Kinetic Modeling of [ 11 C]Raclopride: Combined PET-Microdialysis Studies

Abstract

The in vivo binding of D₂ receptor ligands can be affected by agents that alter the concentration of endogenous dopamine. To define a more explicit relation between dopamine and D₂ receptor binding, the conventional compartment model for reversible ligands has been extended to account for a time-varying dopamine pulse. This model was tested with [¹¹C]raclopride positron emission tomography and dopamine microdialysis data that were acquired simultaneously in rhesus monkeys. The microdialysis data were incorporated into the model assuming a proportional relation to synaptic dopamine. Positron emission tomography studies used a bolus-plus-infusion tracer delivery with amphetamine given at 40 minutes to induce dopamine release. The extended model described the entire striatal time–activity curve, including the decrease in radioactivity concentration after an amphetamine-induced dopamine pulse. Based on these results, simulation studies were performed using the extended model. The simulation studies showed that the percent decrease in specific binding after amphetamine measured with the bolus-plus-infusion protocol correlates well with the integral of the postamphetamine dopamine pulse. This suggests that changes in specific binding observed in studies in humans can be interpreted as being linearly proportional to the integral of the amphetamine-induced dopamine pulse.

Keywords

Raclopride Positron emission tomography Microdialysis Compartmental models

Positron emission tomography (PET) and single-photon emission computer tomography studies demonstrate that administration of pharmacologic agents known to alter synaptic dopamine can affect the binding of D₂ receptor radioligands. It has been shown that substances known to increase synaptic dopamine such as amphetamine (Dewey et al., 1991; Innis et al., 1992), methylphenidate (Volkow et al., 1994), and tetrabenazine (Dewey et al., 1993) lead to a decrease in D₂ ligand binding. Alternatively, stimulation of GABAergic (Dewey et al., 1992) or serotonergic neurons (Dewey et al., 1995), both of which inhibit dopamine release, leads to increased D₂ binding. In addition, pretreatment with reserpine, which depletes dopamine stores, greatly reduces the amphetamine-induced washout of the single-photon emission computer tomography D₂ ligand [¹²³I]iodobenzamide (Innis et al., 1992). Furthermore, it has been shown by analysis of PET studies in primate and microdialysis studies in rat that an induced change in D₂ ligand binding corresponds to an opposing change in dopamine level (Dewey et al., 1995). These studies demonstrate that D₂ tracers can be used to indirectly observe changes in dopamine. However, this effect has not been described quantitatively; that is, a measured change in specific binding has not been related mathematically to an induced change in dopamine. Understanding this relation would lead to better interpretation of dopamine competition studies in humans.

This report develops a mathematical model that describes the relation between tissue dopamine concentration and the binding of [¹¹C]raclopride. Data used for developing and testing the model came from combined PET and in vivo microdialysis studies in rhesus monkey that provided simultaneous measurement of [¹¹C]raclopride and striatal dopamine (Breier et al., 1997). To better observe changes in tracer binding within a single study, bolus-plus-infusion (B/I) tracer delivery (Carson et al., 1993a) was used to obtain a near constant level of [¹¹C]raclopride in plasma and cerebral tissues. For each study, 40 minutes were allowed for the tracer to reach a stable level, at which time amphetamine was administered to induce dopamine release. To describe the [¹¹C]raclopride striatal time–activity curves (TAC), a two-tissue compartment model (Farde et al., 1989) was extended to include the effects of dopamine using the measured microdialysis data. Using the extended model, simulations were performed to determine what feature of the induced dopamine pulse most strongly influences the PET signal. Further simulations were performed to investigate the effect of an amphetamine-induced blood flow change.

THEORY

[¹¹C]raclopride is a reversible D₂ antagonist with kinetics that can be described by a two-tissue compartment model (Farde et al., 1989). This section defines the extended model, which includes the effects of dopamine competition. The compartmental diagram of the model is shown in Fig. 1, and the terminology is defined in Table 1.

Fig. 1.

Compartment model for [¹¹C]raclopride that explicitly accounts for dopamine competition. This model has been extended from that presented in Farde et al. (1989) and includes additional compartments for free (D_f) and bound (D_b) dopamine. The total receptor concentration B_max is the sum of bound raclopride C_b(t), bound dopamine D_b(t), and free receptor B_free(t). The binding rate of [¹¹C]raclopride, k₃(t), which is proportional to B_free(t), thus depends on the level of bound dopamine. In these studies, the binding rate is time-dependent because of the time-varying dopamine concentration after amphetamine. See Table 1 for definition of terms.

Table 1.

Definitions

α	Microdialysis efficiency
B_free(t)	Free D₂ receptor concentration (nmol/L)
B_max	Total D₂ receptor concentration (nmol/L)
c_b(t)	Specifically bound tracer (nCi/cc)
C_f(t)	Free + non-specifically bound tracer in tissue (nCi/cc)
C_p(t)	Radioactivity concentration of unmetabolized tracer in plasma (nCi/cc)
$C_{p}^{T O T} (t)$	Total radioactivity concentration in blood (nCi/cc)
D_b(t)	Specifically bound dopamine (nmol/L)
D_f(t)	Free dopamine (nmol/L)
D^μ(t)	Dopamine curve modeled from the microdialysis data (nmol/L)
f_L	Fraction of receptors in the low affinity state
K_D	Dissociation constant for raclopride (nmol/L)
$K_{D}^{D A}$	Dissociation constant for dopamine (nmol/L)
$K_{D}^{H}$	K_D for dopamine at high affinity binding sites (nmol/L)
$K_{D}^{L}$	K_D for dopamine at low affinity binding sites (nmol/L)
$K_{D}^{μ}$	Apparent dissociation constant for dopamine $(α K_{D}^{D A})$ , (nmol/L)
k_off	Dissociation rate of raclopride from D₂ receptors (min⁻¹)
$k_{o f f}^{D A}$	Dissociation rate of dopamine from D₂ receptors (min⁻¹)
k_on	Bimolecular association rate constant for raclopride (nmol/L⁻¹ min⁻¹)
$k_{o n}^{D A}$	Bimolecular association rate constant for dopamine (nmol/L⁻¹ min⁻¹)
K₁	Plasma to tissue influx constant (mL plasma/min/mL tissue)
k₂	Tissue to plasma clearance (min⁻¹)
k₃(t)	Binding rate of raclopride to D₂ receptors (min⁻¹)
k₄	Dissociation rate of raclopride from D₂ receptors (k_off), (min⁻¹)
SA	Specific activity of [¹¹C]raclopride (mCi/μmol)
ΔSB	Percent reduction in specific binding measured pre- and postamphetamine
V_b	Blood volume (mL blood/mL tissue)
V_e	Free volume of distribution of tracer in striatum (mL plasma/mL tissue)
$V_{e}^{c}$	Free volume of distribution of tracer in cerebellum (mL plasma/mL tissue)

