Kinetic Modeling without Accounting for the Vascular Component Impairs the Quantification of [ 11 C]PBR28 Brain PET Data

Abstract

The positron emission tomography radioligand [¹¹C]PBR28 targets translocator protein (18 kDa) (TSPO) and is a potential marker of neuroinflammation. [¹¹C]PBR28 binding is commonly quantified using a two-tissue compartment model and an arterial input function. Previous studies with [¹¹C]HRJ-PK11195 demonstrated a slow irreversible binding component to the TSPO proteins localized in the endothelium of brain vessels, such as venous sinuses and arteries. However, the impact of this component on the quantification of [¹¹C]PBR28 data has never been investigated. In this work we propose a novel kinetic model for [¹¹C]PBR28. This model hypothesizes the existence of an additional irreversible component from the blood to the endothelium. The model was tested on a data set of 19 healthy subjects. A simulation was also performed to quantify the error generated by the standard two-tissue compartmental model when the presence of the irreversible component is not taken into account. Our results show that when the vascular component is included in the model the estimates that include the vascular component (2TCM-1K) are more than three-fold smaller, have a higher time stability and are better correlated to brain mRNA TSPO expression than those that do not include the model (2TCM).

Keywords

kinetic modeling microglia neuroinflammation PK11195 TSPO

INTRODUCTION

The 18 kDa translocator protein (TSPO)—formerly known as the peripheral benzodiazepine receptor or PBR—is a protein located on the outer mitochondrial membrane and is part of the mitochondrial permeability transition pore. It is involved in a number of mitochondrial functions, including steroidogenesis, Ca²⁺ signaling, generation of reactive oxidative species and apoptosis.¹ TSPO is highly expressed in macrophages, activated microglia and reactive astrocytes and is therefore a putative biomarker of inflammation.² TSPO binds cholesterol with great affinity but also synthetic ligands such as the isoquinoline carboxamide PK11195. PK11195 labelled with [¹¹C] has been extensively used with positron emission tomography (PET) to assay TSPO concentration in vivo in a variety of diseases.³⁵ [¹¹C]-(R)-PK11195 has, however, some unfavorable characteristics for PET imaging, principally a high non-specific binding. This has stimulated the research of new radioligands with increased bio-availability, lower non-specific binding and higher affinity for the target.⁶ Examples include ligands such as phenoxyphenil acetamid derivatives (PBR06, DAA106) or bicyclic linker derivatives (PBR111, DPA713).

Although these new tracers generally displayed very favorable characteristics in animal models, human studies have been plagued by a high intersubject variability in binding affinity.⁷ Genetic differences may partly explain this variability. In fact, a polymorphism of the TSPO protein, due to an Ala-Thr amino-acid substitution in the peptidic chain,⁸ identifies three different classes of binding affinity: high-affinity binders (HAB), mixed-affinity binders (MAB) and low-affinity binders (LAB). Genotype analysis allows the stratification of subjects into more homogeneous human cohorts, thus increasing statistical power.⁹

However, genetic differences do not seem to account entirely for the substantial variability found in humans with these new inflammation ligands. A recent study failed to demonstrate increased neuroinflammation with aging¹⁰ despite the fact that microglial activation is an aging hallmark.¹¹ Another study demonstrated lack of microglial activity in patients with mild depression.¹² Although this particular depression phenotype might not have a neuroinflammatory component, TSPO binding was unexpectedly reduced in the depressed cohorts.¹³

The only positive study that has included genotyping in the design is the one by Kreisl et al,⁴ who showed TSPO upregulation in a relatively large Alzheimer cohort (n = 9), particularly in the temporal cortex. Yet, even with correction for genotype, age and education, regional variability in their cohorts exceeded 30%.

In the present study, we sought to investigate the reasons for the high binding variability of novel TSPO PET ligands. In particular, we focused on the quantitative approach used for these tracers. Kinetic modeling of TSPO binding is generally performed with a two-tissue compartmental model (2TCM). This model includes a tissue compartment representing the free tracer in tissue and the ligand unspecifically bound and a compartment for the ligand specifically bound to the target.¹⁵ One of the basic assumptions of compartmental models is that the ligand concentration in each compartment, in either bound or free form, is homogeneous. However, immunohistochemical studies show a heterogeneous TSPO distribution at the microscopic level in the healthy brain.¹⁶ A much higher microglia density is found on the vessel walls (i.e. the endothelium and smooth wall muscles) than in the brain parenchyma. Also, a concentration gradient is likely to occur through the brain parenchyma, where the ligand binds non-specifically to lipids on its way to the binding sites. This phenomenon should be more important for tracers with high lipophilicity, such as second-generation TSPO ligands.¹⁷

As elegantly demonstrated by Delforge et al,¹⁸ receptor distribution may impact the quantification with kinetic modeling. Because of such gradients, the concentration of the free tracer at the binding site may be lower than its average tissue concentration. The measured in vivo affinity may therefore be lower than the affinity in vitro. This in vivo-in vitro difference was defined by Delforge et al as the reaction volume.¹⁸ TSPO binding sites are present on the endothelial layer of blood vessels, and in the mitochondria of the activated microglia. Lower TSPO concentrations are found in astrocytes but not in neurons¹⁹ (Figure 1). Because of this uneven distribution, the vessels and the microglia will likely have different reaction volumes and thus different apparent affinities for TSPO ligands. This might translate into different PET kinetics.

