Quantification of PET infusion studies without true equilibrium: A tissue clearance correction

Abstract

In some positron emission tomography (PET) studies, a reversibly binding radioligand is administered as a constant infusion to establish true equilibrium for quantification. This approach reduces scanning time and simplifies data analysis, but assumes similar behavior of the radioligand in plasma across the study population to establish true equilibrium in all subjects. Bias in outcome measurements can result if this assumption is not met. This work developed and validated a correction that reduces bias in total distribution volume (V_T) estimates when true equilibrium is not present. This correction, termed tissue clearance correction (TCC), took the form $V_T=V_{T(A)}/(1+\beta \gamma V_{T(A)})$, where β is the radioligand clearance rate in tissue, γ is a radiotracer-specific constant, and V_T(A) is the apparent V_T. Simulations characterized the robustness of TCC across imperfect values of γ and β and demonstrated reduction to false positive rates. This approach was validated with human infusion data for three radiotracers: [¹⁸F]FPEB, (−)-[¹⁸F]flubatine, and [¹¹C]UCB-J. TCC reduced bias in V_T estimates for all radiotracers and significantly reduced intersubject variance in V_T for [¹⁸F]FPEB data in some brain regions. Thus, TCC improves quantification of data acquired from PET infusion studies.

Introduction

Positron emission tomography (PET) imaging of the brain provides powerful in vivo measurements of phenomena including ligand-receptor binding, protein misfolding, and second-messenger systems. A common PET imaging study design features a bolus injection of radiotracer followed by dynamic data acquisition. For reversibly binding radioligands, kinetic analysis of these dynamic (typically non-equilibria) data is used to estimate equilibrium outcome measurements such as binding potential (BP_ND) or distribution volume (V_T).¹ An alternative study design administers radiotracer as a bolus plus continuous infusion (B/I) to experimentally establish radiotracer equilibrium between tissue and arterial plasma.² This approach provides advantages of reducing the required scanning time to a shortened period of equilibrium and simplifying data analysis, since V_T can be estimated directly by the tissue-to-plasma ratio. However, the B/I approach requires an a priori infusion schedule, determined by the parameter K_bol, to achieve equilibrium during the established imaging time period. Variations at either the individual or group level in factors such as radiotracer kinetics or whole-body metabolism may reduce the quality of equilibrium during the imaging time period, which biases estimation of the outcome measurement.

The goal of this work was to characterize bias of equilibrium estimates of V_T in the absence of true equilibrium, and use this characterization to derive a correction that reduces bias under such conditions. This approach was termed tissue clearance correction (TCC). Several simulations were used to illustrate the statistical benefits TCC offers. First, group differences in radiotracer clearance rates assessed improvements in bias. Second, group differences in V_T with intra-group variability in radiotracer clearance rates assessed improvements in effect sizes due to reduced variance. Finally, TCC was validated with human imaging data for three radiotracers with B/I paradigms; [¹⁸F]FPEB, a radiotracer specific to the metabotropic glutamate 5 receptor³; (−)-[¹⁸F]flubatine, a radiotracer specific to α4β2* nicotinic acetylcholine receptors⁴; and [¹¹C]UCB-J, a radiotracer specific to synaptic vesicle glycoprotein 2A.⁵ For these datasets, equilibrium-based values (with and without TCC) were compared to gold-standard values obtained from full kinetic analysis.

Theory

The goal of infusing reversibly binding radioligands for PET studies is to establish equilibrium conditions, or in other words, achieve constant decay-corrected radiotracer concentrations in tissue (C_T) and plasma (C_P). By definition, the tissue-to-plasma ratio (C_T/C_P) at equilibrium is equal to V_T. If radiotracer infusion does not achieve true equilibrium, a state of transient equilibrium can be established where radiotracer clearance occurs at the same fractional rate (β, units of min⁻¹) as tissue clearance. Under such conditions, estimation of C_T/C_P (the apparent volume of distribution, V_T(A)) will be biased relative to the true value of V_T in a β-dependent manner. This theory section aims to derive a simple approach to correct for biased values of V_T(A) under such conditions.

Summarizing the theoretical background of transient equilibrium outlined in Appendix A of Carson et al.,² concentration in plasma (C_P) at late times can approach monoexponential clearance

$C_P(t)\to {\mathit{Be}}^{-\beta t}$ (1)

At this same time, the radiotracer concentration in tissue (C_T) is the convolution of C_P with the impulse response function. In the case of a radiotracer described by a reversible 2-tissue compartment model (2TCM), this is written as

$\begin{array}{c}{C}_T(t)\to {\mathit{Be}}^{-\beta t}\otimes (A_1{e}^{-{\alpha }_{1}t}+A_2{e}^{-{\alpha }_{2}t})\\ =B(A_1\frac{e^{-\beta t}-e^{-{\alpha }_{1}t}}{{\alpha }_1-\beta }+A_2\frac{e^{-\beta t}-e^{-{\alpha }_{2}t}}{{\alpha }_2-\beta })\end{array}$ (2)

where α₁ and α₂ are the tissue eigenvalues

$\begin{array}{c}{\alpha }_1=\frac{\sum _{j=2}^4{k}_j-\sqrt{(\sum _{j=2}^4{k}_j{)}^2-4k_2{k}_4}}{2}\text{\hspace{1em}and\hspace{1em}}\\ {\alpha }_2=\frac{\sum _{j=2}^4{k}_j+\sqrt{(\sum _{j=2}^4{k}_j{)}^2-4k_2{k}_4}}{2}\end{array}$

and A₁ and A₂ are the following

$A_1=K_1\frac{k_3+k_4-{\alpha }_1}{{\alpha }_2-{\alpha }_1}\text{\hspace{1em}and\hspace{1em}}{A}_2=K_1\frac{{\alpha }_2-k_3-k_4}{{\alpha }_2-{\alpha }_1}$

If both tissue eigenvalues are larger than the monoexponential plasma clearance rate β (a condition required for transient equilibrium), then at late times, this simplifies to

$C_T(t)\to B(A_1\frac{e^{-\beta t}}{{\alpha }_1-\beta }+A_2\frac{e^{-\beta t}}{{\alpha }_2-\beta })$ (3)

and V_T(A) approaches a constant value

$\begin{array}{c}{V}_{\text{T}(\text{A})}=\frac{C_T}{C_P}=\frac{B(A_1\frac{e^{-\beta t}}{{\alpha }_1-\beta }+A_2\frac{e^{-\beta t}}{{\alpha }_2-\beta })}{{\mathit{Be}}^{-\beta t}}\\ =\frac{A_1}{{\alpha }_1-\beta }+\frac{A_2}{{\alpha }_2-\beta }=\frac{K_1(k_3+k_4-\beta )}{({\alpha }_1-\beta )({\alpha }_2-\beta )}\end{array}$ (4)

For both one tissue compartment model (1TCM) and 2TCM kinetics, V_T(A) will overestimate V_T during plasma clearance (β > 0). In the 1TCM case, this relationship simplifies to

