Abstract
Conventional functional connectivity (FC), referred to as loworder FC, estimates temporal correlation of the restingstate functional magnetic resonance imaging (rsfMRI) time series between any pair of brain regions, simply ignoring the potentially highlevel relationship among these brain regions. A highorder FC based on “correlation’s correlation” has emerged as a new approach for abnormality detection of brain disease. However, separate construction of the low and highorder FC networks overlooks information exchange between the two FC levels. Such a higherlevel relationship could be more important for brain diseases study. In this paper, we propose a novel framework, namely “hybrid highorder FC networks” by exploiting the higherlevel dynamic interaction among brain regions for early mild cognitive impairment (eMCI) diagnosis. For each sliding windowbased rsfMRI subseries, we construct a wholebrain associated highorder network, by estimating the correlations between the topographical information of the highorder FC subnetwork from one brain region and that of the loworder FC subnetwork from another brain region. With multikernel learning, complementary features from multiple timevarying FC networks constructed at different levels are fused for eMCI classification. Compared with other stateoftheart methods, the proposed framework achieves superior diagnosis accuracy, and hence could be promising for understanding pathological changes of brain connectome.
Introduction
Alzheimer’s disease (AD) is an irreversible serious neurological disease in the elderly population, characterized by progressive perceptive and cognitive deficits^{1}. The incidence of AD doubles every five years after the age of 65^{2}. AD symptoms, such as impaired memory function, get worse over time due to the neurodegenerative processes^{3}. Mild cognitive impairment (MCI) is an intermediate stage of cognitive decline between AD and normal aging^{4}. Recent researches have reported that individuals with MCI tend to progress to AD at a rate of about 10–15% per year^{5, 6}. Such a high conversion rate may possibly be reduced if early interventions could be applied to the early stage of MCI (eMCI)^{7}. Therefore, timely diagnosis of eMCI is of great clinical significance^{8}. Accurate brain imagingbased eMCI diagnosis is still challenging since brain anatomical and functional changes in this stage are considerably subtle^{9, 10}. By far, compared with numerous computeraided diagnosis studies on AD and MCI with various neuroimaging modalities^{11,12,13,14,15,16,17,18,19,20}, those on eMCI diagnosis are still quite few^{8, 21}. Although the accuracy is still not so satisfactory for clinical application, these preliminary studies have already indicated that restingstate functional magnetic resonance imaging (rsfMRI) can serve as a promising imaging technique for eMCI diagnosis.
RsfMRI is an in vivo brain functional imaging modality, measuring blood oxygen leveldependent (BOLD) signals^{22} when subjects are in natural rest. With rsfMRI, temporal synchronization of the spontaneous brain activity among different brain regions can be adopted to measure brain functional connectivity (FC), a metrics reflecting brain intrinsic functional organization^{23}. Based on FC between each pair of brain regions, a wholebrain functional network can be constructed, which opens a new avenue for brain disease study by leveraging brain connectomics and complex network analysis^{24,25,26,27,28,29,30}. For early AD diagnosis, it is usually hypothesized that the FC networkbased biomarkers show up earlier than macroscopic anatomical changes^{8, 31, 32}. However, previous studies on rsfMRI based AD early diagnosis often utilized simply calculated FC by measuring interregional BOLD signal temporal synchronization with Pearson’s correlation or, more generally, with sparse representation^{8, 33,34,35}. This type of networks is loworder by definition because they characterize BOLD signal synchronizations and are insufficient to characterize highlevel interregional interactions. In a recent study^{36}, we proposed a highorder FC network construction method by measuring the similarity between two regions’ FC topographical profiles (i.e., correlation’s correlation). Preliminary group comparison between MCI and health controls has suggested a great potential of using this metrics to provide complementary information to the loworder FC metrics in the context of early AD biomarker detection.
The above two types of FC studies separately calculate loworder and highorder FCs. However, there could be an intriguing relationship and functional association between the two FC levels. Such an interlevel interaction exits in many biological networks, reflecting hierarchical organization and selfresemblance across multiple spatial scales^{37}. Supposing that in human brain the lowlevel connections collect information and the highlevel connections abstract information via the hierarchy, the functions of the interlevel connections could be 1) to facilitate twolevel information “talking” to each other, 2) to let the lowlevel information guide highlevel abstraction and, 3) to change the way of lowlevel information collection for achieving a better highlevel integration. Moreover, from a robust system point of view, a network or a biological system could make itself less fragile and more resistant to targeted pathological attacks through the interlevel connections. Taking brain as an example, via psychophysiological and physiophysiological interactions, highlevel preset of a psychological status (e.g., attention level) may change both sensory information collection and synthesis; their covarying status may indicate such interlevel functional associations. In early AD neuropathological model, one may hypothesize that subtle pathological changes in the stage of eMCI may not only alter highorder FC while leaving the loworder FC largely intact^{21, 36}, but also affect the functional association between the high and loworder FCs. Collectively, the three types of FC networks (i.e., loworder FC, highorder FC and such an interlevel associated FC) complement each other, characterizing brain functional organization from different aspects. By integrating the three types of FCs, eMCI classification may be more accurate compared with that using only a single type of FCs.