Figure 1

Interaction of TSPO ligand with brain parenchyma. The TSPO has a heterogeneous distribution in the normal brain parenchyma. The endothelium expresses high concentrations of TSPO, which has implications for PET kinetics. The free [¹¹C]PBR28 concentration in tissue is not expected to be generally uniform but distributed along a gradient from the capillary space to the microglia, passing through the endothelium into the extracellular space. In the brain parenchyma the ligand exchanges with lipids (molecular bilayer) and possibly other proteins. It then enters the microglial inner cellular space to reach the mitochondria. All these interactions lower the concentration of the [¹¹C]PBR28 amenable to bind the TSPO. This creates a change in the apparent affinity of the tracer of the target. Thus, the apparent affinity would be higher closer to the vascular space, with resulting slower kinetics for the bound fraction (e.g. irreversible trapping).

In a previous work with [¹¹C]HRJ-PKI1195, we have demonstrated a consistent slow TSPO binding component clearly localized in the brain main vasculature, venous sinuses and arteries, associated with a faster equilibrating component that we attributed to microglia.²⁰ We then incorporated this vascular TSPO component into the model and demonstrated that, when vascular sclerosis causes a substantial reduction of vascular TSPO concentration, the neuroinflammatory component of the tracer shows a downward bias.²¹

In this work, we observed that the most commonly used model to describe [¹¹C]PBR28 kinetics does not adequately fit the initial part of the brain time-activity curves. Considering the findings that showed the presence of TSPO receptors in the brain vasculature, we therefore hypothesized the existence of a tissue binding component together with a greater slow binding component of vascular origin.

In this study we used clinical [¹¹C]PBR28 data from healthy volunteers and computer simulations to (1) propose a new kinetic model for [¹¹C]PBR28 that would take into account the slow uptake of the radioligand into the vessels, which we considered as irreversible for modeling purposes, (2) quantify the bias and variability of a model that does not include such irreversible component.

MATERIALS AND METHODS

Compartmental Models

The two-tissue compartmental model. The standard approach proposed in the literature to describe [¹¹C]PBR28 kinetics in brain tissue is a 2TCM.^22,23 The model includes an arterial plasma input function (C_p) and two tissue compartments, one describing the non-displaceable component (C_nd) and one describing the specific binding (C_s) (Figure 2A). The model is described by the following first-order differential equations:

\begin{array}{l} \frac{d C_{nd}}{d t} = K_{1} C_{p} (t) - (k_{2} + k_{3}) C_{nd} (t) + k_{4} C_{s} (t), C_{nd} (0) = 0 \\ \frac{d C_{s}}{d t} = K_{3} C_{nd} (t) - k_{4} C_{s} (t), C_{s} (0) = 0 \end{array}

where the microparameters K₁ (mL/cm³ per minute), k₂ (1/minute), k₃ (1/minute), and k₄ (1/minute) are the rate constants for tracer transport from plasma to tissue and back, and from the non-displaceable to the specific compartment and back, respectively. The principal macropara-meter of interest is the total volume of distribution V_T (mL/cm³), which is calculated as:

V_{T} = \frac{K_{1}}{K_{2}} (1 + \frac{k_{3}}{k_{4}})

(2)

Figure 2

2TCM and 2TCM-1K model structure. (A) The 2-tissue compartmental model (2TCM) model is composed of two exchangeable tissue compartments, one for the non-displaceable component (C_nd) and one for the specific binding (C_s). K₁ and k₂ are the rate constants for transport from plasma to tissue and back, respectively. k₃ and k₄ are the rate constants from the non-displaceable compartment to the specific one and back, respectively. (B) 2TCM-1K includes also a vascular component (C_vasc), with K_b as the rate constant from plasma to the vascular compartment.

The total amount of radioactivity measured by PET, C_measured(t), does not distinguish between the tissue and the blood activity. Therefore the measurement equation in a given volume is

\begin{array}{l} C_{measured} (t) & = & (1 - V_{b}) [tissue activity] + V_{b} [blood activity] \\ = & (1 - V_{b}) [C_{nd} (t) + C_{s} (t)] + V_{b} C_{b} (t) \end{array}

(3)

where C_b, expressed in kBq/mL, represents the total radioactivity concentration in the whole blood, including radiometabolites, and V_b is the fraction of blood volume, which is unitless.

The 2TCM with irreversible vascular trapping (2TCM-1K). The classical 2TCM does not include any trapping component of the tracer, neither in the tissue nor in the blood. Thus, if irreversible tracer retention occurs biologically, it is considered to be negligible in the general picture of the system dynamics.

