1 Introduction
Nowadays, a large amount of magnetic resonance images (MRI) scans are stored across a vast number of clinical centers and institutions. Researchers are currently analyzing these large datasets to understand the underpinnings of brain diseases. However, due to privacy concerns and legal complexities, data hosted in different centers cannot always be directly shared. In practice, data sharing is also hampered by the need to transfer large volumes of biomedical data with the associated bureaucratic burden. This situation led researchers to look for an analysis solution within metaanalysis or federated learning paradigms. In the federated setting, a model is fitted without sharing individual information across centres, but only model parameters. Metaanalysis instead performs statistical testing by combining results from several independent assays[1], for example by sharing p
values, effect sizes, and/or standard errors across centers.
One of the best examples of such a research approach is the Enhancing NeuroImaging Genetics through MetaAnalysis (ENIGMA) consortium (enigma.usc.edu). With a large number of institutions worldwide [2], ENIGMA has become one of the largest networks bringing together multiple groups analyzing neuroimaging data from over 10,000 subjects. However, most of ENIGMA’s secure metaanalytic studies in neuroimaging are performed using massunivariate models.
The main drawback of massunivariate analysis is that they can only model a single dependent variable at a time. This is a limiting assumption in most of the biomedical scenarios (e.g., neighboring voxels or genetic variations are highly correlated). To overcome this problem, multivariate analysis methods have been proposed to better account for covariance in highdimensional data.
In a federated analysis context, a few works proposed generalization of standard neuroimaging multivariate analysis methods, such as Independent Component Analysis
[3], sparse regression, and parametric statistical testing [4, 5]. Since these methods are mostly based on stochastic gradient descent, a largenumber of communications across centers may be required to reach convergence. Therefore, there is a risk of computational and practical bottlenecks when applied to multicentric highdimensional data.
Lorenzi et al.[6, 7] proposed a multivariate dimensionality reduction approach based on eigenvalue decomposition. This approach does not require iteration over centers, and was demonstrated on the analysis of the joint variability in imaginggenetics data. However, this framework is still of limited practical utility in real applications, as data harmonization (e.g., standardization and covariate adjustment) should be also consistently performed in a federated way.
Herein we contribute to the stateoftheart in federated analysis of neuroimaging data by proposing an endtoend framework for data standardization, confounding factors correction, and multivariate analysis of variability of highdimensional features. To avoid the potential bottlenecks of gradientbased optimization, the framework is based on schemes analysis through Alternating Direction Method of Multipliers (ADMM) reducing the amount of iterations.
We illustrate the framework leveraging on the ENIMGA Shape tool, to provide a first application of federated analysis compatible with the standard ENIGMA pipelines. It should be noted that, even though this work is here illustrated for the analysis of subcortical brain changes in neurological diseases, it can be extended to general multimodal multivariate analysis, such as to imaginggenetics studies.
The framework is benchmarked on synthetic data (section 3.1). It is then applied to the analysis of subcortical thickness and shape features across diseases from multicentric, multidatabase data including: Alzheimer’s disease (AD), progressive and nonprogressive mild cognitive impairment (MCIc, MCInc), Parkinson’s disease (PD) and healthy individuals (HC) (section 3.2).
2 Methods
Biomedical data is assumed to be partitioned across different centers restricting the access to individual information. However, centers can individually share model parameters and run pipelines for feature extraction.
We denote the global data (e.g., image arrays) and covariates (e.g., age, sex information) as respectively and , obtained by concatenating respectively data and covariates of each center. Although these data matrices cannot be computed in practice, this notation will be used to illustrate the proposed methodology. In the global setting, variability analysis can be performed by analyzing the global data covariance matrix .
For each center with subjects each, we denote by and the local
data and covariates. The featurewise mean and standard deviation vectors of each center are denoted as
and .The proposed framework is illustrated in Figure 1 and discussed in section 2.1. It is based on three main steps: 1) data standardization, 2) correction from confounding factors and 3) variability analysis.