To this end, we propose a novel approach called “hybrid highorder FC networks” to comprehensively explore the brain’s complicated functional associations and search for subtle early imaging biomarkers which could be related to pathological changes, for better eMCI diagnosis. Besides the three different types of FC, we also need to take advantage of dynamic FC by integrating brain dynamics into the study and to calculate three types of timevarying FC networks for diagnosis. This is because that the pattern of brain FC networks may change along time while brain is switching among different status^{22, 38,39,40,41,42,43,44}, and such a dynamic information may also provide sensitive features for revealing early brain functional abnormalities and for even better improved eMCI diagnosis.
Three main contributions of our study can thus be summarized: 1) A new FC metrics, namely associated highorder FC network, is proposed to characterize previously untouched interlevel interaction between the low and the highorder FC networks; 2) For the first time, we investigate dynamics of the three types of FC networks and utilize them for disease diagnosis; 3) We propose a novel applicable machine learning framework to effectively fuse various types of dynamics FC (thus, it is called hybrid highorder) networks with a multikernel learning strategy for computeraided eMCI diagnosis. Experiments are carried out to compare the accuracy of eMCI diagnosis between our proposed approach and other stateoftheart methods. Experimental results indicate that our method achieves superior performance than those using only the static FC or only the traditional low and highorder FC networks.
Results
Data acquisition
To demonstrate the effectiveness of our method, we apply it to real rsfMRI data from a putative publicaccessible dataset, consisting of eMCI and normal aging subjects. One of our hypotheses is that, with features extracted from our newly developed associated highorder FC networks, eMCI classification could be more accurate, compared with those extracted using either the traditional loworder or highorder FC. Another hypothesis is that our computational framework of the hybrid highorder FC networks could effectively conduct multikernel fusion of the three types of brain dynamic networks and further boost classification performance. In addition, higher diagnosis performance would be achieved using our method compared with those using other stateoftheart methods^{8, 21}.
The rsfMRI data are obtained from Alzheimer’s Disease Neuroimaging Initiative (ADNI) project (http://adni.loni.usc.edu). ADNI was launched in 2003 by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, the Food and Drug Administration, private pharmaceutical companies and nonprofit organizations. The original goal was to define biomarkers for use in clinical trials to determine the most appropriate way to measure treatment effects of AD. The current goal has been extended to discover more effective methods to early detect AD at its predementia stage. We use ADNI phase2 dataset since it includes eMCI subjects.
Data from twentynine eMCI subjects (13 F/16 M, aged 73.7 ± 4.8 years) and 30 agematched (p = 0.6174) normal controls (NCs) subjects (17 F/13 M, aged 74.4 ± 5.7 years) are used. All subjects were scanned with the same protocol using 3.0 T Philips Achieva scanners. The following parameters were used: repetition time (TR) = 3000 ms, echo time (TE) = 30 ms, flip angle = 80°, imaging matrix = 64 × 64, 48 slices, 140 volumes, and slice thickness = 3.3 mm. The rsfMRI data are preprocessed using SPM8 software (http://www.fil.ion.ucl.ac.uk/spm/software/spm8) according to the previous studies^{8, 21}. Briefly, the first 10 volumes of each subject are discarded to ensure magnetization equilibrium. After head motion correction, spatial registration to the standard space, spatial smoothing and temporal filtering (0.01–0.08 Hz), averaged signals from brain ventricle and white matter as well as headmotion parameters are regressed out from rsfMRI data to reduce the nuisance effect on FC estimation. According to the Automated Anatomical Labeling (AAL) brain atlas, the mean regional rsfMRI time series are extracted from each of the 116 brain regions.
Performance evaluation
A nested leaveoneout cross validation (LOOCV) scheme is adopted for performance evaluation of the proposed approach. Specifically, N subjects are involved in our study, N − 1 of them are used for training the classifier while the leftout one is used for evaluating the classification performance. The procedure is repeated N times until each subject serves once for testing. In each repeat of the above procedure, an additional inner LOOCV is carried out on the N − 1 training samples to determine optimal parameters, which include regularization parameters in the LASSO feature selection and also the weighting factors in the multiplekernel learning. The parameter values leading to the best performance on the N − 1 tests are selected and used for learning the optimal classification model. The softmargin parameter in SVM is set as C = 1. For the dynamic FC network construction, the length of sliding window L and the step size S are set to be 70 and 1 (consistent with literature^{21}), respectively, which results in the number of time subseries as K = 61.