We propose a new model, 2TCM-1K, in which an irreversible compartment to account for the binding of the tracer to the TSPO in the blood vessels (C_vasc) is present within the framework of the classical 2TCM (Figure 2B). The model is described by the following first-order differential equations:

\begin{array}{l} \frac{d C_{n d}}{d t} (t) = k_{1} C_{p} (t) - (k_{2} + k_{3}) C_{nd} (t) + k_{4} C_{s} (t), \\ \frac{d C_{s}}{d t} (t) = k_{3} C_{nd} (t) - k_{4} C_{s} (t), C_{s} (0) = 0 \\ \frac{d C_{vasc}}{d t} (t) = K_{b} C_{p} (t) C_{vasc} (0) = 0 \end{array}

(4)

According to this model, the measurement equation becomes:

\begin{array}{l} C_{measured} (t) & = & (1 - V_{b}) [tissue activity] + V_{b} [blood activity] \\ = & (1 - V_{b}) [C_{nd} (t) + C_{s} (t)] + V_{b} [C_{b} (t) + C_{vasc} (t)] \end{array}

where the microparameter K_b (1/minute) is the rate constant for tracer binding to the vascular receptor sites (for the complete definition of the model equation refer to Appendix 1). All six model parameters, K₁, k₂, k₃, k₄, K_b, V_b, are a priori uniquely identifiable (Appendix 2).

The total volume of distribution V_T (mL/cm³) for 2TCM-1K is still calculated according to Equation 2.

Application to Measured PET Data

Radioligand preparation. [¹¹C]PBR28 was synthesized as previously described²⁴ and according to the Investigational New Drug Application #76,441 (http://pdsp.med.unc.edu/snidd/). The radioligand was obtained in high radiochemical purity (>97%).

Image acquisition. Data were obtained from the database of healthy subjects of the Molecular Imaging Branch at the National Institute of Mental Health. In particular two different data sets of healthy subjects (some of them were also included in previous studies^24,25) have been provided: one data set of 11 young healthy subjects (6 men and 5 women; age, 33 ± 9 years); another data set of 8 elderly healthy subjects was used (6 men and 2 women; age, 65 ± 4 years). The protocol was approved by the Ethics Committee of the National Institutes of Health (NIH); all subjects gave written informed consent. The studies were conducted according to the Declaration of Helsinki.

Among the 19 subjects, 10 were heterozygous for the Ala147Thr polymorphism of TSPO, conferring MAB; 8 were homozygous HAB, and for 1 subject no information was available.

All PET images were acquired on an Advance Nxi tomograph (GE Medical Systems, Waukesha, WI, USA), after a bolus injection of 690 ± 13 MBq of [¹¹C]PBR28. An 8-minute ⁶⁸Ge transmission scan of the brain was acquired for subsequent attenuation correction. Two different experimental protocols were employed. For the first data set dynamic scans comprised 33 frames (6 frames of 30 seconds each, then 3 × 60 seconds, 2 × 120 seconds, and 22 × 300 seconds) for a 120-minute experiment duration, while for the second data set dynamic scans lasted 90 minutes (27 frames: 6 frames of 30 seconds each, then 3 × 60 seconds, 2 × 120 seconds, and 16 × 300 seconds). PET data were reconstructed on a 128 × 128 matrix with a pixel size of 2.0 × 2.0 × 4.25 mm. Details regarding image reconstruction and correction are reported in Hines et al²⁴ and Kreisl et al.²⁶ For consistency, all the subjects were analyzed at 90 minutes, i.e. we removed the last 30 minutes of scan in the first data set.

During the acquisition, blood samples (1.0 mL each) were drawn from the radial artery at 15-second intervals until 150 seconds, followed by 3 to 4.5-mL samples at 3, 4, 6, 8, 10, 15, 20, 30, 40, 50, 60, 75, 90, and 120 minutes. The plasma time-activity curve was corrected with the fraction of unchanged radioligand, as previously described.²⁷

Each subject underwent 1.5- or 3-T clinical brain T1-weighted magnetic resonance imaging (MRI). MRI images were used to derive the anatomical information necessary to define the ROIs.

The Anatomic Automatic Labeling template was coregistered on each subject's MRI scan and PET image using Statistical Parametric Mapping 8 (Wellcome Department of Cognitive Neurology). Whole-brain and 10 brain regions were selected: frontal cortex, parietal cortex, occipital cortex, striatum, thalamus, cingulum, hippocampus-amygdala, insula, thalamus, and cerebellum. Data from bilateral regions were the mean of both the left and the right regions.

Model Identification

Both 2TCM and 2TCM-1K model parameters have been identified at the region of interest (ROI) level using the weighted nonlinear least square estimator (WNLLS), as implemented in Matlab (The Mathworks Inc., Natick, MA, USA). Initial microparameter values for 2TCM and 2TCM-1K were set based on the literature.²³ The initial value of K_b for 2TCM-1K was set similarly to the vascular component trapping measured in the [¹¹C]SCH442416 brain PET study, because adenosine A₂A receptors have similar endothelial and tissular distribution.²⁸

The relative weights for the individual data points were defined as the inverse of the variance of the PET measurement error. This was assumed to be additive, uncorrelated, having a Gaussian distribution with zero mean and variance equal to the decay-corrected activity divided by the length of the relative scan interval, multiplied for a scale factor γ. The proportionality constant γ was estimated a posteriori as described by Bertoldo et al.²⁹

Statistical Analysis

Model comparison. The goodness of fit has been evaluated by analyzing (1) the weighted residuals and (2) the precision of the estimated parameters, expressed as coefficient of variation (CV). The two models were compared in terms of parsimony criterion by using the Akaike Information Criterion (AIC). The choice of the best model was made on the basis of parameter precision, weighted residuals inspection and parsimony criteria. The percentage of model failures was also considered for the assessment of the model performance. The results are reported separately for HAB and MAB subjects, in order to take into account the effect of TSPO polymorphism on binding affinity.

