathit{S}}\mathrm{ }+\mathrm{ }{V}_{ND}$$ (1)

$${{\mathit{V}}_{\mathit{T}}}^{\mathit{Drug}}={\mathit{V}}_{\mathit{S}}\mathrm{*}\mathrm{ }\left(1-\mathrm{ }{O}^{\mathit{Drug}}\right)+\mathrm{ }{V}_{ND}$$ (2)

where ${{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}$ and ${{\mathit{V}}_{\mathit{T}}}^{\mathit{Drug}}$ are vectors of the regional $V_T$ values from multiple regions, measured at baseline and after drug administration, respectively. $V_{ND}$ and $O^{\mathit{Drug}}$, non-specific binding and occupancy of the target by the drug, respectively, are scalar values. These are the first two assumptions of the Lassen plot. Namely, that a single value for each parameter applies to the entire brain. That is, $V_{ND}$ and $O^{\mathit{Drug}}$ are uniform everywhere. The Lassen plot itself is derived by subtracting equation (2) from equation (1) and substituting to yield

$${{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}{{- \mathit{V}}_{\mathit{T}}}^{\mathit{Drug}}=O^{\mathit{Drug}}\left(\mathit{Brain}\right)\mathrm{*}\left({{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}-\mathrm{ }{V}_{ND}\left(\mathit{Brain}\right)\right)$$ (3)

Equation (3) is typically solved in a least-squares sense by plotting (${{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}{{-\mathit{V}}_{\mathit{T}}}^{\mathit{Drug}}$) versus ${{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}$ to yield an estimate of $O^{\mathit{Drug}}\left(\mathit{Brain}\right)$. A third assumption is implicit: there is sufficient spread in ${{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}$ to use a linear regression model. Sufficient spread is likely to be achieved when $V_T$ values from different brain regions are used to populate this vector. However, if the data in a region of interest (ROI) contain sufficient true spatial heterogeneity there may be enough spread in $V_T$ to estimate $O^{\mathit{Drug}}$ and $V_{ND}$ in a single region. Extending this logic further, a region could, in principle, encompass just a single voxel and its immediate neighbors.

If $O^{\mathit{Drug}}$ and $V_{ND}$ are allowed to vary between, but not within, brain ROIs, we can estimate $O^{\mathit{Drug}}\left(\mathit{ROI}\right)$ and $V_{ND}\left(\mathit{ROI}\right)$ in a ROI using voxels only within that ROI. The operational equation of this ROI-level Lassen plot would be

$${{\mathit{V}}_{\mathit{T}}}_{\mathit{ROI}}^{\mathit{Base}}{{-\mathit{V}}_{\mathit{T}}}_{\mathit{ROI}}^{\mathit{Drug}}=O^{\mathit{Drug}}\left(\mathit{ROI}\right)\mathrm{ }\mathrm{*}\mathrm{ }\left({{\mathit{V}}_{\mathit{T}}}_{\mathit{ROI}}^{\mathit{Base}}-\mathrm{ }{V}_{ND}\left(\mathit{ROI}\right)\right)$$ (4)

where ${{\mathit{V}}_{\mathit{T}}}_{\mathit{ROI}}^{\mathit{Base}}$ and ${{\mathit{V}}_{\mathit{T}}}_{\mathit{ROI}}^{\mathit{Drug}}$ are vectors populated with all the voxels contained in one specific ROI. By extension, equation (4) can be rewritten to accommodate estimation of $O^{\mathit{Drug}}\left(i,j,k\right)$ and $V_{ND}\left(i,j,k\right)$ at the voxel (i, j, k) using its local neighborhood (LN) as

$${{\mathit{V}}_{\mathit{T}}}_{\mathit{L}\mathit{N}\mathit{(}\mathit{s}\mathit{)}}^{\mathit{Base}}{{- \mathit{V}}_{\mathit{T}}}_{\mathit{L}\mathit{N}\mathit{(}\mathit{s}\mathit{)}}^{\mathit{Drug}}=O^{\mathit{Drug}}\left(i,j,k\right)\mathrm{ }\mathrm{*}\mathrm{ }\left({{\mathit{V}}_{\mathit{T}}}_{\mathit{L}\mathit{N}\mathit{(}\mathit{s}\mathit{)}}^{\mathit{Base}}-\mathrm{ }{V}_{ND}\left(i,j,k\right)\right)$$ (5)

where the number of voxels, s, determines the size of the applied kernel. For example, s = 1 will apply the Lassen plot filter to a LN of (2s + 1)³ = 27 voxels by populating ${{\mathit{V}}_{\mathit{T}}}^{\mathit{Base}}$ and ${{\mathit{V}}_{\mathit{T}}}^{\mathit{Drug}}$ with those 27 values. Throughout the manuscript, s = 1 for every local-neighborhood Lassen plot filter and will thus be called a voxel-level approach. The outputs of equation (5) are two images: a $V_{ND}$ image and an $O^{\mathit{Drug}}$ image. If no spatial variation in $V_{ND}$ is observed, $V_{ND}$ can be fixed in equation (5) to a whole-brain estimate obtained by equation (5) (or another method, such as a blocking experiment). As an additional analysis, we evaluate whether this univariate version of the Lassen plot filter is less sensitive to noise than equation (5).