Extensive experiments are carried out to validate the effectiveness of our proposed method in comparison with other stateoftheart methods. We investigate the diagnosis performance of various methods based on either static FC networks or dynamic FC networks. These compared methods include: (1) Static loworder Network (SN_{L}, where the subscript “L” indicates conventional “loworder” FC); (2) Static highorder Network (SN_{H}, where “H” indicates conventional “highorder” FC used in literature^{36}); (3) Static associated highorder Network (SN_{A}, where “A” denotes “associated”, i.e., our newly proposed higherlevel type of FC); (4) Hybrid Static Networks (SN_{L} + SN_{H} + SN_{A}) which fuse the three aforementioned networks; (5) Dynamic loworder Network (DN_{L}); (6) Dynamic highorder Network (DN_{H}); (7) Dynamic associated highorder Network (DN_{A}); (8) Hybrid Dynamic Networks (DN_{L} + DN_{H} + DN_{A}), which is our proposed framework combining three types of networks in a dynamic way; (9) Sparse temporally dynamic networks (DN_{Wee}) that are constructed using group graphical LASSO^{8}; (10) Highorder network^{21} that is constructed by estimating the correlation between two regions’ loworder FC dynamics and that of another two regions (DN_{Chen}). Among them, SN_{A} and DN_{A} are the novel network modeling methods, while SN_{L} + SN_{H} + SN_{A} and DN_{L} + DN_{H} + DN_{A} are the novel network fusion frameworks for classification.
We evaluate the classification performance based on classification accuracy (ACC), area under ROC curve (AUC), Sensitivity (SEN), and Specificity (SPE). ACC is defined as the ratio of the number of correctly predicted labels to the number of whole samples. AUC measures the probability that a classifier will rank a randomly chosen positive sample higher than a randomly chosen negative one. SEN and SPE are defined as true positive rate and one minus false positive rate, respectively:
Experimental results
Table 1 summarizes the performance on eMCI classification for all of the ten aforementioned methods. Consistent with our hypotheses, the main results are: 1) The highorder FC networks enhanced the classification performance and our associated highorder FC networks gained the highest one if only single type of FC network was used; 2) Integrating all the three types of networks with multikernel learning, eMCI classification yielded better performance compared to that using only single type of networks (even using only static networks, we reached an ACC of 83.1%; while using dynamic networks, we obtained the best ACC of 91.5% amongst all competing methods); and 3) The classifications based on the dynamic FC networks consistently outperformed those based on the static FC networks, indicating the necessity of integrating dynamic FC into classification. Of note, our method always achieved better ACC, AUC, SEN and SPE compared with the most recently developed, stateoftheart methods (DN_{Wee} and DN_{Chen}).
It should be noted that the classification performance of our proposed framework depends on the selection of some parameters. For example, three weighting factors \({\tau }_{1}\in [0,1]\), \({\tau }_{2}\in [0,1]\) and \({\tau }_{3}\in [0,1]\) (\({\tau }_{1}+{\tau }_{2}+{\tau }_{3}=1\)) in the multikernel SVM need to be estimated to fuse the kernel matrices that are derived from the DN_{L}, DN_{H} and DN_{A}, respectively. A larger value of a weighting factor indicates the larger importance of the corresponding kernel matrices for classification. Although we have used inner LOOCV to estimate the best parameters for above classifiers, we can use all of the N subjects to better evaluate the possible dependency on parameter selection. Figure 1 shows accuracy of the classification model based on DN_{L} + DN_{H} + DN_{A} estimated using LOOCV on all of the N subjects, with different combinations of the three weighting factors τ _{1}, τ _{2} and τ _{3} (where is \({\tau }_{3}=1({\tau }_{1}+{\tau }_{2})\)). After the exhaustive searching, the best accuracy of 93.2% is achieved with \({\tau }_{1}=0.3\) (for DN_{L}), \({\tau }_{2}=0.5\) (for DN_{H}), and \({\tau }_{3}=1({\tau }_{1}+{\tau }_{2})=0.2\) (for DN_{A}). The results indicate that DN_{L}, DN_{H} and DN_{A} indeed provide complementary information to each other and all of them are necessary for classification. On the other hand, our method with the parameters estimated by inner LOOCV yielded a 91.5% accuracy, which is close to the best accuracy 93.2% with specific parameters estimated on all subjects.
