A new framework for metabolic connectivity mapping using bolus [ 18 F]FDG PET and kinetic modeling

Abstract

Metabolic connectivity (MC) has been previously proposed as the covariation of static [¹⁸F]FDG PET images across participants, i.e., across-individual MC (ai-MC). In few cases, MC has been inferred from dynamic [¹⁸F]FDG signals, i.e., within-individual MC (wi-MC), as for resting-state fMRI functional connectivity (FC). The validity and interpretability of both approaches is an important open issue. Here we reassess this topic, aiming to 1) develop a novel wi-MC methodology; 2) compare ai-MC maps from standardized uptake value ratio (SUVR) vs. [¹⁸F]FDG kinetic parameters fully describing the tracer behavior (i.e., K_i, K₁, k₃); 3) assess MC interpretability in comparison to structural connectivity and FC. We developed a new approach based on Euclidean distance to calculate wi-MC from PET time-activity curves. The across-individual correlation of SUVR, K_i, K₁, k₃ produced different networks depending on the chosen [¹⁸F]FDG parameter (k₃ MC vs. SUVR MC, r = 0.44). We found that wi-MC and ai-MC matrices are dissimilar (maximum r = 0.37), and that the match with FC is higher for wi-MC (Dice similarity: 0.47–0.63) than for ai-MC (0.24–0.39). Our analyses demonstrate that calculating individual-level MC from dynamic PET is feasible and yields interpretable matrices that bear similarity to fMRI FC measures.

Keywords

dynamic PET Euclidean similarity individual-level metabolic connectivity kinetic modeling

Introduction

Positron emission tomography (PET) is one the earliest imaging modalities used for in vivo assessment of brain function, with very high specificity and increasingly high sensitivity.¹ In clinical practice, single-frame, static PET images are usually acquired, and the total amount of measured tracer activity is converted into a standardized uptake value ratio (SUVR). In experimental research settings, on the other hand, dynamic PET data are often obtained for full characterization of the tracer’s kinetic behavior through the lens of compartmental modeling.²

In the neuroimaging literature of the recent decades, we have witnessed the rise of the field of connectomics, which aims at characterizing the structural and functional connections between brain areas using magnetic resonance imaging (MRI)-based techniques such as diffusion MRI (dMRI) and blood oxygen level- dependent (BOLD) functional MRI (fMRI).³ This field has opened new scenarios for brain PET as well, and in particular for the [¹⁸F]fluorodeoxyglucose ([¹⁸F]FDG) tracer, a glucose analog, which has been increasingly employed to estimate the so-called ‘metabolic connectivity’ (MC), describing the relationships between the metabolic states of different brain regions.

In most studies, only a group-level MC map is obtained. Typically, an atlas with predefined regions of interest (ROIs) is used to extract ROI-wise [¹⁸F]FDG semiquantitative values at individual level,⁴ and then their pairwise correlation across individuals is calculated (labeled here as ‘across-individual MC’ (ai-MC)). PET tracers other than [¹⁸F]FDG have also been employed, with reports typically talking of ‘molecular connectivity’.⁵

The ai-MC approach differs significantly, both in the calculation and in the interpretation, from what is typically done to estimate structural connectivity (SC) from dMRI, or functional connectivity (FC) from fMRI,³ where connectivity matrices are first derived at individual level, and then averaged across individuals to obtain group-level connectivity maps. Importantly, an individual estimate is necessary if one aims to use MC as a biomarker.⁶

Only a handful of studies have attempted to use dynamic data to derive MC at individual level from PET time-activity curves (TACs), either in humans^6–8 or in animal models,⁹ an approach we label as within-individual MC (wi-MC). This is because handling PET time series comes with peculiar challenges as compared to fMRI time series or PET subject series, as we previously discussed:⁸ the strong collinearity amongst TACs, which share the same positive trend related to [¹⁸F]FDG’s irreversible uptake, makes it impossible to directly employ simple correlation analysis as done with fMRI FC. To overcome this, previous works have opted for TAC standardization or detrending, not only in bolus injection protocols, but also for continuous infusion, where MC is becoming popular due to the higher temporal resolution of reconstructed infusion data.^6,9

This approach, however, is problematic, especially with bolus injection studies, as it removes the main signal in PET TACs, and considers only the fluctuations around it, which may be related more to physical and statistical noise (measurement error²) than to biologically informative variability.

Thus, new solutions are highly warranted if wi-MC is to become useful and biologically interpretable.

Turning back to ai-MC approaches, we find that the information they have been based on so far is limiting, since all ai-MC studies use semiquantitative measures of [¹⁸F]FDG uptake (SUVR). Resorting to full kinetic modeling and a biologically validated compartmental description of [¹⁸F]FDG would provide important physiological information, such as the estimates of the tracer’s irreversible uptake (K_i), which can be scaled to obtain the cerebral metabolic rate of glucose, and the single rate constants, or microparameters, i.e., K₁ (tracer inflow, [ml/cm³/min]), k₂ (efflux [min⁻¹]), k₃ (phosphorylation, [min⁻¹]), which provide a kinetic description of the initial steps of glucose metabolism.^2,10

Using kinetic parameters instead of SUVR for ai-MC estimation may give rise to substantially different brain networks: SUVR is correlated but not identical to K_i,¹¹ and may be substantially biased in pathology, and the single rate constants K₁ and k₃ describe the distinct processes of glucose uptake by transporters at the blood-brain barrier, and the phosphorylation by the hexokinase enzyme in brain cells (i.e., the actual first step in glycolysis), respectively.