If non-displaceable binding potential (${BP}_{ND}$) is the measured outcome, receptor occupancy can be estimated as the fractional change in ${BP}_{ND}$ (DBP_ND)

$${\Delta \mathit{BP}}_{\mathit{ND}}=\frac{{\mathit{BP}}_{\mathit{ND}}^{\mathit{Base}}-\mathrm{ }{\mathit{BP}}_{\mathit{ND}}^{\mathit{Drug}}}{{\mathit{BP}}_{\mathit{ND}}^{\mathit{Base}}}=\mathrm{ }\frac{{({\mathit{V}}_{\mathit{T}}-{\mathit{V}}_{\mathit{T}}^{\mathit{Ref}})}^{\mathit{Base}}-{({\mathit{V}}_{\mathit{T}}-\mathrm{ }{\mathit{V}}_{\mathit{T}}^{\mathit{Ref}})}^{\mathit{Drug}}\mathrm{ }}{{({\mathit{V}}_{\mathit{T}}-\mathrm{ }{\mathit{V}}_{\mathit{T}}^{\mathit{Ref}})}^{\mathit{Base}}}$$ (6)

As ${\Delta BP}_{ND}$ is the presumed gold standard in receptor occupancy estimation, it will also be included in our analyses for comparison. ${\Delta BP}_{ND}$ can be calculate at the whole-brain, ROI, and voxel level irrespective of regional variation in $O^{\mathit{Drug}}$. Classically, reference region models assume that $V_{ND}$ is constant (as represented by $V_T$ in the reference region) so to calculated ${\Delta BP}_{ND}$ at the voxel level would require a spatially invariant $V_{ND}$. But with an arterial input function, one could consider estimating ${BP}_{ND}$ locally from the two-tissue compartment model.

Algorithm

An algorithm for Lassen plot filtering was implemented in Matlab (version: 2016b, Mathworks, Natick, USA). The voxel-level Lassen plot is presented below in pseudo-code, annotation is identified with #:

Initialize

Image_base = V_T image obtained at baseline

Image_post = V_T image obtained after drug

s = kernel size

# See note below for choice of s #

For each voxel i,j,k

# Assemble the vectors of LN values #

V_T^Base = Image_base(i-s:i+s,j-s:j+s,k-s:k+s)

V_T^Drug = Image_post(i-s:i+s,j-s:j+s,k-s:k+s)

# Perform Lassen plot on the LN values #

(V_T^Base – V_T^Drug) = O^Drug * (V_T^Base – V_ND)

# Update the output images #

O^Drug(i,j,k) = O^Drug

V_ND(i,j,k) = V_ND

End

The kernel will use all voxels in the LN. In case less voxels have a numerical value (e.g. brain edges), a Lassen plot is constructed as long as more than nine numerical values are available.

*The kernel will use all voxels in the LN. In case less voxels have a numerical value (e.g. brain edges), a Lassen plot is constructed as long as more than nine numerical values are available.

Simulations

Simulations were based on [¹¹C]flumazenil data obtained at baseline, as follows

$${\mathit{V}}_{\mathit{T}}^{\mathit{Base}}=\mathrm{ }\left({\mathit{V}}_{\mathit{ND}}^{\mathit{Sim}}\mathrm{ }.\mathrm{*}{{\mathit{R}}_{\_\mathit{V}}}_{\mathit{ND}}\right)+{\mathit{V}}_{\mathit{S}}^{\mathit{Sim}}+{\mathbf{\epsilon }}_1$$ (7)

$$\begin{array}{l}{\mathit{V}}_{\mathit{T}}^{\mathit{Drug}}=\left({\mathit{V}}_{\mathit{ND}}^{\mathit{Sim}}\mathrm{ }.\mathrm{*}\hspace{0.17em}\mathit{R}\_\mathit{V}\mathit{ND}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+\left(\mathbf{1}-\left({\mathit{O}}^{\mathit{Drug}}\mathrm{ }.\mathrm{*}\hspace{0.17em}\mathit{R}\_\mathit{O}\mathit{Drug}\right)\right).\mathrm{*}{\mathit{V}}_{\mathit{S}}^{\mathit{Sim}}+{\mathbf{\epsilon }}_2\end{array}$$ (8)