In addition, the window length L used in sliding window strategy is an important factor for dynamic FC network analysis. The window length should be large enough to permit a reliable estimation of FC and resolve the lowest frequencies of interest in BOLD signals, while small enough to capture the dynamics of FC^{40}. Leonardi and Van De Ville^{42} have recommended using a window length that exceeds the longest wavelength composing the BOLD signals in order to suppress spurious fluctuations of dynamic FC. According to this criterion, the widow length in our study should be set to be larger than 1/0.01 = 100 s since our highpass cutoff frequency in bandpass filtering is 0.01 Hz. On the contrary, Zalesky and Breakspear^{43} suggested that nonstationary fluctuations in dynamic FC could be fairly robustly detected with a shorter window length (40–60 s). Although these two studies as well as the former review paper^{40} are trying to set up a guideline to decide the window length, it is still far away from consensus. The above two studies are mainly based on simulated data and simulated SNR condition, which might not always be the case in real fMRI data because there is still no “ground truth” of the FC dynamics.
Note that, as a tradeoff, Leonardi and Van De Ville^{42} also mentioned to focus on the frequency interval [0–1/w] where w denotes the window length in second when interpreting the dynamic FC spectrum. To show the dynamic FC spectrum with varied window length, we calculated the separability (as measured by r ^{2}) (see Fig. 2) between NC and eMCI subjects across different frequencies for the discriminative link connecting the left inferior frontal gyrus and left angular gyrus. Consistent with the result in literature^{42}, a longer window length presented a lowpass filtering effect with a lower cutoff frequency. However, we found that useful discriminative information was located at the frequencies higher than the suggested upper limit frequency 1/w, which could contribute to good diagnosis performance of eMCI.
Therefore, we still consider using a window length larger than 100 s (L = 70 volumes, i.e., 210 s). Such a long window length was chosen based on the performance of the eMCI classification using crossvalidation (LOOCV) with training data. Specifically, we compared the eMCI classification accuracies derived by DN_{L} + DN_{H} + DN_{A} using various L (L = 15, 20, 30, 50, 70 and 90) (see Fig. 3). Consistent with the observation in previous eMCI classification studies^{8, 21}, the selection of L = 70 yielded the best classification accuracy, which was thus adopted for the subsequent analysis in our experiment.
Discussion
To investigate the contribution of the associated highorder FC network to the diagnosis, we calculate the grouplevel separability (defined by the differences in the group averaged associated highorder FC networks of the two groups) between the NC and the eMCI subjects in both static and dynamic cases, and compare them with those obtained from the traditional loworder and highorder FC networks. Figure 4 shows the grouplevel SN_{L}, SN_{H} and SN_{A} for NC (first row) and eMCI groups (second row), respectively. The discriminability index, calculated by squared pointwise biserial correlation coefficients (r ^{2} values)^{44, 45} for all connections in each type of the FC networks is shown in the third row. Larger r ^{2} value indicates higher separability of the feature distribution patterns between two classes. From Fig. 4, we can see the separability using the static loworder and highorder FC networks are smaller and involve fewer FC connections. However, the static associated highorder networks reveal more discriminative nodes and higher separability. This explains why SN_{A} yielded better diagnosis performance than SN_{L} and SN_{H}. On the other hand, we found that the three types of FC networks identified several different discriminative FC connections that may serve as complementary features for eMCI diagnosis. This indicates that it is suboptimal to utilize only the new FC network modeling method for disease diagnosis since a better performance can be achieved by combining different types of FC networks. In this sense, our newly developed associated highorder FC metrics does not intend to replace the conventional ones but may provide unique, essential and meaningful information to conventional FC metrics for comprehensive brain connectome research. As a result, further improvement of diagnosis performance can be achieved by integrating these complementary features under our proposed framework of hybrid highorder FC networks.
One of the highlights of our study is that we estimated various types of dynamic FC networks and demonstrated the feasibility of using these dynamic networks to improve classification accuracy. An increasing number of studies^{22, 39,40,41} have suggested that FC network is not stationary but spontaneously changes over time. We, for the first time, use a frequency power spectrum method to effectively take advantage of such discriminative spontaneous changes and demonstrate that such information can be adopted to further improve classification. Figure 5 shows the timevarying FC matrices of DN_{L}, DN_{H} and DN_{A} estimated from different sliding windows, for one randomly selected NC subject and one eMCI subject. We found that all the dynamic FC networks could capture the temporal variation of FC patterns. This assists to explore rich features from timeevolving FC networks, resulting in better (with accuracy improved by about 7–11% if using dynamic networks compared to static ones) diagnosis performance. Most importantly, compared with the DN_{L}, both the DN_{H} and the DN_{A} may enlarge network topology differences along time, which could be used for better differentiation between NC and eMCI. This has actually been proved with profound improvement in diagnosis accuracy (with the increment of about 9–11%) by using the features from dynamic highorder FC networks compared with that using dynamic loworder FC networkbased features.
