Tracking the Progression and Influence of Beta-Amyloid Plaques Using Percolation Centrality and Collective Influence Algorithm: A Study Using PET Images

Abstract

Background

The study of brain networks, particularly the spread of disease, is made easier thanks to the network theory. The aberrant accumulation of beta-amyloid plaques and tau protein tangles in Alzheimer’s disease causes disruption in brain networks. The evaluation scores, such as the mini-mental state examination (MMSE) and neuropsychiatric inventory questionnaire, which provide a clinical diagnosis, are affected by this build-up.

Purpose

The percolation of beta-amyloid/tau tangles and their impact on cognitive tests are still unspecified.

Methods

Percolation centrality could be used to investigate beta-amyloid migration as a characteristic of positron emission tomography (PET)-image-based networks. The PET-image-based network was built utilizing a public database containing 551 scans published by the Alzheimer’s Disease Neuroimaging Initiative. Each image in the Julich atlas has 121 zones of interest, which are network nodes. Furthermore, the influential nodes for each scan are computed using the collective influence algorithm.

Results

For five nodal metrics, analysis of variance (ANOVA; P < .05) reveals the region of interest (ROI) in gray matter (GM) Broca’s area for Pittsburgh compound B (PiB) tracer type. The GM hippocampus area is significant for three nodal metrics in the case of florbetapir (AV45). Pairwise variance analysis of the clinical groups reveals five to twelve statistically significant ROIs for AV45 and PiB, respectively, that can distinguish between pairs of clinical situations. Based on multivariate linear regression, the MMSE is a trustworthy evaluation tool.

Conclusion

Percolation values suggest that around 50 of the memory, visual-spatial skills, and language ROIs are critical to the percolation of beta-amyloids within the brain network when compared to the other extensively used nodal metrics. The anatomical areas rank higher with the advancement of the disease, according to the collective influence algorithm.

Keywords

Brain imaging PET Neurodegenerative disorders Alzheimer’s disease Cognitive assessments Graph theory Percolation centrality Collective influence

Introduction

Globally, Alzheimer’s disease (AD) predominantly stands out as a neurodegenerative disease; it is estimated to cost about two trillion USD by 2030,¹ affecting 75 million individuals by the same year. The indirect costs are about 244 billion USD.² This includes early-onset AD and the old-age (late-onset AD) human population; with no sight of a cure for AD and with increasing cases, early diagnosis and active management are the keys to tackling this disease for now. Predicting the disease’s progression with high accuracy helps improve prognosis, thereby bringing the disease’s management to an affordable cost range.

Noninvasive and invasive investigative procedures range from positron emission tomography (PET) scans or cerebrospinal fluid (CSF) analyses to fundamental psychometric evaluations such as the mini-mental state examination (MMSE) and neuropsychiatric inventory questionnaire (NPIQ), which are pen and paper-based; each has advantages and disadvantages. A universal necessity is to precisely detect whether dementia is caused by AD, followed by the need to show the severity of the condition.

For cognitive and behavioral changes, current approaches include family history and psychiatric history, followed by psychometric evaluations such as the MMSE,³ frontal assessment battery,⁴ and NPIQ.⁵

In addition, depending on the patient’s availability and financial situation, approaches such as genetic testing⁶ for signs of AD, the apolipoprotein-e4 (APOE-e4),⁷ or the use of blood testing/brain imaging to rule out dementia caused by other reasons are used. The use of PET imaging⁸ and lumbar puncture⁹ to evaluate beta-amyloid levels (abnormal levels indicate disease severity in either approach) is the current standard of practice for determining dementia because of AD.^{10, 11}

PET imaging uses radiopharmaceuticals, commonly the 2-[18F], florbetapir-fluorine-18 (AV45), or 11C-Pittsburgh compound B (PiB). AV45 and PiB¹² differ in image construction mechanisms. Both AV45 and PiB bind to beta-amyloid but vary in their half-life. AV45 has a half-life of 109.75 min and PiB has 20 min.¹³ A comparison between PiB and AV45 varies because AV45 shows uptake within the white matter (WM) regions.¹⁴

The use of network theory to describe the different brain connectivity of anatomical neural networks^{15, 16} under various psychological and neurological illness states has been found to be effective. Topological (graphs) analysis of neuroimaging data in terms of functional (electroencephalography, magnetoencephalography, and functional magnetic resonance imaging) and structural analysis (MRI and PET scans) is beneficial for highlighting the variance between a cognitively normal (CN) population and other diagnostic states utilizing various graph-theoretic metrics.^{17, 18}

Common network metrics applied to brain networks of AD patients and controls include characteristic path length, clustering coefficient, modularity, and hubs. Several studies have shown that the progression of mild cognitive impairment (MCI) to dementia owing to AD may be tracked.^{19, 20} To that end, network analysis and graph metrics such as centrality measures have the potential to aid in the understanding of brain networks. The dynamic connection of the Alzheimer’s brain network can be described via network analysis. Connectedness research employing functional magnetic resonance imaging and electroencephalography data yields puzzling indications, particularly when comparing AD patients to the control group,²⁰ that the network’s connectivity is increasing or decreasing. Although a decrease in connectivity could explain network disruption/cortical atrophy, an increase in connectivity could explain the compensatory mechanism.²¹

The majority of learning models or tracers that focus on metabolic networks and their associated variations are used in network analysis on PET images related to AD.^{22, 23} Other ways involve using algorithms to find patterns in raw PET scans to distinguish healthy controls from neurodegenerative patients.²⁴ The most modern innovations include genetic and protein markers to improve illness prediction accuracy.²⁵ Such solutions rely on a multitude of distinct data points as well as equally trustworthy processing technology; standardization of the protocols is currently a hurdle. As previously said, diagnosing AD is a global concern, and finding a method that works in a variety of countries, from wealthy countries like the United States to rural clinics in Southeast Asia or Africa,² is critical. Principle component analysis, for example, has a few downsides; for example, selecting the proper number of principal components and data normalization for numerous PET scans of patients with various tracers requires regulating many factors. Regression analysis is founded on the idea that cause and effect are inextricably linked. Furthermore, a relationship found in a small data set may be overturned in a larger data collection.