where ${\epsilon }_1$ and ${\epsilon }_2$ represent independent random noise components following N(0,(max($V_S$)) * 0.05) and “.*” indicates multiplication of corresponding 2D array elements. Spatial variation was simulated by creating four regions with unique combinations of $O^{\mathit{Drug}}$ and $V_{ND}$. This spatial variation in $V_{ND}$ and $O^{\mathit{Drug}}$ at the region level was introduced by multiplying the $O^{\mathit{Drug}}$ and $V_{ND}$ vectors by scale factors contained in the spatial variation vectors R_O^Drug and R_V^ND, respectively. For simulations without regional variation all values in R_O_Drug and R_V_ND were set to 1. For simulations with spatial variation, voxels in the simulated regions 1 through 4 were assigned the values [0.75, 0.5, 1.5, 1.25] and [0.5, 0.75, 1.25, 2] for R_O^Drug and R_V^ND, respectively. We simulated data under the four conditions (Table S2): (1) no variation in $O^{\mathit{Drug}}$ and $V_{ND}$; {V_ND,O^Drug}, (2) variation in $V_{ND}$ only; {V_ND(X), O^Drug}, (3) variation in $O^{\mathit{Drug}}$ and $V_{ND}$; {V_ND(X), O^Drug(X)}, and (4) variation in $O^{\mathit{Drug}}$ only; {V_ND, O^Drug(X)}.

$O^{\mathit{Drug}}$ was set to 0.5 throughout the brain for every simulated subject, whereas $V_{ND}$ was set to be a constant comparable to the value of the $V_T$ in the pons, which is thought to be an approximation of $V_{ND}$ for flumazenil.^9,10

All methods were used to estimate parametric occupancy images for every condition. The conventional Lassen plot was applied on 29 regions to obtain its estimates. Regional averages were extracted from resulting parametric images and compared to the true value to assess bias and variance for each method.

Biases are reported as mean ± standard deviation and as a range of average regional bias.

Flumazenil data sets

We performed all analyses, except for ${\Delta BP}_{ND}$ (due to the absence of a reliable reference region), on $V_T$ images from [¹¹C]flumazenil PET acquired on humans. Analyses of the data have been published in detail by Nickolls et al.¹¹ In brief, we analyzed scans obtained from five healthy subjects receiving acute doses of either 10 (n = 3) or 65 (n = 2) mg of PF-06372865. PF-06372865 is a GABA inhibitor with high affinity for α₁, α₂, and α₃, but not α₅ subunits of the benzodiazepine binding sites on the GABA-A receptor. Baseline scans occurred 1 day prior to and post-dose scans occurred 1.5 h after PF-06372865 administration. Images were acquired for 120 min on an ECAT EXACT HR+ scanner (Siemens Medical Systems, Knoxville, TN, USA) after injection of 570 ± 141 MBq (injected dose: 2.7 ± 1.3 µg). Each subject had a structural MRI scan of the head for use in image co-registration and to screen subjects for possible anatomical abnormalities. Arterial blood samples were collected to measure the arterial plasma input function and the unmetabolized fraction of the radiotracer in the plasma.

Dynamic PET data were reconstructed with corrections for attenuation, normalization, scatter, randoms and dead time using the ordered subset-expectation maximization algorithm (four iterations and 16 subsets). PET images were corrected for motion using a mutual information algorithm (FSL-FLIRT). After co-registration between the template and each subject’s MRI and PET images, voxel-wise TACs were fitted with the one-tissue compartment model using the metabolite-corrected arterial input function to estimate $V_T$ images.

Two confirmatory analyses were performed for the voxel-wise occupancy method. First, we evaluated the effect of drug dose on the observed $O^{\mathit{Drug}}$. Second, we compared the estimated $O^{\mathit{Drug}}$ levels with the expected $O^{\mathit{Drug}}$ levels based on the literature on receptor-subtype distribution. [¹¹C]flumazenil binds to all subunits, whereas PF-06372865 binds primarily to α₁, α₂, and α₃. The affinity of PF-06372865 for the α₅ subunit is 1–2 orders lower than for the other subunits. PF-06372865 will therefore displace [¹¹C]flumazenil more thoroughly in regions containing predominantly the α₁, α₂, and α₃ subunits. It has been shown that striatal regions, such as caudate and putamen, contain a mix of α₁, α₂, and α₃ and α₅ subunits, whereas other regions such as occipital cortex and cerebellum have mainly contributions from the α₁, α₂, and α₃ subunits.^12,13 Therefore, estimated $O^{\mathit{Drug}}$should be lower in the striatal regions (high α₅ contribution) than in the occipital cortex and cerebellum.