However, the wider use of these physiological rate constants has so far been hampered by the methodological difficulties associated with kinetic modeling (i.e., the need for long acquisitions and, ideally, arterial sampling to obtain an input function, the complexity of parameter estimation with noisy PET acquisitions etc.²) which only very recently have started to be overcome, even allowing for whole-body compartment modeling and microparameter estimation.¹²

The validity and interpretability of MC measures have been and still are objects of debate (for a recent position on this, see¹³) While validation of brain connectivity measures is a complex issue, a first simple readout can be provided by between-modality association: ai-MC maps, in particular, have been related both with SC¹⁴ and with FC.^15,16 From a biological standpoint, since SC represents the underlying axonal pathways between brain areas, we expect it to provide a “scaffold” for metabolic connections (as they should for FC¹⁷) and since FC is interpreted as representative of synaptic-level functional links between the activity of different neural populations,¹⁸ it also expected to be coupled with metabolic links, as glucose consumption happens mostly at the synapses.¹⁹

Nonetheless, the evidence on the coupling between these network estimates is still sparse, and sometimes contradictory (for ai-MC vs. FC, see ^15,16)

With these premises, we set out to provide a more comprehensive framework for [¹⁸F]FDG PET connectivity, using a large dataset of 54 healthy adults undergoing bolus dynamic PET studies.

First, to obtain wi-MC matrices, we compared different TAC standardization strategies and similarity metrics, and after choosing the best available approach, we proceeded to derive wi-MC matrices not only from full tissue TACs, but also from their early (0–10 min) and late part (40–60 min), to characterize MC networks more related to delivery (early part) or metabolism (late part) (Figure 1, bottom).

Figure 1.

Analysis pipeline for inferring metabolic connectivity at the individual (within-individual approach) and group level (across-individual approach). [¹⁸F]FDG PET dynamic data (far left) is the source of all inferences of metabolic connectivity. A static SUVR image (top left) is obtained from frames in the 40–60 min window of the dynamic PET data; in parallel, a two-tissue compartment model is fitted to dynamic PET data to estimate [¹⁸F]FDG kinetic parameters at the voxel level (using Variational Bayesian inference and an image-derived input function as a surrogate for the [¹⁸F]FDG plasma concentration C_p), in particular K_i, K₁ and k₃ (center), and reconstruct the time courses of compartments 1 and 2 (bottom center). From the subject series of SUVR, K_i, K₁ and k₃, parcellated thanks to the chosen atlas ROIs registered to individual PET space, we calculate across-individual MC via Pearson’s correlation (top right), while from the time series of the tissue TACs, compartments 1 and 2, individual-level MC is obtained via Euclidean similarity and averaged across participants (bottom right).

Then, we carried out [¹⁸F]FDG kinetic modeling, using Sokoloff’s two-tissue compartment model^2,10 and an image-derived input function (IDIF) calibrated with venous samples.²⁰ The ai-MC matrices were obtained from SUVR, K_i, K₁, and k₃, describing the physiology of glucose kinetics (Figure 1, top). The reconstructed time series of the first ( $C_{1} (t)$ , [kBq/cm³]) and second compartment ( $C_{2} (t)$ , [kBq/cm³]) of Sokoloff’s model, i.e., the time-varying intracellular concentrations of [¹⁸F]FDG and [¹⁸F]FDG-6P (representative of hexokinase activity), respectively, were also used for wi-MC estimation, as they would provide a way to separate the flow-related exchanges ( $C_{1} (t)$ ) from the actual metabolic activity ( $C_{2} (t)$ ): this echoes what is done with K₁ and k_3, but with the additional advantage of having a time course to use for calculating individual-level networks.

Finally, to assess the validity and interpretability of wi-MC and ai-MC matrices, we compared them at multiple levels, i.e., in terms of similarity in matrix structure and ‘hub’ nodes (i.e., most connected nodes in each network²¹). The match of wi-MC and ai-MC estimates with other connectivity measures was also explored, i.e., SC from an average template²² and FC from resting-state fMRI data of the same individuals. Of note, two different atlas parcellations^23,24 were used to assess the reproducibility of the findings.

Materials and methods

Participants

Fifty-four healthy adults (mean age 57.4 ± 14.8 years, 24 males) underwent [¹⁸F]FDG PET and MRI scans as part of the Adult Metabolism & Brain Resilience (AMBR) study.²⁵ This study was performed according to the principles outlines in the Declaration of Helsinki. All assessments and imaging procedures were approved by Human Research Protection Office and Radioactive Drug Research Committee at Washington University in St. Louis. Written consent was provided from each participant.

Participants were excluded if they had contraindications to MRI, history of mental illness, possible pregnancy, or medication use that could interfere with brain function.

Data acquisition

High-resolution structural MRI images were acquired on a Siemens Magnetom Prisma^fit scanner using a 3D sagittal T1-weighted magnetization-prepared 180° radio-frequency pulses and rapid gradient-echo (MPRAGE) multi-echo sequence (TE = 1.81, 3.6 , 5.39, 7.18 ms, TR = 2,500 ms, TI = 1,000 ms, 0.8 × 0.8 ×0.8-mm voxels). The final T1w image was obtained as the average of the first two echoes.²⁶

On the same scanner, T2* gradient-echo echo planar imaging (GE-EPI) data were acquired (TR/TE = 800/33 ms, flip angle 52°, voxel size 2.4 × 2.4 × 2.4 mm, MB 6, 375 volumes for total scan time of 5 min), together with two spin-echo (SE) acquisitions (TR/TE = 6000/60 ms, flip angle 90°) with opposite phase encoding directions (AP, PA).

[¹⁸F]FDG scans were performed on a Siemens model 962 ECAT EXACT HR + PET scanner (Siemens/CTI), as previously described,²⁷ after i.v. bolus injection of 5.1 ± 0.3 mCi (187.7 ± 12.1 MBq) of [¹⁸F]FDG. Dynamic acquisition of PET emission data continued for 60 min.

All PET images were acquired in the eyes-closed waking state. Participant head movements during scanning were restricted by a thermoplastic mask.

PET data were reconstructed via filtered back-projection (ramp filter, 5 mm FWHM) as 128x128x63 matrices. Attenuation correction was performed using the participant’s transmission scan. The chosen reconstruction grid consisted of 52 frames of increasing duration (24 × 5 s frames, 9 × 20 s frames, 10 × 1 min frames, and 9 × 5 min frames).