We also investigate the potential biological meaning of the machine learning algorithm selected brain regions as biomarkers for early AD detection and compare the results among different types of dynamic networks. Figure 6 presents the principal component coefficients corresponding to the most discriminative features selected by LASSO for DN_{L}, DN_{H} and DN_{A}, respectively. Most of the important information is mainly concentrated in the range of very low frequencies (<0.033 Hz). This is reasonable since the biologically meaningful fluctuations of the dynamic FC time series are believed to be relatively slow^{22}. From the zoomedin areas of Fig. 6, we also found that the spatialfrequency locations of the most discriminative features from different types of FC networks are quite different.
Figure 7 shows spatial locations of the most discriminative ROIs included in the top ten principal component coefficients selected by LASSO for DN_{L}, DN_{H} and DN_{A}, respectively. A total of 22 ROIs are selected from of the different FC networks and they have been consistently reported by previous studies on biomarker detection for AD and MCI, including the left superior temporal gyrus^{46}, right inferior temporal gyrus^{47}, right lobule VIIb of the cerebellar hemisphere^{48}, left rolandic operculum^{48}, right middle temporal pole^{49}, left paracentral lobule^{44}, left putamen^{50}, right paracentral lobule^{51}, right cuneus^{52}, right globus pallidus^{53}, left parahippocampal gyrus^{54, 55}, left inferior frontal gyrus^{56}, left olfactory cortex^{57}, right supplementary motor area^{48}, left transverse temporal gyrus^{58}, right olfactory cortex^{57}, left fusiform gyrus^{59, 60}, left medial orbitofrontal cortex^{56}, left lobule IX of cerebellar hemisphere^{48}, left superior frontal gyrus^{48, 59}, right insula^{52, 55} and the right putamen^{50}.
From Fig. 7, we also found that many selected ROIs were different if using different FC networks for classification. Consistent with our observation of the discriminative static FC connections in Fig. 4, different types of dynamic networks also produced complementary features, which are integrated using the framework of our hybrid highorder FC networks to further improve eMCI identification. This is supported by the superior diagnosis performance (91.5% when using all three networks vs. 72.9–83.1% when using them separately) obtained by the proposed approach of hybrid highorder FC networks. Further investigation on the similarities and the differences among the features selected from different network types will be needed using more data sets and via extensive applications.
To more clearly explain the characteristics of these two new FC metrics beyond the loworder FC, we provide two intuitive examples here. First, let us consider the lowlevel visual area V1 (primal visual cortex) and the highlevel visual processing areas, i.e., posterior parietal cortex (PPC). The wellestablished model for dorsal visual stream begins with V1, goes through secondary and associated visual areas, and finally to the PPC^{61}. Therefore, the BOLD signal synchronization between the V1 and PPC is supposed to be not strong since they are responsible for processing visual information at different levels (instead, the bilateral V1 could have strong BOLD synchronization since they are at the same level). However, due to tight feedforward and feedback between them (e.g., they are all modulated by attention^{62}), their interregion highorder functional association could be strong. That is, there could exist some potentially indirect relationships between these two brain regions, which may not be effectively revealed by the loworder FC. Consistent with the above hypothesized model, based on the data from a randomly selected subject, we found that the loworder FC (direct BOLD signal synchronization) between the left lingual gyrus (encompassing V1) and the left inferior parietal lobule (covering most of the PPC) is low (0.36). However, both highorder FC (FC topographical profilebased similarity) and associated highorder FC are strong (0.61 and 0.74, respectively). This indicates that the lifted functional association, when measured from a high level, could reflex the close relationship between the two regions in the visual pathway.
Another example is from our previous study^{36}, where the three FC metrics are calculated between the left posterior cingulate cortex (PCC) and the anterior cingulate cortex (ACC). Since the PCC is within the default mode networks while the ACC is included in several other attentionrelated functional networks, their loworder FC is observed as weak (0.36). However, enhanced strengths of the highorder FC (0.59) and the associated highorder FC (0.70) between the two regions indicate that the two new highorder FC metrics could be able to capture a close relationship among these highlevel cognitionrelated functional networks^{63}.
Interestingly, both above examples show higher associated highorder FC, compared with the loworder FC, which further indicates that the former could be able to capture more complicated functional interaction between two regions. The associated highorder FC measures the modulatory interaction between the loworder FC and the highorder FC, i.e., a crosslevel functional association. Since our present study aims to demonstrate the feasibility of using highorder FC metrics for disease diagnosis, the biological meaning of these metrics requires more dedicated studies with the aid from existing neurocognitive models in future.
