In this paper, 113 patients with AD, 110 patients with MCI and 117 CU with the age ranged from 55 to 99 years were included. All the data, including the sMRI, FDG-PET, and florbetapir-PET, were downloaded from ADNI 1, ADNI GO, or ADNI 2. For each subject, the data-acquisition interval of the three modalities was within four months. Moreover, the subjects were matched in terms of age, the years of education and gender. The subjects we selected satisfied the following criteria: (1) The MMSE score of each AD subject was between 20 and 26, with a CDR of 0.5 or 1.0. The AD group did not significantly differ with respect to the presence of APOE4 alleles from the MCI group (p = 0.765), but had significantly lower MMSE scores (compared with CU group, p = 1.24 × 10⁻⁹⁰; MCI group, p = 1.61 × 10⁻⁴⁰) and a different presence of APOE4 alleles compared with the CU group (p = 0.014). (2) The MMSE score of each MCI subject was between 24 and 30, and the CDR was 0.5. The MCI group had significantly lower MMSE scores (p = 4.69 × 10⁻³¹) and a different presence of APOE4 alleles (p = 7.34 × 10⁻⁰⁴) compared with CU group. (3) The MMSE score of each CU was between 26 and 30 and their CDR was 0.0. Table 1 shows the demographic information of the subjects.

Images were preprocessed using the VBM8 (Voxel-Based Morphometry 8) Toolbox (http://dbm.neuro.uni-jena.de/vbm8/) in SPM8 (Statistical Parametric Mapping 8) (http://www.fil.ion.ucl.ac.uk/spm/) that running on MATLAB 2010b (The MathWorks, Inc., Sherborn, MA, USA). Based on adaptive maximum posterior and partial volume estimation, every structural image was segmented into rigid-body-aligned gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF) for each subject (Rajapakse et al., 1997; Tohka et al., 2004). Spatially adaptive non-local approach was applied to improve the segmentation. The diffeomorphic anatomical registration through exponential lie algebra (DARTEL) protocol (Ashburner, 2007) in which template creation and image registration were performed to normalize the gray-matter images iteratively by using a diffeomorphic anatomical registration.

All FDG-PET and florbetapir-PET images were co-registered with each individual's sMRI using a rigid body transformation, and subsequently warped to the cohort-specific DARTEL template. Then, the standard uptake value ratio (SUVr) image was calculated for each FDG-PET image and florbetapir-PET image; reference masks for quantification were defined relative to the whole brain (Langbaum et al., 2009; Sabbagh et al., 2015) or cerebellum (Reitan, 1958; Camus et al., 2012), respectively.

Then, based on the Automated Anatomical Labeling (AAL) (Tzourio-Mazoyer et al., 2002), 90 regions of interest (ROIs) (45 for each hemisphere; Table S1) were obtained. The feature of sMRI, FDG-PET, and florbetapir-PET were got by averaging the corresponding value of mean volume of GM, SUVr values of FDG-PET and florbetapir-PET from each ROI that all the voxels within the ROI of each subject.

Suppose n training samples with d-dimension from k classes are represented by A=[a1,a2,…,an]=[A1,…,Al,…,Ak]∈ℜd×n, in which, column vector a_i is the sample i (i = 1, …, n), and submatrix A_j consists of column vectors (samples) from class j (j = 1, …, k), and there are m atoms (each column of the dictionary can be viewed as an atom) in the corresponding dictionary D=[d1,d2,…,dm] ∈ℜd×m(m≤n). The general supervised DL model can be denoted as follows:

where θ is the discriminative parameter and g(θ) represents the discriminative term, X denotes the coding coefficients of training samples A on the dictionary D. g(θ) here indicates the linear classification error function (like ||H-WX||F2 in the DL methods of D-KSVD (Zhang and Li, 2010) and LC-KSVD (Jiang et al., 2013), where H is the class label matrix and W is a classifier).

For classification, the classifier learned with the dictionary may be optimal simultaneously, as in the DL algorithms that incorporate a linear classification error term (Zhang and Li, 2010). However, the inner-structure of representation coefficients between classes has not been considered in such approach. To further enhance the discriminant power of the dictionary, both the linear classifier and the direct restriction of within-Class scatter on coding coefficients in the above discriminant DL scheme in our previous study are indicated (Xu et al., 2016), which is referred to as the SCDDL algorithm.