Regions with V_T estimates higher than 10 mL/cm³ (for HABs and MABs) were excluded as non-physiological. The threshold corresponds to the average value plus three times the standard deviation reported in the literature for [¹¹C]PBR28 data.²² In addition, we discarded V_T values with a CV higher than 50% in order to eliminate unreliable estimates. The statistical difference of the estimated V_T values obtained with 2CTM and 2TCM-1K was assessed by using a one-sample t-test (P<0.05).

Impact of experimental protocol on V_T estimates. We assessed V_T time stability by analyzing the PET data from each young healthy subject (of the first data set) first in their full length (i.e. 0–120 minutes) and then after removing the last 30 minutes (i.e. 0–90 minutes). The stability of V_T estimated with both 2TCM and 2TCM-1K was assessed by comparing the values obtained for a 120-minute scan with those obtained with a 90-minute scan, as follows:

V_{T} r e l a t i v e d i f f e r e n c e = \frac{{\hat{V}}_{T, 90} - {\hat{V}}_{T, 120}}{{\hat{V}}_{T, 120}}

(6)

The comparison was carried out separately for HAB and MAB subjects.

Comparison of [¹¹C]PBR28 PET binding with brain mRNA TSPO expression. Following the approach presented in Rizzo et al,³⁰ we derived the TSPO mRNA mappings from the Allen Human Brain Atlas, made of six healthy human brains,³¹ finely sampled with three different probes, and we compared them with the V_T estimated with 2TCM and 2TCM-1K. Under the assumption that mRNA expression is predictive of the in vivo protein density of unrelated normal subjects, we evaluated whether the [¹¹C]PBR28 binding description obtained with the two models in competition were consistent with TSPO brain gene profiles. The comparison was performed by correlation analysis at the region level within the same ROIs considered for the PET quantification. To match the datasets for age, PET subjects older than 60 years were excluded from the analysis. In total we analyzed 4 HAB and 7 MAB subjects.

Simulation Studies

Simulation studies were performed to:

(1) assess whether the error of applying 2TCM when the system presents a vascular slow component can be considered negligible and to evaluate how the 2TCM reflects the simulated binding variation;

(2) assess whether the use of 2TCM-1K in a system without vascular trapping leads to erroneous results, in particular whether the model identifies an inexistent irreversible compartment.

Simulation study 1: the impact of using 2TCM in the presence of vascular trapping. The simulation included the following steps:

(1) The parameter estimates obtained with 2TCM-1K on the whole-brain 120-minute time-activity curves (TAC) of one HAB and one MAB subject became the baseline values. The baseline V_T values were increased by 10%, 20%, 30%, 40% and 50%, by increasing k₃. The other microparameters (K₁, k₂, k₄, K_b and V_b were left unchanged.

(2) For each V_T value we created 50 TACs, which were consistent with those measured in the whole-brain region of the clinical scans. The TACs were generated by convolving the measured arterial input function of the subject with a model impulsive response function defined by the parameters determined in Step 1. Noise-free simulated TACs were generated according to Equation (5).

(3) Noisy TAC data were then obtained by summing noise-free data and simulated white noise (zero-mean and Gaussian distribution). The simulated noise variance was defined as for the measured data, i.e. frame length over tissue activity, with the proportionality constant y estimated from the real data.

In summary, 300 HAB and 300 MAB kinetics TACs (50 curves × 6 V_T cases) were generated. Simulated data were quantified with 2TCM as done for real data (see the previous section). The ability of 2TCM to detect the simulated V_T variations was assessed by comparing the 2TCM estimates obtained for each jth increase level (10%, 20%, 30%, 40%, 50%) versus the 2TCM V_T value at baseline, as:

% V_{T} var i a t i o n_{j} = \frac{{\hat{V}}_{T . j} - {\bar{V}}_{T}}{{\bar{V}}_{T}}

(7)

where ${\hat{V}}_{T . j}$ represents the estimated value obtained with 2TCM in the jth case and ${\bar{V}}_{T}$ is the estimated value obtained with 2TCM at baseline.

Simulation study 2: reliability of 2TCM-1K in the absence of trapping. This simulation was performed in the same way as the simulation study 1. However, data were generated on the basis of 2TCM results and modeled with a 2TCM-1K. K₁, k₂, k₃ and k₄ values were derived from a 2TCM applied to the reversible tracer [¹¹C](R)-rolipram.^32,33 This has been done in order to avoid using data from a ligand (i.e. [¹¹C]PBR28) that may include a trapping component.

RESULTS

Model Comparison on Clinical Data at 90 minutes

Both 2TCM and 2TCM-1K models described well the descending part of the tissue curves, but 2TCM poorly fitted the initial part of the TAC (Figure 3). In particular, the first 5 minutes of the TAC were constantly underestimated. For this reason, the 2TCM weighted residuals were not randomly distributed, but demonstrated a polarized behavior especially during the first minutes.