Data standardization is a data preprocessing step, aiming to enhance the stability of the analysis and easing the comparison across features. In practice, each feature is mapped to the same space by centering data featurewise to zeromean and by scaling to unit standard deviation. However, this is ideally performed with respect to the statistics from the whole study (global statistics). This issue is addressed by proposing a distributed standardization method in section 2.1.1.
Confounding factors have a biasing effect on the data. To correct for this bias, it is usually assumed a linear effect of the confounders
, that must be estimated and removed. However, for a distributed scenario, computing
is not straightforward, since the global data matrix cannot be computed. We propose in section 2.1.2 to use Alternating Direction Method of Multipliers (ADMM) to estimate a matrix shared among centers, closely approximating . In particular, we show that can be estimated in a federated way, without sharing local data nor covariates .Finally, through federated principal component analysis (fPCA), we obtain a low dimensional representation of the full data without ever sharing any center’s individual information
(section 2.1.3).2.1 Federated Analysis Framework
2.1.1 Standardization
The mean and standard deviation vectors can be initialized to and . They can be iteratively updated with the information of each center by following standard forms [8], by simply transmitting the quantities and from center to center. For each center the scaled data is denoted as and keeps the dimensions of .
2.1.2 Correction from confounding factors
Under the assumption of a linear relationship between data and confounders, the parameters matrix can be estimated via ordinary least squares, through the minimization of the error function .
In a distributed setting, this approach can be performed locally in each center, ultimately leading to independent solutions. However, this would introduce a bias in the correction, as covariates are accounted for differently across centers.
To solve this issue, we propose to constrain the local solutions to a global one shared across centers. In this way, the subsequent correction can be consistently performed with respect to the estimated global parameters. Thus, we can formulate the problem of constrained regression via ADMM [9].
For a given error function associated with each center and constrained to a estimated global matrix of weights we can pose:
As this is a constrained minimization problem, the extended Lagrangian can be calculated as a combination of the parameters from each center (eqn. 1).
(1) 
Where is a penalty factor (or dual update step length) regulating the minimization step length for and . is a dual variable to decouple the optimization of and .
Optimization is performed as follows: i) Each center independently calculates the local parameters and (eqn. 2 and 3); ii) the parameters and are shared to estimate the global parameters (eqn. 4). We note that this last step is performed without sharing either local data or covariates. The parameters are subsequently retransmitted to the centers and the whole procedure is iterated until convergence:
(2) 
(3) 
(4) 
After convergence, is shared across centers, and used to consistently account for covariates by subtracting their effect from the structural data to obtain the corrected observation matrix:
2.1.3 Federated PCA (fPCA)
Principal components analysis (PCA) is a standard approach for dimensionality reduction assuming that the largest amount of information is contained in the directions (components) of greater variability. Data can be thus represented by projecting on the lowdimensional space spanned by the main components: .
From the eigenvalue decomposition of the global covariance matrix , the first eigenmodes provide a lowdimensional representation of the overall variation in . In our federated setting, we note that is the algebraic sum of the local covariance matrices . Based on this observation, Lorenzi et al. proposed to share only the eigenmodes and values of the covariance matrix of each center avoiding the access to individual data [6]. However, sharing the localcovariancematrices can still be prohibitive as the dimension is . For this reason, it was proposed to further reduce the dimensionality of the problem by sharing only the principal eigencomponents associated with the local covariance matrices: . From the practical point of view, computing the eigencomponents can be efficiently performed by solving the eigenproblem associated with the matrix which is usually of much smaller dimension [10].
In what follows, the number of components shared across centers is automatically defined by fixing a threshold of 80% on the associated explained variability contained in .