To understand beta-amyloid dispersion, we propose applying graph theoretic methods on PET images to understand beta-amyloid movement. The main benefits are as follows:

This does not introduce any new steps for data collection from the patient, and

Adds value to the existing data by computing centrality values of a given node at a given time.

The evolution of the illness network can be deduced from the network topology. Understanding the weak linkages within the atlas-based region of interest (ROI) networks can be gained by studying such evolution. The source and sink of beta-amyloid plaque diffusion could be determined using structural connectivity data.

Percolation centrality is defined as the proportion of “percolated paths” that pass through that node; this measure quantifies the relative impact of nodes based on their topological connectivity and percolated states. Or simply put, it is one such graph metric that looks at how a given node within a network has percolated information or can percolate information (Figure 4). The bulk of information transmitted via a given node is provided by values ranging from 0.0 to 1.0.^{26, 27} Prior exploration of percolation centrality on disease networks^27–30 and percolation centrality in disease networks of the brain³¹ is indicative of a promising metric for brain network investigation.

Percolation centrality’s applicability to human PET-image-based networks is still a work in progress. The goal of this project is to fill in the information gaps. In addition, based on the optimal percolation theory, collective impact gives a minimum set of nodes or ROIs that can easily facilitate the development of the disease with optimal spreading³² (Figure 4). The ROIs within the brain that optimally transfer beta-amyloid plaques are found by scanning the network for the smallest number of nodes, which could be affecting the regular functioning of the existing neural networks.

As a result, the capacity to detect the disease and estimate its progression rate at an early stage is critical. The study’s goal is to answer two primary issues in this regard: (a) Is it possible to utilize a percolation centrality measure to determine beta-amyloid percolation in the brain? (b) Can the collective influence (CI) algorithm offer the AD network a minimum set of critical nodes?

Methods

Patient Distribution

The data set is separated into two tracers based on the tracer agents utilized to acquire the PET images: AV45³³ and PiB.¹³ Based on the ADNI study’s psychometric measures, the patients are classified as CN, MCI, or AD based on their clinical state. Next, the PET image is matched with the patient’s diagnostic state at the imaging procedure. This provided a set of observations for each type of tracer for each patient condition clinical group (Table 1). Finally, the resulting patients are matched with the demographic information providing 531 patients.

Table 1.

Patient Distribution.

Cognitively Normal		Mild Cognitive Impairment		Alzheimer’s Disease
AV45	PiB	AV45	PiB	AV45	PiB
262	13	76	65	116	19
Male (M)–122	M–8	M–54	M–45	M–67	M–11
Female (F)–140	F–5	F–22	F–20	F–49	F–8
Total–275; M–130, F–145		Total–141; M–99, F–42		Total–135; M–78, F–57

Abbreviations: AV45, florbetapir; PiB, Pittsburgh compound B.

Network Construction and Processing

PET Image Preprocessing

The image preprocessing procedure is divided into two stages (Figure 1):

Creating a 4D raw activity image by combining individual frames of the PET image. FSL’s fslmerge tool is used to do this.³⁴

Converting a 4D raw activity image to a 4D SUV image by the following formula:

S U V = \frac{C_{i m g}}{C_{i n j}}

(1)

where $C_{i m g}$ (Mbq ml⁻¹) is given by the raw activity image and $C_{i n j} = \frac{I D}{B W}$ .

ID (MB_q) is the injection dose³⁵ and BW (g) is the bodyweight of the patient, considering the equivalency of 1 g = 1 mL.

Correcting for motion by realigning the PET frames spatially. MCFLIRT was used to do this.³⁶ Six degrees of freedom are used to adjust motion. Using the mean image as a template, the PET frames are realigned. The mean image is created by averaging the volumes after applying the motion correction parameters to the time series.

The 4D SUV picture was coregistered from subject space to MNI³⁷ space. This is done with the help of FreeSurfer.³⁸ The image MNI152 T1 2-mm-brain was used for coregistration.

Figure 1.

PET Image Preprocessing Flowchart.

Parallelization of this process is carried out with the help of GNU Parallel.³⁹

PET-Image-Based Network Construction

The Julich Atlas^40–42 is used to build the network. There are 121 ROIs in this atlas, which corresponds to 121 network nodes or vertices (Figure 3). The creation of adjacency matrices is required for building networks from preprocessed pictures. The method described below is used to calculate the adjacency matrix. In "confounding" or "chain" interactions, the Bivariate Pearson correlation performs badly. In these situations, partial correlation estimates the direct link between two nodes after regressing out influences from all other nodes in the network, eliminating false effects in network modeling. A partial correlation may create a misleading correlation in circumstances of "colliding" interactions. As a result, Sanchez-Romero and Cole developed a strategy for combining functional connections.⁴³

The time-averaged voxel intensities in the network are used to represent each brain volume. To construct an initial adjacency matrix, pairwise partial correlations are applied to these time-averaged voxel intensities (matrix_part). In a second matrix (matrix_bivar), the bivariate correlation values of voxel intensities in PET images are used to create this image. Now, matrix_part is modified using matrix_bivar as follows:

m a t r i x_{p a r t} (i, j) = \{\begin{matrix} 0 i f m a t r i x_{b i v a r} (i, j) = 0 \\ n o c h a n g e o t h e r w i s e \end{matrix}

(2)

where matrix_part(i, j) and matrix_bivar(i, j) is the element at (i, j) in the respective matrices. matrix_part is now the combined connectivity adjacency matrix that defines the network.

For each pair of ROIs, the partial correlation is determined using N−2 ROIs as cofactors.⁴⁴ The partial correlation values are used as edge weights in the adjacency matrices and make up the values.

Figure 2.

A connected Network of all the Nodes Using the Julich Atlas. Green Circles Indicate the ROIs, and the Connecting Lines Indicate the Edges Formed Based on Threshold Schemes.

