Reagents
Metformin was provided by Sigma-Aldrich (St. Louis, MO, USA). FDG was produced in-house, according to standard methodology. Daily quality controls always documented adequate standards and, in particular, a radiochemical purity ≥98%.
Animal models
All animal experiments were reviewed and approved by the Licensing and Ethical Committee of the IRCCS San Martino IST, Genova, Italy, and by the Italian Ministero della Salute. A total of 21 six-week-old BALB/c female mice (Charles River Laboratories, Italy) were housed under specific pathogen-free conditions and subdivided into three different groups according to the treatment preceding the imaging study. The first group of controls (CTR) included seven untreated animals that did not receive any treatment and were kept under standard conditions for the whole study duration. The short-term starvation group (STS) included seven mice that were submitted to food deprivation with free access to water within 48 h before imaging. Finally, the last group (MTF) included seven mice treated with metformin for the month preceding the imaging test. In these animals, the drug was administered diluted in autoclaved drinking water at a concentration of 3 mg/mL as to account for a dose of 750 mg/Kg per day [11].
Animal preparation and image processing
In vivo imaging was performed according to a protocol validated in our lab [11]. To ensure a steady state of substrate and hormones governing glucose metabolism, all animals were studied after 6 h of fasting. Mice were weighed, and anesthesia was induced by intraperitoneal administration of ketamine/xylazine (100 and 10 mg/Kg, respectively). Serum glucose level was measured, and animals were positioned on the bed of a dedicated micro-PET system (Albira, Bruker, USA) whose two-ring configuration permits to cover the whole animal body in a single bed position. A dose of 3 to 4 MBq of FDG was then injected through a tail vein, soon after the start of a list mode acquisition lasting 50 min. Acquisition was reconstructed using the following framing rate: 10 × 15 s, 5 × 30 s, 2 × 150 s, 6 × 300 s, and 1 × 600 s. PET data were reconstructed using a maximum likelihood expectation maximization (MLEM) method. Thereafter, each image dataset was reviewed by an experienced observer who recognized three regions of interest (ROIs) encompassing the aortic arc, gut, and liver, respectively (see Figure 2). The same figure also shows the TACs computed from the three ROIs in the case of animals belonging to the analyzed groups (CTR, STS, MTF). We are aware that the determination of the arterial IF is a challenging task in the case of mice. To accomplish it, for each animal model, we have first viewed the tracer first pass in cine mode. Then, in a frame where the left ventricle was particularly visible, we have drawn a ROI in the aortic arc and maintained it for all time points. Therefore, in this study, we used the resulting TAC as arterial IF. We recognize that this procedure may be affected by partial volume effects in the PET data; however, we expect this kind of error to be systematic for all datasets and therefore to affect the results in a way which is essentially independent of the experiment. Cine mode representation was used also to obtain the gut and liver TACs, since especially kidneys and bladder often display a biphasic curve due to tracer filtration-accumulation and possible voiding preventing their recognition in the last frame. For example, in the gut case, a ROI was drawn analyzing in cine mode all 27 frames of acquisition; the ROI volumes ranged between 15 and 30 μL and were placed systematically in the anterior abdomen, paying care to exclude contamination from the liver, spleen, kidney, and large vessels throughout all frames.
Compartmental modeling of FDG kinetics
Our approach for the definition of the dual IF was based on processing a complete set of ROIs drawn on the aortic arc, gut, and liver. In particular, we aimed to describe the gut by means of a compartmental model with one arterial IF and one output directly delivering the tracer in the PV blood. This latter concentration was then used as the venous IF in a second compartmental model focused on the liver. Since both the gut and liver express G-6P-phosphatase, they were described by means of two functional compartments that account for the significant exchange between trapped and free tracer, i.e., between FDG-6P and FDG. This compartmental system is described in Figure 1.
Gut subsystem
This subsystem was considered to have one arterial IF, one output function to the PV, and two compartments: the free tracer in the gut (denoted with g) and the trapped FDG-6P (denoted with t). The mathematical model at the basis of our compartmental analysis relied on the balance of the tracer activities (i.e., concentrations per unit volume) between the different compartments. This balance leads to a number of ordinary differential equations, which are a standard in this framework. In the specific case of the gut tracer kinetics, the compartmental model is made of the following two differential equations with vanishing initial data:
$$ {\dot{C}}_g=-\left({k}_{tg}+k{}_{pg}\right){C}_g+{k}_{gt}{C}_t+{k}_{ga}{C}_a $$
(1)
$$ {\dot{C}}_t={k}_{tg}{C}_g-{k}_{gt}{C}_t. $$
(2)
In these equations, C
g
, C
t
, and C
a
represent the tracer concentrations in the free compartment g, in the phosphorylated compartment t, and in the arterial blood a, respectively; the superposed dot indicates differentiation with respect to time, and the coefficients k
ij
(measured in min−1) denote the rate coefficients to the target compartment i from the source compartment j. Equations (1) and (2) can be formally solved to obtain the analytical expressions of C
g
and C
t
. We observe that such expressions depend on the tracer coefficients that at this stage of the solution process are still unknown.