Venous samples for plasma glucose determination were obtained to verify that glucose levels were within normal range throughout the study. Also, venous samples were collected to assess [¹⁸F]FDG plasma concentration, with sampling occurring at 20, 30, 45 minutes after injection of the radiotracer (for a minority of participants, n = 10, samples were acquired after 30, 40 and 50 minutes), after arteriovenous equilibration. Each sample consisted of about 2 ml, half of which was used to measure radioactivity in plasma. Radioactivity counter measurements was given in counts per 12 seconds. The counter’s efficiency (0.2707 cps/Bq) was experimentally determined.²⁸

Structural MRI preprocessing

Structural T1w images were N4 bias field-corrected,²⁹ skull-stripped,³⁰ and segmented into grey matter (GM), white matter (WM) and cerebrospinal fluid (CSF) using SPM12.³¹

T1w images were normalized to the symmetric MNI152 2009c atlas³² via nonlinear diffeomorphic registration.³³

The Hammers anatomical atlas²³ and the Schaefer functional atlas (100 ROIs, 7 networks)²⁴ were registered to T1w space by inverting the obtained nonlinear transformation.

For the Hammers atlas, 74 ROIs (out of 83) were kept for further analysis, after removing white matter (WM) and cerebrospinal fluid (CSF)-only ROIs. For simpler visualization and interpretation, the regions were divided into 7 anatomical clusters, i.e., 1) frontal lobe, 2) temporal lobe, 3) parietal lobe, 4) occipital lobe, 5) insula and cingulate gyri, 6) subcortical structures, 7) cerebellum. The complete list of the regions (in the same order as in the manuscript’s figures) is reported in the Supplementary Materials.

For the Schaefer atlas, the 100 ROIs were supplemented by 12 subcortical ROIs taken from the Hammers atlas (bilateral caudate, accumbens, putamen, pallidum, thalamus, cerebellum).

Functional MRI preprocessing

The fMRI data were analyzed in a similar way to the Human Connectome Project minimal preprocessing pipeline.³⁴ The first four fMRI volumes were discarded to avoid non-equilibrium magnetization effects. The remaining volumes were corrected for slice timing differences³⁵ and magnetic field distortion,³⁶ and realigned to the median volume.³⁷ A template EPI volume³³ was obtained from realigned fMRI data and used to estimate an affine transform employed to map main T1w tissue segmentations to the EPI space. Motion parameters and their first order derivatives, along with the first 5 temporal principal components obtained after principal component analysis of WM and CSF EPI signals as in,³⁸ were regressed out from all brain voxels in native EPI space. Finally, the BOLD signal was high-pass filtered with a cut-off of 0.008 Hz.

Pre-processed rs-fMRI signals were obtained within each ROIs from the Hammers and Schaefer atlases, which had been linearly mapped from T1w to rs-fMRI space, by averaging over voxels within the T1w grey matter (GM) segmentation (probability >0.8 of belonging to GM).

PET analysis and kinetic modeling

Dynamic PET data were motion-corrected using FSL’s mcflirt.³⁹

A static PET image was obtained by summing late PET frames (40–60 min) after motion correction. The static PET image was linearly registered to T1w space using FSL’s flirt,³⁹ and normalized into SUVR by dividing each voxel’s value by the whole-brain [¹⁸F]FDG average uptake.⁴⁰

To perform PET kinetic modeling, an image-derived input function (IDIF) was extracted from dynamic PET data using a semi-automatic pipeline,⁴¹ including 1) carotid artery segmentation; 2) selection of “hot voxels” within the mask; 3) parametric clustering⁴² to derive the raw IDIF; 4) IDIF model fitting; 5) Chen’s spillover correction²⁰ using three late venous samples. More details are reported in⁴¹ and in the Supplementary Materials.

Voxel-wise estimation of Sokoloff’s model parameters was performed using a Variational Bayesian approach:⁴³ k-means clustering was applied to dynamic PET data, extracting 6 GM and 5 WM clusters (as from T1w tissue segmentations), then weighted nonlinear least squares was used to estimate Sokoloff’s model parameters at the region level, and finally voxel-wise estimation of the parameters was performed using Variational Bayesian inference based on prior distributions derived from cluster-wise estimates. We refer to⁴³ for more complete details on the estimation procedure.

Parametric maps (i.e., voxel-wise maps) of K₁ [ml/cm³/min] (tracer inflow), k₂ [min⁻¹] (efflux), k₃ [min⁻¹] (phosphorylation), V_b [%] (blood volume fraction) were obtained for each participant. The parametric map of K_i [ml/cm³/min] (irreversible tracer uptake) was obtained by the solving the following equation at voxel level:

K_{i} = \frac{K_{1} k_{3}}{k_{2} + k_{3}}

(1)

The group-average maps of SUVR, K_i, K₁, k₃ are shown in Supplementary Fig. 6.

We did not perform partial volume correction on the PET data: using the GM tissue segmentation and avoiding spatial smoothing of PET data during processing allowed to minimize partial volume effects (PVEs),⁴⁴ as also suggested in recent PET analysis works.^45,46 Moreover, there is no gold standard for partial volume correction, especially in dynamic PET studies, as this procedure is known to affect the accuracy of kinetic modeling and potentially alter spatial metabolic patterns.⁴⁷

The voxel-wise predictions of the time-varying free intracellular concentration of the [¹⁸F]FDG tracer ( $C_{1} (t)$ , [kBq/cm³]) and its metabolized (i.e., [¹⁸F]FDG 6-phosphate) intracellular concentration ( $C_{2} (t)$ , [kBq/cm³]), representative of hexokinase activity, were reconstructed from Laplace transform solutions of Sokoloff’s model¹⁰ by using the IDIF as plasma input function ( $C_{p} (t)$ [kBq/ml]):

C_{1} (t) = \frac{K_{1} k_{2}}{k_{2} + k_{3}} e^{- (k_{2} + k_{3}) t} \otimes C_{p} (t)

(2)

C_{2} (t) = K_{i} \int_{0}^{t} C_{p} (τ) d τ

(3)

Within-individual metabolic connectivity (wi-MC)

ROI-level PET signals ([¹⁸F]FDG tissue TACs, $C_{1} (t)$ and $C_{2} (t)$ ) were extracted from the Hammers and Schaefer regions, which had been linearly mapped from T1w to PET space, by averaging over voxels within the GM segmentation (probability > 0.8 of belonging to GM).