Suppose A=[A1,…,Al,…,Ak]∈ℜd×n denotes the n d-dimensional training samples from k classes, D ∈ ℜ^d×m(m ≤ n) is the discriminative dictionary with m atoms that needs to be derived, and X represents the coding coefficients of training samples A on the dictionary D, denoted as X=[X1,…,Xl,…,Xk]∈ℜm×n, same as above. The SCDDL model can be written as follows:

where ||·||F2 represents the Frobenius norm. ∥A-DX∥F2 is the reconstructed error term of the training samples A on the newly constructed dictionary D, α∥H-WX∥F2+β∥W∥F2 is the linear classification error term, and ∑i=1k(∥Xi-Mi∥F2+η∥Xi∥F2) is the within-Class-similar term. W ∈ ℜ^k×m is the parameter of the classifier; each column of H ∈ ℜ^k×m is a vector, corresponds to one training sample with the form as [0, 0, …, 1, …, 0, 0] ∈ ℜ^k, where 1 locates the corresponding class of the training sample; and each column of M_i is the mean vector of the coefficients X_i corresponding to class i. According to the elastic-net theory, the term ∥Xi∥F2 combined with the term ∥X∥₁ might make the solution of Equation (2) more stable (Zou and Hastie, 2005); and η is set as η = 1 for simplicity (Yang et al., 2014). Then Equation (2) can be written as:

The optimization process of Equation (3) has been discussed in our previous study (Xu et al., 2016). In SCDDL, the directly restricted within-Class-similar term makes the coding coefficients similar within one class and the linear classification error term selects the optimal classifier. This combination has been shown to improve the discriminative classification of the dictionary (Xu et al., 2016).

After obtaining the dictionary D and classifier W in the SCDDL model, the test samples can be finally classified.

For a given test sample y, the representation coefficient on D is:

where λ is a scalar constant. The representation coefficient x can be simply combined with the linear classifier W. Then the final identification of the test sample y is obtained in the DL procedure with:

where {·}_l represents the l-th element in the brace, x contains discriminant information for classification.

The SCDDL model is extended to a kernel framework for the further multi-feature fusion in our previous study (Wu et al., 2017). Suppose ϕ(·) is a mapping function from R^N to a higher dimensional feature space. To avoid the explicit high-dimensional mapping procedure, mercer kernels could be helpful. The common mercer kernels include the linear kernel k(x, y) = 〈x, y〉, which equals to non-mapping; the Gaussian kernels k(x,y)=exp(-||x-y||2c); the polynomial kernels k(x, y) = (〈x, y〉 + c)^d (c and d are parameters) and the sigmoid kernels k(x, y) = tanh(a(x^Ty) + r) (a and r are parameters) (Manevitz and Yousef, 2001; Hussain et al., 2011; Liu et al., 2013; Pham and Pagh, 2013; Dyrba et al., 2015).

The training samples A and dictionary D can be mapped to a higher dimensional space by a function of ϕ(·), then A and D in the SCDDL model can be replaced by ϕ(A)∈Rdmap×n and ϕ(D)∈Rdmap×m (d_map is the dimensional number in the mapping space) respectively for the kernel SCDDL framework as follows:

The dictionary can be represented by the training samples as Equation (7), according to the represented theorem (Schölkopf et al., 2001):

where V ∈ R^n×m is the representation matrix. Equation (6) can be transformed to Equation (8) with Equation (7):

The optimization process of Equation (8) has been discussed in our previous study (Wu et al., 2017). Then, the test sample y and dictionary D in Equation (4) can be replaced by ϕ(y)∈Rdmap and ϕ(A)V respectively as:

where λ is a scalar constant as above.

Let T(x)= minx ||ϕ(y)-ϕ(A)Vx||22, then T(x) can be simplified as:

where P = V^Tk(A, A)V, Q = V^Tk(y, A), and S = k(y, y).

Using the conclusions in previous study (Harandi and Salzmann, 2015), Equation (10) is equivalent to:

where ỹ=Σ-12UTQ, D~=Σ12UT, and U Σ U^T is the SVD of P (Nguyen et al., 2012). Then Equation (9) can be denoted as:

The convex problems in Equation (12) can be efficiently solved by plenty of tools such as the L₁-magic software package (Candes and Romberg, 2005), the GPSR package (Figueiredo et al., 2007) and the L₁-homotopy package (Asif and Romberg, 2010).

