Alzheimer's Disease Disrupts Rich Club Organization in Brain Connectivity Networks

Abstract

Diffusion imaging and brain connectivity analyses can monitor white matter deterioration, revealing how neural pathways break down in aging and Alzheimer's disease (AD). Here we tested how AD disrupts the ‘rich club’ effect – a network property found in the normal brain – where high-degree nodes in the connectivity network are more heavily interconnected with each other than expected by chance. We analyzed 3-Tesla whole-brain diffusionweighted images (DWI) from 66 subjects (22 AD/44 normal elderly). We performed whole-brain tractography based on the orientation distribution functions. Connectivity matrices were compiled, representing the proportion of detected fibers interconnecting 68 cortical regions. As expected, AD patients had a lower nodal degree (average number of connections) in cortical regions implicated in the disease. Unexpectedly, the normalized rich club coefficient was higher in AD. AD disrupts cortical networks by removing connections; when these networks are thresholded, organizational properties are disrupted leading to additional new biomarkers of AD.

1. Introduction

Alzheimer's Disease (AD) is a progressive, degenerative brain disease affecting around 1 in 8 people (13%) aged 65 or older [1]. AD profoundly affects the brain's gray matter, leading to extensive cortical and hippocampal atrophy, and the white matter is also affected [2]. White matter fiber tracts lose axons and myelin degenerates in AD [3]. T2-weighted scans are often used to evaluate white matter hyperintensities – a sign of cerebrovascular disease – and there is growing evidence that the breakdown of the brain's fiber networks may explain some of the symptoms as the disease progresses.

Diffusion imaging has recently been added to several largescale neuroimaging studies, including the Alzheimer's Disease Neuroimaging Initiative (ADNI), to monitor white matter deterioration using metrics not available with standard anatomical MRI. Diffusion MRI offers measures sensitive to fiber integrity and microstructure, such as the mean diffusivity and fractional anisotropy of local water diffusion (Clerx et al., 2012); in addition, tractography can infer neural pathways and connectivity patterns, yielding additional, more complex mathematical metrics describing fiber networks.

Here we studied fiber networks in 44 controls and 22 AD subjects using novel mathematical metrics recently proposed in network theory, such as the rich club coefficient [4]. Models of networks in the normal brain suggest that there is a rich club effect – i.e., a core of nodes with a high degree (k) – that are more densely interconnected among themselves than lower-degree nodes in the network. In other words, the high-degree, ‘richly-connected’ nodes form a club. If a network of nodes is connected by edges, the nodal degree is the number of edges a node has. A network's k-core is the subnetwork that remains after deleting all nodes of degree <k. The rich club coefficient, Φ(k), is the ratio of the number of connections among nodes of degree k or higher versus the total possible number of connections if those nodes were fully connected:

$$\mathrm{\Phi }(k)=\frac{{\mathrm{E}}_{>k}}{{\mathrm{N}}_{>k}({\mathrm{N}}_{>k}-\mathit{1})}$$ (Eq. 1)

As higher-degree nodes are more likely to be interconnected with each other simply by chance, Φ(k) is typically normalized relative to Φ calculated on a set of simulated random networks with the same degree distribution, and the same edge distribution, as a function of the nodal degree k. If Φ_norm>1 (i.e., Φ(k) >Φ_rand, for some k), then there is evidence of rich club organization (tests of the rich club effect use randomized networks to create a reference null distribution) [4].

The rich club coefficient assigns a score to the connection density for the structural cores of the brain, called k-cores, thresholded at a range of k values. As k increases, unreliable fibers are removed from the connectivity matrices, but some true connections may also be removed. It is therefore vital to understand which thresholding levels, k, are informative for a given dataset. Given the potential to understand how networks break down in AD, here we assess how AD affects the rich club coefficient, and the nodal degree of the cortical regions that AD affects.

2. Methods

2.1. Subjects and Diffusion Imaging of the Brain

We analyzed diffusion weighted images (DWI) from 66 subjects scanned as part of the Alzheimer's Disease Neuroimaging Initiative (ADNI), a large multi-site longitudinal study to evaluate biomarkers to assist diagnosis and track disease progression. Table 1 shows their demographics and diagnostic information; data collection is ongoing.