The five-second frames of the ROI-wise tissue TACs (120 s in total) were temporally filtered by averaging them in triplets. Denoising was not performed on the TACs of $C_{1} (t)$ and $C_{2} (t)$ , which are noise-free since they are derived from compartmental modeling. All the signals (tissue TACs, $C_{1} (t)$ and $C_{2} (t)$ ) were then interpolated on a uniform virtual grid (five-second step).

To calculate wi-MC, several methods for standardizing PET TACs and evaluating their relationship (Pearson’s correlation, Cosine similarity, Euclidean similarity) were tested and compared (Supplementary Methods, Supplementary Fig. 1) on the basis of a) their modularity, i.e., their capability of retrieving a ‘biological’ network structure having subnetworks along the main diagonal, and inter-hemispheric connections between homologous regions,²¹ b) the between-individual variability of the edges of the MC network.

Modularity of each wi-MC matrix was computed using Newman’s algorithm (gamma hyperparameter = 1).⁴⁸ The between-individual variability of each wi-MC was calculated at edge level as the coefficient of variation (CV%), i.e., the percent ratio of the median absolute deviation (MAD) of each link’s values across individuals divided by the median. An overall index of the between-individual variability was obtained by taking the median ± MAD of the CVs% for each matrix.

While Pearson’s correlation and Cosine similarity are well known, our measure of Euclidean similarity is based on the Euclidean distance $d_{x_{1}, x_{2}}$ between each pair of TACs $x_{i, 1}$ and $x_{i, 2}$ :

d_{x_{1}, x_{2}} = \sqrt{\sum_{i = 1}^{T} {(x_{i, 1} - x_{i, 2})}^{2}} with T = number of time points

(4)

From $d_{x_{1}, x_{2}}$ we derived a measure of Euclidean similarity, as 1 minus the normalized $d_{x_{1}, x_{2}}$ , i.e., divided by the maximum distance among pairs of TACs, and thus scaled to [0, 1]. Due to the markedly heavy-tailed (left-skewed) distribution of Euclidean similarity values, a Fisher z-transformation was applied, and then the values were again rescaled to [0, 1].

wi-MC matrices were calculated from:

the full tissue TACs (0–60 min, all 52 data points),

the early part of the tissue TAC (0–10 min, first 38 data points),

the late part of the tissue TAC (40–60 min, last 4 data points),

the full TACs of $C_{1} (t)$ ,

the full TACs of $C_{2} (t)$ ,

and then averaged across individuals into five group-level wi-MC matrices.

The comparisons between Euclidean, Pearson and Cosine similarity were performed on group-level MC matrices, and only for the full tissue TAC.

The association between each pair of wi-MC matrices was estimated using Pearson’s correlation, after the data were assessed for normality (Shapiro-Wilk test, p value <0.05), calculated between the upper triangular portions of each matrix, both without sparsification and after imposing a threshold (80^th percentile sparsity), as advocated in connectivity studies.⁴⁹ The sparse results are presented in the paper (full matrix results are in the Supplementary Materials), to emphasize the most reliable connections. The significance of the correlation coefficients was assessed using Mantel’s test,⁵⁰ which was tested for significance by 15,000 permutations. Additionally, p-values were Bonferroni corrected for 10 comparisons.

Across-individual metabolic connectivity (ai-MC)

The [¹⁸F]FDG parametric maps were parceled at individual level with the Hammers and Schaefer atlas. The region-wise SUVR, K_i, K₁, k₃ values were within-individual normalized via z-scoring, in accordance with.⁴ ai-MC matrices for SUVR, K_i, K₁, k₃ were computed with Pearson’s correlation (Figure 1, top).

The association between each pair of ai-MC matrices was estimated using Pearson’s correlation (upper triangular, with and without sparsification, with Mantel significance testing, and 6-fold Bonferroni correction), after the data were assessed for normality (Shapiro-Wilk test, p value <0.05). The sparse results are presented in the paper (full matrix results are in the Supplementary Materials), to emphasize the most reliable connections.⁴⁹

Comparing wi-MC vs. ai-MC: matrices, hubs and graph metrics

The information provided by the wi-MC vs. ai-MC approaches was compared via multiple strategies: at edge level by calculating Pearson’s correlation between the matrix elements, and at region level by comparing the hub nodes of each MC matrix.

For direct matrix-to-matrix comparison, Pearson’s correlation coefficients were calculated (upper triangular portions of MC matrices, with and without sparsification, with Mantel significance testing, and 20-fold Bonferroni corrections), after the data were assessed for normality (Shapiro-Wilk test, p value <0.05).

To identify hub nodes, all matrices were thresholded at the 80^th percentile, as previously advocated.⁴⁹ Then, hubs were identified on each matrix as the nodes belonging to the top 20% of the joint graph theory metrics degree (DEG) and eigenvector centrality (EC), thereby identifying nodes with high local and global connectivity.²¹

For comparison of hubs across matrices, the Dice similarity between pairs of binary hub/non-hub vectors of wi-MC and ai-MC was assessed.

In addition, the regional EC values from both wi-MC and ai-MC matrices were plotted against [¹⁸F]FDG parameters (SUVR, K_i, K₁, k₃) averaged across individuals (Supplementary Results).