Finally, the identification of the test sample y can be employed using Equation (5) as follows:

where the {·}_l represents the l-th element in the brace.

As it is shown in the MKL algorithm (Sonnenburg et al., 2006), suppose there are J features for each sample, the kernel can be combined by convex combinations of J kernels, i.e.,

where each sub-kernel k_j corresponds to feature j.

So far, the kernels involved in the solution of Equation (12) can be replaced by Equation (13) for the multi-feature fusion of MKSCDDL. The combination coefficients can be simply set to be equal across all the features or optimized by cross-validation on the training samples. The sub-kernels can be selected from linear kernel, polynomial kernels, Gaussian kernels and sigmoid kernels etc. After the substitution of the kernels involved in the solution of Equation (12), MKSCDDL is realized (Wu et al., 2017).

In MKSCDDL model and the classification scheme, there are several parameters need to be set, including the parameter α for the classification error term, λ for the sparse coding term, λ₁ for the sparsity term, and λ₂ for the with-Class-similar term. Here, for simplify, α was set with α = 1 to make the contribution of the classification error equal (Xu et al., 2016). Furthermore, the parameter in the classification scheme λ made a little effect in the experimental results. So, λ was set with λ = 0.001 in the experiment. For the parameters in the optimization model λ₁ and λ₂, the optimal values were searched from a small set of {0.001, 0.005, 0.01, 0.05, 0.1} with a 5-fold cross-validation on the training set (Wu et al., 2017). For the AD and CU data set: λ₁ = 0.001, λ₂ = 0.1. For the MCI and CU data set: λ₁ = 0.05, λ₂ = 0.05. For the AD and MCI data set: λ₁ = 0.05, λ₂ = 0.005.

The dictionary size in MKSCDDL, mSCDDL, and SCDDL were set as 20 atoms (equivalent to 10 atoms for each class) for AD/CU, MCI/CU and AD/MCI classification; for MKL and JRC algorithms, all the training samples were trained for the model and classification; and for mSRC, all the training samples were used as a dictionary.

In this study, linear kernel was employed for MKSCDDL in the experiment. The combining weight parameters of three modalities for MKSCDDL was derived based on grid search approach with the range of [0,1] at a step size of 0.1 with a 5-fold cross-validation on training set (Zhang et al., 2011; Xu et al., 2015, 2016). Particularly, the combing weight parameters optimized corresponding to sMRI, FDG-PET and florbetapir-PET for classifying AD from CU are 0.5, 0.3, and 0.2; for discriminating MCI from CU are 0.2, 0.7, and 0.1; for detecting MCI from AD are 0.3, 0.6, and 0.1.

To evaluate the performance of all competing methods, their accuracy (the ratio of samples correctly classified among the test samples), sensitivity (the ratio of positive classes that were correctly identified), specificity (the ratio of negative classes that were accurately classified), and the areas under the Receiver Operating Characteristic (ROC) curves (AUC) were employed and compared in classification. For each group (AD, MCI, and CU), samples (subjects) were divided randomly into training and test sets. Sixty samples were selected randomly as the training set, and the rest comprised the test set. The division process was then repeated five times for the results of means and standard deviations, which were reported in this paper. Then, a two-sample t-test was carried out for each comparison pair to obtain the p-value.

In order to find the biomarkers for AD, MCI and CU classification, the 90 features were ranked according to the significance of the two-sample t-test. Then, the classification accuracy with different number (from 1 to 90) of the ranked 90 features has been calculated based on MKSCDDL (Zhang et al., 2011; Xu et al., 2016).

The performance of using single-modality SCDDL (SCDDL-sMRI, SCDDL-FDG-PET, and SCDDL-florbetapir-PET) and MKSCDDL (sMRI + FDG-PET + florbetapir-PET) were evaluated, as shown in Figures 1, 2 and Table 2, the MKSCDDL achieved higher accuracy in classifying AD, MCI, and CU than single-modality SCDDL methods.