Table 1.

Demographic information for 44 controls and 22 AD subjects scanned with diffusion MRI as part of ADNI. 66 subjects were scanned at the time of writing (August 2012). Their minimum age was 55.7 and maximum age was 90.4 years.

	Controls	AD	Total
N	44	22	66
Age	72.7 ± 5.9 SD	75.5 ± 10.0 SD	73.6
Sex	22M/22F	14M/8F	36M/30F

The transformation matrix from linearly aligning the mean b₀ image to the T1-weighted volume was applied to each of the 41 gradient directions to properly re-orient the orientation distribution functions (ODFs). We also performed whole-brain tractography as described in [5] on the sets of DWI volumes. This method detects fibers based on the Hough transform; to better detect crossing fibers, the method uses a constant solid angle orientation density function [5] rather than a diffusion tensor, to model local diffusion.

Our study was conducted at 3T, so connectivity studies at higher fields, or with different protocols, may reveal group differences in additional regions [6]. Due to space constraints, we refer readers to prior papers on image acquisition, pre-processing, cortical extraction and tractography [7,8].

2.2. NxN Matrices Representing Structural Connectivity

For each subject, a baseline 68×68 connectivity matrix was created, 34 right hemisphere ROIs and 34 left hemisphere ROIs (tabulated in [7]). Each element described the estimated proportion of the total number of fibers, in that subject, that passes through each pair of ROIs.

2.3. Brain Network Measures

Topological changes in the brain's networks may be analyzed using graph theory, which represents the brain's components as a set of nodes and edges. The network's nodes are typically defined as ROIs, usually on the cortex, segmented automatically from anatomical MRI. In DTI studies, these network nodes are linked by ‘edges’ whose weights denote some measure of connectivity between the two regions, such as the density or integrity of fibers [9]. In graph theory, an NxN connection matrix may be compiled to describe the network. A square matrix can represent any network of connections, and may also be displayed as a graph, i.e., a discrete set of nodes and edges [9]. In our analysis, the matrix entries store the total number of fibers connecting each pair of regions (the nodes); these could also be considered as the “weights” of the edges that connect a pair of nodes [9]. From the connection matrices, we computed the rich club coefficient for each subject's anatomical network. We analyzed the whole-brain network as well as the left and right hemispheres in all subjects separately. In the single-hemisphere analyses, we did not evaluate fibers that crossed between the hemispheres, as they are present in the whole-brain network; for simplicity, we focused on the subnetwork of nodes and edges that remain entirely within a specific hemisphere.

As noted earlier, the rich club coefficient is the fraction of edges that connect nodes of degree k or higher over a range of k-core values. To create the k-cores, we thresholded the nodal degree for the whole network, to remove lower-degree nodes the rich club coefficients were calculated, for each k. When k was 18, this was the lowest value of k where most (>50%) of the nodes within each hemisphere would remain connected (most nodes would be connected to at least one other remaining node).

We performed a regression, controlling for age and sex, with AD coded as 1 and controls as 0. This tested for differences between controls and AD subjects for the rich club coefficient at each k-core value, first in the whole brain, then separately for the left and right hemispheres.

3. Results

The rich club coefficient increased over an increasing range of k-core values in the whole brain (Fig. 1a and b), left and right hemispheres of controls and AD subjects. The rich club coefficient was normalized with 50 randomized networks of equal size and similar connectivity distribution and was found to be greater than 1, which indicates that a rich club effect was present in both controls and AD subjects (Fig. 1b).

Fig. 1.

a shows the average rich club coefficient for the whole brain in controls (blue) and AD subjects (red), based on thresholding the network to retain only nodes of degree k or higher (i.e., the k- core). Blue and red markers indicate thresholds that yield significant group differences for the rich club coefficient (k between 1-9 and 18-24) based on a regression controlling for age and sex with AD coded as 1 and controls as 0 (FDR p-value=0.023). b shows the average normalized rich club coefficient for the whole brain in controls and AD subjects. c shows the p- values from a regression controlling for age and sex, testing for significant differences between whole-brain rich club coefficients in AD subjects versus controls. Red points highlight p-values that are less than the p-value threshold (0.023) that controls the FDR at 5%. This FDR correction allows us to state that the groups truly differ even though multiple thresholds were tested.