Univariate Lassen plot filter

We also implemented a “univariate” Lassen plot filter to simulated data (condition 4; no variation in $V_{ND}$, but spatial variation in $O^{\mathit{Drug}}$) and to the flumazenil data. In the univariate implementation, $V_{ND}$ was fixed at the whole brain average derived from the Lassen plot filter as described above and only $O^{\mathit{Drug}}$ was estimated at each voxel (see supplemental information).

Results

Simulated data

Parametric $O^{\mathit{Drug}}$ images created for each scenario from every simulation are shown in Figure 1. Note that the ROI-level Lassen plot was applied only to certain ROIs.

Figure 1.

Estimated parametric occupancy ($O^{\mathit{Drug}}$) images by row obtained using ${\Delta BP}_{ND}$ and Lassen plot at the whole-brain, ROI, and voxel level. Different components of the simulated data are varied in the columns. Images show the effects of variation in $V_{ND}$ only ({V_ND(X), O^Drug}) in column 2, variation in $V_{ND}$ and $O^{\mathit{Drug}}$ ({V_ND(X), O^Drug(X)}) in column 3, or variation in $O^{\mathit{Drug}}$ only ({V_ND, O^Drug(X)}) in column 4 compared to no variation in either ({V_ND,O^Drug}) in column 1. Ground truth simulated images are shown in the top row. Notice the similarity between ground truth in the top row and voxel-level estimation in the bottom row.

Parametric $V_{ND}$ images created for each scenario from every simulated situation are shown in Figure 2.

Figure 2.

Estimated parametric $V_{ND}$ images by row obtained the Lassen plot at the whole-brain, ROI, and voxel level. Different components of the simulated data are varied in the columns. Images show the effects of variation in $V_{ND}$ only ({V_ND(X), O^Drug}) in column 2, variation in $V_{ND}$ and $O^{\mathit{Drug}}$ ({V_ND(X), O^Drug(X)}) in column 3, or variation in $O^{\mathit{Drug}}$ only ({V_ND, O^Drug(X)}) in column 4 compared to no variation in either ({V_ND,O^Drug}) in column 1. Ground truth simulated images are shown in the top row. Notice that region-level $V_{ND}$ estimates look most like the ground truth.

Biases in occupancy

In the absence of any variation in $V_{ND}$ or $O^{\mathit{Drug}}$ (column 1 in Figure 1), essentially no bias (<1%) was observed in estimated occupancy with any of the methods.

Variation in $V_{ND}$, but not $O^{\mathit{Drug}}$ (column 2 in Figure 1), introduced a large bias in the whole-brain estimate of $O^{\mathit{Drug}}\left(\mathit{Brain}\right)$ (–10.3 ± 1.0%, range: n.a.), smaller bias in ${\Delta BP}_{ND}$ (1.8 ± 14.3%, range: –12.0 to 26.0) and $O^{\mathit{Drug}}\left(i,j,k\right)$ (0.3 ± 0.8%, range: 0.0–0.7), and no appreciable bias in the $O^{\mathit{Drug}}\left(\mathit{ROI}\right)$ (0.0 ± 0.4%, range: –0.1 to 0.2).

Variation in both $V_{ND}$ and $O^{\mathit{Drug}}$ (column 3 in Figure 1) exacerbated the biases observed in the brain-wide $O^{\mathit{Drug}}\left(\mathit{Brain}\right)$ (63.8 ± 89.3%, range: –33.8 to 164.7), with smaller effects on ${\Delta BP}_{ND}$ (4.7 ± 18.2%, range: –12.2 to 31.5), $O^{\mathit{Drug}}\left(\mathit{ROI}\right)$ (0.0 ± 0.4%, range: –0.1 to 0.2), and $O^{\mathit{Drug}}\left(i,j,k\right)$ (0.3 ± 1.0%, range: 0.0–0.5).

Variation in $O^{\mathit{Drug}}$ alone (column 4 in Figure 1) caused large bias in the brain-wide estimates of $O^{\mathit{Drug}}\left(\mathit{Brain}\right)$ (48.8 ± 82.7%, range: –39.9 to 140.3). ${\Delta BP}_{ND}$ (1.8 ± 5.5%, range: –2.8 to 6.4), $O^{\mathit{Drug}}\left(\mathit{ROI}\right)$ (0.1 ± 0.7%, range: 0.0–0.5), and $O^{\mathit{Drug}}\left(i,j,k\right)$ (0.3 ± 1.2%, range: 0.0–1.3) exhibited very small biases. Figure 3 shows the bias of each method by region for the four simulated regions.