For discriminating AD from CU, MKSCDDL achieved an accuracy of 98.18% (with 99.81% sensitivity and 96.49% specificity) that was much better than the best accuracy of 91.18% with single-modality method (using SCDDL-FDG-PET). Further, the comparison of the ROC curves for classification of AD and CU is shown in Figure 1A, and the comparison of AUCs is shown in Table 2. The ROC curve of MKSCDDL was closer to the top-left corner than that of SCDDL-FDG-PET, SCDDL-florbetapir-PET, and SCDDL-sMRI. The AUC of MKSCDDL was 0.991, which was better than the single-modality methods (AUC = 0.939, p = 0.046 for SCDDL-sMRI; AUC = 0.937, p = 0.028 for SCDDL-florbetapir-PET; and AUC = 0.970, p = 0.151 for SCDDL-FDG-PET, which was not significant in validation, but was numerically greater) as shown in Figure 2A.

For classifying MCI from CU, MKSCDDL achieved an accuracy of 78.50% (with sensitivity of 76.00% and specificity of 81.06%), which was greater than all three single-modality methods (the best classification accuracy was 72.50% when using SCDDL-FDG-PET). The comparison of the ROC curves for classification of MCI and CU are shown in Figure 1B and the comparison of AUCs is shown in Table 2. The ROC curve of MKSCDDL was closer to the top-left corner than that of SCDDL-sMRI, SCDDL-florbetapir-PET, and SCDDL-FDG-PET. Further, based on the significance validation, MKSCDDL was significantly much better than the single-modality methods with AUC, which was 0.839 for the multi-modality method compared with that of the single-modality methods (AUC = 0.762, p = 0.094 for SCDDL-FDG-PET; AUC = 0.742, p = 0.076 for SCDDL-florbetapir-PET; AUC = 0.787, p = 0.315 for SCDDL-sMRI, which were numerically better, though were not significant in validation) as shown in Figure 2B.

For classifying AD from MCI, MKSCDDL achieved an accuracy of 74.47% (with sensitivity of 72.44% and specificity of 78.99%), which was greater than all three single-modality methods (the best classification accuracy was 72.23% when using SCDDL-FDG-PET). The comparison of the ROC curves for classification of AD and MCI are shown in Figure 1C and the comparison of AUCs is shown in Table 2. The ROC curve of MKSCDDL was closer to the top-left corner than that of SCDDL-sMRI, SCDDL-florbetapir-PET, and SCDDL-FDG-PET. Further, based on significant validation, MKSCDDL was significantly much better than the single-modality methods with AUC, which was 0.791 for the multi-modality method compared with that of the single-modality methods (AUC = 0.687, p = 0.091 for SCDDL-sMRI; AUC = 0.694, p = 0.107 for SCDDL-florbetapir-PET; and AUC = 0.742, p = 0.198 for SCDDL-FDG-PET, which was numerically better, though were not significant in validation) as shown in Figure 2C.

The MKSCDDL achieved better classification accuracy and AUC for AD, MCI, and CU classification than the methods based on single-modality SCDDL (SCDDL-sMRI, SCDDL-FDG-PET, and SCDDL-florbetapir-PET), as seen in the results above, either statistically or numerically. The results we derived here were also consistent with those of other studies that have reported fusing multiple modalities could obtain better classification accuracy (Zhang et al., 2011; Westman et al., 2012; Xu et al., 2016).

Notably, on differentiating between MCI and CU, the classification specificity based on SCDDL-FDG-PET was 81.23%, which was slightly higher than that based on MKSCDDL (81.06%), whereas the classification sensitivity based on SCDDL-FDG-PET (62.20%) was much lower than that of MKSCDDL (76.00%). Lower sensitivity with only marginally higher specificity (which could be due to random noise) would result in underdiagnosis. The MKSCDDL method had higher sensitivity and outstanding specificity that was comparable with that of SCDDL-FDG-PET, and much higher than that of the other methods. Therefore, the results suggest the feasibility of using MKSCDDL for neuroimaging classification tasks. These meant that the MKSCDDL method was much or slightly better than SCDDL-florbetapir-PET, SCDDL-sMRI and SCDDL-FDG-PET in differentiating AD or MCI from CU.