Matching wi-MC and ai-MC with structural and functional connectivity

The SC template was derived using 178 tractographies made available in the BCBToolkit as Disconnectome 7 T.⁵¹ Individual tractographies were reconstructed using the HCP 7 T diffusion-weighted imaging dataset.²² Single-individual SC matrices were calculated using DSI studio (https://dsi-studio.labsolver.org/). The final template was derived by averaging the connectivity matrices across controls. For consistency with MC, the sparsity level of the matrix was set to 80%. The MC vs. SC analysis was reproduced on a different publicly available tractography atlas⁵² (Supplementary Fig. 12).

For each participant, the functional connectivity (FC) matrix was obtained by means of Pearson's correlations computed between fMRI time series of each pair of ROIs. Motion-corrupted volumes having frame-wise displacement higher than 0.4 mm were censored before FC computation.⁵³ FC matrices were then Fisher z-transformed and averaged across individuals to obtain the group-level FC. As with MC, the sparsity level was set at 80%.

To assess the agreement between the estimated metabolic connections and the SC and FC structure, the Dice similarity between binarized SC and each binarized wi-MC (group average) and ai-MC network was computed.

Results

Within-individual MC maps from PET time courses

We started our assessment by comparing wi-MC estimation methods (Euclidean similarity, Pearson’s correlation, Cosine similarity) and TAC standardization approaches (no standardization plus five different standardization methods). Of note, only the Euclidean vs. Pearson MC assessments are reported (Supplementary Fig. 1), as Cosine similarity and Pearson’s correlation give virtually identical results.

With regard to the modularity of these networks (here, Newman’s Q values), the best results were achieved for Normalizations 1 (0.42), 2 (0.49), 3 (0.47), for Pearson’s correlation, and non-normalized (0.38) and Normalization 3 (0.41) for Euclidean similarity.

When looking at the networks’ interindividual variability (CV%, median ± MAD of the entries of the upper triangular portion), values were acceptable for non-normalized (12.6 ± 2.7%), Normalization 1 (32.7 ± 5.4%), Normalization 4 (45.8 ± 4.4%) for Pearson’s correlation, and non-normalized (8.3 ± 28.9%) and all normalizations for Euclidean similarity (with the potential exception of Normalization 3 = 37.3 ± 68.9%).

Overall, this assessment tends to select Pearson’s correlation – Normalization 1, and Euclidean similarity – non-normalized, as having the best compromise between high modularity (characteristic of brain networks) and low interindividual variability. However, the use of Pearson’s correlation is limiting for a series of reasons (i.e., it requires to normalize TACs, to use a minimum number of time points for the correlation to be statistically meaningful, and to ideally have a uniform reconstruction grid).

Here, the Euclidean similarity method was the only one capable of retrieving structured MC matrices even without any signal normalization, without incurring in Pearson’s limitations: in particular, in all the matrices reported in Figure 2 (group-average wi-MC for the Hammers atlas, with 80% sparsity; Supplementary Fig. 2, without sparsity-inducing thresholds; Supplementary Fig. 3 for the Schaefer functional atlas), we clearly see both a) a block-diagonal structure along the main diagonal, and b) enhanced secondary diagonals, representing intra-hemispheric within-‘network’ connections and inter-hemispheric homotopic connections (i.e., between homologous regions) respectively. Therefore, we exclusively present wi-MC matrices as obtained by the Euclidean similarity approach.

Figure 2.

Matrix representation of within-individual metabolic connectivity. Matrix elements indicate Euclidean similarity of tracer kinetics from dynamic PET data of different regions. The Hammers parcellation is used to sample dynamic PET time series at the individual level. Following matrix computation for individual subjects, we averaged matrices for the group of 54 individuals. The average matrices for the full tissue TAC (a), its early part (b) and late part (c), the kinetics of $C_{1} (t)$ (d) and $C_{2} (t)$ (e) are displayed. For clarity and for conforming to conventions for identifying network hubs, matrices are visualized with 80% enforced sparsity. Homotopic inter-hemispheric connections are most evident in (a) and (c), i.e., matrices which are more related to metabolic events, while in (b), which highlights the early, flow-related information, there is little difference between inter-hemispheric and intra-hemispheric connections. Pearson’s correlation coefficients (Mantel’s test, all p < 10⁻⁷, Bonferroni corrected) between the upper triangular elements of the five wi-MC matrices are also reported (f), confirming the full tissue TACs associate most with the late part.

When visually evaluating the wi-MC matrices from the full TAC (Figure 2(a)), the late part (Figure 2(c)), and $C_{2} (t)$ (Figure 2(e)), areas of strong within-‘network’ relationships are located in the frontal and occipital cortex, but also in medial temporal lobe.

However, this network structure is clearly modified in the wi-MC from the early part of the TAC (Figure 2(b)), and $C_{1} (t)$ (Figure 2(d)): the occipital lobe loses ‘connectivity’, and the temporal and parietal areas become highly connected both within and between ‘network’. Moreover, the early TAC matrix seems to show a marked decrease of homotopy.

Subcortical structures tend to always display lower wi-MC than cortical areas (see Supplementary Fig. 4).

The between-individual variability of the obtained wi-MC matrices (Supplementary Fig. 5) is overall low, being lowest for the full TAC (CVs% median ± MAD of the entries of the upper triangular portion: 8.3% ± 28.9%), and highest for $C_{2} (t)$ (CVs% median ± MAD of the entries of the upper triangular portion: 47.7% ± 9.4%).

The Pearson’s correlation between the average wi-MC matrices shows that the full TAC wi-MC has strong correlations with the early part, $C_{2} (t)$ and especially the late part of the TAC, as it is expected: the tails of the TACs have higher amplitudes, and thus have the strongest contribution to the between-region similarity in Euclidean sense. Notably, the $C_{1} (t)$ MC has instead weak relationships with the other wi-MC matrices, except for a moderate correlation (r = 0.55, Mantel’s test, p < 10⁻⁷, Bonferroni corrected) with the early TAC (Figure 2(f)).