It should be noted that our introduced highorder FC and associated highorder FC reveal higherlevel functional interactions between FC profiles between any pair of brain regions, yet ignore potential complex relationships among multiple brain regions. A recent method^{64}, called hyper network, has been proposed to reveal more complex FC among multiple regions, and hence provide new approaches to investigate FC, which, however, is fundamentally different from our highorder FC and associated highorder FC metrics. Specifically, this method was developed based on hypergraph theory for exploring the complex interactions among multiple brain regions, where an edge in the hyper network connects with more than two brain regions. A combination of our method and the dynamic hyper network method could further improve the diagnosis performance, which will be investigated in our future work.
In summary, we propose a novel approach, namely hybrid highorder FC networks, to effectively integrate multiple types of FC networks by using multikernel learning strategy for eMCI diagnosis. Associated highorder network, characterizing higherlevel functional interactions between high and lowlevel FC networks, is newly proposed to reveal the previously untouch relationship among brain regions. Three types of dynamic wholebrain FC networks are systematically defined and jointly used to provide complementary discriminative features for early MCI identification. Our method achieves superior performance (accuracy = 91.5%) in this challenging problem, which is even racing ahead of the most recently developed stateoftheart solutions. This study reveals the complexity of our brain connectome, and the feasibility of using it as an effective computeraided individual diagnosis tool for future clinical applications towards precise medicine.
Methods
In our hybrid highorder FC network approach, we combine three types of FC networks and dynamics FC analysis for comprehensive feature extraction. To achieve dynamic FC networks, sliding window strategy is adopted to segment the entire rsfMRI time series into multiple subseries, from each of which the three types of FC networks are constructed. We first construct a traditional loworder FC network and a topographical FC profilebased highorder FC network. This produces two different FC fingerprints for each brain region: the one is the topographical loworder FC profiles between this region and other regions; the other is the highorder FC profiles between the subnetwork centering at this region and those centering at other regions. We then calculate a higherlevel associated FC between the two types of the FC fingerprints for each pair of brain regions, which consequently forms an associated highorder FC network. Finally, all the dynamic FC networks are integrated into a unified model with multikernel learning strategy to make features from one FC networks support those from others, for better classification performance. Each step is detailed in the following subsections.
Loworder FC network construction
With sliding window approach, an rsfMRI time series can be segmented into multiple subseries, each generating one FC matrix (Fig. 8). In particular, \(K=[(PL)/S]+1\) subseries can be generated from an rsfMRI time series with P time points, where L and S are the window length and step size, respectively. Suppose that \({{\bf{X}}}^{k}=[{{\bf{x}}}_{1}^{k},{{\bf{x}}}_{2}^{k},\ldots ,{{\bf{x}}}_{R}^{k}]\in {{\mathbb{R}}}^{L\times R}\) (\(k=1,2,\ldots ,K\)) denotes the kth time subseries for a total of R = 116 ROIs, and \({{\bf{x}}}_{i}^{k}={[{x}_{\mathrm{1,}i}^{k},{x}_{\mathrm{2,}i}^{k},\ldots ,{x}_{L,i}^{k}]}^{T}\in {{\mathbb{R}}}^{L}\) is the kth time subseries corresponding to the ith ROI. The correlation strength \({C}_{ij}^{k}\) between the ith and jth ROIs of the kth time subseries can be typically computed using Pearson’s correlation. Such correlation strength, in graph theoretic analysis of complex brain FC networks, is called the edge weight. By computing such correlation strength of the kth time subseries between each pair of ROIs (or nodes), an FC network can be constructed as a symmetric correlation matrix \({{\bf{C}}}^{k}=[{C}_{ij}^{k}]\in {{\mathbb{R}}}^{R\times R}\). Without loss of generality, we assume that \({{\bf{x}}}_{i}^{k}\) has been centralized by \({{\bf{x}}}_{i}^{k}{\bar{{\bf{x}}}}_{i}^{k}\) and further normalized by \(\sqrt{{({{\bf{x}}}_{i}^{k}{\bar{{\bf{x}}}}_{i}^{k})}^{T}({{\bf{x}}}_{i}^{k}{\bar{{\bf{x}}}}_{i}^{k})}\) for \(i=1,2,\ldots ,R\). The computation of FC network on the kth time subseries can then be equivalently written as:
The dynamic FC networks can be derived by estimating the correlation matrices for all the \(k=1,2,\ldots ,K\) time subseries. Note that Eq. (2) defines dynamic loworder FC networks, while a static network is an extreme case where window length is maximized to the entire time scale (L = P). Thus, multiple FC matrices in Fig. 8 merge into one.
Hybrid highorder FC networks for eMCI diagnosis
The “hybrid highorder FC networks” refer to as a framework which fuses three types of FC networks for improving diagnosis performance. In this framework, a key step is to construct the associated highorder network. Of note, this type of network can be regarded to as a higherlevel FC network as it characterizes the interaction between the conventional loworder and the highorder networks. The construction of associated highorder network will be followed by feature extraction and selection, and a multikernel learning strategy for multitype feature fusion.