$V_{T(A)}=\frac{V_T}{1-\beta /\beta k_2{k}_2}$ (5)

Alternatively, in many 2TCM cases, the approximations β≪(k₃+k₄) and β≪α₂ can be made, and V_T(A) can be simplified to a similar form

$V_{T(A)}=\frac{K_1(k_3+k_4)}{({\alpha }_1-\beta )({\alpha }_2)}=\frac{V_T}{1-\beta /\beta {\alpha }_1{\alpha }_1}$ (6)

Equations (5) and (6) generate relationships between V_T(A) and a corrected V_T if β and α₁ or k₂ are known. However, shorter imaging and blood measurements are typically made during an established equilibrium window for B/I studies, which only allows for estimation of V_T(A) and β. Recognizing that V_T = K₁/k₂ (1TCM) or V_T=(A₁/α₁ + A₂/α₂) (2TCM), we propose the additional approximations 1/k₂ = γV_T or 1/α₁ =γV_T, where γ is a tracer dependent constant with units of min·cm³/mL (inverse of K₁ units). If γ is reasonably uniform across regions, then a single value of γ for a given radiotracer can be validated and used to improve V_T estimation in the absence of true equilibrium. This substitution results in the model-independent correction formula:

$V_{T(A)}=\frac{V_T}{1-\beta \gamma V_T}$ (7)

Inverting this equation to solve for the corrected version of V_T, V_T(C), yields the TCC operational equation

$V_{T(C)}=\frac{V_{T(A)}}{(1+\beta \gamma V_{T(A)})}$ (8)

The assumption of γ uniformity depends on how little K₁ varies across regions. For radioligands described by the 2TCM, there is additional dependence on regional heterogeneity of kinetic microparameter values. Since estimation of β from plasma data can yield estimates with poor precision owing to the noisy nature of the measurement, estimation of β from PET data was used to implement this correction. However, the clearance rate of radiotracer from tissue only approximates β under transient equilibrium conditions. Possible limitations of these assumptions were explored in simulations.

Methods

Simulations

Simulations were used to evaluate benefits and limitations of TCC. Synthetic data were created approximating the behaviors of [¹⁸F]FPEB and [¹¹C]UCB-J with 2TCM and 1TCM, respectively. Simulations were performed in MATLAB with the Compartment Modeling Kinetic Analysis Tool.⁶

Simulation 1: Bias characterization

This simulation characterized the bias in estimation of apparent V_T (V_T(A)) and corrected V_T (V_T(C)) relative to true simulated values of V_T (V_T(True)). The bias was characterized across a range of β and V_T values. The value of β was varied by simulating infusion protocols using a range of K_bol values with a metabolite-corrected input function from a representative bolus study, resulting in non-equilibrium curves (non-zero β). For [¹¹C]UCB-J, infusion input functions with β ranging from −0.0025 to 0.0038 min⁻¹_, in addition to a bolus curve with clearance rate of β = 0.0081 min⁻¹, were created and combined with simulated V_T values spanning 12 to 24 mL/cm³, accomplished with K₁ = 0.33 mL·cm⁻³·min⁻¹ and k₂ values spanning 0.027 to 0.013 min⁻¹. The ‘true’ value of γ (γ = 1/K₁) was 3.0 min·cm³/mL. For [¹⁸F]FPEB, infusion input functions with β ranging from −0.0012 to 0.0039 min⁻¹ (β = 0.0039 min⁻¹ for a bolus injection) were created and combined with V_T values spanning 10 to 30 mL/cm³, accomplished with the microparameters of K₁ = 0.4 mL·cm⁻³·min⁻¹_, k₂ = 0.067 min⁻¹, and k₄ = 0.05 min⁻¹, while k₃ values ranged from 0.033 to 0.2 min⁻¹. The ‘true’ value of γ (γ ≡ 1/mean(α₁V_T)) was set as 3.2 min·cm³/mL based on a brain mean V_T of 24 mL/cm³ (γ range across simulated V_T values: 3.1–3.7 min·cm³/mL). Noiseless time-activity curves (TACs) were generated for all permutations of V_T and β. Simulated curves were analyzed during equilibrium windows of 90–120 min post-injection for [¹⁸F]FPEB and 60–90 min post-injection for [¹¹C]UCB-J. Values of V_T(A) were estimated as the ratio of simulated tissue curves to simulated input functions, averaged during the equilibrium window. Regionally specific values of β were estimated from monoexponential fits to output curves. Note that values of β used to calculate V_T(C) are based on simulated TACs and may not correspond to ‘true’ values of β due to deviations from transient equilibrium. V_T(C) was then estimated as described in equation (8) using the ‘true’ values of γ listed above. Heat maps illustrating variations in V_T(A)/V_T(True) and V_T(C)/V_T(True) as functions of V_T(True) and β were created to illustrate bias.

Simulation 2: Errors in β and γ

Next, Simulation 1 was extended to assess the robustness of V_T(C) to incorrect values of β and γ. Estimates of β could be incorrect due to noise in the measurement. In these simulations, the value of β used to calculate V_T(C) was adjusted from the true noiseless β value (β_Sim) by Δβ, where Δβ is the maximal uncertainty in estimated β from tissue data as described in the Human Data section (Δβ = 8.0 × 10⁻⁴min⁻¹). Values of γ could be incorrect if kinetic microparameters deviate from the values used to establish γ. For example, a radiotracer may have slightly altered kinetic parameters (such as K₁) in a patient group compared to a control group, creating a scenario of error in γ. The value of γ used to calculate V_T(C) was also adjusted up and down by 25% of the ‘true’ values based on the microparameters described in Simulation 1. Combinations of the two errors were simulated to explore the interaction of the two effects, resulting in a unique heat map of V_T(C)/V_T for each permutation of incorrect β and γ.

Simulation 3: Bias causing artificial group differences

This simulation illustrated how between-group differences in β cause group bias in V_T(A) that can be eliminated by V_T(C). Two ‘study populations’ of n = 20 were created for each of [¹⁸F]FPEB and [¹¹C]UCB-J, simulated in six different regions capturing the diversity of kinetics across the brain based on previously described parameters,^7,8 detailed in Supplementary Table A. For each group, V_T values were sampled from the same normally distributed population (i.e. no true group differences) centered at literature values and with 8% coefficient of variation (CoV, σ/μ = 0.08), consistent with reported CoV.^7,8 Individual variation in V_T was simulated by altering k₂ ([¹¹C]UCB-J) or k₃ ([¹⁸F]FPEB). For each radiotracer, a population-averaged input function of bolus injections was fit to an analytic function including at least two exponential terms. The slowest exponential term was manipulated to simulate variability in plasma clearance rates. The terminal clearance rates of the ‘control’ group were sampled from a normal distribution centered at the population average (β_FPEB = 0.0039 min⁻¹; β_UCBJ = 0.0081 min⁻¹) with 25% CoV. The β value of the ‘patient’ group was sampled from a normal distribution centered at 2β_True (i.e. twice as fast) with 25% CoV. Infusion input functions were then created by convolving the bolus input with a B/I function,² defined by K_bol (K_bol = 190 min and 150 min for [¹⁸F]FPEB and [¹¹C]UCB-J, respectively). Once simulated time activity curves were produced, normally distributed noise was introduced as C_T(Noise) = C_T(True) + c₁*N_(0,.1), where N_(0,1) is a number randomly sampled from the normal distribution.⁹ ROI-level noise was created by adjusting c₁ to match the average residual sum of squares estimated from optimal compartment model fits of ROI data. Apparent (V_T(A)) and corrected (V_T(C)) distribution volumes were calculated. The ‘population’ averages and standard deviations were estimated and compared to the V_T(True) to assess bias and variability with and without correction. Effect sizes were estimated with Cohen’s d, the difference in group means divided by the pooled standard deviation.¹⁰ Values of 0.2, 0.5, and 0.8 are considered small, medium, and large, respectively.

