A method to quantify at late imaging a release rate of <sup>18</sup>F-FDG in tissues

Éric Laffon; Michèle Allard; Roger Marthan; Dominique Ducassou

doi:10.1016/j.crvi.2005.06.004

Medical sciences / Sciences médicales

A method to quantify at late imaging a release rate of ¹⁸F-FDG in tissues
[Une méthode de quantification aux temps tardifs d'une libération du ¹⁸F-FDG par les tissus]

Présenté par : Jean Rosa

Éric Laffon ^1,² ; Michèle Allard ¹ ; Roger Marthan ² ; Dominique Ducassou ¹

¹ Service de médecine nucléaire, hôpital du Haut-Lévêque, centre hospitalo-universitaire de Bordeaux, av. de Magellan, 33600 Pessac, France
² Laboratoire de physiologie cellulaire respiratoire, Inserm E356, université Bordeaux-2, 33076 Bordeaux, France

Comptes Rendus. Biologies, Volume 328 (2005) no. 8, pp. 767-772.

Résumés

Anglais
Français

This theoretical work shows that the rate constant for the ¹⁸F-FDG release in tissues can be assessed without needing any arterial blood sampling. The method requires that the clearance of ¹⁸F-FDG from plasma has occurred, whereas ¹⁸F-FDG is still present in the tissue. This condition can be met dating from 3 h after ¹⁸F-FDG injection, when hydration and/or phlorizin injection are applied after the routine static acquisition. The release rate constant can be obtained from a graphical analysis performed at the later decreasing phase of the tissue tracer activity. A two-compartment and a three-compartment model are developed, both in accordance with one another.

Le principe d'une quantification du taux de libération du ¹⁸F-FDG par un tissu ne nécessitant pas de prélèvement de sang artériel est présenté. La méthode est valide lorsque la disparition du traceur du plasma est intervenue, alors que le traceur est toujours présent dans le tissu. Cette condition peut être remplie 3 h après l'injection du traceur, avec une hydratation et/ou l'injection de phlorizine pratiquées après l'acquisition statique de routine. Le taux de libération peut être alors obtenu par une analyse graphique de la phase tardive de la courbe d'activité du ¹⁸F-FDG dans le tissu. Deux modèles sont proposés, à deux puis à trois compartiments, qui sont en accord.

Métadonnées

Reçu le : 2004-11-16
Accepté le : 2005-06-21
Publié le : 2005-08-01

PMID

DOI : 10.1016/j.crvi.2005.06.004

Keywords: PET, ¹⁸F-FDG release, Kinetic modelling, k₄
Mot clés : TEP, Relaxation du ¹⁸F-FDG, Modèle cinétique, $ {k}_{4}$

Affiliations des auteurs :

Éric Laffon ^{1, 2} ; Michèle Allard ¹ ; Roger Marthan ² ; Dominique Ducassou ¹

¹ Service de médecine nucléaire, hôpital du Haut-Lévêque, centre hospitalo-universitaire de Bordeaux, av. de Magellan, 33600 Pessac, France
² Laboratoire de physiologie cellulaire respiratoire, Inserm E356, université Bordeaux-2, 33076 Bordeaux, France

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     title = {A method to quantify at late imaging a release rate of {\protect\textsuperscript{18}F-FDG} in tissues},
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Éric Laffon; Michèle Allard; Roger Marthan; Dominique Ducassou. A method to quantify at late imaging a release rate of ¹⁸F-FDG in tissues. Comptes Rendus. Biologies, Volume 328 (2005) no. 8, pp. 767-772. doi : 10.1016/j.crvi.2005.06.004. https://comptes-rendus.academie-sciences.fr/biologies/articles/10.1016/j.crvi.2005.06.004/