The conventional model for [¹¹C]raclopride includes compartments for free (C_f) and specifically bound (C_b) tracer in tissue. The plasma–tissue exchange constants across the blood-brain barrier are K₁ and k₂, and V_e is the free volume of distribution (K₁/k₂). The exchange between free and bound tracer is described by the binding rate k₃ and the dissociation rate k₄. The general form of the model differential equation is

\begin{aligned} \frac{d C_{f}}{d t} = & K_{1} C_{p} - (k_{2} + k_{3}) C_{f} + k_{4} C_{b} \\ \frac{d C_{b}}{d t} = k_{3} C_{f} - k_{4} C_{b} \end{aligned}

(1)

The binding rate is proportional to the free receptor concentration,

k_{3} (t) = k_{o n} B_{f r e e} (t)

(2)

where k_on is the bimolecular association rate constant for raclopride, and B_free is the free receptor concentration. The total D₂ receptor concentration B_max is equal to the sum of bound raclopride, bound dopamine (D_b), and free receptor. The free receptor pool can thus be expressed as follows:

B_{f r e e} (t) = B_{m a x} - \frac{C_{b} (t)}{S A} - D_{b} (t)

(3)

where SA is specific activity of [¹¹C]raclopride.

For high-specific activity studies, the receptor occupancy of raclopride is negligible. Therefore, the free receptor concentration and the binding rate of raclopride are assumed to be constant. However, in these experiments, amphetamine causes both the free and bound dopamine concentrations to change; therefore, the free receptor concentration and the binding rate are time-dependent. To account for the time-dependent binding rate, dopamine must be included in the model.

We extend the conventional model by including additional compartments for free (D_f) and bound (D_b) dopamine. We ignore the binding of amphetamine to D₂ receptors, which has been found to be negligible (Innis et al., 1992). The following assumptions regarding dopamine D₂ receptor binding were made: (1) dopamine receptor binding kinetics are sufficiently fast that bound dopamine is always in equilibrium with free synaptic dopamine, regardless of how rapidly free dopamine changes; thus, we do not add an additional differential equation to account for the binding of dopamine; and (2) high- and low-affinity binding sites for dopamine are not included; that is, dopamine is assumed to bind to D₂ receptors with a single affinity. The effects of violations of these assumptions are presented in the discussion.

With these assumptions, an equilibrium equation can be derived to relate free and bound dopamine,

k_{o n}^{D A} (B_{m a x} - \frac{C_{b} (t)}{S A} - D_{b} (t)) D_{b} (t) = k_{o f f}^{D A} D_{b} (t)

(4)

where $k_{o n}^{D A}$ and $k_{o f f}^{D A}$ are the D₂ binding constants for dopamine. Solving for D_b(t) in Eq. 4, then substituting into Eqs. 2 and 3, yields for the time-dependent binding rate

k_{3} (t) = \frac{k_{o n} (B_{m a x} - \frac{C_{b} (t)}{S A})}{1 + \frac{D_{f} (t)}{K_{D}^{D A}}}

(5)

where $K_{D}^{D A} = k_{o f f}^{D A} / k_{o n}^{D A}$ , the equilibrium dissociation constant for dopamine. In the absence of dopamine, Eq. 5 reduces to the conventional expression for the binding rate.

To apply the model, the ratio $D_{f} (t) / K_{D}^{D A}$ in Eq. 5 must be related to the microdialysis data. We assume that the microdialysis dopamine concentration curve D^μ(t) is proportional to the free dopamine concentration, that is,

D^{μ} (t) = α D_{f} (t)

(6)

In Eq. 6, α is a constant scale factor. We define a new parameter $K_{D}^{μ}$ as follows:

K_{D}^{μ} = α K_{D}^{D A}

(7)

When α is equal to 1, which corresponds to perfect measurement of D_f(t), then $K_{D}^{μ}$ equals $K_{D}^{D A}$ . In these studies, D^μ(t) is expected to underestimate D_f(t) (see Discussion), i.e. α is less than 1, therefore, $K_{D}^{μ}$ underestimates $K_{D}^{D A}$ .

Upon substitution of Eqs. 5 –7, the model equations (Eq. 1) can be expressed in terms of the microdialysis data D^μ(t) as follows:

\begin{aligned} \frac{d C_{f}}{d t} = & K_{1} C_{p} (t) - \\ (\frac{K_{1}}{V_{e}} + k_{o n} \frac{B_{m a x} - \frac{C_{b} (t)}{S A}}{1 + \frac{D^{μ} (t)}{K_{D}^{μ}}}) C_{f} (t) + k_{4} C_{b} (t) \end{aligned}

(8)

\frac{d C_{b}}{d t} = k_{o n} \frac{B_{m a x} - \frac{C_{b} (t)}{S A}}{1 + \frac{D^{μ} (t)}{K_{D}^{μ}}} C_{f} (t) - k_{4} C_{b} (t)

(9)

There is no analytical solution to Eqs. 8 and 9 in the presence of an arbitrary time-varying dopamine concentration; therefore, we used a fourth-order Runge-Kutta routine to solve the differential equations. The final tissue model also included a term for intravascular radioactivity, $V_{b} C_{p}^{T O T} (t)$ , where V_b is the blood volume and $C_{p}^{T O T} (t)$ is the total radioactivity concentration in blood.

METHODS

Animal studies

A complete description of the PET methodology used here is given by Carson et al. (1997). Adult male rhesus monkeys (n = 3, 7 to 11 kg) were initially sedated with ketamine (10 mg/kg intramuscularly) then anesthetized for the remainder of the study with isoflurane (1% to 2%, to effect). Each animal was studied twice on the same day with approximately 3 hours separating the start of each study. Bolus-plus-infusion delivery of [¹¹C]raclopride was used with a specific activity of 161 to 710 mCi/μmol and injected dose of 2.0 to 6.0 mCi. The bolus volume was equal to 60 minutes of constant infusion; therefore, 40% of the total volume was delivered as a bolus for a 90-minute study. The bolus volume was given over 1 minute. Amphetamine was given as an intravenous bolus, usually at 40 minutes, with doses of 0.2 mg/kg (study 1) and 0.4 mg/kg (study 2), respectively.