The performance of using MKL, JRC, mSRC, mSCDDL, and MKSCDDL were evaluated and compared, including recognition rate, ROC curve and testing time. As shown in Figures 3–5 and Table 3, the MKSCDDL achieved higher accuracy in classifying AD or MCI from CU than other multimodal methods, and outperforms in testing time.

For differentiating AD from CU, MKSCDDL achieved an accuracy of 98.18% accuracy CU that was higher than MKL (93.64%), JRC (94.55%), mSRC (94.55%), and mSCDDL (97.36%). The comparison of the ROC curves for classification of AD and CU is shown in Figure 3A and the comparison of AUCs is shown in Table 3. The ROC curve of MKSCDDL was closer to the top-left corner than that of MKL, JRC, mSRC, and mSCDDL. The areas under the ROC curves for differentiation of AD and CU based on the five different methods are displayed in Figure 4A, in which the MKSCDDL method (AUC = 0.991) performed equally well statistically or numerically better than the other three multi-modality methods (AUC = 0.963, p = 0.095 for MKL; AUC = 0.971, p = 0.291 for JRC; AUC = 0.978, p = 0.429 for mSRC; and AUC = 0.985, p = 0.603 for mSCDDL). Figure 5 has shown the computational time for classification of per test sample with the corresponding methods. As shown, MKSCDDL consumed much less testing time than JRC (p = 0.007), mSRC (p = 0.010), and mSCDDL (p = 0.036), and was comparable with the MKL (p = 0.208) method.

For classifying MCI from CU, MKSCDDL achieved an accuracy of 78.50% (with sensitivity of 76.00% and specificity of 81.06%), which was greater than MKL (74.77%), JRC (73.83%), mSRC (75.70%), and mSCDDL (77.66%). The comparison of the ROC curves for classification of MCI and CU are shown in Figure 3B and the comparison of AUCs is shown in Table 3. The ROC curve of MKSCDDL was closer to the top-left corner than that of MKL, JRC, mSRC, and mSCDDL. Further, based on significant validation, MKSCDDL was numerically better than the corresponding methods with AUC, which was 0.839 for the MKSCDDL method compared with that of the corresponding methods (AUC = 0.804, p = 0.534 for MKL; AUC = 0.793, p = 0.331 for JRC; AUC = 0.785, p = 0.223 for mSRC; and AUC = 0.828, p = 0.843 for mSCDDL), as shown in Figure 4B. As shown in Figure 5, MKSCDDL consumed much less testing time than JRC (p = 0.009), mSRC (p = 0.015) and mSCDDL (p = 0.047), and was comparable with the MKL (p = 0.389) method.

For classifying AD from MCI, MKSCDDL achieved an accuracy of 74.47% (with sensitivity of 72.44% and specificity of 78.99%), which was greater than MKL (72.94%), JRC (72.05%), mSRC (68.55%), and mSCDDL (73.20%). The comparison of the ROC curves for classification of AD and MCI are shown in Figure 3C and the comparison of AUCs is shown in Table 3. The ROC curve of MKSCDDL was closer to the top-left corner than that of MKL, JRC, mSRC, and mSCDDL. Further, based on significant validation, MKSCDDL was numerically better than the corresponding methods with AUC, which was 0.791 for the MKSCDDL method compared with that of the corresponding methods (AUC = 0.779, p = 0.600 for MKL; AUC = 0.772, p = 0.477 for JRC; AUC = 0.693, p = 0.120 for mSRC; and AUC = 0.780, p = 0.593 for mSCDDL), which shown in Figure 4C. As shown in Figure 5, MKSCDDL consumed much less testing time than JRC (p = 0.011), mSRC (p = 0.019) and mSCDDL (p = 0.059), and was comparable with the MKL (p = 0.352) method.

To characterize the classification performance for AD, MCI, and CU with all 90 features (without feature selection), the classification accuracy has been investigated under feature selection with 1, 2, 3, …, or 90 features for each of the ranked 90 features. The results of classification performance for different numbers of ranked features are shown in Figure 6.