Simulation 4: Effect size

This simulation assessed the impact of TCC on effect size for a 2-group comparison where a true between-group difference in V_T exists. For each radiotracer, two ‘study’ populations of n = 20 were created, featuring a ‘control’ group sampled from V_T = V_T(True) and a ‘patient’ group sampled from V_T = 0.9*V_T(True), with 8% CoV for both groups. Unlike Simulation 3, the plasma clearance rates were sampled from the same ‘control’ distribution for both populations. Values of V_T(A) and V_T(C) were compared to V_T(True) for all simulated subjects to assess bias. Effect sizes were estimated between the simulated ‘control’ and ‘patient’ groups for both apparent and corrected analyses in each region.

Simulation 5: K₁ effects

This simulation assessed the performance of TCC if γ was incorrect because K₁ was systematically different between patient and control groups. For each radiotracer, one control population and two patient populations of n = 20 were created. All groups were sampled from populations where V_T = V_T(True) and β = β_True; however, the two ‘patient’ group curves were simulated with K₁ values 25% lower and higher than the control group (see Supplementary Table 1). A single value of γ was used to estimate V_T(C) for all groups. V_T(C) bias was estimated between the simulated ‘control’ and ‘patient’ groups in each region.

Human data

All scanning procedures were approved and overseen by the Yale University Human Investigation Committee and the Yale New Haven Hospital Radiation Safety Committee in accordance with the United States federal policy for the protection of human research subjects in Title 45 Part 46 of the Code of Federal Regulations (45 CFR 46). Written informed consent was obtained from all subjects prior to participation after complete explanation of study procedures. TCC was applied to B/I scans with three different tracers:

[¹⁸F]FPEB

A set of 12 [¹⁸F]FPEB scans previously acquired from 7 healthy male subjects^8,11 was re-analyzed, including 5 subjects scanned 3–11 weeks apart. Doses of 165 ± 28 MBq high molar activity [¹⁸F]FPEB were given with K_bol = 190 min. Imaging data and arterial blood samples to measure the metabolite-corrected input function were acquired continuously for at least 120 min.

(−)-[¹⁸F]Flubatine

A set of nine (−)-[¹⁸F]flubatine scans previously acquired from five healthy non-smoker subjects¹² was re-analyzed, including four subjects scanned two to five weeks apart. Doses of 242 ± 58 MBq high molar activity (−)-[¹⁸F]flubatine were given with K_bol = 360 min. Imaging data and arterial blood samples to measure the metabolite-corrected input function were acquired from 0–60 min to 90–120 min.

[¹¹C]UCB-J

A set of 10 [¹¹C]UCB-J scans were acquired from 10 healthy subjects. Doses of 569 ± 100 MBq high molar activity [¹¹C]UCB-J were given with K_bol = 150 min. Imaging data and arterial blood samples to measure the metabolite-corrected input function as previously described⁷ were acquired for 120 min.

For all human PET scans, TACs were extracted from the following 10 regions (R = 10) of interest (ROIs) using subject-specific MRIs nonlinear transformed to the AAL MR template¹³: occipital, cingulum, frontal, parietal, and temporal cortices; thalamus, cerebellum, hippocampus, putamen, and caudate.

Estimation of K₁, V_T and V_T(A)

A ‘gold-standard’ compartment modeling analysis (2TCM for [¹⁸F]FPEB,⁸ and (−)-[¹⁸F]flubatine,¹² and 1TCM for [¹¹C]UCB-J⁷) was used for all regions to estimate V_T, K₁, and α₁ (for [¹⁸F]FPEB and (−)-[¹⁸F]flubatine). Estimates of V_T(A) were made in each region by estimating the ratio of average radiotracer concentrations in tissue to that in plasma during equilibrium windows (90–120 min for [¹⁸F]FPEB and (−)-[¹⁸F]flubatine; 60–90 min for [¹¹C]UCB-J).

Estimation of γ

The radiotracer specific value of γ was defined based on kinetic parameter values from R = {1,…,10} gray matter regions where r∈R as γ ≡ mean(1/(α_1,rV_T,r)) for [¹⁸F]FPEB and (−)-[¹⁸F]flubatine (2TC), and γ ≡ 1/mean(K_1,r) for [¹¹C]UCB-J (1TCM).

Estimation of β

The radiotracer clearance rate β was estimated in three different ways: (1) a single value determined from the plasma input function; (2) a single whole brain value from a tissue curve averaged across whole-brain gray matter; (3) individual values from tissue curves extracted for each analyzed brain region. For all approaches, values of β were estimated using only data from the equilibrium windows described above by fitting curves to a single decaying exponential function.

Estimates of V_T(C) from infusion data

Corrected values of V_T (V_T(C)) were estimated with equation (8) using the estimated values of V_T(A), γ, and β. For V_T(A) and each V_T(C) evaluation (the three approaches to estimate β as described above), the bias (relative to V_T estimated with compartment modeling) and group CoV were calculated. Group CoV was also calculated for compartment modeling estimates of V_T. The Morgan-Pitman test^14,15 was used to compare the variances of V_T(A) and V_T(C) at the regional level (uncorrected for multiple comparisons) with significance level α = 0.05. Morgan-Pitman tests for unequal variances of two dependent variables by determining if the Pearson’s correlation between the two variables (r) significantly differs from zero according to the test statistic

$T=r\sqrt{\frac{n-2}{1-r^2}}$ (9)

which follows a Student’s T distribution with n = 2 degrees of freedom.

Results

Simulation 1: Bias characterization

Simulation 1 compared the bias of TCC with uncorrected data in cases where β and γ are perfectly known. In the results (Figure 1), V_T bias is shown in color where red indicates a positive bias, blue indicates a negative bias, and white indicates no bias. The estimated values of V_T(A) resulted in similar bias characteristics for both [¹⁸F]FPEB and [¹¹C]UCB-J (see Figure 1(a) and (b)). As expected, V_T(A) was relatively unbiased for when β was close to 0 (near true equilibrium conditions). As β increased (became more bolus-like), V_T(A) was increasingly overestimated, while negative values of β resulted in underestimation of V_T(A). For fixed values of β, the bias exhibited complex dependence on V_T. These results do not agree with behavior predicted by equations (5) and (6), which predict that larger V_T values (i.e., smaller k₂ values) increase the magnitude of V_T(A) bias (underestimated V_T(A) when β is negative or overestimated V_T(A) when β is positive) assuming ideal transient equilibrium conditions. However, transient equilibrium is not strictly achieved for a given fixed time point where the magnitude of the deviation from transient equilibrium depends on k₂. The magnitude of V_T(A) bias was slightly greater for [¹⁸F]FPEB than [¹¹C]UCB-J given similar values of β.