The PET studies were acquired using a Scanditronix PC2048-15B scanner (Uppsala, Sweden), which acquires 15 slices separated by 6.5 mm with reconstructed resolution of 6 to 7 mm full width at half maximum. A stereotactic head holder was used that positioned the head so as to obtain coronal slices. Scan frames acquired were 6 × 30 seconds, 3 × 60 seconds, 2 × 120 seconds, 4 × 300 seconds, 8 × 150 seconds, and 8 × 300 seconds for a total study duration of 90 minutes. Arterial blood samples (0.5 to 1.0 mL) were collected throughout the study, and plasma was counted on a sodium iodide gamma counter calibrated so that counts per minute were converted to nanocuries per cubic centimeter. Blood samples collected at 5, 12, 25, 40, 60, and 90 minutes were analyzed for radioactive metabolites (de Bartolomeis, 1996). The unmetabolized fractions were fit to a model (Carson et al., 1997) based on the method of Huang et al. (1991) to estimate the continuous plasma raclopride time course.

Irregular regions of interest were drawn on three adjacent slices for both the left and right striatum, and on two adjacent slices for the cerebellum. Regions of interest were applied to all scan frames to generate TAC. Final striatum and cerebellum TAC were derived from an area-weighted mean of the individual TAC. Variance in the TAC was estimated (Carson et al., 1993b) and provided weights for the model fits.

Microdialysis

Rhesus monkeys were prepared for in vivo microdialysis at least 3 months before PET scanning, as previously described (Kolachana et al., 1994). The surgical procedure involved a small cranial opening and stereotactic positioning of probe guide holders that were then fixed to the skull using bone cement. Each guide holder features a 4 × 15 array of holes (<1 mm diameter, 1 mm center-to-center distance) that was positioned over an area (6 to 8 mm × 15 to 20 mm) of removed skull through which dialysis probes could be inserted. To determine which holes were most accurately aligned with the left and right caudate, vitamin E–filled silica barrels were placed through individual guide holes, but not into the brain, during magnetic resonance scanning.

On the day of the experiment, microdialysis probes were positioned in the left and right caudate at least 30 minutes before the PET study. The microdialysis outlet tubing was sufficiently long (20 cm) that the dialysate collection vials were outside of the scanner's field of view. Probes were perfused by means of a microinfusion pump (Harvard Apparatus, Model 2400-006, South Natick, MA, U.S.A.) at a flow rate of 1.5 μL/min with artificial cerebrospinal fluid (147 mmol/L Na⁺, 3 mmol/L K⁺, 1.3 mmol/L Ca²⁺, 1.0 mmol/L Mg²⁺, 155 mmol/L Cl⁻; buffered with 1.0 mmol/L phosphate, pH 7.3 to 7.4). Microdialysate samples were collected over 10-minute intervals for a total sample volume of 15 μL, and immediately frozen on dry ice. Samples were thawed, and 13 μL of each sample were analyzed for dopamine content using high-performance liquid chromatography with electrochemical detection (Saunders et al., 1994). External dopamine standards (10⁻⁸ mol/L) also were analyzed (n = 3, for each study) to generate a calibration factor that was used to convert the high-performance liquid chromatography measurements to units of dopamine concentration (nmol/L).

Utilization of microdialysis data

A microdialysis data set consisted of three to four preamphetamine baseline samples, one dialysate sample that overlapped the amphetamine injection, and three or more additional postamphetamine samples. The amphetamine injection was not aligned in time with the microdialysis samples because of the time delay (∼13 minutes) needed for the dialysate to flow from the probe through the outlet tubing to the collection vial. The long time delay resulted from the length of outlet tubing (∼20 cm) necessary to keep the dialysate collection apparatus outside of the PET scanner's field of view.

Because the microdialysis samples were collected over 10-minute intervals, the transient peak of the dopamine pulse was not adequately characterized by the data. To derive a continuous curve D^μ(t) from the data, the pulse was modeled as a monoexponential curve (Fig. 2). The microdialysis data were fit to a model with five parameters: the initial baseline B₁, the pulse height H, the clearance rate of the pulse R, the start time of the dopamine pulse Δt, and the postamphetamine baseline B₂, that is,

Fig. 2.

Dopamine microdialysis data. The microdialysis samples were collected over 10-minute intervals and are shown as a bar graph. A continuous dopamine curve was fit to the data by modeling the dopamine pulse as a monoexponential curve D^μ(t) (Eq. 10, solid line). The peak of the curve begins at the time of the amphetamine bolus, usually 40 minutes.

\begin{matrix} D^{μ} (t) = B_{1} & t < Δ t \\ = B_{2} + H e x p (- R (t - Δ t)) & t ⩾ Δ t \end{matrix}

(10)

B₁ is the endogenous dopamine level in the absence of a stimulus and provides a reference to evaluate the size of the pulse. In measurements taken more than 2 hours after amphetamine, the dopamine concentration did not return to the initial baseline, so the curve was modeled with a second baseline (B₂). For the second study of each study pair, B₂, was set equal to B₁ because there were insufficient postamphetamine data to reliably detect a change in baseline. Since each dialysate sample is an integral over 10 minutes, these data were fit to the integral of the model curve (Eq. 10). This ensured that the fitted curve would conserve the dopamine content measured by microdialysis. For use in modeling the PET data, the fitted curve D^μ(t) was shifted in time so that the dopamine pulse began at the time of amphetamine delivery, typically 40 minutes. Shifting D^μ(t) to begin 1 minute after amphetamine delivery produced little effect on the modeling results.

The microdialysis data underestimate the dopamine concentration in vivo and ideally should be corrected by the in vivo recovery, that is, the ratio of the concentration of dopamine in the collected dialysate to the tissue concentration in vivo. This value is difficult to measure, however, so to approximate this correction, the microdialysis data were corrected by the in vitro recovery. To measure the in vitro recovery, the dialysis probe is placed in a stirred dopamine solution with a known concentration. The in vitro recovery equals the ratio of the dopamine concentration in the collected dialysate to the known concentration and will be greater than the in vivo recovery (see Discussion). The probes used in these experiments have a measured in vitro recovery of 30% to 40%. Here, the microdialysis data D^μ(t) were scaled assuming a fixed in vitro recovery of 30%. Since the microdialysis data enter the model equations (Eqs. 8 and 9) as the ratio $D^{μ} (t) / K_{D}^{μ}$ , errors in estimating the in vivo recovery will produce matching errors in $K_{D}^{μ}$ , with no change in the modeled tissue concentration curve.