Partial correlations are computed as the correlation of residuals. The first-order partial correlation (ρ_ij._k) of x_i and x_j, controlling for x_k is given by⁴⁵

c o r r (r e s i d (i | k), r e s i d (j | k)) = \frac{C_{i j} - C_{i k} v_{k} C_{k j}}{\sqrt{v_{i -} c_{i k} c_{k i}} \sqrt{v_{j} - c_{j k} v_{k} c_{k j}}}

(3)

where c_ij = cov (x_i,x_j) and v_k = var (x_k). Further,

ρ_{i j . k} = \frac{ρ_{i j} - ρ_{i k} ρ_{j k}}{\sqrt{1 - ρ_{i k}^{2}} \sqrt{1 - ρ_{j k}^{2}}}

(4)

Since we are controlling for (N−2) ROIs for each pair of ROIs, ROI_i and ROI_j, we calculate the (N−2)th-order partial correlation. This is calculated recursively as follows:

For each ROI_k ∈ ROIs

ρ_{i j . R O I s} = \frac{ρ_{i j . R O I s \ \{k\} -} ρ_{i k . R O I s \ \{k\}} ρ_{k j . R O I s \ \{k\}}}{\sqrt{1 - ρ_{i k . R O I s \ \{k\}}^{2}} \sqrt{1 - ρ_{k j . R O I s \ \{k\}}^{2}}}

(5)

Equation (3) is the base case of this recursive method. The precalculation of residuals before computing cross-correlation makes estimating partial correlations a computationally demanding job. Because there are so many covariates, this calculation is done in a time-efficient manner with the R package ppcor.⁴⁵

Figure 3.

Analysis Pipeline.

Next, adjacency matrices are obtained using (a) data-driven thresholding scheme based on orthogonal minimal spanning trees (OMSTs)^{24, 46, 47}; network thresholding removes inconsequential (or low-impact) edges and reduces network complexity, and (b) shortest path thresholding scheme.⁴⁸ This is done to compare the two common schemes used for thresholding.

The NetworkX⁴⁹ Python library is used for network construction from adjacency matrices with thresholds, subsequent percolation centrality computation, and other graph metrics.

Percolation Centrality Computation

Percolation centrality is a nodal metric and is calculated for each node. The percolation centrality for each node v at time t is calculated as shown below:

P C^{t} (v) = \frac{1}{(N - 2)} \sum_{s \neq v \neq r} \frac{σ_{s, r} (v)}{σ_{s, r}} \frac{x_{s}^{t}}{[\sum x_{i}^{t}] - x_{v}^{t}}

(6)

where, $σ_{s, r}$ is the number of shortest paths between nodes $s$ and $r;$ $σ_{s, r} (v)$ is the number of shortest paths between nodes $s$ and $r$ that pass through node $v$ ; $x_{i}^{t}$ is the percolation state of node $i$ at time $t$ ; $x_{i}^{t} = 0$ indicates a nonpercolated node; and $x_{i}^{t} = 1$ indicates a fully percolated node.

The percolation centrality value (PCv) is calculated for each network using the inbuilt function of NetworkX.

Percolation centrality is defined for a given node at a given time, as the proportion of “percolated path” is the shortest path between a pair of nodes where the source node is percolated.

Figure 4.

Generic Example of a Percolation Network and Percolation Centrality.

Collective Influence Algorithm

The fraction of occupied sites (or nodes) belonging to the giant (largest) linked component is defined as G(q). According to percolation theory,⁵⁰ if we choose these q fractions of nodes at random, the network will collapse at a critical fraction where the probability of the presence of the huge linked component will vanish, G = 0. Finding the least proportion qc of nodes to delete such that G(qc) = 0, that is, the minimum fraction of “influencers” to fragment the network, is the optimal percolation issue.

We look for the lowest nonzero giant linked component G for any fixed fraction q < qc. Readers are invited to read more about how the challenges of optimal immunization and spreading (optimal influencer problem) are related to the problem of minimizing the network’s gigantic component, that is, the optimal percolation problem.⁵¹

Given a network, the method assumes that the flow of information within the network is optimal with a small number of nodes that obstruct the flow of information through the network.³² Small groups of nodes/ROIs would be critical in the migration of beta-amyloid plaques in the context of this study.

We discover that the list of important nodes we get for each brain scan is not exhaustive in the sense that we do not get every ROI in the output of influential nodes. Rather, each brain scan generates a unique set of nodes that are arranged in a specific sequence. To facilitate the ordinal comparison of these ROIs across the clinical conditions and tracer types, a ranking strategy is implemented, which is as follows:

For each ROI, the rank is calculated as:

\frac{\sum_{a l l i n f l u e n c t i a n l n o d e l i s t s} p o s i t i o n i n l i s t}{n u m b e r o f o c c u r r e n c e i n i n f l u e n t i a l n o d e s l i s t s}

The key premise is that a network’s general functioning in terms of information propagation (or, in our instance, migration of beta-amyloid plaques) is dependent on a certain collection of nodes known as influencers. Activating influential nodes in social networks to propagate information⁵² or de-activating or immunizing influential nodes to prevent large-scale pandemics are examples of how this concept of locating the most influential nodes has been employed in the past.^{29, 53} This method has recently been utilized in neuroscience to uncover nodes required for the global integration of a memory network in rodents.⁵⁴ To the best of our knowledge, this is the first time it has been used to analyze the course of AD. These small clusters of influential nodes/ROIs would be critical in the migration of beta-amyloid plaques in the context of our study.

The CI method is more efficient than prior heuristic strategies at pinpointing the most influential nodes. CI is an optimization procedure that seeks to determine the smallest number of nodes that will allow the network to fragment in the most efficient way possible. Their removal would, in a sense, disassemble the network into many isolated and nonextensive components. According to percolation theory, removing nodes at random will cause the network to collapse at a critical proportion where the probability of the huge connected component is G = 0. The optimal percolation is a problem in which the goal is to discover the smallest number of influencers q in order to get G(q) = 0.

Other Graph Centrality Metrics

Four different nodal metrics of a graph are calculated in addition to the PCv. The four centrality measures derived for the PET-image-based graphs for comparison are listed below.