Figure 1.

Simulation 1 results: Characterization of bias for V_T(A) (a, b) and V_T(C) (c, d) with the radiotracers [¹¹C]UCB-J and [¹⁸F]FPEB. The dashed horizontal line indicates β = 0. Note the different heat-map scales for V_T(A)/V_T (0.7–1.5) and V_T(C)/V_T (0.95–1.05).

Correction for non-equilibrium conditions diminished bias in V_T(C) to less than 4% for both radiotracers across the full range of simulated conditions (see Figure 1(c) and (d), note change in the display scale). For [¹¹C]UCB-J, described by the 1TCM, if transient equilibrium was achieved, no bias would be expected based on equations (5) and (7). Due to slight lack of transient equilibrium, bias of V_T(C) was strictly dependent on β. This relationship was more complex for [¹⁸F]FPEB owing to its 2TCM nature.

A clearance rate simulating bolus data was also included. V_T(A) of bolus data were positively biased by 61–85% and 58–73% for [¹⁸F]FPEB and [¹¹C]UCB-J, respectively. Estimates of V_T(C) resulted in positive biases of 1–7% and 2–3% for the respective radiotracers over the ranges of V_T values tested.

Simulations 2: Errors in β and γ

Simulation 1 was expanded to characterize bias in V_T(C) with incorrect values of β and/or γ for [¹¹C]UCB-J (Figure 2) and [¹⁸F]FPEB (Figure 3). Overestimates of β (top rows, Figures 2 and 3) generally resulted in a negatively biased V_T(C) (blue color), while underestimates of β (bottom rows, Figures 2 and 3) generally resulted in a positively biased V_T(C) (red color). For both radiotracers, errors in β generally resulted in greater magnitude of bias as V_T increased (compare along x-axes across a given column in Figures 2 and 3). Values of γ that were too large (middle row, right column, Figures 2 and 3) overcorrected V_T(C) (e.g. combinations of V_T and β with positive bias in top row of Figure 1 were now negatively biased), while values of γ that were too small (middle row, left column, Figures 2 and 3) undercorrected V_T(C). The magnitude of bias was smaller for overcorrection (γ too large) compared to undercorrection (γ too small). When errors in both β and γ occurred, the effect on bias was cumulative. The largest biases occurred in extreme conditions of non-equilibrium for concurrent errors in β and γ (top or bottom sections of corner panels).

Figure 2.

Bias in [¹¹C]UCB-J V_T(C) with respect to errors in γ and β. The middle row shows results for errors only in γ, i.e. with the correct β. The top and bottom rows show cases with β over- and underestimated, respectively. The middle column shows results for errors only in β, i.e. with the correct γ. The left and right columns show cases with γ under- and overestimated, respectively. The corner heat maps show the effects of combined errors. Dashed horizontal lines in each panel indicate β = 0.

Figure 3.

Bias in [¹⁸F]FPEB V_T(C) with respect to errors in γ and β. The middle row shows results for errors only in γ, i.e. with the correct β. The top and bottom rows show cases with β over- and underestimated, respectively. The middle column shows results for errors only in β, i.e. with the correct γ. The left and right columns show cases with γ under- and overestimated, respectively. The corner heat maps show the effects of combined errors. Dashed horizontal lines in each panel indicate β = 0.

In aggregate, the simulated errors in β or γ alone resulted in small but acceptable bias at near-equilibrium conditions. There was little benefit in cases of large clearance rates combined with imprecise values of both β and γ. Thus, TCC is robust to errors in β and γ when the true value of β is small, but relies more strongly on precise values of β and γ as the true value of β increases (i.e. becomes more bolus-like).

Simulation 3: Bias causing artificial group differences

Two ‘populations’ were simulated with the same tissue kinetics but different plasma clearance rate distributions. Values of β estimated from noisy TACs for ‘controls’ were 1.1 × 10⁻³± 1.7 × 10⁻³min⁻¹ for [¹⁸F]FPEB and 5.5 × 10⁻⁴± 1.1 × 10⁻³min⁻¹ for [¹¹C]UCB-J, in good agreement with values observed in control data (results below). Values from the ‘patient’ groups were 2.0 × 10⁻³± 1.5 × 10⁻³min⁻¹ for [¹⁸F]FPEB and 9.6 × 10⁻⁴± 1.3 × 10⁻³min⁻¹ for [¹¹C]UCB-J. In [¹⁸F]FPEB simulations, V_T(A) estimates were biased by 1–11% for the ‘control group’ and 5–18% for the ‘patient group’ (higher bias due to higher β), resulting in small to medium artificial effect sizes of 0.2–0.5 when no true effect of V_T exists. Using TCC, V_T(C) estimates were unbiased for both groups (<3.6%), reducing artificial V_T effect sizes to no greater than 0.33. For [¹¹C]UCB-J, V_T(A) estimates were biased by 1–8% for the ‘control group’ and 7–16% for the ‘patient group’, resulting in large artificial effect sizes of 0.6–0.8. Correction resulted in unbiased V_T(c) estimates for both groups (<3.5%), yielding V_T effect sizes no greater than 0.33. These data, summarized in Table 1 (Simulation 3 section) demonstrate that group differences in plasma clearance can cause artificial between-group differences in V_T(A) that are eliminated by V_T(C).

Table 1.

Simulated population level data for [¹⁸F]FPEB (top) and [¹¹C]UCB-J (bottom).