Tracer kinetic modeling

To achieve sufficient statistics over the course of a 90-minute study, we gave large doses of [¹¹C]raclopride: 0.57 ± 0.21 mCi/kg. Given the specific activity at time of injection (200 to 800 Ci/mmol), there was sufficient mass in these doses that specific binding of raclopride could not be ignored, thus the term C_b/SA was not discarded from model Eqs. 8 and 9. As a result, instead of having a single parameter equal to the product of k_on and B_max, these terms must be treated separately. In initial attempts to simultaneously fit both parameters, large coefficients of variation (exceeding 100% in three of six cases) were found. Thus, the model was constrained by fixing k_on to 0.015 min⁻¹ (see Discussion).

Six parameters (K₁, V_e, B_max, $K_{D}^{μ}$ , k_off, V_b) remain in the model (Eqs. 8 and 9) for striatum data. Two approaches were taken to fit the striatal TAC. In the first approach, all six parameters were estimated simultaneously. In the second, the cerebellum TAC was initially fit to a two-compartment, five-parameter model $(K_{1}^{c}, k_{2}^{c}, k_{5}^{c}, k_{6}^{c}, V_{b}^{c})$ , where the c superscript refers to the cerebellum. $K_{1}^{c}$ and $k_{2}^{c}$ are plasma–tissue exchange constants, and $k_{5}^{c}$ and $k_{6}^{c}$ describe the exchange between free and nonspecifically bound tissue compartments. The free volume of distribution (free plus nonspecific) in cerebellum $(V_{e}^{c})$ was calculated from

V_{e}^{c} = \frac{K_{1}^{c}}{K_{2}^{c}} (1 + \frac{k_{5}^{c}}{K_{6}^{c}})

(11)

The free volume of distribution for the striatum V_e was set equal to $V_{e}^{c}$ , and the remaining five parameters K₁, B_max, $K_{D}^{μ}$ , k_off, V_b) were estimated from the striatal TAC.

Two studies were performed in each animal on a single day. If there were no physiologic changes between studies, it would ideally be possible to model the data from both studies with a single set of floated parameters. This was tested in combined fits using the five-parameter model. For the two studies in each animal, V_e for each study was fixed to its corresponding $V_{e}^{c}$ , and K₁, B_max, $K_{D}^{μ}$ , k_off, and V_b were estimated from striatal TAC of both studies simultaneously. To test if this combined fit could be improved with an additional parameter, the estimation procedure was repeated, allowing each study to have independent K₁ values (i.e., a six-parameter fit).

Simulation studies

The study design used here—B/I delivery of [¹¹C]raclopride with amphetamine displacement—has been used to investigate the response to dopamine challenge in schizophrenic patients (Breier et al., 1997). This B/I paradigm allows changes in specific binding to be determined within a single study without blood sampling (Carson et al., 1997). These patient studies were analyzed by calculating the percent change in specific binding. Briefly, the B/I protocol achieves near-equilibrium conditions before amphetamine delivery; the control level of specific binding (SB₁) is calculated from SB₁ = (C_BG − C_CER)/C_CER, where C_BG and C_CER are the radioactivity concentrations averaged for basal ganglia and cerebellum, respectively. After amphetamine, the striatal activity clears and a new level of specific binding (SB₂) is then estimated from C_BG and C_CER. The percent reduction in specific binding from preamphetamine to postamphetamine (ΔSB) then is calculated, where ΔSB = 100(SB₁ − SB₂)/SB₁.

Through simulations, we applied the extended model to characterize the relation between changes in specific binding measured in this manner and the underlying dopamine pulse. In particular, the dopamine pulse (Fig. 2) is characterized by its pulse height and clearance rate. Dopamine pulses (D^μ(t)) were simulated with six different pulse heights (167, 333, 500, 667, 833, 1000 nmol/L), and five different clearance rates (0.03, 0.06, 0.09, 0.12, 0.15 min⁻¹). These parameters correspond to the range of dopamine pulses found in these experiments (peak: 74 to 839 nmol/L, rate: 0.03 to 0.13 min⁻¹). Each pulse was used to simulate a [¹¹C]raclopride TAC using a measured plasma input function and the fitted model parameters from one of the PET studies. Specific binding was computed preamphetamine and postamphetamine using scan frames from 20 to 40 minutes and 60 to 90 minutes, respectively. For each time activity curve, ΔSB was calculated, and these values then were correlated with the dopamine pulse height, its clearance rate, or the integral of the dopamine pulse. The integral of each pulse above baseline was computed over 40 minutes.

In addition to increasing synaptic dopamine, amphetamine may induce a change in blood flow that could affect [¹¹C]raclopride TAC (Detre et al., 1990; Daniel et al., 1991; Russo et al., 1991). Using the measured plasma input function and the fitted model parameters from one of the PET studies, a change in blood flow was simulated as a step change in K₁ at 40 minutes. Since V_e was held constant, k₂ effectively changed as well. The effect of a transient blood flow change also was tested by simulating a step change in flow followed by exponential return to the initial flow rate with a halftime of 14 minutes (mean of DA pulse halftimes). Time–activity curves were generated for K₁ changes up to ±25%, and the effects on ΔSB were calculated.

RESULTS

Microdialysis

For five of six amphetamine injections, the microdialysis data were well described by the exponential model (Fig. 2). In one study (with 0.2 mg/kg of amphetamine), the dopamine pulse had a poorly defined peak and was modeled as a polynomial curve. Based on these fitted curves, baseline dopamine was 17 ± 4 nmol/L and 43 ± 14 nmol/L before 0.2 mg/kg and 0.4 mg/kg of amphetamine, respectively. The estimate of baseline dopamine is consistent with a previously reported value of 31 nmol/L in rhesus monkey (Moghaddam et al., 1993). In the three animals, baseline dopamine measured before the 0.4 mg/kg study was 92%, 267%, and 108% larger than baseline dopamine measured before the 0.2 mg/kg study, possibly because of residual amphetamine effects. Dopamine pulses described by an exponential curve had a peak height of 489 ± 302 nmol/L and a clearance rate of 0.06 ± 0.04 min⁻¹ (halftime 14 ± 6 min). To compare the magnitude of dopamine release for different amphetamine doses, the average dopamine concentration was computed over the first 40 minutes after amphetamine, then expressed as a percent of the baseline dopamine concentration in the first study of each pair. For the three animals, dopamine increased by 620%, 1100%, and 96%, respectively, over baseline after 0.2 mg/kg of amphetamine. After 0.4 mg/kg of amphetamine, dopamine increased 1600%, 2500%, and 260%, respectively. Dopamine release after 0.4 mg/kg of amphetamine more than doubled over that after 0.2 mg/kg of amphetamine.