Associated highorder FC network construction
Loworder FC network on the kth time subseries can be rewritten as \({{\bf{C}}}^{k}=[{{\bf{c}}}_{1}^{k},{{\bf{c}}}_{2}^{k},\ldots ,{{\bf{c}}}_{R}^{k}]\in {{\mathbb{R}}}^{R\times R}\), where the ith column \({{\bf{c}}}_{i}^{k}\) (or the ith row due to the symmetry of C ^{k}) defines the connectivity pattern between the ith ROI and all other ROIs. Therefore, we regard \({{\bf{c}}}_{i}^{k}\) as a loworder “subnetwork” between node i and other regions. Of note, it is quite important to have this “subnetwork” definition based on the loworder FC network, since, similarly, we can define a highorder FC by the topographical similarity between any pair of these loworder subnetworks. Then, a highorder FC network can be constructed by calculating the FC between every pair of the loworder subnetworks. Figure 9 illustrates the construction of the highorder FC network.
Assuming \({{\bf{c}}}_{i}^{k}\) (\(i=1,2,\ldots ,R\)) has been centralized and normalized, a highorder network construction on the kth time subseries can be similarly written as:
where a certain element of H ^{k}, \({H}_{ij}^{k}\), denotes the topographical similarity (measured by Pearson’s correlation) between the i and jth loworder subnetworks, and \({{\bf{C}}}^{k}={({{\bf{X}}}^{k})}^{T}{{\bf{X}}}^{k}\). Based on the same form of Eqs (2) and (3), both low and highorder FC calculations can be mathematically unified. By using the whole length of rsfMRI time series, a static highorder network can similarly be calculated.
To construct associated highorder FC network, we further refer the highorder FCs corresponding to the same “node” i (here the “node” is actually a loworder subnetwork centering at region i) as a highorder subnetwork \({{\bf{h}}}_{i}^{k}\), and \({{\bf{H}}}_{i}^{k}=[{{\bf{h}}}_{1}^{k},{{\bf{h}}}_{2}^{k},\ldots ,{{\bf{h}}}_{R}^{k}]\in {{\mathbb{R}}}^{R\times R}\). Supposing that the \({{\bf{h}}}_{i}^{k}\) (\(i=1,2,\ldots ,R\)) is centralized and normalized, we can characterize the interlevel interactions between the loworder subnetworks \({{\bf{c}}}_{i}^{k}\) (\(i=1,2,\ldots ,R\)) and highorder subnetworks \({{\bf{h}}}_{i}^{k}\) (\(i=1,2,\ldots ,R\)), which can be written in the following form:
where A ^{k} defines the associated highorder FC network, a higherlevel FC network, using the kth time subseries. The element \({A}_{ij}^{k}\) in A ^{k} denotes the interaction between ith loworder subnetwork and jth highorder subnetwork (Fig. 10). From Eq. (4), we can see that the associated correlation matrix \({{\bf{A}}}^{k}\in {{\mathbb{R}}}^{R\times R}\) is an asymmetrical matrix. To improve interpretation, we further transform the asymmetrical A ^{k} to symmetrical by using \({{\bf{A}}}^{k}=({{\bf{A}}}^{k}+{({{\bf{A}}}^{k})}^{T})/2\), similarly to that also used in a previous study^{32}. Of note, our experiment has shown that this additional symmetry operation does not significantly affect final classification accuracy. Similar to the other two types of dynamic FC networks, a static associated highorder FC network can be estimated using the whole length of rsfMRI time series.
Feature extraction and selection
In this section, we introduce the procedure of feature extraction and selection for the dynamic loworder FC networks. Please note that the same procedure is also carried out to extract and select features from the dynamic highorder and the dynamic associated highorder FC networks.
Because of the unconstrained mental activity during resting state, features directly extracted from each sliding windowbased FC network for a subject do not have temporal correspondence with those extracted from the same sliding window for other subjects. Therefore, for one subject, the different features extracted from different sliding windows cannot be concatenated along time. To allow feature concatenation, one must have a hypothesis that features extracted from the same temporal window for different subjects belong to the same instantaneous brain network, which cannot be guaranteed^{65}. To enforce the feature correspondence across subjects, we transform the temporally dynamic FC networks for each subject into the frequency domain, obtaining multiple frequencyspecific FC networks. Specifically, for the dynamic loworder networks \({{\bf{C}}}^{1},{{\bf{C}}}^{2},\ldots ,{{\bf{C}}}^{K}\), where \({{\bf{C}}}^{k}=[{C}_{ij}^{k}]\in {{\mathbb{R}}}^{R\times R}\), dynamic FC time series between regions i and j can be obtained by concatenating the elements \({C}_{ij}^{k}\) across K temporal windows as \({{\bf{g}}}_{ij}=[{C}_{ij}^{1},{C}_{ij}^{2},\ldots ,{C}_{ij}^{K}]\in {{\mathbb{R}}}^{K}\), representing how the FC fluctuates along time. Fast Fourier transform (FFT) is then applied to transform this dynamic FC time series into power spectrums \([{Z}_{ij}^{1},{Z}_{ij}^{2},\ldots ,{Z}_{ij}^{Q}]\), where Q is the number of effective frequency bins. Thus, we can construct timeinvariant FC networks for all spectrums as \({{\bf{Z}}}^{1},{{\bf{Z}}}^{2}\ldots ,{{\bf{Z}}}^{Q}\), where \({{\bf{Z}}}^{q}=[{Z}_{ij}^{q}]\in {{\mathbb{R}}}^{R\times R}\). These timeinvariant FC networks characterize the frequency characteristics of the temporally dynamic FC networks. Note that the FFT is not implemented for the static network.