Figure 3

Model fit comparisons. Example of model fits to the parietal (A) and occipital cortex (B) for a representative MAB subject. Black circles represent the measured time-activity curves, while dotted and solid gray lines represent the model description provided by 2TCM and 2TCM-1K, respectively. The zoom of the initial phase of the curve (in the interval 0–20 minutes) is reported in the center of each panel. On the right side are the weighted residuals of 2TCM and 2TCM-1K (dotted and solid gray lines, respectively).

Nevertheless, 2TCM-1K improved the curves fit to the tissue data points in all the regions and increased the randomness of the weighted residuals (Figure 3). To quantify this improvement, we calculated with both models the mean relative difference of the weighted residual sum of squares (WRSS) and the mean relative difference of the residual sum of squares (RSS). 2TCM-1K gave lower values (i.e. a better fit) than 2TCM using both criteria (−64% ± 15% and –74% ± 13%, expressed as mean ± s.d., respectively). Supplementary Figures 1 and 2 report the single-compartments time courses for both 2TCM and 2TCM-1K for the same regions of Figure 3. Supplementary Figures 3 and 4 report the 2TCM and 2TCM-1K fit comparison in other regions of interest for a representative HAB subject (thalamus and insula).

Figure 4

Distributions and correlation of regional V_Tand K_b estimates. Results are reported for 2TCM (V_T, panel A) and 2TCM-1K (V_Tand K_b panels A and B, respectively). Homozygous high-affinity binders (HAB, 8 subjects) and heterozygous mixed-affinity binders (MAB, 10 subjects) are shown separately. (C) Correlation analysis of 2TCM versus 2TCM V_T estimates of HAB subjects (black diamonds) and MAB subjects (gray squares). The equations of the regression lines and the Pearson's correlation coefficients (R²) are reported in the same chart.

In terms of outliers the two models performed similarly: considering all the subjects and all the brain regions, 2TCM-1K properly identified all the TACs, while 2TCM showed a poor fit in only one brain region of one subject (frontal cortex of a HAB subject). The outlier region was excluded from further statistical analyses, which were performed on the intersection of the regions where both models returned physiological and precise estimates.

The precision of the remaining individual V_T estimates was consistent between both models (on average CV V_T = 5% ± 7% for 2TCM-1K and 3% ± 1% for 2TCM). Also, the microparameters were precisely estimated with both models: for V_b, K₁ and k₂ the precision was lesser than 14% ± 7% and for k₃, and k₄ lesser than 20% ± 9%. The precision of K_b with the 2TCM-1K model was 11% ± 8%.

In terms of parsimony criteria, 2TCM-1K outperformed 2TCM (AIC for 2TCM-1K was smaller in 94% of the regions), thus confirming the identification of 2TCM-1K as the optimal model among the tested ones to describe [¹¹C]PBR28 brain data. Notably, the remaining 6% of the regions in which the 2TCM model had a smaller AIC were concentrated in the striatum, the hippocampus/amygdala, insula and the thalamus.

The regional mean V_T estimates across the subjects, computed with 2TCM and 2TCM-1K, respectively, were statistically different (P<0.05) (Table 1). The V_T values obtained with 2TCM-1K were more than three-fold smaller. Considering the TSPO genotype, HABs had on average 57% greater [¹¹C]PBR28 binding than MABs with 2TCM-1K, but this difference was of only 39% with the classic 2TCM estimates.

Table 1

Between-subject mean and variability of V_T estimates obtained with 2TCM and 2TCM-1K

V_T (mL/cm³)	HAB				MAB
	2TCM		2TCM-1K		2TCM		2TCM-1K
	Mean	s.d.	Mean	s.d.	Mean	s.d.	Mean	s.d.
WHOLE	4.49	1.02	1.56	0.31	3.47	0.82	0.98	0.18
Frontal^a	4.43	1.04	1.59	0.36	3.58	0.92	0.98	0.21
Parietal	4.44	0.95	1.58	0.32	3.50	0.80	0.96	0.19
Occipital	4.62	0.92	1.77	0.28	3.69	1.07	1.05	0.20
Striatum	4.73	1.03	1.45	0.25	3.09	0.82	0.99	0.24
Temporal	4.68	1.21	1.59	0.30	3.45	0.79	1.04	0.24
Cingulum	4.34	0.83	1.62	0.32	3.36	0.91	0.95	0.18
HipAmy	4.88	1.42	1.53	0.26	3.01	0.74	1.16	0.36
Insula	4.74	1.06	1.79	0.51	3.37	0.78	1.14	0.26
Thalamus	5.25	1.20	1.85	0.48	3.22	0.86	1.30	0.19
Cerebellum	4.35	0.75	1.72	0.20	3.16	0.72	1.02	0.17

Abbreviations: HAB, high-affinity binder; MAB, mixed-affinity binder; 2TCM, two-tissue compartmental model. The values are reported as mean (s.d.) for all the subjects, except where indicated, in all the regions where both models returned physiological and reliable estimates. The values are reported separately for homozygous high-affinity binders (HAB, 8 subjects) and heterozygous mixed-affinity binders (MAB, 10 subjects). ^aValues are mean ± s.d. for 7 subjects.