Version originale du texte intégral

1 Introduction

Assuming that the 2-[¹⁸F]fluoro-2-deoxy-d-glucose (¹⁸F-FDG) is trapped in cancer cells in an irreversible manner, PET-scan allows quantification of the metabolic rate of glucose in tumours, by several methods that aimed at quantifying the ¹⁸F-FDG uptake rate [1–5]. Is the irreversibility of the ¹⁸F-FDG uptake an oversimplification? Usually, most biological reactions do not occur in a completely irreversible manner, which has been clearly established in neuroreceptor quantification models [6]. Moreover, in the particular case of ¹⁸F-FDG PET imaging, the tracer metabolism by hepatic tumours challenges this notion of irreversibility [7]. A first quantification method of the ¹⁸F-FDG release rate constant has been proposed by Phelps et al. [8], as an extension of Sokoloff's model [1]. Then, Patlak has proposed a graphical method for reversible tracers [2], as an extension of his previous method for irreversible ones. These latter methods both require a dynamic acquisition, which is achieved during the first hour after the tracer injection, and an invasive blood tracer sampling.

The aim of this theoretical work is to propose a graphical method allowing for the assessment of the ¹⁸F-FDG release in tissues, on certain conditions, but without needing any blood tracer sampling. It is an extension of a recently published method that aimed at quantifying the rate constant for an irreversible tracer uptake [5]. It requires the acquisition of a dynamic PET scan at the later decreasing phase of the tissue tracer activity. Both a two-compartment and a three-compartment model are developed, and simulated data are provided.

2 Two-compartment model

Let us consider the ¹⁸F-FDG uptake from blood to an intracellular compartment (Fig. 1a). The ¹⁸F-FDG uptake rate constant is K, and the ¹⁸F-FDG release rate constant is $k_{B}$ . At the steady state, the rate of trapped tracer per tissue volume unit $d C_{T} / d t$ is:

d C_{T} / d t = K C_{P} (t) - k_{B} C_{T} (t) - λ C_{T} (t)

(1)

where

C_{P} (t)

represents the tracer concentration in the plasma, and λ is the ¹⁸F physical decay constant

(λ = \ln 2 / 110 \min^{−1})

.

Fig. 1
(a) The two-compartment model roughly considers compartments (A) and (B), which involve blood free ¹⁸F-FDG and ¹⁸F-FDG trapped in intracellular cytoplasm, respectively. The ¹⁸F-FDG uptake and release rate constants, K and $k_{B}$ , respectively, are the transport rate constants between the two compartments. (b) In comparison with the two-compartment model, the three-compartment model introduces an intermediate reversible compartment (R), which considers free ¹⁸F-FDG in interstitial fluid and intracellular cytoplasm. The rate constants $k_{1}$ and $k_{2}$ account for forward and reversed transport between compartments $(A^{'})$ and (R), respectively. The rate constants $k_{3}$ and $k_{4}$ account for forward and reversed transport between compartments (R) and $(B^{'})$ , i.e. for ¹⁸F-FDG phosphorylation by hexokinase and ¹⁸F-FDG-6-PO₄ dephosphorylation by G-6-phosphatase, respectively.

In patients with a normal renal and hepatic function, the time duration required to reach the steady state is estimated to last about 5 min [3], and the amount of the tracer that accumulates in the tissue during this time duration is assumed to be negligible in comparison with that which accumulates during the whole examination. Moreover, for simplicity, let us assume that the ¹⁸F-FDG blood time-activity curve (TAC) exponentially decays with a mean constant α, after the peak (see simulated data for a multiexponential decay). In a nearly similar manner to that used in a previous work on ¹⁸F-FDG (irreversible) uptake [5], when $t = 0$ is set at the blood tracer peak, the solution of Eq. (1) is:

C_{T} (t) = K C_{P} (t = 0) [e^{- (λ + k_{B}) t} - e^{- α t}] / (α - λ - k_{B})

(2)

Eq. (2) is very similar to that of a radioactive chain equation, but with a whole decay constant equal to $λ + k_{B}$ . When $e^{- (λ + k_{B}) t}$ is much greater than $e^{- α t}$ , i.e. when the clearance of ¹⁸F-FDG from plasma has occurred whereas ¹⁸F-FDG is still present in the tissue, the later phase of the trapped tracer TAC can be approximated by:

λ C_{T} (t) ≅ λ K C_{P} (t = 0) e^{- (λ + k_{B}) t} / (α - λ - k_{B})

