Both amyloid and tau imaging data have been downloaded from the ADNI website as preprocessed. Briefly, amyloid PET used florbetapir (18F) as a tracer to measure amyloid-β (Aβ) plaques (Okamura and Yanai, 2010). For each subject, brain regions of interest (ROIs) were defined from structural MRI through segmentation and parcellation using Freesurfer (Fischl, 2012). Then, each florbetapir scan was coregistered to the corresponding MRI and calculated the mean florbetapir uptake within the predefined ROIs. All the regional amyloid deposition was re-normalized using the whole cerebellum as a reference region. Tau PET used flortaucipir as a tracer to detect the aggregated tau (Fleisher et al., 2020), and the regional tau aggregation was obtained similarly as amyloid. All the regional tau tangle accumulation was re-normalized using inferior cerebellar as reference region. Finally, we have amyloid measurement in 68 cortical ROIs and tau measurement in 73 ROIs. More detailed image processing information can be found in Landau et al. (2013) and Landau et al. (2016). To remove potential bias, both amyloid and tau measures were pre-adjusted using baseline age, gender, and the weight derived from healthy controls. Finally, they were normalized to zero mean and unit variance for subsequent analysis.

Genotype data of both ADNI-1 and ADNI-2/GO phases were also obtained from the ADNI cohort (adni.loni.usc.edu). We focused our analysis on top SNPs from the International Genomics of Alzheimer's Project (IGAP), a large-scale genome-wide association study of AD (Schellenberg and IGAP, 2012). It tested the association of 7,055,881 single nucleotide polymorphisms (SNPs) of 17,008 Alzheimer's disease cases and 37,154 controls. SNPs with p ≤ 5 × 10⁻⁶ in their meta analysis were used as our candidates and their genotypes were extracted based on the quality controlled and imputed genetic data in the ADNI using PLINK (Purcell et al., 2007). Finally, we have 1,080 SNPs for the subsequent imaging genetics association.

In this study, we proposed to use a fused similarity network as a new prior knowledge, as inspired by Wang et al. (2014) and hypothesize that it will help improve the performance of imaging genetics association. First, we have original SNP data and imaging data showed in Figure 1A, we build a sample-sample similarity matrix for imaging and genetic data, respectively (Figure 1B) and their subject similarity network look like in Figure 1C. This similarity matrix can be seen as a similarity network G = (V,E,W), where nodes V represent subjects {x₁, x₂, ..., x_n}, the weighted edges E represent similarities of a subject to others and W is a n × n similarity weighted matrix representing the similarity between subjects x_i and x_j. Suppose ρ (x_i,x_j) is euclidean distance between subjects x_i and x_j. Then a scaled exponential similarity kernel was used to determine the weight of the edge: (1)W(i,j)=exp(-ρ2(xi,xj)μεi,j) where μ is a hyper parameter that can be empirically set. It was recommended from [0.3, 0.8], and we set it as 0.5 by default (Wang et al., 2014). and ε_{i, j} is used to eliminate the scaling problem. Here we define: (2)εi,j=mean(ρ(xi,Ni))+mean(ρ(xj,Nj))+ρ(xi,xj)3 where N_i denote a set of x_i's neighbors including x_i in G, and ρ(x_i, N_i) is the average value of the distance between x_i and each of its neighbors. Each row of W was then normalized as below: (3)P(i,j)={W(i,j)2Σk≠iW(i,k),j≠i1/2,j=i Given a graph G, we use K nearest neighbors (KNN) to measure local affinity as: (4)S(i,j)={W(i,j)Σk∈NiW(i,k),j∈Ni0otherwise

P offers the similarity information of each subject to all others and S encodes the similarity to the K most similar neighbors for each subject. In this article, we have two types of data, genomics data and imaging data. We first calculated the status matrices P⁽¹⁾ and P⁽²⁾ following equation (3), and then the kernel matrices S⁽¹⁾ and S⁽²⁾ following in equation (4). For both amyloid and tau data, we tested the model performance with varying K values from 5 to 50. Association performance was found highly stable across varying K values, therefore we set K = 20 as default.