		‘Control’ group			Simulation 3: 2 × faster β ‘patient' group				Simulation 4: 10% lower VT ‘patient' group					Simulation 5: altered K1 ‘patient’ groups
														0.75 K1 group	1.25 K1 group
	Region	Mean VT	VT(A) Bias (%)	VT(C) Bias (%)	VT(A) Bias (%)	VT(A)d	VT(C) Bias (%)	VT(C)d	Mean VT	VT(A) Bias (%)	VT(A)d	VT(C) Bias (%)	VT(C)d	VT(C) Bias (%)	VT(C) Bias (%)
[18F]FPEB	Putamen	26.6	6.3	−1.7	15.1	0.34	−3.0	0.11	24.1	7.6	0.35	−1.0	0.47	1.2	−3.2
	Cerebellum	9.4	10.1	1.2	15.7	0.22	3.0	0.16	8.5	9.2	0.81	1.0	0.83	3.3	0.6
	Cingulate Cortex	29.1	1.4	−1.2	10.5	0.46	0.7	0.15	27.6	3.4	0.39	0.7	0.61	0.8	−1.3
	Hippocampus	23.2	−4.2	−0.6	4.8	0.46	3.6	0.33	20.2	1.2	1.30	0.0	1.40	−4.4	−0.4
	Occipital Cortex	22.5	6.4	0.3	15.2	0.33	0.6	0.02	19.9	8.3	0.76	0.1	0.90	0.1	−2.3
	Thalamus	16.8	11.0	−0.6	18.5	0.24	−1.2	0.05	15.1	11.1	0.80	−1.1	0.87	0.4	−2.9
[11C]UCBJ	Putamen	21.9	1.8	0.2	10.7	0.85	0.0	0.03	20.6	4.0	0.48	−0.2	0.59	1.3	−6.4
	Cerebellum	14.3	6.3	−0.2	15.1	0.64	2.3	0.18	13.1	8.9	0.62	1.8	0.69	3.3	−1.5
	Cingulate Cortex	20.7	−2.0	−0.6	7.7	0.71	0.1	0.05	18.4	2.3	0.61	0.9	0.67	1.5	−2.2
	Hippocampus	14.0	7.7	0.2	9.4	0.82	3.5	0.33	12.9	3.5	0.54	1.9	0.75	1.0	0.3
	Occipital Cortex	19.6	1.0	0.0	9.8	0.83	1.6	0.14	17.4	5.0	0.96	−0.1	1.09	0.4	−2.7
	Thalamus	15.8	7.1	0.5	15.8	0.71	0.7	0.09	13.8	10.2	0.91	0.2	1.04	1.6	−3.6

Note: Bias for V_T(A) and V_T(C) is shown for five simulated populations: ‘Control’ group, a ‘patient’ group with 2β of ‘control’ group (but no different V_T; ‘Simulation 3’), a ‘patient’ group with 0.9V_T of ‘control’ group (but no different β; ‘Simulation 4’), and ‘patient’ groups with 0.75K₁ and 1.25K₁ (but no different V_T or β; ‘Simulation 5’). Cohen’s d effect sizes comparing V_T(A) and V_T(C) the control group with the patient groups in Simulations 3 and 4 are listed.

Simulation 4: Effect size

Two ‘populations’ were simulated with input functions sampled from the same plasma clearance rate distribution, but with different V_T distributions where the mean V_T was 10% lower in the ‘patient group’ to assess improvements in effect size estimation. These conditions resulted in V_T(A) estimates biased by 1–11% and 2–10% in the ‘patient’ groups for [¹⁸F]FPEB and [¹¹C]UCB-J, respectively. Using the infusion correction, V_T(C) estimates were unbiased (within 2.5%). Effect sizes modestly increased across all regions for both radiotracers (see Table 1, Simulation 4 section).

Simulation 5: K₁ effects

Two ‘patient populations’ were simulated where both V_T and β were sampled from the same distributions as controls, but K₁ was 25% lower or higher than the control group. For the 25% lower K₁ group, biases were reduced from uncorrected values but slightly more positive than the ‘control’ group (biases of −4.4 to 3.3% and 0.4 to 3.3% for [¹⁸F]FPEB and [¹¹C]UCB-J, respectively). This is consistent with the use of a γ value that is too small, resulting in an undercorrection. For the 25% higher K₁ group, biases were generally negative but of smaller magnitude than uncorrected values (biases of −3.2 to 0.6% and −6.4% to 0.3% for [¹⁸F]FPEB and [¹¹C]UCB-J, respectively). This is consistent with the use of a γ value that is too large, resulting in an overcorrection. In all cases, the magnitude of bias was the same or smaller than that of V_T(A).

Human data: [¹⁸F]FPEB

After 90–120 min infusion, stable rates of change in [¹⁸F]FPEB concentration were achieved in blood (8.9±23.0%/h) and tissue (6.6 ± 12.8%/h). The variability in blood values likely results from higher estimation uncertainty. Based on 2TCM results, a value of γ = 4.3 min·cm³/mL was determined. Compared to 2TCM estimates of V_T, uncorrected equilibrium estimates exhibited greater variance with generally positively bias, as previously shown.¹¹ Performance of TCC is shown for [¹⁸F]FPEB in Table 2. Blood estimates of β (1.5 × 10⁻³± 3.6 × 10⁻³min⁻¹) to correct V_T(A) reduced the bias averaged across regions from 12.6% to 0.7%; however, variance increased in all regions. Whole brain estimates of β (7.5 × 10⁻⁴±1.2 × 10⁻³min⁻¹) to correct V_T(A) reduced the average bias from 12.6% to 2.8%, with comparable variance to uncorrected values. ROI-specific estimates of β (1.1 × 10⁻³± 2.0 × 10⁻³min⁻¹) to correct V_T(A) reduced the average bias from 12.6% to 1.9% and reduced variance in all regions, with significant reductions in six regions. ROI-specific β estimation therefore provided the optimal combination in reducing bias and variance (Supplementary Figure 1).

Table 2.

Average regional values of bias and coefficient of variation for the different approaches to estimating [¹⁸F]FPEB V_T.

	2TCM		Uncorrected		Corrected, blood β		Corrected, whole brain β		Corrected, ROI β
	VT (mL/cm3)	1/(α1VT) (minċcm3/mL)	VT(A) (mL/cm3)	Avg. bias (%)	VT(C) (mL/cm3)	Avg. bias (%)	VT(C) (mL/cm3)	Avg. bias(%)	β (min−1)	VT(C) (mL/cm3)	Avg. bias (%)
Occipital Cx	22.8 ± 3.5	4.0 ± 0.7	26.1 ± 5.4	14.6	23.2 ± 9.2	1.9	23.9 ± 5.9	4.6	1.1 × 10−3 ± 9.9 × 10−4	23.1 ± 4.7	1.2
Cingulum Cx	31.3 ± 5.2	3.3 ± 0.6	33.3 ± 6.9	6.9	29.6 ± 13.2	−4.5	29.7 ± 6.5	−4.4	7.7 × 10−4 ± 1.6 × 10−3	29.7 ± 4.7*	−4.2
Frontal Cx	23.5 ± 3.5	3.9 ± 0.7	26.8 ± 4.5	14.9	24.0 ± 8.8	2.0	24.5 ± 4.7	4.7	1.4 × 10−3 ± 1.1 × 10−3	23.0 ± 3.1	−1.6
Parietal Cx	23.2 ± 3.2	3.1 ± 0.6	26.7 ± 5.1	15.0	23.9 ± 10.4	3.1	24.5 ± 5.9	5.0	1.1 × 10−3 ± 8.1 × 10−4	23.6 ± 3.8*	1.8
Temporal Cx	27.7 ± 4.6	4.9 ± 0.9	29.7 ± 5.9	7.9	26.5 ± 11.3	−4.2	26.9 ± 6.1	−2.4	7.7 × 10−4 ± 9.0 × 10−4	26.7 ± 3.8*	−2.7
Thalamus	17.3 ± 1.9	6.2 ± 0.8	20.9 ± 3.5	21.2	18.6 ± 4.2	7.3	19.3 ± 2.9	11.8	1.4 × 10−3 ± 1.5 × 10−3	18.3 ± 2.1*	6.4
Cerebellum	10.2 ± 1.4	6.3 ± 0.8	12.3 ± 2.5	21.3	11.4 ± 2.3	11.9	11.7 ± 2.0	15.3	2.3 × 10−3 ± 2.2 × 10−3	11.0 ± 2.0	8.4
Hippocampus	23.9 ± 3.2	3.7 ± 0.4	23.7 ± 4.2	− 0.3	21.4 ± 9.9	−10.3	21.8 ± 3.9	−8.0	−7.9 × 10−4 ± 1.5 × 10−3	25.8 ± 3.5	8.8
Putamen	26.8 ± 4.3	3.4 ± 0.4	30.5 ± 6.0	14.5	27.0 ± 10.4	0.6	27.2 ± 4.7	2.7	8.7 × 10−4 ± 1.5 × 10−3	27.1 ± 3.8*	3.0
Caudate	26.4 ± 3.7	4.2 ± 0.7	28.9 ± 5.5	9.5	26.1 ± 13.2	−1.0	25.8 ± 4.2	−1.4	1.0 × 10−3 ± 1.8 × 10−3	25.5 ± 4.5*	−2.5