Betweenness Centrality

The basic definition of betweenness centrality is defined as:

C_{B} (u) = \frac{\sum_{s \neq u \neq t} \frac{\partial_{s t} (u)}{\partial_{s t}}}{(n - 1) (n - 2) / 2}

(7)

This centrality information provides its uniqueness within the network. This study provides the ROIs that play a vital role in the information flow in the network, the information being beta-amyloid or tau proteins accumulation in those regions.

Closeness Centrality

The closeness centrality of a node denotes how close a node is in the given network. It is inversely proportional to the farness of the node. Freeman defined the closeness centrality as:

C_{c} (u) = \frac{n - 1}{\sum_{\forall u, v \neq u} d (u, v)}

(8)

The distance between two nodes/ROIs u and v in a network, denoted d (u,v), is defined as the number of hops made along the shortest path between u and v. In this case, the lesser the hops; the closer are the two ROIs and the ease with which the misformed proteins can travel.

Current Flow Betweenness Centrality

Given a source ROI (s) and a target ROI (t), the absolute current flow through the edge (i,j) is the quantity $A_{i, j} |v_{i}^{(s, t)} - v_{j}^{(s, t)}|$ . By Kirchhoff’s law, the current that enters a node equals the current that leaves the node. Hence, the current flow $F_{i}^{(s, t)}$ through a node i different from the source s and a target t is half of the absolute flow on the edges incident in i:

F_{i}^{(s, t)} = \frac{1}{2} \sum_{j} A_{i, j} |v_{i}^{(s, t)} - v_{j}^{(s, t)}|

(9.1)

Moreover, the current flows $F_{s}^{(s, t)}$ and $F_{t}^{(s, t)}$ through both s and t are set to 1, if end-points of a path are considered part of the path, or to 0 otherwise. Since the potential $v_{i}^{(s, t)} = G_{i, s}^{+} - G_{i, t}^{+}$ , with G⁺ the generalized inverse of the graph Laplacian, equation (9.1) can be expressed in terms of elements of G⁺ as follows:

F_{i}^{(s, t)} = \frac{1}{2} \sum_{j} A_{i, j} |G_{i, s}^{+} - G_{i, t}^{+} + G_{j, t}^{+} - G_{j, s}^{+}|

(9.2)

Finally, the current-flow betweenness centrality b_i of node i is the flow through i averaged over all source-target pairs (s, t):

b_{i} = \frac{\sum_{s < t} F_{i}^{(s, t)}}{(1 / 2) n (n - 1)}

(9.3)

Eigenvector Centrality

Eigenvector centrality measures the influence a node has on a network. If a node is pointed to by many nodes (which also have high eigenvector centrality), that node will have high eigenvector centrality.

For a given graph G:= (V, E) with |V| vertices, let A = (a_v,t) be the adjacency matrix. Where a_v,t = 1 if the vertex v is linked to vertex t, and a_v,t = 0 otherwise.

The relative score x_v of the vertex v is calculated using the equation below:

x_{v} = \frac{1}{λ} \sum_{t \in M (v)} x_{t} = \frac{1}{λ} \sum_{t \in G} a_{v, t} x_{t}

(9.4)

Here M(v) is a set of neighboring ROIs of v and λ is a constant.

Each brain volume is represented as time-averaged voxel intensities during each PET scan. Pairwise partial correlations are applied to these time-averaged voxel intensities, yielding an initial adjacency matrix. The eigenvector centrality computation is then applied to this matrix.

Another interpretation of the centrality metric is that it identifies key regions in the brain network hierarchy and aids in the detection of localized variations between patient groups.⁵⁵ Eigenvector centrality has been shown to be effective in the treatment of AD patients by comparing with healthy people.⁵⁶

Statistical Analysis

The null hypothesis in this study is that percolation centrality does not reflect beta-amyloid propagation within the brain network.

A comparison with the ROIs from the brain atlas is performed using multiple linear regression (MLR) analysis to assess the impact of the percolation value over each PET scan. The multiplicity problem is substantial, and this work is exploratory. Furthermore, using numerous test procedures does not solve the challenge of drawing accurate statistical inferences from data-driven hypotheses. However, it aids in the description of a potential mechanism.

Pairwise Analysis of Variance

To obtain pairwise group differences, we perform a post-prior (post-hoc) analysis using the scikits-post-hocs package; the student t-test pairwise gives us the respective P values. The ANOVA test is performed for each node in the network with the null hypothesis that the mean percolation centrality of that node is the same across the three-stage null hypothesis test; ANOVA with a significance level (α) of 0.05 is used.

Error Correction

To control for multiple comparisons of 121 nodes, the Scheffe test and control for experiment-wise error rate (EER) are carried out. A single-step procedure calculates the simultaneous confidence intervals for all pairwise differences between means.

Multivariate Linear Regression

A correlation between the PCvs for all 121 nodes and psychometric test scores – MMSE and NPIQ – is computed to identify the ROIs as reliable predictors (see Figure 5 and Table 19). Multivariate regression analysis uses regularization techniques instead of performing multiple correlations across all three diagnoses. The target variable is the MMSE or NPIQ score, and the characteristics are nodal PCvs. The goal is to quantify each node’s contribution in discriminating between clinical states for interpretation purposes, not to develop a predictive model. If the goal was to create a machine learning model, it would necessitate the creation of complicated feature sets (not merely percolation centrality) and the use of advanced machine learning architectures (which provide less room for interpretability).

Figure 5.

Illustrates the ROIs that Correspond to MMSE and NPIQ. The Green Circles Represent ROIs Associated with the MMSE Psychometric Assessment, the Red Circles Represent ROIs Associated with the NPIQ Psychometric Assessment, and the Blue Circles Represent ROIs Associated with Both MMSE and NPIQ.

