Residual vectors for Alzheimer disease diagnosis and prognostication

Abstract

Alzheimer disease (AD) is an increasingly prevalent neurodegenerative condition and a looming socioeconomic threat. A biomarker for the disease could make the process of diagnosis easier and more accurate, and accelerate drug discovery. The current work describes a method for scoring brain images that is inspired by fundamental principles from information retrieval (IR), a branch of computer science that includes the development of Internet search engines. For this research, a dataset of 254 baseline 18-F fluorodeoxyglucose positron emission tomography (FDG-PET) scans was obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI). For a given contrast, a subset of scans (nine of every 10) was used to compute a residual vector that typified the difference, at each voxel, between the two groups being contrasted. Scans that were not used for computing the residual vector (the remaining one of 10 scans) were then compared to the residual vector using a cosine similarity metric. This process was repeated sequentially, each time generating cosine similarity scores on 10% of the FDG-PET scans for each contrast. Statistical analysis revealed that the scores were significant predictors of functional decline as measured by the Functional Activities Questionnaire (FAQ). When logistic regression models that incorporated these scores were evaluated with leave-one-out cross-validation, cognitively normal controls were discerned from AD with sensitivity and specificity of 94.4% and 84.8%, respectively. Patients who converted from mild cognitive impairment (MCI) to AD were discerned from MCI nonconverters with sensitivity and specificity of 89.7% and 62.9%, respectively, when FAQ scores were brought into the model. Residual vectors are easy to compute and provide a simple method for scoring the similarity between an FDG-PET scan and sets of examples from a given diagnostic group. The method is readily generalizable to any imaging modality. Further interdisciplinary work between IR and clinical neuroscience is warranted.

Figures

Figure 1

Geometric interpretation of ordinary least squares regression. A vector N (representing the PET scan of an MCI nonconverter) is projected onto a space, C, which is composed of PET scans from MCI patients who converted to AD within 2 years of being scanned. Although C is depicted as being planar, in actuality it has as many dimensions as the number of PET scan vectors that compose it. The projection vector, P, can be computed by means of multiplying a “hat” matrix by the original vector, N. The hat matrix is derived from the matrix C by the equation , where the −1 superscript represents the matrix inverse and the T superscript represents the matrix transpose. The residual vector, R, is then calculated by subtracting the projection P from N. The residual is orthogonal to all vectors in the column space of C, but retains some similarity to the original vector, N.

Figure 2

Grand average residual vector created by (1) projecting each AD PET scan onto a space defined by 90% of the NC PET scans, (2) subtracting the projection from the original AD PET scan to obtain a residual vector, and (3) averaging together all of the residuals. Voxels with the lowest residual values are located in the lateral temporal lobes, lateral parietal lobes, precuneus, and posterior cingulate, corresponding to the default mode network.

Figure 3

Grand average residual vector created by the same general method as in Fig. 2, but projecting MCI-n PET scans onto a space defined by MCI-c PET scans. Voxels with the highest residual values are topographically similar to those with the low residual values in Fig. 2.

Figure 4

ROC curves showing performance of a simple logistic regression model for classification of subjects into elderly control and AD groups. The independent variable was a cosine similarity score computed from vectors corresponding to each subject's PET scan and residual vectors like the one depicted in Fig. 2.

Figure 5

ROC curves showing performance of logistic regression models for separation of MCI subjects into a group that converted to AD within 2 years and a group that went 2 years without converting. (A) ROC curve using only cosine similarity scores for classification. These scores were derived by computing cosine similarity between each subject's PET scan and a residual vector like the one depicted in Fig. 3. (B) ROC curve using both cosine similarity scores, FAQ score, and their interaction. Addition of FAQ substantially improves the classifier.