(3)

Eq. (3) shows that a single exponential fitting to the later phase of the ¹⁸F-FDG TAC can provide the release rate constant $k_{B}$ , since λ is known, without needing any arterial blood sampling. (If an arterial blood sampling is carried out, a graphical analysis can also provide the ¹⁸F-FDG uptake rate constant K [5].) Let us note that, at the later phase of the tracer activity in the tissue: (i) neglecting the part of the tracer amount accumulated in the tissue during the time duration to reach the steady state is justified, since both ¹⁸F physical decay and ¹⁸F-FDG release have occurred, and (ii) $λ C_{T} (t)$ actually corresponds to the measured trapped ¹⁸F-FDG activity, since the blood ¹⁸F-FDG is negligible (see simulated data). Furthermore, Eq. (2) allows us to predict that the time $T_{m}$ corresponding to the maximal ¹⁸F-FDG activity in the tissue is lower than that estimated when no ¹⁸F-FDG release occurs $(T_{m}^{'})$ [5]:

T_{m} = \ln [α / (λ + k_{B})] / [α - (λ + k_{B})]

(4)

T_{m}^{'} = \ln [α / λ] / [α - λ]

(5)

3 Three-compartment model

In a three-compartment model (Fig. 1b), the ¹⁸F-FDG transport rate constants between the compartments are different from those of a two-compartment model (Fig. 1a). However, relationships can be established [2]:

K = k_{1} k_{3} / (k_{2} + k_{3})

(6)

k_{B} = k_{4} k_{2} / (k_{2} + k_{3})

(7)

Following Patlak [2], let us suppose that $k_{4}$ is much lower than the other rate constants of the system. Furthermore, let us consider measurements are started up at the later phase of the ¹⁸F-FDG activity in the tissue, when the clearance of ¹⁸F-FDG from plasma has occurred whereas ¹⁸F-FDG is still present in the tissue, in other words, when the tracer uptake from the blood into the tissue is much lower than the tracer release from the tissue. Thereby, when the tracer uptake from the blood into the tissue can be ignored, the rate of change of trapped tracer per tissue volume unit $d C_{T} / d t$ is:

d C_{T} (t) / d t = k_{3} C_{rev} (t) - k_{4} C_{T} (t) - λ C_{T} (t)

(8)

where

C_{rev}

is the free ¹⁸F-FDG in the tissue. The rate of change of the free tracer per tissue volume unit

d C_{rev} (t) / d t

is:

d C_{rev} (t) / d t = k_{4} C_{T} (t) - k_{3} C_{rev} (t) - k_{2} C_{rev} (t) - λ C_{rev} (t)

(9)

The whole solutions of Eqs. (8) and (9) are, respectively:

C_{T} (t) = C_{T_{0}} e^{- (λ + k_{4}) t} + k_{3} e^{- (λ + k_{4}) t}] \int_{0}^{t} C_{rev} (τ) e^{(λ + k_{4}) τ} d τ

(10)

C_{rev} (t) = C_{R_{0}} e^{- (λ + k_{2} + k_{3}) t} + k_{4} e^{- (λ + k_{2} + k_{3}) t}] \int_{0}^{t} C_{T} (τ) e^{(λ + k_{2} + k_{3}) τ} d τ

(11)

where

C_{T 0}

and

C_{R 0}

are constants representing trapped and free ¹⁸F-FDG concentration in the tissue at the beginning of the measurement, respectively. This measurement start, i.e.

t = 0

in the three-compartment model, occurs at the steady state of the later decreasing phase of the tissue tracer TAC, i.e. when the tracer uptake from the blood into the tissue is much lower than the tracer release from the tissue, hence when

C_{rev}

mainly depends on the release of trapped tracer. Consequently, (a) in Eq. (10) the second term of the right-hand side is much lower than the first one (also see below), (b) in Eq. (11) (representing the whole solutions of Eq. (9)), the first term of the right-hand side is not relevant and can be cancelled. Thus, introducing Eq. (10) in Eq. (11) yields the following solution for free ¹⁸F-FDG in the tissue, when the second order term is neglected:

C_{rev} (t) ≅ k_{4} e^{- (λ + k_{2} + k_{3}) t} \int_{0}^{t} C_{T_{0}} e^{- (λ + k_{4}) τ + (λ + k_{2} + k_{3}) τ} d τ

(12)

Then, integrating Eq. (12) leads to:

C_{rev} (t) ≅ [k_{4} / (k_{2} + k_{3} - k_{4})] C_{T_{0}} [e^{- (λ + k_{4}) t} - e^{- (λ + k_{2} + k_{3}) t}]

(13)

hence, since

k_{4} ≪ k_{2} + k_{3}

:

C_{rev} (t) ≅ [k_{4} / (k_{2} + k_{3})] C_{T_{0}} e^{- (λ + k_{4}) t}

(14)

This result justifies that in Eq. (10) the second term of the right-hand side can be considered much lower than the first one. Furthermore, introducing Eq. (14) into Eq. (10) and rearranging yields the following expression for the trapped tracer activity:

λ C_{T} (t) ≅ λ C_{T_{0}} e^{- (λ + k_{4}) t} (1 + k_{R} t)

(15)

k_{R} = k_{4} k_{3} / (k_{2} + k_{3})

(16)

where

k_{R}

can be considered a ¹⁸F-FDG ‘retrapping’ rate constant, by comparison with Eq. (6). Eq. (15) also approximates the measured tissue activity, since the part of the free tracer activity can be considered to be negligible in comparison with that of the trapped tracer.

Furthermore, Eq. (17) hereafter shows that the transport rate constants available from Eqs. (15) and (16), $k_{4}$ and $k_{R}$ in a three-compartment model, are in agreement with that available from Eq. (3), $k_{B}$ in a two-compartment model:

e^{- (λ + k_{B}) t} = e^{- (λ + k_{4} - k_{R}) t} ≅ e^{- (λ + k_{4}) t} [1 + k_{R} t]

(17)

4 Simulated data

The proposed two-compartment model assumes, for simplicity, that the ¹⁸F-FDG blood TAC decays with a mean constant α. However, it has been shown that the ¹⁸F-FDG plasma clearance was multi-exponential, i.e. that the input function (IF) was three-exponential [9]. When such an IF is taken into account, Eq. (2) becomes:

C_{T} (t) = \sum_{i = 1}^{3} K A_{i} (e^{- (λ + k_{B}) t} - e^{- α_{i} t}) / (α_{i} - λ - k_{B})

(18)

Fig. 2 shows simulated data from Eq. (18) and references [9,10], for different values of $k_{B}$ and $α_{3}$ . At the later phase of the ¹⁸F-FDG TAC, when $\exp [- (λ + k_{B}) t]$ is much greater than $\exp [- α_{i} t]$ , a single exponential fitting can provide the ¹⁸F-FDG release rate constant $k_{B}$ , since λ is known, without needing any arterial blood sampling. One hour after injection, in comparison with $λ + k_{B} = 0.0113 \min^{−1}$ [10], the two first IF exponential constants satisfy this condition, $α_{1} = 1.568 {min}^{−1}$ and $α_{2} = 0.1438 {min}^{−1}$ [9], but not $α_{3} = 0.0109 {min}^{−1}$ [9].

Fig. 2
Simulation of the tissue ¹⁸F-FDG TAC in a two-compartment model (Eq. (18)), for different values of $α_{3}$ and $k_{B}$ [9–11]: (1) $α_{3} = 0.0109$ , $k_{B} = 0 \min^{−1}$ ; (2) $α_{3} = 0.0109$ , $k_{B} = 0.0050 \min^{−1}$ ; (3) $2.25 \times α_{3} = 0.0245$ , $k_{B} = 0.0050 \min^{−1}$ . The axis origin is set at the blood tracer TAC peak (not shown).