Regularization and Cross-Validation

In MLR, we utilize regularization to ensure that our regression model generalizes better to unknown data. Controlling overfitting requires regularization. Both Lasso regression (L1 penalty) and Ridge regression (L2 penalty) are investigated in this paper, and both produce similar root mean squared errors (RMSE) and desired results. For reporting our results, we use Lasso with a value of 0.1 (Figure 6). A leave-one-out cross-validation (LOOCV) is used to assess the resilience and dependability of our model before and after regularization. Because it is unbiased and better suited to our smaller sample sizes, we chose this cross-validation technique (especially in the PiB tracer subset). With an increase in regularization (parameter), we see an improvement in validation RMSE. However, we find that penalizing weights excessively at very high values can cause the regression model to converge to the mean of the output MMSE/NPIQ scores. To account for this, we plot the standard deviation in anticipated MMSE/NPIQ outputs and set a regularization factor of one for adequate but not excessive regularization.

Figure 6.

Regularization Using Lasso regression with L1.

Results

Pairwise ANOVA

The student t-test is performed on the resulting five centrality values for each tracer type and clinical conditions. There was a significant effect of the beta-amyloid accumulation on the five centrality values at P < .05 level for the three clinical groups [F (3, 454) = 3.002 for AV45 and F (3, 97) = 3.027 for PiB; see Table 2].

Table 2.

Number of Scans Per Tracer Type and Corresponding Critical F-Values.

Tracer	AV45	PiB
No. of scans	454	97
Critical F-value	3.002	3.027

Abbreviations: AV45, florbetapir; PiB, Pittsburgh compound B.

A one-way ANOVA between clinical groups was conducted to compare the effect of beta-amyloid accumulation/tau protein on five centrality values in the CN, MCI, and AD patients (Tables 3 and 4).

Table 3.

Pairwise ANOVA for AV45 and PiB Tracers for Percolation Centrality.

Tracer Type Thresholding scheme	AV45			PiB
Tracer Type Thresholding scheme	ROI	f-value	P-value	ROI	f-value	P-value
OMST	GM premotor cortex BA6 R	5.87528	0.003	GM Broca’s area BA45 R	5.14527	0.008
	GM hippocampus subiculum L	4.85879	0.008	GM Broca’s area BA45 L	4.82934	0.01
	GM superior parietal lobule 7M L	4.48843	0.012	GM anterior intra-parietal sulcus hIP3 R	3.87541	0.024
	GM visual cortex V5 L	4.38407	0.013	GM Broca’s area BA44 L	3.42666	0.04
	GM anterior intraparietal sulcus hIP1 L	4.14044	0.017	GM mamillary body	3.33030	0.04
	GM superior parietal lobule 7P R	3.83170	0.022	–	–	–
	GM primary motor cortex BA4a L	3.77499	0.024	–	–	––
	GM primary somatosensory cortex BA1 R	3.65131	0.03	–	–	–
	GM primary auditory cortex TE1.2 R	3.34477	0.04	–	–	–
SPT	GM insula Ig2 L	4.64660	0.01	GM Broca’s area BA44 R	6.30152	0.003
	WM acoustic radiation R	4.38388	0.013	GM primary auditory cortex TE1.2 R	5.03631	0.008
	GM hippocampus subiculum L	3.93752	0.02	WM cingulum R	4.57068	0.013
	GM mamillary body	3.54618	0.03	GM inferior parietal lobule PFcm R	3.94533	0.023
	GM inferior parietal lobule Pga R	3.17572	0.045	–	–	–
	GM lateral geniculate body L	3.05960	0.05	–	–	–

Abbreviations: OMST, orthogonal minimum spanning tree; SPT, shortest path threshold; PiB, Pittsburgh compound B; ROI, region of interest; GM, gray matter; WM, white matter.

Table 4.

Pairwise ANOVA for AV45 and PiB Tracers for Betweenness Centrality.

Tracer Type Thresholding scheme	AV45			PiB
Tracer Type Thresholding scheme	ROI	f-value	P-value	ROI	f-value	P-value
OMST	GM premotor cortex BA6 R	5.94059	0.003	GM Broca’s area BA45 R	5.17524	0.007
	GM hippocampus subiculum L	4.96072	0.007	GM Broca’s area BA45 L	4.75064	0.02
	GM visual cortex V5 L	4.39253	0.013	GM anterior intra-parietal sulcus hIP3 R	3.79312	0.03
	GM superior parietal lobule 7M L	4.34316	0.014	GM Broca’s area BA44 L	3.41149	0.04
	GM primary motor cortex BA4a L	3.73311	0.025	GM mamillary body	3.28404	0.042
	GM primary somatosensory cortex BA1 R	3.56446	0.03	–	–	–
	GM primary auditory cortex TE1.2 R	3.31334	0.04	–	–	–
SPT	GM insula Ig2 L	4.58607	.01065	GM Broca’s area BA44 R	6.22323	0.003
	WM acoustic radiation R	4.20148	.01554	GM primary auditory cortex TE1.2 R	5.14313	0.008
	GM hippocampus subiculum L	3.89671	.02097	WM cingulum R	4.61277	0.012
	GM mamillary body	3.63729	.02707	GM inferior parietal lobule PFcm R	3.96185	0.022
	GM Inferior parietal lobule Pga R	3.04114	.04872	–	–	–

Abbreviations: OMST, orthogonal minimum spanning tree; SPT, shortest path threshold; PiB, Pittsburgh compound B; ROI, region of interest; GM, gray matter; WM, white matter.

Cross-Validation and Regularization

Increasing regularization (α) improves the validation RMSE, making it more robust and generalizing unseen data. But at higher values of α, it is observed that the standard deviation of predicted MMSE scores decreases to less than 2, irrespective of clinical condition. This could mean that it saturates predicting the mean MMSE value when regression weights are extremely penalized. Thereby choosing a reasonably small yet effective α value (<2), the validation RMSE and the standard deviation in output predicted MMSE.

Multivariate Linear Regression

A linear regression model between the PCvs for all 121 nodes and psychometric test scores – MMSE and NPIQ – is computed to identify the ROIs that can be used as reliable predictors (Figure 4). Instead of performing multiple correlations across all three diagnoses, a multivariate regression analysis using LOOCV is carried out. The features are the nodal PCvs and the psychological assessment scores’ target variable (see Tables 8 –10 in the supplementary material).