The regional variability of V_T (s.d. of regional values across subjects) was smaller in absolute value with 2TCM-1K than with 2TCM (0.33 and 1.04 mL/cm³ for HABs; 0.22 and 0.84 mL/cm³ for MABs, respectively) (Table 1). These results were confirmed also when comparing the individual regional V_T estimates (i.e. 208 values), estimated with both models, in terms of mean (± s.d.) and median (± median absolute deviation, MAD) (Table 2). The magnitude of individual HAB V_T estimates was still higher than MAB V_T (Figure 4A) with both models, but was clearly in a different range: the mean relative V_T difference between 2TCM and 2TCM-1K was –64% ± 7% (range: –87% to –49%) for HABs and –67% ± 10% (range: –85% to –44%) for MABs. Importantly, the V_T values obtained with 2TCM and 2TCM-1K were not only significantly different, but also poorly correlated, with an R² = 0.18 for HAB and R²<0.001 for MAB subjects (Figure 4B).

Table 2

Statistical description of the distribution of V_T and K_b estimates (across all regions and subjects)

V_T (mL/cm³)	HAB		MAB		K_b (1/min)	HAB	MAB
V_T (mL/cm³)		2TCM	2TCM-1K	2TCM	2TCM-1K	2TCM-1K
Mean	4.63	1.64	3.35	1.05	Mean	0.29	0.21
s.d.	1.02	0.34	0.83	0.24	s.d.	0.12	0.10
CV (%)	22	21	25	23	CV (%)	42	46
Min	2.59	0.85	1.93	0.63	Min	0.14	0.10
Max	7.77	2.60	5.75	1.75	Max	0.79	0.76
Median	4.61	1.64	3.20	0.99	Median	0.26	0.20
MAD	0.81	0.26	0.65	0.19	MAD	0.09	0.06
CV (%)	18	16	20	20	CV (%)	32	32

Abbreviations: CV, coefficient of variation; HAB, high-affinity binder; MAB, mixed-affinity binder; 2TCM, two-tissue compartmental model. V_T values have been obtained with 2TCM and 2TCM-1K. K_b Values have been obtained with 2TCM-1K. The values are reported separately for homozygous high-affinity binders (HAB, 8 subjects) and heterozygous mixed-affinity binders (MAB, 10 subjects), s.d. and MAD refer to standard deviation and median absolute deviation respectively.

The between-subject variability of V_T values across brain regions was smaller in absolute value for 2TCM-1K compared with 2TCM (0.34 and 1.02 mL/cm³ for HABs; 0.24 and 0.83 mL/cm³ for MABs, respectively; Table 2). A smaller variability with 2TCM-1K was also found in terms of median ± MAD (Table 2).

The vascular component estimated with 2TCM-1K was consistent across the brain and precisely estimated (average precision between regions: 11% ± 8%). The between-subject mean value of K_b in the whole brain was 0.263 ± 0.134 per minute. Considering the individual K_b estimates of all the regions (i.e. 208 values), the mean value was 0.29 per minute (s.d. K_b = 0.12; median K_b = 026) and 0.21 per minute (s.d. K_b = 0.10; median K_b = 0.20) for HAB and MAB subjects, respectively (Figure 4C and Table 2).

We also considered separately the model performance in the two different data sets (i.e. young versus elderly subjects). Two main findings resulted from the comparison: (1) 2TCM-1K better described brain [¹¹C]PBR28 data compared with 2TCM in elderly subjects (smaller AIC in 100% of ROIs; higher mean relative difference of WRSS and RSS, equal to –67% ± 17% and –79% ± 1%, respectively); (2) it was not possible to determine a unique trend of V_T changes with age in HAB and MAB subjects with either models. Interestingly, only K_b showed an increasing trend for higher age. However, the limited number of subjects per category (young: 3 HABs and 7 MABs; elderly: 5 HABs and 3 MABs) prevented a more robust statistical comparison. An example of 2TCM and 2TCM-1K model fitting is reported in Supplementary Figures 5 and 6 for one MAB and one HAB subject.

Figure 5

Application of 2TCM in simulated tissues with vascular trapping. We simulated different scenarios of V_T with the 2TCM-1K model (baseline and from 10% to 50% V_T increases) for both HAB and MAB cases (dark and light gray bars, respectively). The figure reports the V_T variation (mean ±s.d.) estimated by the 2TCM model when applied to these data.

The remaining subject of the data set (not tested for genetic polymorphism) showed very high values of V_T (2.50 ± 0.24 mL/cm³) with 2TCM-1K, higher than those obtained in both HAB and MAB subjects. Also the vascular estimate of this subject (K_b = 0.55 ± 0.13 per minute) was much higher than that measured in both HAB and MAB subjects.

Finally, V_b estimation is also affected by taking the vascular irreversible component into account in the kinetic model. With the classic 2TCM V_b values were 0.064 ± 0.013, and with 2TCM-1K V_b values were 0.055 ± 0.014, which is closer to the standard 5% value commonly used as brain blood fraction.