Next, we performed the network fusion of two kernel matrices using a message-passing theory (Pearl, 1988) non-linear method. This is an iterative process where both networks keep getting updated until they converge (i.e., not change much). The final network, known as the fused network, is expected to represent the subject relationships supported by both brain image data and genotype data. Let Pt=0(1)=P(1) and Pt=0(2)=P(2) be the initial two status matrices when t = 0. The fusion process will iteratively update two similarity matrices corresponding to two data types as follows: (5)Pt+1(1)=S(1)×Pt(2)×(S(1))T (6)Pt+1(2)=S(2)×Pt(1)×(S(2))T where Pt+1(1) and Pt+1(2) are the status matrix of these two data types after t iterations. After each iteration, we performed normalization on Pt+1(1) and Pt+1(2) as in equation (3). This step ensures that subject self-similarity is always higher than the similarity to other neighbors. Here, the alternating multiplication of the squared KNN similarity of the two modalities essentially combines the local information of the two modalities in a way that reinforces their shared information. By multiplying the initial similarity matrix of one modality with the squared KNN similarity matrix of the other modality, the shared information between the two modalities is amplified and the unique information in each modality is retained. This process is then repeated in an alternating manner to ensure that both modalities contribute equally to the final similarity matrix, thereby achieving a balanced fusion of the two modalities (The fused network as showed in Figure 1D). This approach is expected to result in a more informative similarity matrix that captures the shared and unique features of both modalities, which in turn can improve the performance of downstream analysis such as association identification between brain imaging and genetic features.

In this part, let X={x1,x2,...,xn}⊆Rp be the imaging data and Y={y1,y2,...,yn}⊆Rq be the genotype data, where n is the number of patients, p and q are the numbers of ROIs and SNPs, respectively. Sparse canonical correlation analysis (SCCA) aims to find the maximal correlation between Xu and Yv by adjusting these two weights, u and v, which indicates the significance of each feature of the imaging genetic associations. As shown in this formula: (8)maxu,vuTXTYv s.t.uTXTXu=1,vTYTYv=1,P1(u)≤c1,P2(v)≤c2 where P₁(u) ≤ c1andP₂(v) ≤ c2 are two penalty terms to control the sparsity of selected features. In this study, we used the PMA package (Witten et al., 2009) that applied the L₁ norm penalty for P₁ and P₂ constraints to perform the SCCA method. To ensure the selection of disease-relevant features, we used a novel discriminative SCCA (DSCCA) algorithm (Yan et al., 2017) to integrate imaging data, SNPs data and the prior knowledge for imaging genetics association. Prior knowledge can be diagnosis network, fusion network or ROI–ROI co-expression network. As such, we can not only identify disease-relevant multi-modal biomarkers, but also reveal a strong association between them. Finally, we compare the performance of multiple DSCCA models guided by different prior knowledge.

For the original DSCCA algorithm, there are two constraints, P₁ and P₂, which are added for the multi-class discrimination, inspired by the application of locality preserving projection (LPP) in linear discriminative analysis (Ghamisi et al., 2018). (9)P1(u)=||u||D=uTXTLwXuP2(v)=||v||D=vTYTLwYv Here, L_w is the Laplacian graphs of prior knowledge graph.