In summary, by using PET TACs and the Euclidean similarity metric, it is possible to obtain individual-level MC matrices showing both within-‘network’ and homotopic connections, characterized by a low between-individual variability, and with the additional possibility of highlighting different physiological events (by considering the full kinetics, a portion of it, or the kinetics of the model compartments). Importantly, these features are present for both Hammers and Schaefer atlas.

Across-individual MC maps: SUVR and kinetic parameters

The ai-MC matrices are displayed in Figure 3 (full matrix version in Supplementary Fig. 7; Schaefer version in Supplementary Fig. 8).

Figure 3.

Matrix representation of across-individual metabolic connectivity. Matrix elements indicate Pearson’s correlation coefficients between [¹⁸F]FDG parameter values of different regions across our group of 54 individuals. The Hammers parcellation is used to sample individual-level [¹⁸F]FDG parametric maps. The across-individual MC matrices for SUVR (a), K_i (b), K₁ (c) and k₃ (d) are reported. For clarity and for conforming to conventions for identifying network hubs, matrices are visualized with 80% enforced sparsity. The Pearson’s correlation coefficients (Mantel’s test, all p < 10⁻⁷, Bonferroni corrected) between the upper triangular elements of the four ai-MC matrices are also reported (e), highlighting how k₃ MC has the lower similarity to traditionally calculated SUVR MC.

The ai-MC of [¹⁸F]FDG kinetic model parameters (K_i, K₁, k₃) is presented here for the first time (see also k₂ and V_b ai-MC matrices in Supplementary Fig. 9) while ai-MC based on SUVR is the most frequently used approach in literature.

Some similarities are shared between the different parameters, especially between MC of SUVR and K_i (r = 0.72, Mantel’s test, p < 10⁻⁷, Bonferroni corrected), with strong within-‘network’ connections in temporo-limbic areas. The k₃-based ai-MC is instead quite different from the others, with enhanced connections in frontal areas and subcortical structures, and is in fact the least correlated with the others, especially with SUVR MC (r = 0.44, Mantel’s test, p < 10⁻⁷, Bonferroni corrected).

Comparing within-individual vs. across-individual MC: similarity of matrices and hubs, match with [¹⁸F]FDG parameters

When the wi-MC and ai-MC networks were related via Pearson’s correlation (Table 1), some significant relationships are found, especially between wi-MC matrices and K₁ and k₃ ai-MC. However, the correlation values are generally low, with a maximum r = 0.37; even lower values are found for the Schaefer atlas (maximum r = 0.28). SUVR ai-MC seems to carry no meaningful relationships with wi-MC approaches.

Table 1.

Within-individual vs. across-individual MC correlations.

Pearson’s r	SUVR ai-MC	K_i ai-MC	K₁ ai-MC	k₃ ai-MC
Full TAC wi-MC	0.17*	0.2*	0.29*	0.29*
First 10’ wi-MC	0.06	0.11*	0.26*	0.25*
Last 20’ wi-MC	0.16*	0.17*	0.26*	0.25*
$C_{1} (t)$ wi-MC	0.18*	0.22*	0.37*	0.33*
$C_{2} (t)$ wi-MC	0.2*	0.23*	0.29*	0.33*

Across-edge Pearson’s correlations between within-individual (group-average, rows) and across-individual (columns) MC matrices (upper triangle, 80th percentile threshold). Significant correlations (Mantel’s test, p < 0.05, Bonferroni corrected) are reported as *.

We then identified ‘hub’ nodes, i.e., highly connected nodes in each network, as is typically done in the field of connectomics.

wi-MC hubs are mainly located in frontal and temporal areas, with the exception of $C_{1} (t)$ with more parietal involvement (Figure 4(a)).

Figure 4.

Comparison of wi-MC (a) vs. ai-MC (b) ‘hubs’, identified as the top DEG and EC nodes for each matrix. Hub nodes are shown on the Hammers atlas regions in red (a) and blue (b) respectively. A matrix of Dice similarity values between wi-MC and ai-MC hubs is shown in the central panel. Early-time and late-time dynamic PET MC have the highest association with across-individual covariation of k₃.

As to ai-MC hubs, while SUVR, K_i and K₁ have a similar hub distribution, mainly in temporal, insular and cingulate cortices, k₃ hubs fall in frontal and subcortical areas (Figure 4(b)).

When we look at the Dice similarity between hub vectors of wi-MC vs. ai-MC matrices, again we find a lack of match between SUVR MC and wi-MC hubs, with higher overlap in the case of K₁ and especially k₃.

We also related the regional EC values from both wi-MC and ai-MC matrices to the average [¹⁸F]FDG SUVR, K_i, K₁, k₃ (Supplementary Results).

Matching within-individual and across-individual MC with structural and functional connectivity

Finally, we assessed the similarity between the wi-MC and ai-MC matrices and a) a SC template (Supplementary Fig. 11 A), b) the group-average FC from the same individuals (Supplementary Fig. 11B), to understand if a) structural or b) BOLD-based functional ‘connections’ could support the identified metabolic networks.

When looking at SC (Figure 5(a)), Dice similarity coefficients (DSC) tend to be higher for wi-MC matrices, especially for the early part of the TAC (DSC = 0.42), full TAC (DSC = 0.40) and $C_{1} (t)$ (DSC = 0.40). Amongst the ai-MC matrices, k₃ has the highest similarity (DSC = 0.40), while SUVR the lowest (DSC = 0.31). Notably, the results are highly similar with another SC template (Supplementary Fig. 12).

Figure 5.

Dice similarity (stem plots) of the wi-MC (red) and ai-MC (blue) binarized matrices (80th percentile) with the SC template (a) and group-average FC matrix (b).