Impact of Experimental Protocol

Reducing the experimental scanning time caused an important reduction in V_T values for 2TCM in both HABs and MABs (–23% ± 10% and –31% ± 12%, respectively). However, 2TCM-1K V_T remained constant also at 90 minutes, with a relative difference of 0 %± 18% and 1% ± 17% for HABs and MABs, respectively (Supplementary Figure 7). Similarly, K_b was also insensitive to scan duration (relative difference 4% ± 22% and 2%± 13% for HABs and MABs, respectively).

Genomic versus PET Analysis

When we considered the comparison mRNA-PET binding for all the subjects (HABs and MABs), 2TCM-1K V_T showed a higher correlation with gene expression than 2TCM V_T (depending on the probe: from R² = 0.31 to R² = 0.49 for 2TCM-1K versus R²<0.01 for 2TCM). For both models the estimates were directly proportional to mRNA values (Supplementary Figure 8).

We also assessed separately HABs and MABs, even though no genotype effect was expected in the mRNA profiles. Nevertheless, we found that the V_T estimated by 2TCM in MABs was not only less correlated (R²<0.1 versus R² = 0.45 for 2TCM-1K), but also inversely proportional, highlighting a spatial expression pattern inconsistent with the genomic distribution.

In general, the mRNA-PET cross-correlation values for 2TCM-1K were comparable with the mRNA auto-correlation of TSPO gene expression: from R² = 0.37 to R² = 0.45.

Notably, K_b was always well correlated with TSPO mRNA expression (R² = 0.30 ± 0.28).

Simulation Studies

Simulation study 7. Figure 5 reports the results of the use of 2TCM in tissues with vascular trapping, for different levels of V_T, for both HAB and MAB scenarios.

2TCM converged in all MAB curves, yielding reliable and precise but not accurate estimates. In HAB curves there were however varying percentages of outliers. In particular, when considering a 30% increase of V_T compared to the baseline, 9% of estimates had to be eliminated. The percentage of outliers followed the increase of V_T (up to 34% of outliers for a 50% V_T increase).

Importantly, 2TCM greatly underestimated the increase of simulated V_T values. For example, when MAB V_T values were increased by 20%, the V_T estimated with 2TCM was increased by only 5% ± 5%. With 50% V_T increase, the variation of the estimated V_T was only +15% ± 6%. This bias was even greater in HAB curves, where the estimated variation on V_T was even smaller and the variability higher (e.g. 7% ± 11% variation for a 50% increase).

Simulation study 2. When 2TCM-1K was applied to tissues devoid of any irreversible component, the estimator converged to unreliable K_b estimates for both HAB and MAB tissues. In fact, in 98% of curves CV K_b was higher than 50%, and it was higher than 100% in 84% of curves. Moreover, despite a good fit of the data, the irreversible compartment fraction was negligible with regard to the other identified components (Supplementary Figure 9 of Supplementary Material).

DISCUSSION

In this work we proposed a new modeling approach for [¹¹C]PBR28 brain PET studies, 2TCM-1K, by adding an irreversible compartment to the standard 2TCM. In particular, we demonstrated that the inclusion of the vascular binding component in the kinetic modeling of this tracer always provided a more parsimonious description of the data. Compared with 2TCM, 2TCM-1K was always selected as the optimal model to describe the data, according to all the indexes evaluated (residuals, parsimony criteria, estimates' values and precisions).

In addition, the new model showed to be insensitive to reduce scan duration (from 120 to 90 minutes), with an average relative difference smaller than 2%. On the contrary, 2TCM applied to 90-minute tissue curves showed a reduction in V_T of up to 30%.

The V_T values estimated by the new model were consistent with the known TSPO brain distribution in humans: the highest binding was observed in the thalamus, as previously reported also with [¹¹C]-(R)-PK11195.²⁰ Notably, this pattern of homogeneous cerebral V_T in the control subjects was in agreement with the pattern of TSPO gene expression. On the contrary, 2TCM V_T showed to be more heterogeneous across brain regions and less correlated with genomic information.

Among the anatomical regions considered, none could be used as the reference region, as specific binding was found all over the brain.

Importantly, microglial V_T values estimated by 2TCM-1K were low, which is very much in line with the very low activated microglial density in healthy individuals.³⁴

Using the novel model, the known relationship between V_Tand binding affinity (owing to the Ala147Thr polymorphism of TSPO) was maintained: HAB subjects had ~60% greater [¹¹C]PBR28 binding than MAB subjects in all regions measured (compared to the ~40% reported by Kreisl et al²⁵).

Notably, the V_T values of the subject whose genotype was unknown was twofold the average value of HAB subjects. This is not related to an age effect as the [¹¹C]PBR28 binding is significantly higher than the one observed in elderly subjects.

In this work, the plasma free fraction of the radioligand (f_p) was not used to correct V_T, because of the high variability of the measurements (f_p was 4.0 ± 1.2, coefficient of variance 29%). As reported in Hines et al,²⁴ the source of this variability could have resulted from technical differences in blood processing, such as time between obtaining and processing the sample and operator variability. Therefore, only V_T (as defined in Equation (2)) was presented.

The use of the new kinetic model consistently reduced the variability of the tissular V_T, indicating that most of the intersubject variance previously reported in human cohorts may be associated with a variable binding of the tracer to the vascular TSPO component. This finding is of biological interest and will be subject to further investigations.