The final objective function of DSCCA can be written as follows: (10)maxu,vuTXTYv-β12P1(u)-β22P2(v)s.t.uTXTXu=1,vTYTYv=1,||u||1≤c1,||v||1≤c2 Using Lagrange multipliers, Equation (10) can be reformulated as follows: (11)maxu,vuTXTYv-γ12||Xu||22-γ22||Yv||22-β12P1(u)-β22P2(v)-λ1||u||1-λ2||v||1 Equation (11) is known as a bi-convex problem, which can be solved using an alternating algorithm as discussed in Witten et al. (2009). By fixing u and v, respectively, we will have the following two minimization problems shown in Equations (12) and (13). (12)minu-uTXTYv+γ12uTXTXu+β12P1(u)+λ1||u||1 (13)minv-uTXTYv+γ22vTYTYv+β22P2(v)+λ2||v||1 We used the Nesterovs accelerated proximal gradient optimization algorithm to solve this objective function following original DSCCA paper (Liu et al., 2012; Yan et al., 2017). The convergence is based on the value changes of the objective function and we use 10⁻⁶ as stop criteria. A five-fold nested cross-validation was applied to automatically tune the parameters β₁, β₂, λ₁, and λ₂. According to Chen et al. (2012), the learned pattern and performance are insensitive to γ₁ and γ₂ settings. Therefore, in this article, we set both of them to 1 for simplicity.

We first tested the association between brain-wide amyloid deposition and top AD-risk SNPs. The performance of test data including the DSCCA algorithm with fused network, diagnosis network, co-expression network as prior knowledge, and the original SCCA method are shown in Table 2. As expected, DSCCA algorithms with the guidance of prior knowledge all outperformed traditional SCCA, confirming that prior knowledge does help reveal stronger brain imaging genetics associations. More specifically, out of all three types of prior, fused network and diagnosis network led to similar association performance, which is much better than the co-expression network.

For all ROIs and SNPs, we averaged their weights across five-folds for feature selection. Figure 2 shows the top 10 ROIs selected by DSCCA guided by the fused network, including left precentral, right parahippocampal, right rostral middle frontal, right precentral, right bankssts, left rostral anterior gingulate, right caudal middle frontal, left postcentral, right postcentral, and left parstriangularis. Among these, right bankssts, right caudal middle frontal, parstriangularis, and left rostral anterior gingulate are part of the default mode network (DMN), and right rostral middle frontal is part of the frontoparietal network. Both of them have consistently shown early accumulation of cortical Aβ fibrils in previous studies (Palmqvist et al., 2017). Amyloid deposition of these regions is strongly associated with only one SNP, APOE SNP (rs429358), which is known as the major risk factor for Alzheimer's disease. There are strong evidences suggesting that APOE could inhibit amyloid-β (Aβ) clearance and promote Aβ aggregation to increase AD risk (Polvikoski et al., 1995; Kim et al., 2009; Kok et al., 2009; Wirths, 2010).

We next tested the association between brain-wide tau deposition and top AD risk SNPs. The performance on test data across all the methods is shown in Table 3. Similar to amyloid, DSCCA guided by three types of prior knowledge performed significantly better than SCCA. Also, all tree DSCCA models showed very similar performance. Considering that the co-expression network only showed moderate performance in amyloid data, we speculate that the selection of candidate genes has a major effect on the prior co-expression network and later lead to the fluctuation of association performance.

After averaging the weight across 5-folds, we found the tau deposition in the left and right amygdala are strongly associated with 56 SNPs (Figure 3). Amygdala is one of the earliest sites showing tau deposition and neurofibrillary tangles, as reported in previous post-mortem and neuroimaging studies (Vogt et al., 1990; Abiose et al., 2020; Insel et al., 2020). Early tau position in Amygdala is associated with reduced volume and worse cognition performance as well in the preclinical stage of AD (Abiose et al., 2020; Berron et al., 2021). Selected 56 SNPs associating with the amygdala tau deposition are located in or near genes TOMM40, AC011481.3, NECTIN2, APOC1, AC011481.2, AC015687.1, CLU. TOMM40 poly-T lengths have a significant relationship with the higher medial temporal plaque and tangle burden in the living brain of non-demented older adults within individuals not carrying the APOE-4 allele (Siddarth et al., 2018). We further performed pathway enrichment analysis of these genes using EnrichR (Kuleshov et al., 2016). Top enriched biological process in gene ontology include regulation of receptor-mediated endocytosis (adjusted p = 0.01) and cholesterol transport (adjusted p = 0.01).