Figure 3.

Estimated $O^{\mathit{Drug}}$ levels in the simulated regions 1–4 (different panels, clockwise) using each method. Bars within each panel represent the average of all simulations estimated by ${\Delta BP}_{ND}$, and Lassen plot on whole-brain, ROI, and voxel level. Groups of bars represent estimates from simulations with variation in $V_{ND}$ only ({V_ND(X), O^Drug}), variation in $V_{ND}$ and $O^{\mathit{Drug}}$ ({V_ND(X), O^Drug(X)}), or variation in $O^{\mathit{Drug}}$ only ({V_ND, O^Drug(X)}). Horizontal reference lines indicate the simulated occupancy level. The condition of the simulation is on the X-axis. The estimated occupancy is on the Y-axis. The brackets above the bars indicate which parameters vary in the phantom and by what average amount.

Biases in V_ND

In the absence of any variation in $V_{ND}$ or $O^{\mathit{Drug}}$, the smallest bias in estimated $V_{ND}$ was achieved with the conventional whole-brain Lassen plot’s $V_{ND}\left(\mathit{Brain}\right)$ (–0.1 ± 0.4%, range: n.a.). $V_{ND}\left(\mathit{ROI}\right)$ (0.3 ± 1.8%, range: 0.0–0.9) and $V_{ND}\left(i,j,k\right)$ (0.7 ± 2.2%, range: 0.6–0.8) exhibited small biases.

Variation in $V_{ND}$, but not $O^{\mathit{Drug}}$, introduced a large bias into the conventional Lassen plot estimates for $V_{ND}\left(\mathit{Brain}\right)$ (–46.5 ± 20.7%, range: –68.5 to –5.4), no appreciable bias in the $V_{ND}\left(\mathit{ROI}\right)$(0.0 ± 2.0%, range: –0.4 to 0.8), and a small bias in $V_{ND}\left(i,j,k\right)$ (1.1 ± 3.4%, range: 0.2–1.9).

Variation in both $V_{ND}$ and $O^{\mathit{Drug}}$ exacerbated the biases in the conventional Lassen plot estimates of $V_{ND}\left(\mathit{Brain}\right)$(82.3 ± 70.2%, range: 7.4–222.2) but had little effect on the $V_{ND}\left(\mathit{ROI}\right)$ (–0.3 ± 3.6%, range: –0.9 to 0.3) or $V_{ND}\left(i,j,k\right)$ (1.1 ± 5.0%, range: 0.0–2.7).

Variation in $O^{\mathit{Drug}}$ only resulted in the largest bias in $V_{ND}\left(\mathit{Brain}\right)$ (23.2 ± 38.5%, range: n.a.). Negligible bias was observed for the $V_{ND}\left(\mathit{ROI}\right)$ (0.3 ± 2.9%) with a small bias in $V_{ND}\left(i,j,k\right)$ (1.2 ± 4.9%, range: 0.2–2.0) in this condition. Figure 4 shows the bias of each method for the four simulated regions.

Figure 4.

Estimated $V_{ND}$ levels in all four simulated regions (different panels) using each method. Sets of bars within each panel represent estimates from simulations with variation in $V_{ND}$ only ({V_ND(X), O^Drug}), variation in $V_{ND}$ and $O^{\mathit{Drug}}$ ({V_ND(X), O^Drug(X)}), or variation in $O^{\mathit{Drug}}$ only ({V_ND, O^Drug(X)}). Horizontal reference lines indicate the simulated $V_{ND}$ level. The condition of the simulation is on the X-axis. The estimated occupancy is on the Y-axis. The brackets above the bars indicate which parameter vary in the phantom and by what average amount. Note largest biases for three of four cases in the conventional whole brain Lassen plot.

Flumazenil results

Generated parametric $O^{\mathit{Drug}}$ and $V_{ND}$ images are shown in Figure 5. Representative voxel-level Lassen plots are shown in Figure S1.

Figure 5.

$O^{\mathit{Drug}}$ and $V_{ND}$ images created from human data based on PF-06372865 displacement of ¹¹C-Flumazenil. Voxel- (top row) and ROI-level (middle row) Lassen plot filters were applied. The conventional Lassen plot was also performed and is portrayed as images (bottom row). Color bar represents the estimated $O^{\mathit{Drug}}$ (values on left) and estimated $V_{ND}$ (values on right). The images shown are for a single subject administered 10 mg of PF-06372865.