PET Data and modeling

Figure 3 shows typical TAC data for striatum and cerebellum with model fits. Radioactivity in cerebral tissues became nearly constant by 20 to 30 minutes by using a B/I protocol. After amphetamine, activity cleared from the striatum with no change in cerebellum. There was no apparent late increase in striatal activity within the 50-minute period postamphetamine.

Fig. 3.

The positron emission tomography data with model fits after bolus-plus-infusion injection of 5.4 mCi of [¹¹C]raclopride. First, the cerebellum time–activity curve (TAC) (▴) was fit to a two compartment, five-parameter model, and the volume of distribution $V_{e}^{c}$ was computed from the fitted model parameters (Eq. 11). The free volume of distribution in striatum V_e was set equal to $V_{e}^{c}$ , and five parameters were estimated to model the striatum TAC (▪). This model described the break in the striatal curve after amphetamine (0.4 mg/kg at 40 minutes) and the prolonged depression of [¹¹C]raclopride concentration. AMP, amphetamine.

The striatum data were fit to the six- and five-parameter model, and a typical fitted curve for the five-parameter model is shown in Fig. 3. Parameter values are shown in Table 2. The values obtained using both models were similar; however, the five-parameter model, with V_e fixed from a cerebellar fit, gave smaller standard errors for all parameters, particularly B_max. In four of six studies, the five-parameter model had a lower Akaike information criterion (Akaike, 1976) than the six-parameter model. Therefore, we chose the five-parameter model for further testing and performance of simulations.

Table 2.

Fitted parameter values

	6-Parameter model			5-Parameter model
	Mean	%SD	%SE	Mean	%SD	%SE
K₁ (mL/min/mL)	0.16	25	8	0.17	29	3
V_e (mL/mL)	0.70	31	19	0.64	21	3
B_max (nmol/L)	16.4	43	21	17.6	37	6
K_D^μ (nmol/L)	347	58	34	424	69	27
k_off (min⁻¹)	.072	51	12	.074	49	10
V_b (mL/mL)	.038	55	24	.036	61	22

Parameter values obtained from fitting striatal [¹¹C]raclopride TAC to the extended model Eqs. 8 and 9 (see Table 1 for definitions). The 6-parameter model estimated all parameters from a fit of the striatal TAC. For the 5-parameter model, the free volume of distribution in striatum V_e was set equal to that of the cerebellum, as determined from a fit of the cerebellar TAC to a two-compartment model. With V_e fixed, the remaining 5 striatal parameters were then estimated in fits to the extended model. Shown here are the mean values (n = 6) with percent standard deviations (%SD). The column %SE is the average uncertainty (standard error) of each parameter as predicted by the fit. V_e values for the 5-parameter model are obtained from the cerebellum fit.

The new parameter $K_{D}^{μ}$ was determined in the five-parameter fit to be 424 nmol/L. $K_{D}^{μ}$ had a large interanimal coefficient of variation of 69%, which could be because of its dependency on the efficiency of microdialysis sampling (Eq. 7). Among other factors, the microdialysis efficiency depends on in vivo recovery and probe positioning within the tissue, and thus will be different in studies with different probes and different animals. However, we performed paired studies in each animal with dialysis probes that remained in the same position throughout both studies. If there were no significant physiologic changes between the studies, then the parameter values should be the same for both studies. Therefore, we attempted combined fits to each pair of studies using the five-parameter model with a single set of floated parameters. In two of three animals, a single set of five floated parameters (K₁, B_max, $K_{D}^{μ}$ , k_off, V_b) was sufficient to describe each pair of striatal TAC. Results for the third animal are shown in Fig. 4. Here, the combined fit was statistically improved (F test, P < .001) by using a six-parameter model that allowed both studies to have their own K₁ ( $K_{1}^{1}$ , $K_{1}^{2}$ , B_max, $K_{D}^{μ}$ , k_off, V_b). In this fit, $K_{1}^{1}$ and $K_{1}^{2}$ were 0.146 and 0.117 mL/min/mL, respectively. The success of the combined fits gives support to the adequacy of the model and the reliability of the parameters.

Fig. 4.

Simultaneous fit of striatal time–activity curves (TAC) from consecutive studies (with 6.0 and 5.6 mCi of [¹¹C]raclopride, respectively) in the same animal. A single set of five parameters was sufficient to model both studies for two of the three animals studied. For the striatal TAC shown in the figure (animal 3), a statistically better fit (F test, P < .001) was achieved by allowing both studies to have their own blood to tissue influx rates ( $K_{1}^{1}, K_{1}^{2}$ ). For this animal, 0.2 mg/kg of amphetamine was delivered at 40 minutes in the first study (▴), and 0.4 mg/kg of amphetamine at 47 minutes in the second (▪). AMP, amphetamine.

Simulation studies

In Fig. 5, simulated values of the percent reduction in specific binding ΔSB (calculated from preamphetamine and postamphetamine radioactivity levels) are plotted versus three features of the dopamine pulse: pulse height (5A), clearance rate (5B), and integral (5C). It is found that ΔSB does correlate with both the pulse height (r = 0.63) and the clearance rate (r = 0.69) (plots A and B), but there is considerable scatter. For example, in Fig. 5A, at a given dopamine pulse height there is considerable variation in ΔSB, with the largest ΔSB value corresponding to the smallest dopamine clearance rate. Clearly, the best correlation (r = 0.99) is found between ΔSB and the integral of the dopamine pulse (Fig. 5C). This suggests that changes in specific binding measured in patient studies can be interpreted as linearly proportional to the integral of the dopamine pulse. Notice that $K_{D}^{μ}$ is a scale factor for the dopamine pulse (Eq. 8), so that ΔSB also is inversely proportional to $K_{D}^{μ}$ .

Fig. 5.

