Multivariate models of inter-subject anatomical variability

Abstract

This paper presents a very selective review of some of the approaches for multivariate modelling of inter-subject variability among brain images. It focusses on applying probabilistic kernel-based pattern recognition approaches to pre-processed anatomical MRI, with the aim of most accurately modelling the difference between populations of subjects. Some of the principles underlying the pattern recognition approaches of Gaussian process classification and regression are briefly described, although the reader is advised to look elsewhere for full implementational details. Kernel pattern recognition methods require matrices that encode the degree of similarity between the images of each pair of subjects. This review focusses on similarity measures derived from the relative shapes of the subjects' brains. Pre-processing is viewed as generative modelling of anatomical variability, and there is a special emphasis on the diffeomorphic image registration framework, which provides a very parsimonious representation of relative shapes. Although the review is largely methodological, excessive mathematical notation is avoided as far as possible, as the paper attempts to convey a more intuitive understanding of various concepts. The paper should be of interest to readers wishing to apply pattern recognition methods to MRI data, with the aim of clinical diagnosis or biomarker development. It also tries to explain that the best models are those that most accurately predict, so similar approaches should also be relevant to basic science. Knowledge of some basic linear algebra and probability theory should make the review easier to follow, although it may still have something to offer to those readers whose mathematics may be more limited.

Fig. 1

A two dimensional illustration of a regression model, whereby the horizontal and vertical positions of the squares denote the values of pairs of features, and the numbers in the squares indicate labels to be predicted. After fitting the model, prediction is achieved by projecting the data.

Fig. 2

Left: A dot-product can be conceptualised as projecting one vector on to another. Projecting on to the discriminating direction is by a^Tx = |a||x| cos(θ), where θ is the angle between a and x. Right: A logistic function is used for squashing the results into the range of zero to one (because they are probabilities).

Fig. 3

A two dimensional illustration of the generative model used by Fisher's linear discriminant analysis.

Fig. 4

This figure shows a selection of some of the approaches that can be used for linear discrimination. Top-left: Ground truth is based on the probability densities of the two Gaussians from which data were simulated. The line shows the discriminant direction for the underlying model. Top-right: Fisher's Linear Discrimination, also including the resulting discriminant direction. Bottom-left: Linear Support Vector Classification results. Bottom-right: A simple logistic ridge-regression model.

Fig. 5

This figure illustrates a Bayesian approach to logistic regression. Top-left: Contours of probability from a naïve implementation of logistic regression, where the contours remain parallel (see Fig. 4). Top-right: The discriminating direction is estimated with uncertainty, which is illustrated by a random sample of possible separating hyperplanes. Accurate inference requires this uncertainty to be integrated into the predictive model. Bottom-left: Predictive probabilities are made more accurate by incorporating uncertainty. Bottom-right: By integrating out the uncertainty, the contours properly reflect the loss of accuracy further away from the training data.

Fig. 6

A poorly-determined general linear model may have a number of columns in the design matrix that is high, compared to the number of rows (left). The conditioning of the problem may be improved by augmenting the design matrix (centre and right). In this situation, fitting the GLM involves a trade-off between fitting the data and keeping the coefficients small. If the regularisation part is small (centre), the trade-off is towards the former, whereas if it is large (right), then model will tend towards the latter.

Fig. 7

An illustration of allometric relationships using BMI as an example.

Fig. 8

The original images used for this illustration are shown in (a). After alignment with their common average, they are shown in (b). Note that exact alignment is not achieved, especially for the white hole in the middle of two of the images. Decreasing the amount of regularisation used by the registration would have allowed the hole to be closed further, but its area would never reach exactly zero (a singularity). The Jacobian determinants indicate the relative volumes before and after non-linear registration. Lighter colours indicate areas of expansion, where the Jacobians are smaller. Darker colours indicate contraction, and larger Jacobians. A Jacobian determinant of one would indicate no volume change. The Jacobians of the mapping from the original images to the warped versions are shown in (c). The diffeomorphic framework allows deformations to be invertible, so mappings from the warped images to the originals can also be generated. The Jacobians of these mappings are shown in (d). The deformations and their inverses are shown in (e) and (f). Spatially normalised versions of the individual images were generated by resampling them according to (f), whereas (e) could be used to overlay the template on to the original images. The forward and inverse mappings can be composed together, in which case the results should be identity transforms (which would appear as a regular grid).

Fig. 9

The small deformation framework is not accurate for larger deformations. This figure shows the sum of the forward and backward displacement fields shown in Fig. 8. The results are clearly not identity transforms.

Fig. 10

Deformations can be generated from the residuals as illustrated here (Younes, 2007). The top row shows the initial state of the system, and each subsequent row shows it at the next time point during the evolution. The bottom row shows the final state. The first three columns show the template and its spatial gradients as it evolves to match the individual image. The next column shows the residual difference between the template and warped image, scaled to account for contraction and expansion.

Fig. 11

This figure shows residual differences between the warped images and the template, which are scaled at each point by the Jacobian determinant. In conjunction with the template, these residuals encode the information needed to reconstruct the original images (apart from a small amount of information lost through inexact interpolation). Dividing the residuals by the Jacobian determinants and adding the template will give warped versions of the originals, which can then be unwarped by resampling with the appropriate deformation. The deformations and Jacobians needed to perform these operations are actually encoded by the residuals (illustrated in Fig. 10).

Fig. 12

Average of 450 T1-weighted scans from the IXI dataset, which have been aligned using a geodesic shooting model. The left side of the brain is shown towards the left of the image.

Fig. 13

Exaggerated versions of female (left) and male (right) average brains, which correspond to 99.99999% probabilities. Note that the caricatures were generated by warping the average brain shown in Fig. 12, and that the deformations outside the brain are less accurate (so skull thicknesses etc are not accurately represented). The left side of the brain is shown towards the left of the image.