Comparison of Threshold Schemes

The two schemes are compared on the number of ROIs that can be considered based on ANOVA analysis (P ≤ .05). The Juelich atlas has five clusters: frontal, parietal, temporal, occipital lobes, and the WM regions. Also, the performance of the threshold schemes between the two tracers (see Tables 5 –10) are compared.

Table 5.

Multivariate Linear Regression Analysis – Number of Region of Interest Across Clinical Conditions for Both Threshold Schemes for Betweenness Centrality.

Clinical Condition			CN		MCI		AD
Graph Metric	PT	Tracer	OMST	SPT	OMST	SPT	OMST	SPT
Betweenness centrality	MMSE	AV45	4	5	13	8	4	7
	MMSE	PiB	5	4	5	1	2	9
	NPI-Q	AV45	0	3	1	1	0	0
	NPI-Q	PiB	2	14	1	0	3	10

Abbreviations: OMST, orthogonal minimum spanning tree; SPT, shortest path threshold; MMSE, mini-mental state examination; NPI-Q, neuropsychiatric inventory questionnaire; PT, psychometric assessment; PiB, Pittsburgh compound B; CN, cognitively normal; AV45, florbetapir.

Table 6.

Multivariate Linear Regression Analysis – Number of Region of Interest Across Clinical Conditions for Both Threshold Schemes for Percolation Centrality.

Clinical Condition			CN		MCI		AD
Graph Metric	PT	Tracer	OMST	SPT	OMST	SPT	OMST	SPT
Percolation Centrality	MMSE	AV45	4	6	12	2	4	7
	MMSE	PiB	5	3	5	0	2	9
	NPI-Q	AV45	6	3	2	1	0	0
	NPI-Q	PiB	2	14	0	1	3	10

Abbreviations: PT, psychometric assessment; OMST, orthogonal minimum spanning tree; SPT, shortest path threshold; MMSE, mini-mental state examination; NPIQ, neuropsychiatric inventory questionnaire; AD, Alzheimer’s disease; PiB, Pittsburgh compound B; CN, cognitively normal; AV45, florbetapir.

Table 7.

Multivariate Linear Regression Analysis – Region of Interest Across Clinical Conditions for Both Threshold Schemes for Current Flow Betweenness Centrality.

Clinical Condition			CN		MCI		AD
Graph Metric	PT	Tracer	OMST	SPT	OMST	SPT	OMST	SPT
Current flow betweenness centrality	MMSE	AV45	3	0	1	0	0	0
	MMSE	PiB	0	0	1	0	0	0
	NPI-Q	AV45	0	0	0	0	0	0
	NPI-Q	PiB	0	0	0	0	0	0

Table 8.

Multivariate Linear Regression Analysis – Region of Interest Across Clinical Conditions for Both Threshold Schemes for Eigenvector Centrality.

Clinical Condition			CN		MCI		AD
Graph Metric	PT	Tracer	OMST	SPT	OMST	SPT	OMST	SPT
Eigenvector Centrality	MMSE	AV45	2	2	11	4	7	10
	MMSE	PiB	2	2	6	6	8	3
	NPI-Q	AV45	1	4	1	1	0	0
	NPI-Q	PiB	3	0	3	3	2	9

Table 9.

Multivariate Linear Regression Analysis – Region of Interest Across Clinical Conditions for Both Threshold Schemes for Closeness Centrality.

Clinical Condition			CN		MCI		AD
Graph Metric	PT	Tracer	OMST	SPT	OMST	SPT	OMST	SPT
Closeness Centrality	MMSE	AV45	0	5	1	2	2	1
	MMSE	PiB	2	4	46	0	6	0
	NPI-Q	AV45	1	0	0	2	0	0
	NPI-Q	PiB	3	30	1	0	0	10

The OMST provides 112 ROIs across the five nodal metrics. Sixty ROIs are obtained for AV45 and 52 PiB based on the tracers. Further, the ranking of ROI based on the threshold scheme is listed for the three clinical groups for the respective tracers.

Shortest path threshold (SPT): Here, a total of 66 ROIs are obtained; 31 and 35 for AV45 and PiB tracers, respectively. The ROIs are ranked based on the CI algorithm for the three clinical groups and their respective tracers.

Further, comparing the number of statistically valid (MLR) ROIs across the five centrality values with the psychometric tests, MMSE and NPI-Q are performed. This helps compare the function of the ROI with the assessment carried out on the test (see Tables 11–18 in supplementary material).

Other Graph Metrics

The closeness centrality provides the highest ROI overall, 80 in total. Eigenvector centrality provides 33 ROIs, whereas percolation centrality has 24 ROIs followed by the betweenness centrality measure with a total of 23 ROIs and current flow betweenness centrality with 19 ROIs across the two tracers (Table 10).

Table 10.

Distribution of ROIs Across Graph Metrics and Tracer Type Based on ANOVA Test.

Threshold Scheme Tracer/Centrality Measure	OMST		SPT
Threshold Scheme Tracer/Centrality Measure	AV45	PiB	AV45	PiB
BC	9	5	5	4
CC	33	29	10	6
CFBC	5	2	1	12
EVC	4	11	9	9
PC	9	5	6	4

Abbreviations: BC, betweenness centrality; CC, closeness centrality; CFBC, current flow betweenness centrality; EVC, eigenvector centrality; PC, percolation centrality; SPT, shortest path threshold; PiB, Pittsburgh compound B; ROI, region of interest; AV45, florbetapir.

Collective Influence Ranking

The CI algorithm ranks the ROIs; the rank list is generated for the two tracers – AV45 and PiB. When a comparison of the rank is carried out between the clinical groups and tracers in the case of PiB, the ranking increases when moving from CN clinical condition to MCI, and then ranking decreases from MCI to AD. Overall, the ranking increases by 50% from CN to AD.

A comparison of the ROIs across clinical conditions and tracers does not provide any new information regarding a common or group of common ROIs across the data (see Tables 6 and 7).