Regionally averaged $O^{\mathit{Drug}}$ and $V_{ND}$ estimates were extracted from each set of parametric images and are summarized in Figure 6a and b, respectively. Significantly lower estimates for $O^{\mathit{Drug}}$ in the caudate and putamen, compared to the standard Lassen plot, were obtained with the ROI-level (p = 0.003 and p = 0.0002, respectively) and voxel-level (p = 0.030 and p = 0.027, respectively) Lassen plot filters. Higher $O^{\mathit{Drug}}$ levels than in the striatal regions were found in the occipital cortex and cerebellum with both the ROI-level and voxel-level methods, but only the voxel-level estimate of cerebellar $O^{\mathit{Drug}}$ achieved significance (p = 0.01). ROI-level and voxel-level $V_{ND}$ estimates were significantly lower than the conventional Lassen plot in the caudate (p = 0.01 and p = 0.0002, respectively). In addition, the ROI-level $V_{ND}$ estimates were lower in the putamen (p = 0.004) and cerebellum (p = 0.003).

Figure 6.

(a) Occupancy ($O^{\mathit{Drug}}$) and (b) $V_{ND}$ of PF-06372865 with [¹¹C]flumazenil in different regions by the Lassen plot and the ROI- and voxel-level implementations. *p < 0.05, **p < 0.01, and ***p < 0.0001.

As expected, estimated $O^{\mathit{Drug}}$ levels were higher for participants who received a 65 mg dose of PF-06372865 (92 ± 7%) than for participants that received 10 mg (75 ± 13%). The difference between the 10 and 65 mg doses was larger for regions with low estimated $O^{\mathit{Drug}}$, such as the caudate (57% vs. 88%) and putamen (67% vs. 86%), compared to regions with higher $O^{\mathit{Drug}}$ such as the occipital cortex (83% vs. 97%) and cerebellum (88% vs. 95%).

Univariate Lassen plot filter

Application of the univariate Lassen plot filter, with $V_{ND}$ fixed to the whole-brain estimate, showed a reduction in spatial heterogeneity of occupancy images in the simulated data (Figure 7a). This effect was also observed for the flumazenil data sets, while the observed spatial patterns of $O^{\mathit{Drug}}$ remained (Figure 7b).

Figure 7.

Univariate Lassen plot filter with $V_{ND}$ fixed at a whole-brain estimate (bottom row) shows a reduction in variance in $O^{\mathit{Drug}}$ compared to simultaneous estimation of both $V_{ND}$ and $O^{\mathit{Drug}}$ at the local-neighborhood level. Note that the spatial pattern, visible on these sagittal and coronal slices, remains intact after fixation of $V_{ND}$ in both the simulated (left) and [¹¹C]flumazenil data (right).

Discussion

Simulations showed that in the absence of spatial variation in $O^{\mathit{Drug}}$ and $V_{ND}$ all evaluated methods returned unbiased estimates of $O^{\mathit{Drug}}$ and $V_{ND}$. Spatial variation in either variable introduced biases in conventional Lassen plot estimates. ${\Delta BP}_{ND}$ was shown to be robust against variation in $O^{\mathit{Drug}}$, but highly sensitive to spatial variation in $V_{ND}$. ROI-level Lassen plots showed negligible bias when applied to all types of simulated conditions. Voxel-wise $O^{\mathit{Drug}}\left(i,j,k\right)$ estimates were able to recover simulated $O^{\mathit{Drug}}$and $V_{ND}$ but did show small biases and larger variances than the ROI-level implementation. ROI- and voxel-level Lassen plots estimated large displacement of [¹¹C]flumazenil by PF-06372865 in regions with high α₁–α₂–α₃ receptor-subtype purity. The $O^{\mathit{Drug}}$ was estimated to be lower by both Lassen plot filter methods in regions with known α₅ receptor subtypes, as would be expected. Regional differences were more pronounced for participants receiving a lower dose of PF-06372865. Fixing $V_{ND}$ to a whole-brain estimate reduces variance but retains sensitivity to spatially varying $O^{\mathit{Drug}}$.