The greater the constant $α_{3}$ , the faster the tracer plasma clearance 1 h after injection, the earlier the implementation of the method. This feature highlights the major part of hydration and/or phlorizin injection [11,12], which should be made in practice from the end of the routine static acquisition to the beginning of the dynamic acquisition, in order to increase $α_{3}$ . It has been shown indeed that phlorizin injection resulted in an increase by a factor of 2.25 in the plasma clearance rate of ¹⁸F-FDG in rats, and that urinary excretion of ¹⁸F-FDG in phlorizin-treated rats was very near that of hydrated rats [11]. Consequently, assuming that at least the condition:

\exp [- (λ + k_{B}) t] = 10 \times \exp [- α_{3} t]

(19)

is required, when

α_{3}

is increased to a value of

2.25 \times 0.0109 = 0.0245 {min}^{−1}

, the time landmark

t_{l}

to apply the proposed method is about 3 h after ¹⁸F-FDG injection.

From data involved in curve 3 of Fig. 2 ( $k_{B} = 0.0050$ and $α_{3} = 0.0245 {min}^{−1}$ , i.e. with hydration and/or phlorizin injection), single exponential fittings to the 180–240, 240–300, 300–360, and 360–420-min time ranges lead to estimates of $k_{B}$ equal to 0.0042, 0.0046, 0.0048 and 0.0049 min⁻¹, respectively, which should be compared with the value of 0.0050 min⁻¹. Note that curve 2 in Fig. 2 ( $k_{B} = 0.0050$ and $α_{3} = 0.0109 {min}^{−1}$ , i.e. without hydration and/or phlorizin injection), leads in the 360–420-min time range to an (under)estimate of 0.0023 min⁻¹.

5 Discussion

Without needing any blood tracer sampling, when hydration and/or phlorizin injection are applied after the routine static PET acquisition, a two-compartment model analysis allows us to quantify the ¹⁸F-FDG release rate constant $k_{B}$ from a dynamic acquisition occurring at about 3 h after ¹⁸F-FDG injection. The later the dynamic acquisition, the more accurate the $k_{B}$ estimate, but, the greater the ¹⁸F-FDG release from the tissue, the more underestimated the value of $k_{B}$ . Moreover, this theoretical work presents a three-compartment model analysis, which supports the two-compartment one, and allows us to differentiate between the rate constants for ¹⁸F-FDG dephosphorylation and rephosphorylation, $k_{4}$ and $k_{R}$ , respectively. The principle of the present model analysis could be applied to other ¹⁸F-labelled tracers than the ¹⁸F-FDG, at the later decreasing phase of the tissue TAC, when the clearance of the tracer from plasma has occurred whereas it is still present in the tissue.

From Eq. (18) and literature data [9–11], Fig. 2 shows quite different ¹⁸F-FDG tissue TACs, related to different values of $k_{B}$ and of the late IF exponential constant $α_{3}$ . To the very best of our knowledge, the later decreasing phase of ¹⁸F-FDG tissue TACs and the significant parts of $k_{B}$ and $α_{3}$ have not been studied so far. It is suggested that, given the limited understanding of tumour metabolism, in addition to the quantification of the ¹⁸F-FDG uptake, the quantification of the ¹⁸F-FDG release could be investigated to characterize tumours, to differentiate malignant processes from infectious or inflammatory processes [13], and to assess the effects of treatment [7]. Furthermore, Patlak's graphical analysis for assessing ¹⁸F-FDG release rate constant requires an invasive arterial blood sampling and the acquisition of a dynamic PET scan that are performed during the first hour after the tracer injection [2–4]. In contrast, a similar assessment that can be obtained from the proposed graphical analysis appears to be easier to implement in a busy clinical environment, since it is non-invasive and requires a dynamic acquisition that occurs far from the static acquisition.

In conclusion, the principle of a graphical analysis that aimed at quantifying the rate constant for ¹⁸F-FDG release in tissues is presented, which is an extension of a recently published method for assessing the tracer uptake rate [5]. This whole method emphasizes the interest of the later decreasing phase of the ¹⁸F-FDG activity in tissues.

Acknowledgement

The authors wish to thank Schering-CISbiointernational for helpful discussion, and H. Dupouy and I. Hessling for technical assistance. They also thank an anonymous reviewer for his comments on the manuscript.

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