Relative to controls, AD patients had a lower rich club coefficient for low k-core values, but it became higher when the network degree threshold reached k=14 or higher. Similarly, in AD, the rich club coefficient became greater at k=11 for the left and k=9 for the right hemisphere. As k increases, a stricter threshold is applied to the connectivity matrix and low-degree nodes are deleted. This enhances the influence of nodes with greater degrees. A rich club coefficient was computed for k-core values ranging from 1-24 in controls and AD subjects and a normalized rich club coefficient measure for k-core values ranging from 1-19 in controls and AD.

We performed a regression controlling for age and sex with AD coded as 1 and controls as 0, to test for group differences in the whole-brain rich club coefficients. For this test, we also varied the nodal degree threshold, k, as the rich club coefficient depends on it. Significant effects of Alzheimer's disease were found for nodal degree threshold values in the ranges k=1-9 and 18-24 (Fig.1c). Similarly, a regression model was designed to predict the rich club coefficient for the left and right hemispheres in AD patients and controls. It detected significant differences in k-cores created using nodal degree thresholds ranging from 11-14 and 16-18 in the right hemisphere (Fig. 2a). In the left hemisphere, the significantly differing k-core values lay in the ranges 1-7 and 14-19 (Fig. 2b).

Fig. 2.

a shows the average number of nodes with degree greater than k for the whole brain in 44 controls (blue) and 22 AD subjects (red). b shows the average number of edges remaining in a subgraph with degree greater than k for the whole brain in 44 controls (blue) and 22 AD subjects (red).

We computed the nodal degree (i.e., number of edges) for each of the 68 cortical areas of the k-core at level k=18 for the whole brain, then assigned the cortical regions to the left and right hemispheres (Table 2). We tested for group differences in the nodal degree, in brain regions that first show atrophy in AD (temporal and parietal lobes). As expected, AD subjects had an average nodal degree that was smaller by 5 ± 1 SD edges per node (on average) than controls over all 68 cortical regions. Controls had an average nodal degree of 21 ± 9 SD for the left hemisphere and 20 ± 8 SD for the right; AD patients had an average of 15 ± 7 SD edges per node in the left hemisphere and 16 ± 7 SD edges per node on the right. To assess the reliability of our results, we created 20 groups of randomly selected controls, each consisting of 22 subjects and computed the average degree for each group. The average nodal degree in the overall brain of the 20 control groups was 20 ± 8 SD, which is close to the average nodal degree for the whole brain in all 44 controls analyzed here (21 ± 8 SD). This indicates that there is little variability in computing the nodal degree from randomly selected groups and supports the idea that the lower nodal degree in AD may be attributable to disease effects on the brain network.

Table 2.

The average nodal degree for the k-cores of the whole brain (k=18, applied on the connectivity matrices) in 44 controls and 22 AD subjects. As expected, the mean nodal degree in both left and right hemispheres (LH, RH) was lower in AD than controls. The p-values are derived from a linear regression testing for differences between AD and controls (controlling for age and sex), for the right and left hemispheres. These cortical areas were selected as being among those that show atrophy in AD. Greatest nodal degrees were found in the precuneus for both controls (LH, 36 and RH, 34) and AD subjects (LH, 28 and RH, 29).