Demographics

There were 531 patients available for this study based on the selection criteria. There were 48% of females who were CN, 25% who had MCI, and 27% who had AD.

About 43% of the patients received more than 12 years of education in contrast to only 16% who received less than 12 years of education, 31% received more than 12 years of education in the MCI group as opposed to 69% with less than 12 years of education.

Left-handed patients made up 47% of the AD clinical group, compared to 26% of right-handed patients. Multilingualism was found in one patient in the MCI group, two in the CN group, and four in the AD group.

Discussion

To have a deeper understanding of the exploration, five nodal metrics are compared: betweenness centrality, closeness centrality, current flow betweenness centrality, eigenvector centrality, and percolation centrality. It is feasible to show AD advancing through the beta-amyloid plaque networks utilizing variance analysis and multivariate regression testing, as well as a comparison to the percolation centrality computed using PET imaging.

Across the three clinical circumstances, tracer types, and threshold schemes, the student t-test yields nodes for each of the five centrality measures. When utilizing the OMST scheme for threshold, the current flow betweenness centrality measure delivers the fewest ROIs across all conditions (Tables 6 –10), with only three in CN conditions and two in mild cognitively impaired conditions.

For the tracer PiB, the PCvs of some brain areas, such as the inferior and superior parietal lobules, are credible. Meanwhile, in most other cases, the brain areas for each tracer showed significant differences. The tracers may cause the variance because AV45 and PiB attach to the amyloids. The percolation centrality of Broca’s area is also revealed to be a reliable key differentiator between CN and AD clinical situations, corroborating prior results that cognitive decline impairs speech production.⁵⁷

The MLR analysis for each of the centrality measures across the clinical situations for both tracer types reveals one to two ROIs that could be used as markers for AD research on average. When compared to the other centrality measures, the current flow betweenness centrality metric yields the least ROIs, with just four (3–CN and 1–MCI) ROIs for both threshold schemes and only one (MCI) ROI for AV45 and PiB tracers, respectively.

Further, MLR for the NPI questionnaire provides fewer ROIs for each centrality measure. When comparing the contribution of ROIs for MMSE-related tasks and NPIQ-related tasks, percolation centrality has the highest percentage (41%) of ROIs, followed by closeness centrality with 40.5%, and betweenness centrality with 30%. While the betweenness centrality describes the influence of a node (ROI in this case) on the flow of information, the percolation centrality indicates the proportion of percolated paths of the beta-amyloid plaques or tau tangles in the PET-based networks. Further, ROIs based on the percolation centrality are from brain regions involved in memory, visual-spatial, and language functions, such as the ROIs from the premotor and visual cortex, hippocampus, and insula. These ROIs are also common to betweenness centrality, demonstrating that percolation centrality is nearly and equally as efficient as betweenness centrality.

The CI algorithm yields six rank lists. These tables contain the ranking of the nodes, from most influential to least. Furthermore, MLR gives fewer ROIs for each centrality measure for the NPI questionnaire. When comparing the ROI contributions for MMSE and NPIQ tasks, with 41% of ROIs, percolation centrality has the highest percentage. Closeness centrality came second with 40.5% and betweenness centrality came third with 30%.

The betweenness centrality points out the importance of a node (ROI in this example) on the flow of information, whereas the percolation centrality reflects the fraction of percolated routes of beta-amyloid plaques or tau tangles in PET-based networks, depending on the patient’s state and the type of tracer utilized (Figure 5). Furthermore, because the influential nodes are identified using optimal percolation theory, percolation centrality is validated as a viable screening metric.

By increasing the number of samples, ROIs that could be a possible predictor of AD development across clinical circumstances and tracers can be found. When comparing the rank lists for the two threshold schemes, it becomes clear that the OMST scheme has more ROIs than the SPT strategy.

The Scheffe test results provide a way to validate and raise confidence in the results – we used the LOOCV technique to test the regression’s robustness and reliability. Because it is impartial and better suited to our smaller sample size, the cross-validation technique is used here. It is noticed that increasing regularization on validation RMSE when using regularization (L1 – Lasso or L2 – Ridge) control for overfitting. The ROIs obtained from the pairwise t-test for the clinical conditions show that the OMST scheme provides more valid ROIs across the five centrality measures. AV45 provides 92 ROIs, whereas PiB gives 88 ROIs compared to the two tracer types and threshold schemes. Of this, 33.7% and 39.8% of the ROIs are based on the SPT scheme.

Previous research has shown that amyloid-beta seeding occurs in the neocortical and subcortical regions⁵⁸; however, in this investigation, the following WM ROI was identified for PiB. Both the neocortical and subcortical regions of the brain contain the superior occipitofrontal fascicle R. Apart from that, the AV45 tracer has GM Medial geniculate body L ROI in the subcortical region and the following ROIs in the neocortical region: GM Superior parietal lobule 7P L, GM Anterior intraparietal sulcus hIP3 R, GM Superior parietal lobule 7A L, and GM Superior parietal lobule 5L L.

Prior research shows that damage to the parietal lobe is common in AD, leading to apraxia,^{57, 59} which is attested by these results. AD is associated with atrophy of the cornu ammonis, the subfield of the hippocampus, and deficits in episodic memory and spatial orientation.^60–62

And the following in the subcortical region—GM Amygdala-laterobasal group L, GM Amygdala-laterobasal group R, and GM Hippocampus hippocampal-amygdaloid transition area R. Age factor is not so important, however, the presence of beta-amyloid deposits is.63 These ROIs stand out irrespective of the clinical condition or demographic backgrounds; the percolation centrality has the potential to be a reliable value for AD diagnosis and both tracers and threshold schemes pick up these ROIs.

The demographics of this study do not provide adequate insight into the potential benefits or drawbacks of environmental or lifestyle factors on beta-amyloid percolation. This is because of the small sample size and the unequal distribution of data points within the clinical groups. The intensity of a voxel/node is used to identify whether it is percolated or not; a higher intensity indicates that the node is percolated, whereas zero or lower intensity indicates that the node can easily permeate (Figure 2).