Simulation study results. Correlation of the percent reduction in specific binding (ΔSB) measured in bolus-plus-infusion studies with parameters of the dopamine pulse. A variety of dopamine pulses were simulated (pulse heights 167, 333, 500, 667, 833, 1000; clearance rates 0.03, 0.06, 0.09, 0.12, 0.15 min⁻¹), then using the blood data and fitted parameter values from a positron emission tomography study, striatal time–activity curves were generated and ΔSB was computed. Here, ΔSB is plotted versus three features of the dopamine pulse: (A) pulse height, (B) clearance rate, and (C) pulse integral (40 minutes). The plots show that ΔSB best correlates with the integral of the DA pulse.

Simulations of changes in blood flow showed a minor effect on specific binding changes. Adding a 25% step increase (decrease) in K₁ and k₂ at the time of amphetamine administration altered ΔSB by +5.5% (−7.2%). If blood flow exponentially returns to the initial flow rate, a smaller change in ΔSB is observed.

DISCUSSION

We have extended the conventional compartment model for [¹¹C]raclopride (Farde et al., 1989) to include dopamine competition. With a single additional parameter $K_{D}^{μ}$ , the model was able to fit striatal TAC from these experiments and adequately described the amphetamine-induced washout of tracer. Simulations using the extended model predict that changes in specific binding measured with the B/I method are highly correlated with the integral of the induced dopamine pulse and are virtually unaffected by changes in blood flow.

The parameter $K_{D}^{μ}$ is linearly proportional to $K_{D}^{D A}$ , the dissociation constant for dopamine (Eq. 7), and thus should provide meaningful information about dopamine. However, the value of $K_{D}^{μ}$ depends on the accuracy of the microdialysis data as well as the characteristics of dopamine physiology. We must carefully consider these issues to understand what these experiments reveal about endogenous dopamine.

Microdialysis measurements

Extracellular dopamine measured with microdialysis was used in these studies to estimate D_f(t), the concentration of dopamine in the vicinity of D₂ receptor sites. Since there are both synaptic and extrasynaptic D₂ receptor sites (Levey et al., 1993; Sesack et al., 1994; Yung et al., 1995), it is not clear if sampling from the extracellular space provides a good measure of D_f(t). In particular, we need to consider the relation between synaptic and extrasynaptic dopamine. Garris et al. (1994) suggests that dopamine uptake by the presynaptic transporter could produce a dopamine concentration gradient between the synapse and extracellular fluid. However, based on a model of the efflux of dopamine from the synapse, they predicted that extracellular dopamine rapidly equilibrates with synaptic dopamine. Thus, even if there is a concentration gradient, it is reasonable to assume that the concentrations of synaptic and extracellular dopamine are linearly proportional. Thus, we approximated the microdialysis data D^μ(t) as being linearly proportional to D_f(t).

To accurately estimate the extracellular dopamine concentration, the microdialysis data D^μ(t) should be corrected for the in vivo recovery. To obtain an approximate estimate, D^μ(t) was scaled by the in vitro recovery (see Methods). However, the in vivo recovery is less than the in vitro recovery, because the in vivo recovery is affected by tissue clearance mechanisms such as reuptake and metabolism (Bungay et al., 1990; Morrison et al., 1991). Therefore, even after scaling by the in vitro recovery, D^μ(t) underestimates extracellular dopamine and $K_{D}^{μ}$ underestimates $K_{D}^{D A}$ .

An additional concern is that the in vivo recovery may not be constant throughout an experiment. In particular, amphetamine has been found to reduce the in vivo recovery of dopamine by up to 37% (Olson and Justice, 1993), perhaps because of inhibition of the presynaptic dopamine transporter. Cocaine also has been found to lower the in vivo recovery of dopamine (Smith and Justice, 1994). The in vivo recovery depends on the rate that the surrounding tissue replenishes molecules that have been depleted by the dialysis probe. Molecules with a fast turnover rate are replaced more quickly than molecules with a slow turnover rate. Therefore, faster turnover corresponds to higher in vivo recovery (Bungay et al., 1990; Morrison et al., 1991). Blocking presynaptic uptake slows dopamine turnover and reduces in vivo recovery. As a consequence, the microdialysis measurement before amphetamine may have a higher in vivo recovery compared with the postamphetamine measurement, which would result in an underestimation of the size of the dopamine pulse relative to baseline dopamine. This would violate the assumption that D^μ(t) and D_f(t) are linearly proportional (Eq. 6). To test the effect of violating this assumption, we simulated a change in the in vivo recovery and determined its effect on the fitted parameter values. For this simulation, a noise-free TAC from the five-parameter model fit was used as data, and fit to the model with a modified dopamine pulse. A 33% decrease in postamphetamine in vivo recovery (which is mathematically equivalent to a 50% increase in the postamphetamine dopamine pulse), produced a 51% increase in $K_{D}^{μ}$ , with less than 2% changes in the other parameters. In contrast, a 33% increase in the preamphetamine recovery caused $K_{D}^{μ}$ to increase by only 5%, and no other parameter changed by more than 2%. This demonstrates that the estimated $K_{D}^{μ}$ value is most sensitive to the in vivo recovery during the postamphetamine phase.

Even if we could accurately correct for in vivo recovery, quantification of the dopamine pulse still may be problematic. Consider that after 0.4 mg/kg of amphetamine, the peak dopamine concentration in the three animals was 74, 672, and 839 nmol/L, respectively. This large range most likely results from differences in probe positioning, that is, a larger dopamine signal may-indicate that the dialysis probe was more optimally located among dopamine releasing sites during the experiment.

Model assumptions

Dopamine binding kinetics. The compartmental model (Fig. 1) shows the association and dissociation of dopamine to the receptor. To simplify the model, we assumed that dopamine binding is in rapid equilibrium (i.e., $k_{o f f}^{D A}$ is infinitely fast), which allowed bound dopamine D_b(t) to be expressed explicitly in terms of $K_{D}^{D A}$ and D_f(t). However, it is not clear that dopamine binding remains in secular equilibrium during the transient dopamine pulse induced by amphetamine. Based on a value of k_on for neurotransmitter receptors of 0.06 nmol/L⁻¹ min⁻¹ and a lower bound for $K_{D}^{D A}$ of 5 nmol/L (high affinity), Logan et al. (1991) estimated $k_{o f f}^{D A}$ to be as low as 0.3 min⁻¹. To test the effect of a finite value of $k_{o f f}^{D A}$ , the differential equation for dopamine binding was added to the model. Using typical microdialysis data and parameters, noise-free striatal TAC were generated with different $k_{o f f}^{D A}$ values and $k_{o n}^{D A}$ equal to 0.06 nmol/L⁻¹ min⁻¹. These TAC were compared with the TAC generated with the original extended model. For $k_{o f f}^{D A}$ equal to 0.3 min⁻¹, there was less than a 2% difference in radioactivity concentration at all time points, with the greatest difference found within the first 10 minutes after amphetamine. Fitting this TAC to the five-parameter model yielded less than 4% changes in the fitted parameter values. For $k_{o f f}^{D A}$ values greater than 0.3 min⁻¹, the differences were smaller. Thus, the assumption that dopamine binding is in continuous secular equilibrium should not affect the accuracy of the model.