3 Experiments
Database (total)  ADNI (802)  MIRIAD (68)  PPMI (232)  UK Biobank (208)  

Group  HC  MCInc  MCIc  AD  HC  AD  PD  HC 
N (females)  109 (115)  62 (119)  78 (130)  89 (100)  11 (12)  26 (19)  85 (147)  116 (92) 
Age sd  75.79 (4.99)  74.93 (7.72)  74.54 (7.09)  75.19 (7.48)  69 (7.18)  69.17 (7.06)  60.69 (8.95)  60.72 (7.52) 
3.1 Synthetic Data
We randomly generated and matrices. The data matrix was subsequently computed as , and corrupted with Gaussian noise , with set to 20% of . Then, and were split in centers of equal sample size. Our federated framework was then applied for each scenario across 200 folds, and convergence analyzed as shown in Figure 2.
3.2 Real Data: Neuroimaging
Data. T1weighted MRI scans at baseline were analyzed from several research databases (table 1). In total, we included data for 455 controls (HC), 181 with nonprogressive MCI (MCInc), 208 progressive (MCIc), 234 Alzheimer’s disease (AD), 232 with Parkinson’s disease (PD).
Feature extraction. ENIGMA Shape Analysis was applied to the MRI data of each center [11, 12]. In our analysis we extracted: a) radial distance (an approximate measure of thickness) and, b) the of the Jacobian determinant (surface area dilation/contraction) for each vertex of the following subcortical regions: hippocampi, amygdalae, thalami, pallidum, caudate nuclei, putamen and accumbens nuclei. The overall data dimension is of 54,240 features.
Federated analysis. Each database of table 1 was modeled as an independent center. , and were used to correct the vertexwise shape data according to 2.1.1 and 2.1.2. For ADMM, convergence was ensured through 10 iterations. Finally, the analysis of the variability was performed according to 2.1.3.
Results. The projection in the latent space spanned by the federated principal components is shown in Figure 3. To ease visualization, the projection for MCI converters and those who remained stable is shown in the bottom panel. Figure 4 shows the weight maps associated to the first principal component. We note that principal components 1 to 3 identify a variability from healthy to AD consistent across centers. Moreover, healthy ADNI participants are in between the AD subjects and the rest of the population. This result may denote some residual effect of on the resulting imaging features, even after correction. Interestingly, the issue of “leaking” spurious variability of confounders after correction has been already reported in a number of multicentric studies, and is matter of ongoing research [13, 14]. Finally we note that PD subjects are generally similar to the healthy individuals with respect to the modelled subcortical information.
4 Conclusions
In this work we proposed, tested, and validated a fully consistent framework for federated analysis of distributed biomedical data. Further developments of this study will extend the proposed analysis to largescale imaging genetics data, such as in the context of the ENIGMA metastudy.
5 Acknowledgments
This work was supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR15IDEX01 (project Meta ImaGen). A.A. holds an Medical Research Council eMedLab Medical Bioinformatics Career Development Fellowship. This work was supported by the Medical Research Council [grant number MR/L016311/1]. Additional support is from NIH grants RF1AG051710, R01MH116147 and R56AG058854. The full list of acknowledgments for the data providers is available at the supplementary material in section 6.
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6 Supplementary material: Acknowledgements
6.1 Funding
This work was supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR15IDEX01 (project Meta ImaGen). A.A. holds an Medical Research Council eMedLab Medical Bioinformatics Career Development Fellowship. This work was supported by the Medical Research Council [grant number MR/L016311/1]. Additional support is from NIH grants RF1AG051710, R01MH116147 and R56AG058854.
6.2 The Alzheimer’s Disease Neuroimaging Initiative (ADNI)
Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wpcontent/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf
6.3 The Parkinson’s Progression Markers Initiative (PPMI)
Data used in the preparation of this article were obtained from the Parkinson’s Progression Markers Initiative (PPMI) database (www.ppmiinfo.org/data). For uptodate information on the study, visit www.ppmiinfo.org.
PPMI – a publicprivate partnership – is funded by the Michael J. Fox Foundation for Parkinson’s Research and funding partners, including [list the full names of all of the PPMI funding partners found at www.ppmiinfo.org/fundingpartners.
6.4 UK Biobank
This research has been conducted using the UK Biobank Resource. Additional information can be found at: https://www.ukbiobank.ac.uk.
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