When we look at the match with FC, the wi-MC matrices have even higher Dice (early TAC: DSC = 0.63, $C_{1} (t)$ : DSC = 0.54, $C_{2} (t)$ : DSC = 0.55), while ai-MC maintain lower values (k₃: DSC = 0.39, SUVR: DSC = 0.24) (Figure 5(b)). This result is reproduced also with the Schaefer atlas (full TAC: DSC = 0.44; early TAC: DSC = 0.38, $C_{1} (t)$ : DSC = 0.42, $C_{2} (t)$ : DSC = 0.40; SUVR: DSC = 0.35; k₃: DSC = 0.33).

Discussion

In this work, we reassessed the concept of metabolic connectivity from a PET kinetic modeling perspective, trying to capitalize on the multifaceted physiological information provided by dynamic [¹⁸F]FDG data.

The first issue we wanted to tackle was to select a good approach to estimate individual-level MC (i.e., wi-MC).

The methods used in the fMRI literature to assess individual-level FC conventionally rely on variance-based methods (correlation/covariance), designed to identify signals that vary together over time.³ These approaches are also borrowed in the few studies which attempt to estimate individual-level MC from dynamic PET.^6,7,9 However, variance-based methods are problematic with dynamic PET data: while without any signal standardization no clear network structure is evident, the use of standardization produces different networks depending on the way the global signal is removed (Supplementary Fig. 2). Importantly, signal fluctuations, which are at the core of resting-state fMRI measurements, are of unclear physiological meaning in the case of dynamic PET, even though in previous work^6,7,9 structured networks have been extracted from these fluctuations (reconstructing data with a uniform time grid which does not truly respect PET count statistics).

While it is remains unclear whether fluctuations contain relevant biological information or are more affected by non-physiological noise, the amplitude and shape of PET time-activity curves are known to be biologically informative, which is why they are used for kinetic modeling.²

With respect to this, our method of choice (Euclidean similarity) identifies signals that are close in a Euclidean distance sense. When used on voxel-wise dynamic data, Euclidean distance already proved to effectively identify biologically meaningful clusters.⁵⁴ Here, we have repurposed this metric from a hard cluster assignment to a continuous space of pharmacokinetic similarities across brain regions, which, interestingly, display the hallmarks of brain connectivity studies, i.e., 1) a small-world network structure with 2) strong homotopy.³

Moreover, the between-individual variability of the Euclidean similarity-based wi-MC matrices is low, which highlights the robustness of the chosen approach. This is the reason why we feel confident with looking at the group-average matrices, without risk of overlooking strong individual peculiarities as in fMRI FC.⁵⁵

We have proposed several types of MC, both for the within-individual and for the across-individual approaches.

To try and disentangle the multifaceted physiological information of dynamic PET data we have, for the first time, derived five different wi-MC matrices: the MC networks based on the early TAC and the time course of the free intracellular tracer concentration, $C_{1} (t)$ , are expected to be more related to inflow and delivery, while the late TAC and the metabolized intracellular tracer concentration, $C_{2} (t)$ , are more directly associated with metabolic exchanges. This overcomes the limitation of using only the raw tissue TAC, which combines the delivery and the actual metabolism.⁴ We find good overlap between the late TAC and $C_{2} (t)$ , while the $C_{1} (t)$ wi-MC is the least similar to the other matrices, implying that kinetic modeling might provide non-redundant information for individual-level MC calculation.

Notably, this framework could easily be extended to other PET tracers, separating delivery, specific and non-specific binding to estimate individual-level connectivity for receptor systems.⁴

We have also provided a new perspective on the dominant across-individual MC approach,⁴ always based on SUVR, due to it being the easiest parameter to obtain. In the case of other PET tracers, kinetic model parameters (e.g., V_T, BP_ND) have already been employed for ai-MC analyses,^4,56 but this is a new attempt for [¹⁸F]FDG. As we have shown, relevant differences emerge when the chosen parameter is not SUVR, especially with k₃. SUVR and K_i are typically considered to be the most important (and convenient) [¹⁸F]FDG parameters to summarize [¹⁸F]FDG metabolism, but this adds to the evidence that the microparameters (e.g., K₁ and k₃) might bear important meaning, both in physiology⁵⁷ and pathology.^12,58

Our final aim was to enhance the interpretability of MC measures by relating the within- vs. across-individual MC frameworks to one another and to other connectivity measures, in a first rudimentary form of validation. With the SUVR ai-MC being the most employed approach, it is highly relevant to understand whether the across-individual estimates also reflect individual-level information.

When we assessed the match between the ai-MC and wi-MC, weak-to-moderate correlations are found for K₁- and k₃-based MC vs. wi-MC matrices, consistently with microparameters being more sensitive to the physiological processes probed by wi-MC, as both are a function of the same dynamic PET signal. The correlation with SUVR MC is instead low. Also, wi-MC hubs are more concentrated in frontotemporal areas, while ai-MC hubs of SUVR, K_i and K₁ in temporal and limbic cortex. Notably, wi-MC hubs from infusion protocols were also in frontotemporal regions,⁶ unlike SUVR MC hubs.^6,59

Finally, we compared MC to typical measures of brain connectivity, i.e., SC and FC. There is not a high overlap between MC and SC (Dice similarity ∼ 0.3–0.4). However, we found a higher match for wi-MC, and SUVR ai-MC (previously related to SC in ¹⁴) is the one with the least similarity with the underlying structural network. Importantly, while we did not have individual diffusion MRI acquisitions for our participants, these results are consistent across different SC templates. To some extent, we can expect structural and functional/metabolic connections to have only a partial overlap, as already shown for SC vs. FC,¹⁷ since structural connectivity highlights only direct connections (plus technical issues which may cause SC estimates to miss some tracts). A more fine-grained investigation, focusing on short- vs. long-range or within- vs. between-network connections, might bear additional insight.¹⁴

The match between FC and wi-MC is good (Dice ∼ 0.5) with both atlases, and higher than the FC vs. ai-MC match and the wi-MC vs. ai-MC match itself (Table 1). This is in line with bolus and infusion findings.^6,9 While further investigation of the FC-MC coupling at a finer scale is required, our findings show how individual-level MC captures more of the BOLD-based functional network information than its across-individual counterpart. It is intriguing to wonder which reasons might be behind the correspondence between pharmacokinetic similarity among brain regions and FC, but it seems the metabolic properties underlying “hubness” in the MC network might inherently be linked to a region’s capacity to connect with other regions, both locally and globally. This also indicates that the coupling between glucose metabolism and fMRI FC, which we found to be somewhat limited when considering only local metabolic measures like SUVR,⁶⁰ may become stronger when both PET and rs-fMRI are framed into a large-scale connectivity environment.