Relation between k_on and B_max. Because of the large doses of [¹¹C]raclopride used in these studies, we included the nonnegligible receptor occupancy of raclopride in the model. This required separation of the terms k_on and B_max. To constrain the model, k_on was fixed to 0.015 min⁻¹, which is at the upper range of values measured in humans (Farde et al., 1989). To determine how a smaller value of k_on would affect the fitted parameter values, noise-free striatal TAC were generated with k_on equal to 0.015 min⁻¹, then fit to the model with k_on set to 0.0075. In these fits, B_max, $K_{D}^{μ}$ , and k_off increased by 92%, 14%, and 8%, respectively. If receptor occupancy by raclopride was negligible, then k_on and B_max would be perfectly correlated, so that changes in k_on would produce matching changes in B_max (so k_onB_max remains constant) and $K_{D}^{μ}$ and k_off would be unaffected. The weak dependence of $K_{D}^{μ}$ and k_off on the assumed value of k_on is caused by the dependence of k₃(t) on the receptor occupancy of raclopride (Eq. 5).

Multiple-affinity states. Although D₂ receptors have high- and low-affinity states for agonist binding (Sibley et al., 1982), we assumed that dopamine binds with a single affinity. To investigate the effects of violating this assumption, we performed simulations where the D₂ receptors have two affinity states (or equivalently where raclopride binds to two receptor subtypes having different affinities). Let f_L be the fraction of receptors in the low-affinity state $K_{D}^{L}$ , with the remaining fraction (1 − f_L) in the high-affinity state $K_{D}^{H}$ . We assumed that the fraction f_L remained constant during these studies. Assuming that dopamine binding is in equilibrium at both affinities, and that raclopride binds at a single affinity, two equilibrium equations for dopamine binding, analogous to Eq. 4, can be written for each affinity state, and the raclopride binding rate becomes

k_{3} (t) = k_{o n} (B_{m a x} - \frac{C_{b} (t)}{S A}) (1 - D_{f} (t) [\frac{f_{L}}{K_{D}^{L} + D_{f} (t)} + \frac{1 - f_{L}}{K_{D}^{H} + D_{f} (t)}])

(12)

Equation 12 replaces Eq. 5 and was used to derive differential equations analogous to Eqs. 8 and 9.

Simulations were performed to determine how the presence of two affinity states affects parameter values. A wide range of values have been determined experimentally for $K_{D}^{H}$ (2.8 to 474 nmol/L) and $K_{D}^{L}$ (1705 to 2490 nmol/L) (Gingrich and Caron, 1993), and for the fraction of receptors in each state (f_L = 10% to 77%) (Sibley et al., 1982; Richfield et al., 1986; Ross, 1991). Initial simulations were performed with a range of f_L values and a 10:1 affinity ratio $K_{D}^{L} / K_{D}^{H}$ using K_D values that yielded TAC similar to those obtained in these studies. Noise-free TAC generated using the two-affinity model were fit to the original five-parameter model. As f_L varied from 0 to 1, the estimated $K_{D}^{μ}$ value varied between $K_{D}^{H}$ and $K_{D}^{L}$ . B_max was slightly underestimated, more so for low f_L values. A second set of simulations was performed with a 100:1 affinity ratio where $K_{D}^{H}$ was reduced; therefore, baseline dopamine occupied a significant fraction of the high-affinity sites. Compared with using a 10:1 affinity ratio, the estimated value of $K_{D}^{μ}$ was much closer to $K_{D}^{L}$ . In addition, the estimate of B_max was close to f_LB_max, that is, the fraction of receptors in the low-affinity state. Apparently, as $K_{D}^{H}$ gets increasingly small (leading to greater baseline dopamine occupancy), the high-affinity sites become less accessible to raclopride, which leads to an underestimate of B_max. Since competition then would take place primarily at low-affinity sites, the estimated value of $K_{D}^{μ}$ becomes closer to $K_{D}^{L}$ . Thus, in the presence of two affinity states for dopamine, the estimated value of $K_{D}^{μ}$ lies between $K_{D}^{H}$ and $K_{D}^{L}$ .

Measurement of the input function. The model assumes that the input function is measured perfectly. However, with metabolite measurements at only a few sample times, the limited statistics of the late data will introduce nontrivial errors in the input function that can affect the quality of model fits. We assessed the sensitivity of the fitted parameters to errors in the metabolite correction by simulations. In fits to the extended model, a 10% bias in the input function (beyond 60 minutes) could lead to errors in the estimate of $K_{D}^{μ}$ of 40% or more, with less than 10% errors in the remaining parameters. Underestimating the late input function caused the model to overestimate $K_{D}^{μ}$ because, in this case, some of the postamphetamine drop in the tissue curve (Fig. 3) was attributed to a drop in the input function level. Thus, $K_{D}^{μ}$ is sensitive to the accuracy of the metabolite correction, and this likely contributes to the large variability observed in $K_{D}^{μ}$ . However, the ability to perform combined fits of paired studies with a single set of parameter values suggests sufficient reliability in the measurement of the input functions in these studies.

Inferences from the model

Interpretation of ΔSB. With the B/I protocol, the percent reduction in specific binding (ΔSB) can be measured directly from the preamphetamine and postamphetamine tissue concentration values. The simulations showed that ΔSB is linearly proportional to the integral of the dopamine pulse. Notice that simulations with biexponential pulses also showed strong correlation between the pulse integral and ΔSB. The effect of an induced blood flow change is minor. Thus, for multiple studies in a particular subject, assuming there are no changes in dopamine physiology, the relative changes in specific binding should predict the magnitude (integral) of the amphetamine-induced dopamine pulse. However, caution must be used in extending this result to the comparison of studies in different subjects or under different physiologic conditions. Differences in ΔSB also could be attributed to different $K_{D}^{μ}$ values; that is, an increase in ΔSB could result from an increase in the dopamine pulse, or a decrease in $K_{D}^{μ}$ , or a combination of the two. Differences in $K_{D}^{μ}$ values between patient groups could occur by having different proportions of receptors in highland low-affinity states.