The standard Lassen plot is not equipped to account for variation in $O^{\mathit{Drug}}$ throughout the brain. It over-/under-estimates the actual $O^{\mathit{Drug}}$where it is lower/higher than the volume-weighted whole-brain average. Although regional variation in brain receptor $O^{\mathit{Drug}}$ is still controversial, published studies have observed and explored this phenomenon.^4,5 Martinez et al.⁴ proposed several potential mechanisms: (1) Regions containing a high fraction of high activity state receptors might exhibit larger displacement of pure antagonist tracers by agonists leading to higher than average occupancies. (2) The radioligand and the competing drug might have different in vivo affinities for internalized versus externalized receptors. Regions with different ratios of internalized/externalized receptors would then demonstrate different occupancy levels if the radioligand and drug had differential capabilities of crossing the cell membrane.^14,15 (3) As was the case in the presented [¹¹C]flumazenil data, the radioligand binds to more sites than the drug under investigation. This results in less apparent occupancy in regions with an abundance of those sites bound by the radioligand, but not the investigational drug. Independent of the particular mechanism underlying regional differences in $O^{\mathit{Drug}}$, these differences could have important clinical implications. If the true $O^{\mathit{Drug}}$ in the therapeutic target region were lower than the whole-brain estimate, some patients would be effectively underdosed. This might result in poor response to treatment. A second possible application relates to testing of new tracers and/or drugs. If the tracer is known to be non-selective and the drug or compound is believed to be selective, the Lassen plot filter could be used as a screening tool. A flat $O^{\mathit{Drug}}$ image would suggest that the drug lacks any greater selectivity than the tracer. This screening method would be objective. The current method of Lassen plot requires the user to eyeball the data to identify regions that may not lie on the line of best fit.

Non-displaceable binding is not an outcome of great interest in the majority of PET studies. However, for proper analysis with various kinetic models, it is a crucial assumption that $V_{ND}$ is uniform throughout the brain.¹ Non-displaceable binding is defined as the sum of nonspecifically bound and free radioligand. The former is interpreted as radioligand associated with macromolecular components in a way that cannot be displaced. Radioligands are designed to have high selectivity for their target receptor but can still be trapped by other structures.¹⁶ For example, regions with large volumes of white matter might trap a larger proportion of lipophilic radioligands. If the exogenous drug does not displace the radioligand at this secondary trapping site, the non-displaceable binding is a function of a secondary factor that cannot be assumed to be spatially invariant. We show that occupancy estimated using either the Lassen plot or ${\Delta BP}_{ND}$ is biased if regional variation in $V_{ND}$ is present. Therefore, even if a reference region exists, it would be desirable to investigate $O^{\mathit{Drug}}$ with methods (other than ${\Delta BP}_{ND}$) that allow for regional variation in $V_{ND}$. If $V_{ND}$ is known not to vary throughout the brain, we show that a univariate Lassen plot filter can be used. The advantage of this approach is the reduction in sensitivity to noise, while existing spatial patterns in $O^{\mathit{Drug}}$ will still be detected.

A region-level Lassen plot using all voxels within a ROI to estimate a ROI-specific $O^{\mathit{Drug}}$ and $V_{ND}$ can accurately and precisely recover simulated values. Due to the many data points available for the regression per region, this method performs robustly in our simulations. It should be noted however, that our simulated differences in $V_{ND}$ and occupancy were confined to a single set of ROI boundaries. In essence, this means the simulations were ideal for the ROI-level Lassen plot method under consideration. If different occupancy levels were to exist within one ROI, our region-based method would introduce a bias similar to the one observed for the standard Lassen plot estimate of the whole brain. Therefore, the aptness of this method depends on the within-ROI variability in $O^{\mathit{Drug}}$ and $V_{ND}$. For example, a template containing a uniform striatum will miss existing differences in the caudate and putamen and provide one biased average value for both sub-regions.

We show that a Lassen plot of every voxel and its 26 nearest-neighbors is able to recover regional differences in $O^{\mathit{Drug}}$ and $V_{ND}$. Although the variance of the estimates was higher than with the ROI-based method, observed biases in $O^{\mathit{Drug}}$ were minimal. A major advantage of the Lassen plot filter method is the ability to construct parametric occupancy and $V_{ND}$ without user intervention. This is in contrast to the present use of the classic whole-brain Lassen plot. In common practice, regions or groups of regions that seem to fall outside the primary regression line are arbitrarily assigned to separate classes and separate regression lines.³ The Lassen plot filter eliminates the need for user intervention. The Lassen plot filter extends the now-classic Lassen plot to the voxel level. As with most voxel-level analyses, there is likely to be more susceptibility to noise. Poorer signal to noise ratio in voxel-level time-activity curves will lead to outlier values for some $V_T$ estimates and this will deleteriously affect the occupancy images. We are working on ways to deal with $V_T$ outliers. Voxel-level parametric images of any sort that are built from more than one image are heavily dependent on the accuracy and reproducibility of registration algorithms. In our case, the need is to properly align the $V_T$ images from the baseline and drug conditions. Algorithms for spatial normalization of human brain scans are a long-standing area of research and they have achieved quite high levels of performance. Nevertheless, this is yet another aspect of the Lassen plot filter performance that must be investigated. Finally, there is statistical variation in all PET data and thus in all derived images. If occupancy images are to prove useful, we will need to find appropriate and reliable tests for identifying significant spatial variation in the images. This is also an area of ongoing research.