^*Significant reduction in variance compared to V_T(A) (Morgan-Pitman test, α < 0.05).

(−)-[¹⁸F]Flubatine

After 90–120 min infusion, stable rates of change in (−)-[¹⁸F]flubatine concentration were achieved in blood (3.5 ± 6.8%/h) and extrathalamic brain tissue (−3.4 ± 7.1%/h). For (−)-[¹⁸F]flubatine, equilibrium is not established in thalamus, due to its much higher V_T, thus the following summaries are restricted to extrathalamic gray matter regions. Based on 2TCM results, a value of γ = 3.3 min·cm³/mL was selected. Compared to 2TCM estimates of V_T, uncorrected equilibrium estimates exhibited comparable variances with a slight negative bias, as previously shown.¹² Performance of TCC is shown for (−)-[¹⁸F]flubatine data in Table 3. Blood estimates of β (5.8 × 10⁻⁴± 1.1 × 10⁻³min⁻¹) to correct V_T(A) negligibly altered the bias (relative to V_T estimation with 2 T) from −5.0% to −5.6%, and the variances were comparable. Whole brain estimates of β (2.4 × 10⁻⁵± 6.1 × 10⁻⁵min⁻¹) to correct V_T(A) did not alter bias or variance. ROI-specific estimates of β (−5.6 × 10⁻⁴± 1.2 × 10⁻³min⁻¹) to correct V_T(A) reduced the average bias from −5.0% to −3.8%, while the variance did not significantly change. In thalamus, both blood and whole-brain corrections did not remove the large bias; however, a region-specific estimate of β successfully reduced bias from −18.8% to −1.1%. Based on this observation and otherwise comparable results, the ROI-based approach was selected as providing the best correction approach for (−)-[¹⁸F]flubatine (Supplementary Figure 2).

Table 3.

Average regional values of bias and coefficient of variation for the different approaches to estimating (−)-[¹⁸F]flubatine V_T.

	2TCM		Uncorrected		Corrected, Blood β		Corrected, Whole Brain β		Corrected, ROI β
	VT (mL/cm3)	1/(α1VT) (minċcm3/mL)	VT(A) (mL/cm3)	Avg. bias (%)	VT(C) (mL/cm3)	Avg. bias (%)	VT(C) (mL/cm3)	Avg. bias (%)	β (min−1)	VT(C) (mL/cm3)	Avg. bias (%)
Occipital Cx	8.8 ± 1.2	3.1 ± 0.5	8.5 ± 1.2	−2.6	8.5 ± 1.1	−3.1	8.5 ± 1.2	−2.6	8.1 × 10−5 ± 9.1 × 10−4	8.5 ± 1.1	−2.8
Cingulum Cx	9.9 ± 1.0	3.4 ± 0.5	9.5 ± 1.1	−4.5	9.4 ± 1.0	−5.1	9.5 ± 1.0	−4.6	−3.5 × 10−4 ± 8.9 × 10−4	9.6 ± 1.1	−3.7
Frontal Cx	9.8 ± 1.0	3.0 ± 0.4	9.5 ± 1.0	−2.4	9.4 ± 1.0	−3.1	9.5 ± 1.0	−2.5	1.4 × 10−4 ± 8.6 × 10−4	9.5 ± 1.0	−2.8
Parietal Cx	9.6 ± 1.2	3.3 ± 0.5	9.3 ± 1.3	−3.4	9.2 ± 1.2	−4.0	9.2 ± 1.3	−3.5	1.5 × 10−4 ± 8.9 × 10−4	9.2 ± 1.4	−3.7
Temporal Cx	8.9 ± 1.0	3.6 ± 0.6	8.6 ± 1.1	−3.4	8.6 ± 1.1	−3.9	8.6 ± 1.1	−3.4	−1.8 × 10−4 ± 6.2 × 10−4	8.7 ± 1.1	−3.0
Cerebellum	11.6 ± 1.2	3.4 ± 0.6	11.1 ± 1.3	−4.2	11.0 ± 1.3	−4.9	11.1 ± 1.3	−4.3	−4.0 × 10−4 ± 4.7 × 10−4	11.3 ± 1.3	−2.9
Hippocampus	10.3 ± 1.2	3.7 ± 0.5	9.2 ± 0.7	−10.3	9.1 ± 0.7	−10.8	9.2 ± 0.7	−10.4	−1.4 × 10−3 ± 9.5 × 10−4	9.7 ± 1.0	−6.5
Putamen	10.8 ± 1.3	3.3 ± 0.5	10.0 ± 1.1	−6.6	10.0 ± 1.1	−7.2	10.1 ± 1.1	−6.7	−7.6 × 10−4 ± 1.1 × 10−3	10.4 ± 1.3	−4.4
Caudate	10.1 ± 1.7	3.6 ± 0.6	9.4 ± 1.5	−7.4	9.3 ± 1.5	−7.9	9.3 ± 1.5	−7.4	−1.1 × 10−3 ± 9.5 × 10−4	9.8 ± 1.8	−4.1
Thalamus	27.4 ± 3.6	3.1 ± 0.5	22.1 ± 2.1	−18.8	21.8 ± 2.5	−19.8	22.1 ± 2.1	−18.9	−2.6 × 10−3 ± 7.6 × 10−3	27.7 ± 4.5	−1.1

The thalamus, with very high V_T, does not achieve near-equilibrium conditions with the implemented infusion protocol; its results are shown separately in the bottom row.

