Lumped constant (LC)

The lumped constant was formulated by Sokoloff et al (1977), to allow quantitative assessment of D-glucose utilization by measuring the tissue uptake of ¹⁴C-labelled 2-deoxy-D-glucose. The concept of lumped constant is based on the biochemical principles of competitive substrate kinetics, and it can be used with any metabolism tracer which competes with a natural substrate, for example to estimate the rates of amino acid and and fatty acid uptake.

LC of [¹⁸F]FDG

The lumped constant of [¹⁸F]FDG accounts for the differences in transport and phosphorylation rates between D-glucose and 2-fluoro-2-deoxy-D-glucose, and is used to transform the [¹⁸F]FDG uptake rate to glucose uptake rate. LC is not actually a constant, because of variable contribution of different glucose transporters to the transport step, and relative importance of the transport and phosphorylation step. [¹⁸F]FDG itself is a good substrate only for facilitated glucose transporters (GLUTs), but not for Na⁺-dependent glucose cotransporters (SGLTs). Especially in tumours the LC is highly variable and [¹⁸F]FDG PET may not allow accurate assessment of glucose utilization (Barrio et al., 2020).

Brain

A common assumption in FDG PET studies is that lumped constant is uniform over the whole brain and in all subject and patient groups, but this is not strictly true: Hexokinase favours glucose over FDG, and transport favours FDG over glucose. Although there are several different estimates on the normal value of LC in the brain, the estimates are always less than 1.0, representing that in normal condition the phosphorylation is the rate-limiting step in glucose uptake. In supply limited conditions (hypoglycemia and ischemia) LC increases as shown by Crane et al (1981 and 1983) and Hawkins et al (1981). In healthy subjects, LC decreases after oral glucose loading (Ishibashi et al., 2016). In traumatic brain injury, LC is reduced (Hattori et al., 2003; Hermanides et al., 2021).

Lumped constant is a function of the rate constants (Sokoloff et al., 1977). The variability of LC results primarily from changes in k₃/k₂ for glucose and for FDG (k₃*/k₂*) as is expressed in below (Phelps et al., 1983):

${LC} \propto \frac{1 + ({p}/{q}) \times \frac{k_3^*}{k_2^*}}{1 + \frac{k_3^*}{k_2^*}}$

Phelps et al (1983) used values p=0.50 and q=1.67 (q/p=3.34) to estimate the changes in LC between normal and ischemic brain regions. This equation was used by Sasaki et al (1986) to estimate whether LC is uniform over the whole brain.

Kuwabara and Gjedde (1990 and 1991) derived equation for LC which is independent of the model used:

${LC} = \varphi + (\tau - \varphi) \times \frac{K_i^*}{K_1^*}$

, where K_i* is the unidirectional clearance from the circulation to the metabolic compartment (net influx rate), τ is the ratio between FDG and glucose clearances (K₁*/K₁), and φ is the phosphorylation ratio between FDG and glucose (k₃*/k₃). Kuwabara and Gjedde (1991) used estimates τ=1.10 and φ=0.30. Hasselbalch et al (1999) used estimates τ=1.48 and φ=0.38, based on their previous studies (Hasselbach et al., 1996 and 1998). Before that, Crane et al (1983) have used estimates τ=1.67 and φ=0.55 in rat studies.

In irreversible 2-tissue compartment model K_i=K₁*k₃/(k₂+k₃), and thus (Gejl et al., 2012)

${LC} = \varphi + (\tau - \varphi) \times \frac{k_3^*}{k_2^{*} + k_3^{*}}$

Recommended values for using as LC

The recommended LC for brain [¹⁸F]FDG studies is 0.65, if irreversible uptake is assumed (3-parameter model or Patlak plot without k_LOSS), and LC=0.81, if dephosphorylation is considered (4-parameter model or Patlak plot with k_LOSS) (Wu et al. 2003). Graham et al (2002) obtained value LC=0.89±0.08 for normal brain, and 0.78±0.11 for cerebellum, using reversible two-tissue compartmental model, but, in comparison to other studies, suggested that 0.80 should be used.