The novel 2TCM-1K includes a vascular irreversible component, which is associated to the parent plasma C_p. We assessed whether this component, instead of being due to endothelial TSPO, might have been caused by the trapping of radioactive metabolites, but the use of the metabolites' input function did not support this. The use of the tracer concentration of metabolites or of the total concentration in plasma as input for the irreversible component did not allow recovering the initial part of the tissue curve. Therefore these new models were never deemed to have the optimal structure to describe the data (data not shown). In addition, as described by Turkheimer et al,²⁰ we also considered the possibility of a slowly reversible (rather than irreversible) vascular component. However, this alternative model could not identify the kinetic parameters. This does not mean however that the biological interaction of the tracer with the vasculature is truly irreversible, but just that there is not enough information in the data to properly identify a possible vascular washout.

Further evidence of the need to model the tracer binding to the vasculature was obtained by applying the 2TCM-1K to a gray matter region and to a vascular tissue curve. While in the gray matter the 2TCM-1K was reliably identified, in the vascular tissue the model collapsed to a single-tissue compartment and the vascular trapping remained unaltered (Supplementary Figure 10.

The brain vascular trapping K_b was estimated accurately and it was homogeneous across the brain regions, as expected from a data set of healthy subjects. Also, this parameter correlated well with the TSPO polymorphism, with higher K_b values for HABs, and with TSPO mRNA expression.

Given that the standard model to quantify [¹¹C]PBR28 binding is a 2TCM, including when different populations of subjects (healthy versus pathological) are compared, we investigated the impact of not including the endothelial component in the quantification and the correlation between the estimates of the two competing models. The analysis of the clinical data demonstrated that 2TCM and 2TCM-1K V_T estimates are significantly different and, more importantly, poorly correlated (R²<0.25). The simulation analysis not only confirmed these findings, but also showed that 2TCM is not very sensitive to variations in tissular V_T, as only differences greater than 20% could be properly identified for simulated MAB curves. This effect was even more critical when considering HAB where even a 50% increase of V_T translated into a 7% variation in the estimated value. This finding might substantially change the results of previously published clinical studies.

Further investigations of the 2TCM-1K performance in clinical models with neuroinflammation are required. Notably, for the validation of this new model in pathological conditions, correlation with biological evidence (such as immunohistochemical staining) becomes even more essential than in the healthy-subject conditions.

CONCLUSION

The use of a model that explicitly accounts for the endothelial TSPO binding is fundamental for a precise and accurate quantification of [¹¹C]PBR28 brain PET data.

When this component is neglected, the receptor binding in tissue is severely underestimated and binding changes due to neuroinflammation may be lost.

Footnotes

The authors declare no conflict of interest.

ACKNOWLEDGMENTS

We thank Dr Robert Innis and Dr Masahiro Fujita at the Molecular Imaging Branch of the National Institute of Mental Health for providing the [¹¹C]PBR28 images and for the interesting scientific discussions regarding the results.

APPENDIX 1

Derivation of the measurement equation of 2TCM-1K model

To derive the measurement equation for 2TCM-1K in terms of concentrations, we start by formulating it in terms of mass.³⁵ The total amount measured by PET, q_measured(t), is the summation of all tracer amounts present in the volume of observation. The measurement mass equation is:

where q_nd(t) and q_s(t) represent the amount of the tracer in the nondisplaceable and specific space (tissue compartments) and q_vasc(t) and q_b(t) are the amounts of tracer in the vascular space and whole blood, respectively (blood pool).

Denoting with V_tissue, V_blood, and V_total the tissue and the blood volume, and the total volume (given by V_total = V_tissue + V_blood), respectively, one can express Equation (A1.1) in terms of concentration, i.e. q_measured(t) = V_totalC_measured(t):

Expressing V_tissue = V_total = V_blood and dividing by the total volume, one has:

From A1.3 it is easy to derive the blood volume fraction as V_b = V_blood/V_total, i.e. the fraction of total volume occupied by the blood pool, thus obtaining Equation (5).

APPENDIX 2

Demonstration of a priori identifiability of 2TCM-1K model

The proposed 2TCM-1K model, as defined in , can be represented by the following system of first-order differential equations:

The total amount of radioactivity measured by PET, C_measured(t), is given by

where C_nd(t) + C_s(t) represents the activity of the tracer in the target tissues, C_b the activity in the whole blood, C_vasc the tracer trapped in the blood vessel endothelium, and V_b the fraction of volume occupied by blood over the tissue within the considered field of view. To demonstrate that the proposed model 2TCM-1K is uniquely identifiable a priori we used a transfer function approach.³⁵ Rearranging the Laplace transforms of Equation (A1) gives

Solving for C_nd, C_s and C_vasc,

Thus, the Laplace transform of C_measured(t) can be written as

which leads to

The exhaustive summary of the model is given by

where φ₁, …, φ₆ are the known observational parameters. It is easy to show that all the six parameters K₁, k₂, k₃, k₄, K_b and V_b are now identifiable a priori. In fact the solution of the system of Equation (A7) gives

Notes

Supplementary Information accompanies the paper on the Journal of Cerebral Blood Flow & Metabolism website ()

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