Cortical Area	Nodal Degree
	Controls	AD
Superior Temporal (LH p=2.0E-03, RH p=0.02)	LH: 22	LH: 17
	RH: 19	RH: 17
Middle Temporal (LH p=8.2E-05, RH p=1.0E-03)	LH: 19	LH: 15
	RH: 20	RH: 17
Inferior Temporal (LH p=9.0E-04, RH p=5.0E-04)	LH: 22	LH: 17
	RH: 21	RH: 16
Transverse Temporal (LH p=0.20, RH p=0.65)	LH: 13	LH: 10
	RH: 13	RH: 12
Banks of Superior Temporal Sulcus (LH p=5.0E-03, RH p=2.0E-03)	LH: 18	LH: 14
	RH: 20	RH: 17
Superior Parietal (LH p=5.5E-06, RH p=2.2E-06)	LH: 32	LH: 26
	RH: 33	RH: 27
Inferior Parietal (LH p=1.4E-05, RH p=9.1E-06)	LH: 26	LH: 21
	RH: 28	RH: 23
Superior Frontal (LH p=1.7E-06, RH p=1.0E-04)	LH: 34	LH: 24
	RH: 30	RH: 24
Rostral Middle Frontal (LH p=1.5E-06, RH p=7.5E-05)	LH: 26	LH: 16
	RH: 26	RH: 19
Lateral Orbitofrontal (LH p=6.0E-03, RH p=3.2E-05)	LH: 20	LH: 17
	RH: 24	RH: 15
Medial Orbitofrontal (LH p=1.0E-04, RH p=4.0E-03)	LH: 21	LH: 15
	RH: 14	RH: 9
Caudal Middle Frontal (LH p=6.0E-05, RH p=1.0E-03)	LH: 20	LH: 13
	RH: 22	RH: 17

4. Discussion

Here we analyzed brain connectivity in cognitively healthy elderly people and patients with Alzheimer's disease. As predicted, the AD group had, on average, a lower nodal degree (fewer connections), as is expected for a disease that erodes connectivity. We also predicted that the rich club effect – an organizational feature of normal brain networks – would be lower or lost in AD, but in fact it was greater. The rich club coefficient describes the density of connections for subnetworks, or k-cores, created by thresholding the network at a range of nodal degrees, k, starting with k=1 and ending when no more nodes of degree k or higher remain. High values of k only leave the primary core of the network, keeping the highly connected nodes and removing less reliable nodes. For low k-levels, the unnormalized rich club coefficient was greater in controls, but, at higher k (starting at k=14 for the whole brain, k=11 for the left and k=9 for the right hemisphere), the rich club coefficient was greater in AD. One plausible explanation is that the node counts and edge counts, N(k) and E(k), fall off more rapidly with increasing k in AD subjects (Figs. 3 and 4), than controls. This leads to a smaller denominator in the calculation of the rich club coefficient, and a more pronounced rich club phenomenon in AD, even for the same k, as a smaller fraction of nodes are left. In general, for both AD and controls, the rich club coefficient is higher when a smaller percentage of nodes are retained. Applying the same numerical threshold to the nodal degree will retain a smaller fraction of nodes from AD networks, making their rich club coefficient appear higher as less of the network is left. Even so, at low k, the higher rich club coefficient in controls fits with the hypothesis that some aspects of normal network organization may be lost or impaired in disease.

Fig. 3.

a shows the average rich club coefficient for the left hemisphere in controls (blue) and AD subjects (red). Blue and red markers indicate thresholds yielding significant group differences for the rich club coefficient (k between 1-7 and 14-19) based on a regression as described in Fig. 1 (FDR p-value=0.009). b shows the average rich club coefficient for the right hemisphere in controls (blue) and AD subjects (red). Similarly, blue and red markers indicate thresholds that yield significant group differences for the rich club coefficient (k between 11-14 and 14-16, FDR p-value=0.013). c shows the average normalized rich club coefficient for the left and right hemispheres (LH, RH) in controls (CTL) and AD subjects (AD).

To see if rich club coefficient findings might relate to differences in specific cortical areas in the brain, we assessed the most basic mathematical element of the rich club coefficient, the nodal degree. The rich club coefficient assigns an overall value to each k-core for a range of k values. It is a global descriptor of the architecture of the entire brain or hemisphere, not necessarily reducible to effects in more specific cortical regions. In Table 2 we showed the average nodal degree for controls and AD subjects. There is some evidence that the left hemisphere is more affected than the right hemisphere in AD, consistent with prior findings [7].

The rich club phenomenon takes into account the fiber density for the white matter connections [4], and intuitively, this is expected to be lower in disease, relative to controls. On the other hand, based on its mathematical definition, the rich club coefficient increases with increasing k and decreasing nodal degree (when smaller percentage of nodes are retained). AD reduces the fraction of connections in the brain network and therefore, this may induce a higher rich club coefficient.