In brain tumours the LC is so variable that no LC can be used, and FDG tumour studies should refer to the result as MR_FDG (Spence et al., 1998; Graham et al., 2002).

In rat brain studies, lumped constant of 0.71 is recommended (Tokugawa et al., 2007; Krohn et al., 2007).

Myocardium

Lumped constant for [¹⁸F]FDG in myocardium is dependent on the serum insulin concentration (Hariharan et al., 1995; Bøtker et al., 1997 and 1999; Ng et al 1998). LC was estimated to be 1.44±0.14 in fasting state and 0.99±0.07 during insulin infusion (Ng et al 1998). Values for τ and φ in myocardium in humans were determined to be 2.26 and 0.43, respectively (Bøtker et al., 1997), and these values have been used later by the same group (Wiggers et al., 1999; Bøtker et al., 2000; Gejl et al., 2002). In isolated working rat hearts values were 1.73 and 0.15, respectively (Bøtker et al., 1999). If LC is not determined from the dynamic FDG PET data, the recommended LC for Patlak and FUR analysis of heart FDG studies is 1.

Skeletal muscle

Peltoniemi et al (2000) measured the LC in skeletal muscle to be 1.16.

Adipose tissue

LC in human adipose tissue is 1.14 (Virtanen et al., 2001).

Liver

LC in the liver of pigs is close to one (Iozzo et al., 2007).

LC of [¹⁸F]FTHA

Lumped constant for [¹⁸F]FTHA represents the ratio of the probability that arterial tracer molecule, [¹⁸F]FTHA, will be activated to [¹⁸F]FTHA-CoA, to the probability that an arterial long-chain fatty acid (palmitate and other FFAs) molecule will undergo activation to fatty acyl-CoA. If the kinetics of the tracer and of the average native compound would be identical, LC would equal 1.

Because LC is difficult to measure in human subjects, it is often assumed to equal 1. This may cause bias in results and should be taken into account in interpretation of the results.

Alternatives to LC

Correction using Michaelis constant

Williams et al (2012a and 2012b) have proposed an alternative method to traditional measure of metabolic rate of glucose (MR_gluc). In this method, the metabolic rate is extrapolated to the hypothetical condition of glucose saturation:

${MR}_{glu}^{max} = K_i^* \times (K_M + C_{glu})$

This method has been shown in mice studies to provide results with smaller variation than the traditional method. Michaelis-Menten constant K_M=130 mg/dL was used in the mice studies.

Literature

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Hasselbalch SG, Holm S, Pedersen HS, Svarer C, Knudsen GM, Madsen PL, Paulson OB. The ¹⁸F-fluorodeoxyglucose lumped constant determined in human brain from extraction fractions of ¹⁸F-fluorodeoxyglucose and glucose. J Cereb Blood Flow metab. 2001; 21(8): 995-1002. doi: 10.1097/00004647-200108000-00012.

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Williams S-P, Flores-Mercado JE, Baudy AR, Port RE, Bengtsson T. The power of FDG PET to detect treatment effects is increased by glucose correction using a Michaelis constant. EJNMMI Res. 2012b; 2:35. doi: 10.1186/2191-219X-2-35.

Wu H-M, Bergsneider M, Glenn TC, Yeh E, Hovda DA, Phelps ME, Huang S-C. Measurement of the global lumped constant for 2-deoxy-2-[¹⁸F]fluoro-D-glucose in normal human brain using [¹⁵O]water and 2-deoxy-2-[¹⁸F]fluoro-D-glucose positron emission tomography imaging: a method with validation based on multiple methodologies. Mol Imaging Biol. 2003; 5: 32-41. doi: 10.1016/S1536-1632(02)00122-1.

Tags: FDG, Fatty acids, Lumped constant

Updated at: 2023-06-14
Created at: 2009-01-09
Written by: Vesa Oikonen