[¹¹C]UCB-J

After 60–90 min infusion, rates of change in [¹¹C]UCB-J concentration in blood were not small (15.4 ± 9.7%/h), while tissue concentrations were more stable (8.5 ± 7.5%/h). Based on 1TCM results, a value of γ = 3.1 min·cm³/mL was determined. Compared to 1TCM estimates of V_T, uncorrected equilibrium estimates exhibited a positive bias with slightly greater variance. Performance of TCC for [¹¹C]UCB-J data is shown in Table 4. Blood estimates of β (2.5 × 10⁻³± 1.6 × 10⁻³min⁻¹) to correct V_T(A) reduced the bias averaged across regions (relative to V_T estimation with 2T) from 10.7% to −1.27% with comparable variance to uncorrected values. Whole brain estimates of β (1.2 × 10⁻³± 0.8 × 10⁻³min⁻¹) to correct V_T(A) reduced the average bias from 10.7% to 3.5%, with non-significant reductions in variance. ROI-specific estimates of β (1.4 × 10⁻³± 1.2 × 10⁻³min⁻¹) to correct V_T(A) reduced the average bias from 10.7% to 3.7% with non-significant reductions in variance. Both tissue-based approaches to estimate β provided improvement. For consistency in analysis with the previous radiotracers, the ROI-based approach selected for this radiotracer (Supplementary Figure 3).

Table 4.

Average regional values of bias and coefficient of variation for the different approaches to estimating [¹¹C]UCB-J V_T.

	1TCM		Uncorrected		Corrected, Blood β		Corrected, Whole Brain β		Corrected, ROI β
	VT (mL/cm3)	1/(K1) (minċcm3/mL)	VT(A) (mL/cm3)	Avg. bias (%)	VT(C) (mL/cm3)	Avg. bias (%)	VT(C) (mL/cm3)	Avg. bias (%)	β (min−1)	VT(C) (mL/cm3)	Avg. bias (%)
Occipital Cx	16.4 ± 2.9	2.9 ± 0.4	18.3 ± 3.4	11.0	16.1 ± 3.3	−2.3	17.0 ± 3.1	3.2	1.3 × 10−3 ± 8.3 × 10−4	17.0 ± 3.2	3.5
Cingulum Cx	16.0 ± 2.2	3.3 ± 0.4	17.4 ± 2.7	8.5	15.3 ± 2.3	−4.3	16.2 ± 2.4	1.1	6.1 × 10−4 ± 8.7 × 10−4	16.9 ± 2.7	5.1
Frontal Cx	15.4 ± 2.2	3.0 ± 0.3	17.2 ± 2.8	11.9	15.2 ± 2.6	−1.1	16.1 ± 2.6	4.4	1.5 × 10−3 ± 8.4 × 10−4	15.9 ± 2.6	3.6
Parietal Cx	16.7 ± 3.0	2.9 ± 0.4	18.3 ± 3.3	9.6	16.1 ± 3.2	−3.6	17.0 ± 3.0	1.9	1.3 × 10−3 ± 7.5 × 10−4	17.1 ± 3.2	2.3
Temporal Cx	16.9 ± 2.5	3.2 ± 0.3	18.1 ± 2.8	7.1	15.9 ± 2.7	−5.9	16.8 ± 2.5	−0.4	5.8 × 10−4 ± 7.4 × 10−4	17.6 ± 2.8	3.8
Thalamus	12.2 ± 1.6	2.9 ± 0.3	14.1 ± 2.2	15.2	12.7 ± 2.1	3.9	13.3 ± 2.0	8.8	2.3 × 10−3 ± 1.2 × 10−3	12.7 ± 1.8*	4.7
Cerebellum	12.1 ± 1.7	3.3 ± 0.3	14.1 ± 2.2	15.9	12.7 ± 2.0	4.5	13.3 ± 2.0	9.4	1.6 × 10−3 ± 1.2 × 10−3	13.2 ± 2.1	8.6
Hippocampus	11.8 ± 1.8	4.2 ± 0.4	13.0 ± 2.1	10.0	11.8 ± 1.9	0.0	12.3 ± 2.0	4.3	1.5 × 10−3 ± 1.1 × 10−3	12.2 ± 2.0	3.7
Putamen	18.3 ± 2.6	2.7 ± 0.3	19.9 ± 3.2	9.0	17.3 ± 2.8	−5.6	18.3 ± 2.7	0.5	1.5 × 10−3 ± 1.5 × 10−3	18.4 ± 3.3	0.4
Caudate	14.2 ± 2.5	3.5 ± 0.5	15.5 ± 2.8	8.5	13.8 ± 2.5	−3.0	14.5 ± 2.6	2.0	1.4 × 10−3 ± 1.5 × 10−3	14.4 ± 2.2*	1.6

^*Significant reduction in variance compared to V_T(A) (Morgan-Pitman test, α < 0.05).

Discussion

Bolus/infusion studies will ideally produce true equilibrium in blood and all brain regions so that V_T can be determined directly from the tissue-to-plasma ratio. However, if true equilibrium is not reached, the apparent ratio (V_T(A)) will be biased. This work derived and implemented a correction approach (TCC) to improve equilibrium estimation of V_T. This correction has the form $V_T=V_{T(A)}/(1+\beta \gamma V_{T(A)})$ to leverage measurements practically available during a short equilibrium window (β and V_T(A)). Simulations confirmed the unbiased performance of this correction. Additional simulations demonstrated that (1) accounting for individual variation in equilibrium conditions can reduce false positive rates if radiotracer clearance were systematically different between study groups, and (2) statistical power to detect group differences can modestly be increased by suppressing this bias. Analysis of human imaging data confirmed that TCC reduces bias in V_T estimation. This approach thus has practical benefits to PET imaging studies.

The hypothesized benefits of TCC were two-fold: reducing bias in V_T estimates, and reducing group-level variance by correcting for individual differences in radiotracer clearance. Both simulations and real data clearly demonstrate that correction for non-equilibrium effectively reduces bias. For example, in Simulation 1, V_T(C) estimates reduced simulated biases from 8% for [¹⁸F]FPEB and 10% for [¹¹C]UCB-J to less than 4% across a wide range of clearance rates. These improvements mirrored the analysis of real human data, where correction reduced average V_T bias from 12.6% to 1.9% for [¹⁸F]FPEB and from 10.7% to 3.7% for [¹¹C]UCB-J. In contrast, only slight reductions in group-level V_T variance were observed with TCC. Simulated studies of 10% group differences in V_T including variance in clearance rates (Simulation 4) revealed modest improvements in effect sizes by correcting for non-equilibrium conditions, mirroring the slight improvement in group variance for healthy control data for [¹⁸F]FPEB and [¹¹C]UCB-J. These unexpected results can be explained by inspecting the top row of heat maps in Figure 1, where red and blue colors denote over- and underestimation, respectively. Scanning horizontally for any given β value shows that lower V_T values are overestimated, and higher V_T values are underestimated. This causes group variance in V_T values to be artificially reduced by the bias present in V_T(A) estimation. We conclude that reductions in population variance were more modest than expected because bias due to non-equilibrium in V_T(A) estimation artificially reduces the true population variance. In summary, TCC reduces bias in V_T estimation and may more accurately depict population variance.