Graph theorybased feature extraction and selection are implemented based on the FC spectrum networks. In this study, we adopt weighted local clustering coefficient (WLCC)^{66} as a nodal feature for each brain region and each frequency. The WLCC quantifies the “cliqueness” of each node in a weighted network. The cliqueness is originally a graph theoretic concept, which characterizes a network’s local topology for every node. This metrics has been widely used as a sensitive feature in eMCI diagnostic studies^{8, 21}. For each network Z ^{q} (\(q=1,2,\ldots ,Q\)), the WLCC for the ith node can be defined as:
where \({{\rm{\Omega }}}_{i}\) is a set of nodes directly connected to the ith node and v _{ i } denotes the number of elements in \({{\rm{\Omega }}}_{i}\). After extracting the features from all the nodes at all Q frequency bins, we concatenate them to form a feature vector according to:
This feature vector is of a relatively high dimension and may contain irrelevant or redundant features which need to remove. To do this, we construct a feature vector for each subject according to Eq. (6), thereby obtaining a feature vector set as \({\bf{F}}={[{{\bf{f}}}_{1},{{\bf{f}}}_{2},\ldots ,{{\bf{f}}}_{N}]}^{T}\), where N is the number of subjects. Principal component analysis (PCA)^{67} is implemented on F to reduce feature dimension. The original features with the dimensionality of Q × R are transformed into a new feature space defined by all N − 1 principal components with nonzero eigenvalues. Subsequently, a supervised feature selection strategy based on the least absolute shrinkage and selection operation (LASSO)^{68, 69} is adopted to select discriminative features from the N − 1 principal components. The features corresponding to nonzero LASSO regression coefficients are retained as crucial features for classification.
Multikernel SVM for classification
With the abovementioned feature extraction and selection, we obtain three feature vector sets for the three FC network types, respectively. Since one of our hypotheses is that these FC networks could provide complementary information to each other for classification, we can fuse all features to generate better classification performance. The simplest way for this is to concatenate all features from different types of FC networks into a longer feature vector. However, such simple concatenation may not be optimal for achieving effective feature combination^{13}. On the other hand, a kernelbased feature combination using multikernel learning offers more flexibility for feature fusion by estimating different weights on the features from different modalities^{70,71,72}, which could provide a better way to integrate the features derived from different types of FC networks.
Therefore, we adopt multikernel learning to fuse the features by a linear combination of kernels that are estimated from the loworder, the conventional highorder and the novel associated highorder FC networks, respectively. An SVM classifier with a linear kernel K(a, b) = a ^{T} b based on LIBSVM^{73} is used for the multikernel learning based classification. Specifically, we first perform normalization on each feature vector to make sure that all the features from different types of FC networks are comparable. Based on the normalized features, a linear kernel is calculated across subjects for each type of the FC networks. Effective feature fusion is then achieved by computing a composite kernel through an optimal linear combination of the multiple kernels. Finally, classification is carried out using SVM with the composite kernel. Figure 11 summarizes the overall framework of our proposed eMCI diagnosis method based on hybrid highorder FC networks.
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Acknowledgements
This study is partially supported by NIH grants (EB006733, EB008374, EB009634, MH107815, AG041721, and AG042599). This study is also partially supported by Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (No. 2017000451).
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Y.Z. implemented the code and experimental study. Y.Z. and H.Z. drafted the manuscript. X.C., S.W.L. and D.S. participated in idea discussion and revised the manuscript.
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Zhang, Y., Zhang, H., Chen, X. et al. Hybrid Highorder Functional Connectivity Networks Using Restingstate Functional MRI for Mild Cognitive Impairment Diagnosis. Sci Rep 7, 6530 (2017). https://doi.org/10.1038/s41598017065090
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DOI: https://doi.org/10.1038/s41598017065090
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