The figure shows that the MKSCDDL method could reach strong classification accuracy even with fewer than 5 features (the top 5% ranked features on sMRI, FDG-PET, and florbetapir-PET) for AD/MCI/CU classification. In particular, there was higher than 90% accuracy for classifying AD from CU, higher than 78% accuracy for distinguishing MCI from CU, and higher than 61% accuracy for discriminating AD and MCI. The MKSCDDL method was stable (with less ups and downs) for the classification of AD/MCI from CU, which indicated that redundant features likely introduced little interference of classification. For classification of AD and MCI, though the accuracy was also acceptable, it was not as stable as the classification accuracy for AD/MCI with CU, which may be due to the biomarkers for AD and MCI having very high similarity. When the top 10% features were used, the accuracy for classification of AD and MCI was higher than 64%.

As shown in Figure 6, the MKSCDDL could achieve a promising or acceptable accuracy even with less than 5 features (the top 5% ranked features). Thus, for convenience, one could apply a small set of features to effectively discriminate AD, MCI, and CU. Here, the top 5–10% ranked features (4–9 features) consisted of sMRI, FDG-PET, and florbetapir-PET data and could be chosen as biomarkers for further classification (Xu et al., 2016).

The biomarkers of different modalities for classification of the AD, MCI, and CU groups are displayed in Table 4 and Figure 7. For classification of AD and CU, the Hippocampus, Inferior Temporal, and ParaHippocampal may be the discriminating biomarkers on sMRI; the Angular, Posterior Cingulum, and Inferior Parietal may be the important regions on FDG-PET; and the Hippocampus and ParaHippocampal may be the key regions on florbetapir-PET. For discriminating MCI from CU, the Hippocampus, Middle Temporal, and ParaHippocampal may be the discriminating biomarkers on sMRI; the Angular and Posterior Cingulum may be the important regions on FDG-PET; and the Hippocampus, Posterior Cingulum, and Middle Frontal (Orbital part) may be the key regions on florbetapir-PET. For differentiating AD and MCI, the SupraMarginal, Angular, and left Superior Frontal (Orbital part) were the discriminating biomarkers on sMRI; the Angular, Inferior Parietal, and SupraMarginal may be the important regions on FDG-PET; and the Calcarine, Heschl, and Lingual may be the key regions on florbetapir-PET.

For AD and CU classification, the Hippocampus (Wisse et al., 2014; de Flores et al., 2015; Voineskos et al., 2015), Inferior Temporal (Seo et al., 2017), ParaHippocampal (Guo et al., 2014; Peng et al., 2016), Angular (Sanabria-Diaz et al., 2013), Posterior Cingulum (Nakata et al., 2009; Demirhan et al., 2015), and Inferior Parietal (Murray et al., 2015; Zhang et al., 2015) have been proposed in several studies to be effective biomarkers. The Hippocampus (Wee et al., 2011; Zhou et al., 2011; Liu et al., 2014a), Middle Temporal (Lenzi et al., 2011; Jiang et al., 2014), ParaHippocampal (Cerami et al., 2015; Kato et al., 2016), Angular (Nobili et al., 2010; Martlno et al., 2013; Zu et al., 2015), Posterior Cingulum (Choo et al., 2010; Yu et al., 2017), and Middle Frontal (Orbital part) (Xiang et al., 2013) have been reported as the important regions for discriminating MCI and CU. For differentiating AD and MCI, the SupraMarginal (Esposito et al., 2013; Moretti, 2015), Angular (Hirao et al., 2005; Griffith et al., 2010; Li et al., 2016), Superior Frontal (Orbital part) (Liu et al., 2012), Inferior Parietal (Desikan et al., 2009; Triplett et al., 2016), Calcarine (Liu et al., 2012), Heschl (Hanggi et al., 2011), and Lingual (Li et al., 2016) may be the key biomarkers for diagnosis.

Therefore, MKSCDDL was proved as a very efficient method for classifying AD or MCI from CU, and had potential to discriminate AD from MCI, as compared to the single-modality method and several state-of-art multi-modality methods. The MKSCDDL method performed better than MKL, JRC, mSRC, and mSCDDL in terms of accuracy rate and AUC, often significantly on validation but at least numerically for AD, MCI, and CU classification. In addition, the MKSCDDL method took less computation time than did JRC, mSRC, and mSCDDL, and was comparable to MKL in terms of computation time. Together, this indicates that the MKSCDDL method could potentially play an important role in AD and MCI diagnosis.

Informed consent was obtained from all individual participants included in the study.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.