In order to describe the output of the gut subsystem, we introduced a further compartment p, anatomically represented by the PV. Assuming that in such a compartment the blood flow is constant (which implies k
pg
= k
fp
), activity conservation leads to a third differential equation for the concentration C
p
:
$$ {\dot{C}}_p={k}_{pg}{C}_g-{k}_{pg}{C}_p. $$
(3)
Also, in the case of Equation (3), the analytical solution can be formally determined and depends on the kinetics parameter k
pg
.
Liver subsystem
According to standard results [17], also the liver can be described as a two-compartment model, one consisting of non-metabolized, free, tracer (compartment f), and one consisting of phosphorylated, metabolized tracer (compartment m), where, as in the case of the gut, dephosphorylation is explicitly allowed. Tracer inputs through the HA and PV (compartments a and p, respectively) are modeled in an independent manner, i.e., there is no mixing of blood from the two vessels before entrance into the liver. On the contrary, possible heterogeneities in perfusion source may occur as to consider the most general input model. Accordingly, exchange coefficients for the HA blood (k
fa
) and the PV blood (k
fp
) are considered as independent. On the other hand, FDG delivery to the PV blood is considered to occur only from the gut, tracer exchange throughout the vein is negligible, and thus, the whole tracer amount entered into the PV is available for liver uptake.
We now denote with C
f
and C
m
the concentration of the free and metabolized FDG pools, respectively, with k
sf
the rate coefficient from the free compartment to the venous efflux to the suprahepatic vein s, k
mf
the exchange coefficient from the FDG to the FDG-6P pool, and k
fm
the exchange coefficient for the inverse process. Then, the usual assumption on the conservation of activities provides:
$$ {\dot{C}}_f=-\left({k}_{mf}+{k}_{sf}\right){C}_f+{k}_{fm}C{}_m+{k}_{fa}{C}_a+{k}_{pg}{C}_p $$
(4)
and
$$ {\dot{C}}_m={k}_{mf}{C}_f-{k}_{fm}{C}_m. $$
(5)
Equations (4) and (5) can be again formally solved to obtain the analytical expressions of C
f
and C
m
. We observe again that such expressions depend on the tracer coefficients that at this stage of the solution process are still unknown.
Data optimization
Micro-PET data provide information on the overall concentrations in ROIs drawn on the gut and liver throughout the whole acquisition. Therefore, denoting with \( {\tilde{C}}_{\mathrm{gut}} \) and \( {\tilde{C}}_{\mathrm{liver}} \) such experimental concentrations, we can write the following two equations for the micro-PET data:
$$ {\tilde{C}}_{\mathrm{gut}}={C}_g+{C}_t $$
(6)
$$ {\tilde{C}}_{\mathrm{liver}}-V\left(0.11{C}_a+0.89{C}_p\right)=\left(1-V\right)\left({C}_f+{C}_m\right) $$
(7)
where the numerical coefficients 0.11 and 0.89 indicate the rate of arterial and venous contributions to the hepatic blood content V per unit volume [8,17]. Further, we assumed for V the physiologically sound value of 0.3 [9]. In principle, these values may change between the different groups; therefore, we made the same computation for different pairs of values (0.15 to 0.85, 0.25 to 0.75, 0.5 to 0.5, respectively). The mean values of the tracer coefficients did not change significantly while the corresponding uncertainties increased with respect to the choice 0.11 to 0.89.
In order to numerically solve Equations (6) and (7) and therefore to determine the tracer coefficients, we applied, separately and in cascade, a regularized multi-dimensional Newton algorithm [19], where a nice trade-off between the numerical stability of the problem solution and an appropriate fitting of the measured data were obtained by means of an optimized selection of the regularization parameter. To this aim, we first observed using simulations that the regularized Newton algorithm is rather robust with respect to the choice of the regularization parameter. In fact, in the case of a synthetic dataset, there exists a unique value of the regularization parameter that minimizes the distance between the reconstructed and ground-truth tracer coefficient vector. For all simulations performed, this value had an order of magnitude of around 104, and tuning such value in the range of 103 to 105 changed the reconstructed coefficients of less than 0.5%. In the case of experimental data, for each mouse, we applied a discrepancy approach: we chose as optimal value of the regularization parameter the value for which the discrepancy between the experimental data and the data predicted by the regularized solution coincided with the uncertainty on the measurement [20]. This uncertainty was computed by assuming that the noise on the activity is Poisson and the values of the regularization parameter we obtained were around 104, as in the simulated cases. We are aware that the Poisson assumption on the uncertainty does not account for correlation and other effects induced by the reconstruction algorithm; however, the robustness of regularization guarantees for the reliability of this approach.
From a computational viewpoint, the codes implementing the reduction of these compartmental models are extremely fast (less than 1 min for each analysis). Further, in order to assess the numerical robustness of the approach, for each animal model we have perturbed 50 times the PV TAC (obtained from the reduction of the gut subsystem) with 50 different Poisson random components, and accordingly, we have computed the tracer coefficients for the liver. For each coefficient, the standard deviation corresponding to the mean value over the several runs of the algorithm represents a reliable measure of the robustness of the approach with respect to uncertainties in the computation of the PV TAC. We found out that all standard deviations were below 4% of the mean values.