[¹¹C]flumazenil is a radioligand that binds with high affinity to benzodiazepine binding sites of GABA-A receptors containing α₁, α₂, α₃, and α₅ subunits (K_i ≈ 1 nmol/L) but has lower affinity for α₄ and α₆ (K_i ≈ 150 nmol/L).¹⁷ The exogenous drug used in our example data-set, PF-06372865, has high affinity for α₁, α₂, and α₃ (K_i ≈ 0.2, 2.92, and 1.1 respectively) and low affinity for α₅ (K_i ≈ 18.04).¹¹ Therefore, we expected higher apparent $O^{\mathit{Drug}}$ in regions with α₁, α₂, and α₃ receptor-subtype purity, and lower apparent $O^{\mathit{Drug}}$ in regions with a considerable fraction of α₅-containing binding sites (i.e. those that [¹¹C]flumazenil binds, but PF-06372865 does not displace). The caudate and putamen contain large contributions of GABA-A receptors with the α₅ subunit, whereas the cerebellum and occipital cortex contain a mix of only α₁, α₂, and α₃ receptors.^12,13 A visual representation of this distribution has been published by Myers et al.¹³ Our parametric images of $O_{i,j,k}^{\mathit{Drug}}$ are consistent with this distribution; $O^{\mathit{Drug}}$ of PF-06372865 was low in the caudate and putamen compared to the occipital cortex and cerebellum.

We also showed that the univariate Lassen plot filter preserved the expected spatial variation in $O^{\mathit{Drug}}$ for the subtype-selective GABA-A receptor antagonist PF-06372865. Figure 7 demonstrates the trade-off between bias and variance. The occupancy images on the lower row (fixed $V_{ND}$) are smoother and less heterogeneous than those made by the one-pass algorithm. Fixing $V_{ND}$ will provide an anchoring point for the voxel-level regressions. The bias introduced into the occupancy values may yet be found to be intolerable. But at least we can understand its source. Fixing $V_{ND}$ (the x-intercept on the Lassen plot) to a whole-brain average will alter the slope of each local Lassen plot in a predictable way. If the fixed $V_{ND}$ is higher than the estimated one, the slope (i.e. the $O^{\mathit{Drug}}$) will likely increase. If the fixed $V_{ND}$ is lower than estimated, the slope is likely to be lower in the two-pass result. In the case of Figure 6, fixing the $V_{ND}$ to a whole-brain average will raise the $V_{ND}$ in the caudate and lower it in the putamen with opposing effects on the occupancies in those regions. The optimal selection of a fixed $V_{ND}$—perhaps on a region-wide basis rather than whole brain—is the subject of ongoing research. In practice, the univariate method could obtain a fixed $V_{ND}$ in one of two ways. In human studies, we envision a two-pass algorithm: Lassen plot filter to obtain a $V_{ND}$ image, then the univariate Lassen plot filter with the average $V_{ND}$ from the first pass. In the case of non-human primates, a self-block experiment with a large cold dose of the radioligand molecule could yield a $V_{ND}$ image.

We have introduced an adaptation of the classic Lassen plot—a local Lassen plot filter—which enables ROI- or voxel-level estimation of $O^{\mathit{Drug}}$ and $V_{ND}$. Using simulated $O^{\mathit{Drug}}$ and $V_{ND}$ images, we showed that both ROI- and voxel-level estimation can recover regional differences in both outcomes with only minimal bias. In contrast, both the standard Lassen plot and the standard, point-by-point calculation of ${\Delta BP}_{ND}$ were unable to cope with variation in $O^{\mathit{Drug}}$ and/or $V_{ND}$, thus providing significantly biased estimates of both $O^{\mathit{Drug}}$ and $V_{ND}$. Application to a [¹¹C]flumazenil data-set with an expected regional variation in $O^{\mathit{Drug}}$ showed that the estimated distribution seems to adhere to that expectation based on the known differential spatial distribution of α subunits of the benzodiazepine binding sites on the GABA-A receptor. Additional studies with a priori hypotheses regarding regional variation in occupancy will be required to validate and optimize multiple aspects of the Lassen plot filter that affect bias and accuracy such as choice of kernel size, and registration of $V_T$ images to an anatomical template. In certain circumstances, parametric images of $O^{\mathit{Drug}}$ could be used as a screening tool for new tracers or drugs. Parametric images of $V_{ND}$ could help identify valid reference regions for new tracers. Parametric images of $O^{\mathit{Drug}}$ could indicate whether or not $O^{\mathit{Drug}}$ is spatially varying throughout the brain. It will be of particular interest, going forward, to investigate if regionally varying $O^{\mathit{Drug}}$ images can explain lack of intended responses to certain subtype-selective drugs.