Limitations

This research provides no evidence of disease progression in terms of ROIs or patient clinical groups. However, increasing the number of observations within each patient’s clinical group can help to overcome this issue.

The PET tracers, PiB and AV45, are compared to evaluate which one among the two provides consistent or reliable PCv. Here, the AV45 tracer binds with a high affinity to the beta-amyloid plaque, whereas PiB binds to oligomers or protofibrils. A possible explanation for the difference in PCv generated using these tracers would be their binding targets. The pipeline would need to be tested with second-generation tracers to test the applicability of percolation centrality to other neurodegenerative diseases and the possibility of using it in metastatic cancer scenarios.

An extensive dataset with more patients that takes into account healthy aging brain shrinkage, which reduces the distances between brain networks, can assist increase the PCv’s reliability. This can then be used to test various psychological tests that could be used as early markers for dementia caused by AD, allowing it to be tailored to certain demographics or population subsets.

This pipeline is built for tracers such as AV45 and PiB, which indicate beta-amyloid plaque concentrations directly and as a post-hoc implementation. However, the pipeline can work with second-generation tracers and tracers like FDG with some appropriate modifications, for example, taking the multiplicative inverse of the percolation states of each of the ROIs to reflect the behavior of the FDG tracer.

Conclusion

The findings suggest that percolation centrality is a promising predictor and that it can be utilized to identify and track disease progression by identifying the nodes that regulate the mobility of beta-amyloid plaque.

This topological analysis reveals that standard neuroimaging methods, such as PET-CT, can provide value with a relatively short calculation time if the infrastructure is capable. The capacity to provide a metric for the severity of the disease state is useful in today’s Alzheimer’s era. Patients are pushed into a range of unconventional medical encounters as a result of modern medicine’s ability to extend life. The use of a value like percolation centrality to show the difference has potential applications.

Concerns about the number of patients within clinical groups and the overall sample size, data-collecting time points, demographics, and the PET tracers utilized were limiting variables that may be addressed to increase the reliability of percolation centrality. As a result, this research establishes the utility of PCv in determining the patient’s condition and paves the way for future research into additional neurodegenerative disorders.

The CI algorithm provides a minimum set of nodes in the network that are key to the beta-amyloid plaque movement, which can provide information about a particular pathway that is susceptible to the disease, contrary to network metrics such as hub centrality or betweenness centrality, which provide information about a vital vertex/node.

According to the threshold schemes used in this study, a data-driven strategy such as the OMST outperforms the shortest path approach. Finally, we use the CI algorithm to rank the ROIs based on network influence. We evaluate the outcomes of two thresholding methodologies, SPT and OMST, as well as the influential nodes, to determine the dependability of various threshold schemes. For each network, the CI algorithm generates a ranked list of influential nodes (or scan).

A node’s rank is then derived for each category as the total of that node’s individual ranks for each scan-wise list, divided by the number of scans it appears in. The nodes are then sorted into categories and graded accordingly. Instead of looking at influential nodes independently in each scan, this provides a general rating of nodes in a category (AD/MCI/CN). The results differ slightly depending on the thresholding strategy used, but they largely correspond to the earlier MLR results. Because this is an exploratory study, enhancing resilience across various thresholding systems could be a future project. ROIs that could be a viable predictor of AD development across clinical contexts and tracers can be found by increasing the number of samples. When the ROIs for the two threshold schemes are compared, it is evident that the OMST scheme has more ROIs than the SPT approach.

Supplemental Material

Supplemental material for this article is available online. Supplementary Information Please see Supplementary Material for Tables 11–18. Table 19. Juelich Atlas for MMSE and NPI-Q, the table shows the node details and the anatomical region that may be involved in both or either of the psychometric assessment of the patient. Please follow the link for the full set of tables. Table 20. The rank list of two tracers across three clinical conditions. Given the large volume of data, carrying out computation sequentially would be highly time-consuming. Thus, GNU Parallel [46] was used to parallelize this operation. This could give a maximum speedup of up to 12 times on the system we used for computation (6-core, hyperthreaded Intel i7). The PCv is calculated for each network using the inbuilt function of NetworkX. This has a worst-case time complexity of O(n3), where n is the number of nodes in the network. Using a modified form of Brandes’ fast algorithm for betweenness centrality, the complexity can be reduced to O(nm), where m is the number of edges. However, percolation centrality calculation with target nodes cannot take advantage of this optimization and has a worst-case time complexity of O(n3).

Supplemental Material for Tracking the Progression and Influence of Beta-Amyloid Plaques Using Percolation Centrality and Collective Influence Algorithm: A Study Using PET Images by Gautam Kumar Baboo, Raghav Prasad, Pranav Mahajan and Veeky Baths, in Annals of Neurosciences

Footnotes

Acknowledgements

Data usage and analysis for this study are enabled by the AD Neuroimaging Initiative – National Institutes of Health Grant U01 AG024904, DOD ADNI, Department of Defense award number W81XWH-12-2-0012.

Authors’ Contribution

GKB conceived the hypothesis and the design of the study. PM explored the potential of CI on the network. Both RP and PM carried out the analysis and the necessary scripting for computing PC values, the listing of the influential nodes, and statistical analyses, and VB helped in the manuscript preparation.

Authors’ Note

The manuscript’s authors attest that the secondary analyses using shared data (ADNI – Alzheimer disease neuroimaging initiative database) are under the terms agreed upon their receipt. The source of the data is acknowledged in respective sections of the manuscript. The analysis carried out in this study explores the application of network analysis on PET-image-based networks, which is unique and shows the potential to be used as a feature in learning models. A copy of the manuscript has been submitted to the ADNI portal for their kind perusal.

Statement of Ethics

Informed consent from the patients is obtained before the assessment is carried out by the ADNI study team (see ADNI website for details), and this study is a secondary data analysis of the ADNI data collection, which aims at providing a simplified metric for an already diagnosed patient. The data access and usage are within the ADNI data use agreements.

Declaration of Conflicting Interests

The authors declare that the research was conducted without any commercial or financial relationships construed as a potential conflict of interest.

Funding

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

ORCID iD

Gautam Kumar Baboo

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