Further simulations were performed to test if the values of k_off, B_max, or the baseline dopamine level also could affect ΔSB. A 20% increase (decrease) in k_off changed ΔSB by +6.6% (−12.1%). A faster k_off causes raclopride to clear more rapidly in response to a dopamine pulse, and therefore leads to a greater change in specific binding. A ±20% change in B_max had a negligible effect on ΔSB; thus, dopamine displaces a similar fraction of specifically bound raclopride, regardless of changes in B_max. The baseline dopamine level also had little effect on ΔSB unless some of the receptors were in the high-affinity state. In the case of two affinity states with $K_{D}^{L} / K_{D}^{H} = 100$ and f_L = 50%, a 20% decrease (increase) in baseline dopamine changed the control ΔSB by +8.1% (−6.4%). As baseline dopamine increases and occupies more of the high-affinity sites, the subsequent dopamine pulse occupies fewer receptors (and displaces less raclopride) because most of the remaining sites are in the low-affinity state. Thus, the model predicts that the dopamine pulse and $K_{D}^{μ}$ are the primary determinants of ΔSB, with lesser effects attributed to k_off and baseline dopamine.

D₂ receptor occupancy. Based on the model parameters and the baseline microdialysis data (B₁), it is theoretically possible to calculate the D₂ receptor occupancy. Specifically, inserting Eqs. 6, 7, and 10 into Eq. 4, the occupancy is predicted to be $B_{1} / (K_{D}^{μ} + B_{1})$ . Baseline dopamine receptor occupancy in these studies was only 3% to 15%, because baseline dopamine (17 nmol/L) was small compared with $K_{D}^{μ}$ (424 nmol/L). This estimate of the receptor occupancy appears to be inconsistent with the results of Dewey et al. (1995) and Laruelle et al. (1997), which showed that dopamine depletion can increase D₂ ligand specific binding by more than 30%. Such an increase would require at least a 25% occupancy of D₂ receptors by baseline dopamine (Laruelle et al., 1997). A possible explanation is that the estimate of $K_{D}^{μ}$ from this model reflects a combination of high and low affinities. For example, if 50% of receptors were in the high-affinity state and baseline dopamine is comparable with $K_{D}^{μ}$ , then about 50% of the high-affinity receptors and about 25% of all receptors would be occupied by baseline dopamine.

The peak occupancy of D₂ receptors by dopamine after amphetamine can be similarly estimated based on the peak of the dopamine curve (B₂ + H; Eq. 10), and is given by $(B_{2} + H) / (K_{D}^{μ} + B_{2} + H)$ . To estimate the peak occupancy, $K_{D}^{μ}$ values from the combined fits were used. The predicted peak occupancy was 52 ± 19% and 62 ± 12% for 0.2 mg/kg and 0.4 mg/kg of amphetamine, respectively. Notice that the large increase in dopamine produced by 0.4 mg/kg of amphetamine, in comparison with 0.2 mg/kg, produced only a moderate increase in saturation because of the nonlinearity of receptor binding.

Return of radioactivity to preamphetamine level. In these studies, an amphetamine-induced dopamine pulse displaced [¹¹C]raclopride from the receptors. Because of the continuous infusion, it would be expected that raclopride will reoccupy receptors as dopamine clears. However, despite the rapid clearance of dopamine (Fig. 2), the striatal radioactivity concentration did not rapidly return to its preamphetamine level, but rather, it remained depressed throughout postamphetamine data acquisition. This demonstrates that the response to an amphetamine-induced dopamine pulse is driven primarily by the tissue kinetics of raclopride (k₂, k_on, k_off), which are not fast enough to follow transient changes in dopamine.

To estimate when the radioactivity concentration after amphetamine would approach the preamphetamine equilibrium value, TAC were simulated using an input function that remained constant beyond 40 minutes. With no dopamine pulse, the simulated striatal radioactivity value at 40 minutes was within 0.5% of the true equilibrium value. Time–activity curves were then simulated with a range of dopamine pulses (pulse height: 333 to 1000 nmol/L; clearance rate: 0.025 to 0.1 min⁻¹). All of these TAC reached their minimum value at about 70 minutes, and from the minimum had returned less than 20% to the equilibrium value by 90 minutes. This is consistent with not observing an increase in striatal activity toward the end of these studies because of noise. On average, the TAC returned halfway (50%) to the equilibrium value by 110 minutes, with slower clearance rates of dopamine corresponding to longer return times (the magnitude of the dopamine pulse had no effect on return time). A doubling of baseline dopamine postamphetamine, as was typically found in these studies, further delayed the 50% return time to 120 minutes. Thus, if these studies were extended to 120 minutes, the model predicts that the striatal TAC would return partially to the original equilibrium level.

CONCLUSION

Recently, there has been great interest in in vivo imaging studies of receptor systems using the competition of radioligands with endogenous neurotransmitter. However, interpretation of these studies is complex because of the combined effects of tracer and neurotransmitter kinetics. Thus, it is helpful in interpreting patient studies to have a model that can predict how various physiologic parameters can affect a given measurement. Here, we have extended the [¹¹C]raclopride model to include dopamine competition. The extended model was able to describe the striatal TAC in B/I studies with amphetamine challenge. It also permitted a useful interpretation of changes in specific binding measured with the B/I protocol (ΔSB). In particular, the model predicts that ΔSB is proportional to the integral of the amphetamine-induced dopamine pulse. Similar modeling studies are required for other neurotransmitter systems to determine if this result is generally applicable. Use of these modeling approaches also can be helpful in elucidating in vivo neurotransmitter information ( $K_{D}^{μ}$ ) and in the selection of tracers with the most appropriate characteristics for these competition studies.

Acknowledgments: The authors thank the National Institutes of Health PET Department cyclotron, production, quality control, and technologist staff for their excellent technical assistance. We also thank Drs. Peter Bungay, Robert Dedrick, and Peter Herscovitch for their helpful comments and discussion.

Footnotes

Abbreviations used

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