Summing up, can within-individual MC be seen as a preferable approach? Overall, our comparison between wi-MC and ai-MC shows that individual-level estimates provide non-redundant networks which cannot be inferred from ai-MC, in agreement with what was found for ai-MC vs. infusion wi-MC.⁶ Moreover wi-MC seems to be more interpretable in terms of match with fMRI FC. Together, these properties suggest a potential for individual PET connectivity measures to be used as a biomarker, which warrants further exploration.

There are limitations to this work. Absolute quantification of Sokoloff’s model parameters requires an input function ideally obtained via arterial sampling.² Bias in the estimates might have been introduced by the IDIF approach, especially since the data come from a low-spatial-resolution HR+ scanner, but we believe our multi-step IDIF extraction algorithm with Chen’s correction²⁰ (the reference standard in IDIF extraction with older scanners, and the only approach with potential for calculating microparameters⁶¹) allows to retrieve sufficiently accurate inputs. Moreover, for MC estimates, it is not the absolute parameter value that is of interest, but their relative spatial distribution, which is likely to be preserved. Also, our Bayesian estimation framework minimizes noise.⁴³ Future work should reassess these findings using PET data with higher spatiotemporal resolution.⁵⁸

Our approach does not solve the inherent problem of functional network analyses: connectomes may have plausible structure, but their biological underpinnings and interpretation remain more elusive since a “ground truth” is rarely available. The attempts at validation presented here (i.e., relating MC to other connectivity measures) provide only a partial understanding of the underlying mechanisms. Further efforts are highly warranted, possibly using interventional approaches in animal models to elucidate causative links.

In conclusion, after developing a new approach to estimate individual-level wi-MC using considerations of tracer kinetics, and estimating across-individual ai-MC from [¹⁸F]FDG kinetic parameters, we thoroughly assessed the relationships between wi-MC and ai-MC at multiple levels, i.e., in terms of matrix and hub similarity, and their match with FC and SC matrices. We found the two MC frameworks to provide different and somewhat complementary information, with wi-MC having higher similarity with the FC network. Future work should consider applying wi-MC approaches to clinical populations, to verify if they can provide useful biomarkers.

Supplemental Material

sj-pdf-1-jcb-10.1177_0271678X231184365 - Supplemental material for A new framework for metabolic connectivity mapping using bolus [¹⁸F]FDG PET and kinetic modeling

Supplemental material, sj-pdf-1-jcb-10.1177_0271678X231184365 for A new framework for metabolic connectivity mapping using bolus [¹⁸F]FDG PET and kinetic modeling by Tommaso Volpi, Giulia Vallini, Erica Silvestri, Mattia De Francisci, Tony Durbin, Maurizio Corbetta, John J Lee, Andrei G Vlassenko, Manu S Goyal and Alessandra Bertoldo in Journal of Cerebral Blood Flow & Metabolism

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Funding for this research was provided by the McDonnell Center for Systems Neuroscience and the NIH/NIA R01AG053503 (AGV) and R01AG057536 (AGV, MSG). Some of the MRI sequences used were obtained from the Massachusetts General Hospital. MC was supported by Fondazione Cassa di Risparmio di Padova e Rovigo (CARIPARO) – Ricerca Scientifica di Eccellenza 2018 – (Grant Agreement number 55403).

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

TV and AB designed the research. AGV and TD collected the data. TV, GV, ES, and MDF analyzed the data. TV, ES, MC, JJL, AGV, MSG and AB interpreted the results. TV wrote the manuscript. All authors revised the manuscript.

Supplementary material

Supplemental material for this article is available online.

ORCID iDs

Tommaso Volpi

John J Lee

Manu S Goyal

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Supplementary Material

Please find the following supplemental material available below.

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A new framework for metabolic connectivity mapping using bolus [ 18 F]FDG PET and kinetic modeling

Abstract

Keywords

Introduction

Materials and methods

Participants

Data acquisition

Structural MRI preprocessing

Functional MRI preprocessing

PET analysis and kinetic modeling

Within-individual metabolic connectivity (wi-MC)

Across-individual metabolic connectivity (ai-MC)

Comparing wi-MC vs. ai-MC: matrices, hubs and graph metrics

Matching wi-MC and ai-MC with structural and functional connectivity

Results

Within-individual MC maps from PET time courses

Across-individual MC maps: SUVR and kinetic parameters

Comparing within-individual vs. across-individual MC: similarity of matrices and hubs, match with [18F]FDG parameters

Matching within-individual and across-individual MC with structural and functional connectivity

Discussion

Supplemental Material

sj-pdf-1-jcb-10.1177_0271678X231184365 - Supplemental material for A new framework for metabolic connectivity mapping using bolus [18F]FDG PET and kinetic modeling

Footnotes

Funding

Declaration of conflicting interests

Authors’ contributions

Supplementary material

ORCID iDs

References

Supplementary Material

Comparing within-individual vs. across-individual MC: similarity of matrices and hubs, match with [¹⁸F]FDG parameters

sj-pdf-1-jcb-10.1177_0271678X231184365 - Supplemental material for A new framework for metabolic connectivity mapping using bolus [¹⁸F]FDG PET and kinetic modeling