TCC was derived using assumptions of transient equilibrium. Surprisingly, the correction remained unbiased (>5%) under bolus conditions for both simulated radiotracers (see Figure 1(c) and (d)), even when transient equilibrium was not achieved. Online Appendix A demonstrates that the assumption of transient equilibrium is not necessary for unbiased correction assuming accurate knowledge of the variables β and γ. Simulations characterizing the robustness of this approach revealed that when clearance rates are large, modestly imprecise values of β and γ resulted in little benefit of the correction approach, and larger errors in these values can result in a greater bias with correction than if uncorrected. Therefore, applying this correction to bolus data is only appropriate in conditions with very low noise (small errors in β) and where γ (which is highly K₁ dependent) has well-characterized homogeneity across analyzed regions and the study population. In contrast, near-equilibrium conditions allow for small but acceptable bias with modest errors in β (consistent with ROI-level noise) and γ (25% difference), identifying conditions for robust performance of this correction.

Three approaches to estimate β were evaluated, since larger errors in β increase bias in V_T(C). Errors in β are driven by noise in the measurement. This likely explains the poorer performance of estimation of β from blood data in the human studies, which exhibited greater variance than those from tissue. Regionally specific tissue-based estimation of β proved superior in terms of reducing bias and variance for [¹⁸F]FPEB and (−)-[¹⁸F]flubatine. Interestingly, both tissue-based methods yielded similar results for [¹¹C]UCB-J. We speculate that this may result from the relatively uniform [¹¹C]UCB-J k₂ values across brain regions, as well as the higher noise level in ROI-based β estimates for a C-11 tracer. Since equilibrium conditions may vary across regions at a given time, resulting in variations in both the magnitude of bias and the apparent clearance rate, accounting for these differences at the regional level likely leads to improved performance, supported by online Appendix A. Since β must be reliably estimated, ROI-level data are the likely current limit for this approach because the level of statistical noise for voxel data may not yield reliable estimation of β. Denoising algorithms^16,17 may prove useful to improve β estimation, particularly for regions with small volumes. In these data, 30 min of dynamic data were acquired in 5 min bins, allowing for 6 data points to estimate β. While imaging for a shorter time would reduce subject discomfort, it would also reduce the time scale and temporal sampling available to estimate radiotracer dynamics. Shortening frame length to increase sampling frequency could help mitigate this, but shorter frame durations increase noise, so the net effect may be small. These considerations underscore the need to accurately and precisely estimate β to effectively implement TCC.

A key assumption in this derivation hinged on using γV_T(A) to approximate 1/k₂ or 1/α₁. A single γ value greatly simplified this approach, but relied on how uniform γ is across analyzed regions. We note that γ is dependent on K₁. Simulation 5 characterized the robustness to 25% errors in γ at the population level caused by group differences in K₁. Errors in γ increased V_T(C) bias compared to values generated with the true γ, but these biases were still substantially lower than V_T(A) bias. In the human data results, values of γ were based on modeling analysis (referred to as analytic values). An alternative approach is to determine empirical values of γ estimated with the available data to confirm the use of analytic γ values. To perform empirical γ estimation, the operational equation $V_T=\frac{V_{T(A)}}{(1+\beta \gamma V_{T(A)})}$ was reformulated as a linear equation with γ as the slope

$V_{\text{T}(\text{A})}/V_{\text{T}}-1=\gamma *\beta *V_{\text{T}(\text{A})}$ (10)

Using model estimated (‘true’) V_T and corresponding V_T(A) values, plots of this form (referred to as γ plots) were constructed with V_T(A)/V_T-1 on the y-axis and β*V_T(A) on the x-axis. The slope of these plots (combining all regions across all subjects) was estimated with linear regression to yield a single empirical estimate of γ for comparison with analytic γ values. These plots demonstrated reasonable agreement between empirical and analytic γ estimates based on ROI-estimation of β for all radiotracers (Supplementary Figures 4 to 6). The use of the same human data sets both to estimate γ and to validate TCC represents a limitation of the work, since this analysis likely yields a best-case scenario in terms of reducing bias. Values of γ may vary across patient groups; therefore, careful consideration of kinetic microparameters (especially K₁) between groups is important for implementing TCC. Validation of a γ value should be performed for other radiotracers when implementing this approach, which is possible with the data from standard kinetic studies used to validate the K_bol value and appropriateness of an infusion approach.

Equilibrium conditions may not be present, despite a previously validated infusion protocol, for numerous reasons. Changes in radiotracer metabolism and/or bioavailability due to the presence of drugs¹⁸ or disease state¹⁹ may systematically alter equilibrium conditions between groups. This possibility is a critical condition for infusion studies. Simulation 3 demonstrated that artificial group differences in V_T can result from group differences in clearance rates if not appropriately corrected, and that TCC can suppress bias from such a scenario (see Table 1). Alternatively, large variation in radiotracer kinetics across brain regions can require different periods of time for equilibration. Targets with densities that are dramatically greater in certain brain regions, such as the thalamus for α4β2* nAChRs or basal ganglia for dopamine D₂/D₃ receptors, can impose practical logistic constraints that preclude equilibrium conditions across all brain regions in a single scan session.^12,20 For example, the established schedule for (−)-[¹⁸F]flubatine (specific for α4β2* nAChRs) requires 4 h of infusion to achieve equilibrium in thalamus, an approach not practical for typical human studies. However, the introduced correction approach reduced bias in thalamus with an equilibrium approach from −18.8% to −1.1%, providing reliability sufficient for future analysis of this region using this correction approach. Such compensation for differences in clearance rates across different brain regions suggests a possible extension of this correction approach to correct a tissue-to-reference region ratio (SUVR) to an accurate estimate of distribution volume ratio (DVR), an application left to future work. Overall, this approach provides the important practical improvement of reducing bias from tissue to plasma ratios in PET studies where equilibrium is not present.

The recommended implementation of TCC is to first evaluate tissue clearance rates across study conditions or groups. If clearance rates significantly differ between groups, then implementing this correction is necessary to eliminate systematic bias in the data, similar to the scenario in Simulation 3. However, if clearance rates are comparable across groups, then correction for nonequilibrium is best applied under low noise conditions. If clearance rates are known to systematically vary over different conditions, it may be better to adjust the infusion schedule to establish true equilibrium instead of relying on this correction post hoc. To this end, feedback systems have been proposed to ensure equilibrium.²¹ Thus, this correction is best suited to the analysis of collected data with systematic variation in non-equilibrium.

Conclusion

This work derived and validated a simple correction that reduces bias of V_T estimation with equilibrium analysis for infusion studies with non-equilibrium conditions. TCC is robust to noise in estimation of the clearance rate β at the ROI level, and therefore provides an important correction to improve quantification of infusion PET data, particularly if systematic variation in clearance rates is present.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: We gratefully acknowledge the funding support of the National Institute on Alcohol Abuse and Alcoholism (K01 AA024788).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Authors’ contributions

ATH and REC conceived and developed TCC. ATH performed simulations, data analysis, and drafted the initial manuscript. ATH and REC edited the manuscript

Supplementary material

Supplemental material for this article is available online.