Multivariate longitudinal statistics for neonatal-pediatric brain tissue development

pre-printThe topic of studying the growth of human brain development has become of increasing interest in the neuroimaging community. Cross-sectional studies may allow comparisons between means of different age groups, but they do not provide a growth model that integrates the continuum of time, nor do they present any information about how individuals/population change over time. Longitudinal data analysis method arises as a strong tool to address these questions. In this paper, we use longitudinal analysis methods to study tissue development in early brain growth. A novel approach of multivariate longitudinal analysis is applied to study the associations between the growth of different brain tissues. In this paper, we present the methodologies to statistically study scalar (univariate) and vector (multivariate) longitudinal data, and demonstrate exploratory results in a neuroimaging study of early brain tissue development. We obtained growth curves as a quadratic function of time for all three tissues. The quadratic terms were tested to be statistically significant, showing that there was indeed a quadratic growth of tissues in early brain development. Moreover, our result shows that there is a positive correlation between repeated measurements of any single tissue, and among those of different tissues. Our approach is generic in natural and thus can be applied to any longitudinal data with multiple outcomes, even brain structures. Also, our joint mixed model is flexible enough to allow incomplete and unbalanced data, i.e. subjects do not need to have the same number of measurements, or be measured at the exact time points

A framework for longitudinal data analysis via shape regression

pre-printTraditional longitudinal analysis begins by extracting desired clinical measurements, such as volume or head circumference, from discrete imaging data. Typically, the continuous evolution of a scalar measurement is estimated by choosing a 1D regression model, such as kernel regression or fitting a polynomial of fixed degree. This type of analysis not only leads to separate models for each measurement, but there is no clear anatomical or biological interpretation to aid in the selection of the appropriate paradigm. In this paper, we propose a consistent framework for the analysis of longitudinal data by estimating the continuous evolution of shape over time as twice differentiable flows of deformations. In contrast to 1D regression models, one model is chosen to realistically capture the growth of anatomical structures. From the continuous evolution of shape, we can simply extract any clinical measurements of interest. We demonstrate on real anatomical surfaces that volume extracted from a continuous shape evolution is consistent with a 1D regression performed on the discrete measurements. We further show how the visualization of shape progression can aid in the search for significant measurements. Finally, we present an example on a shape complex of the brain (left hemisphere, right hemisphere, cerebellum) that demonstrates a potential clinical application for our framework

Detection of influential observations in longitudinal multivariate mixed effects regression models

The purpose of this dissertation is to detect possible influential observations in longitudinal data with more than one observation per subject at each time point, that is, in multivariate longitudinal data. An influential observation is an observation which has large effect on the parameter estimation of a given model. Influential observations are important because: (1) removal of the observation(s) from the data set can substantially change the values of the estimated parameters; (2) in multivariate longitudinal mixed effect models, influential observations can affect the population and subject-specific trajectories; (3) influential observation(s) of one response may affect the predicted effects of the other response within the same individual; (4) an influential observation may indicate an abnormal or misdiagnosed subject. This research was motivated by opthalmological clinical research in glaucoma. In many ophthalmology studies, both eyes are repeatedly measured. Sometimes one eye can be measured by different devices or measured for different quantities (retina thickness for different quadrants, OCT, VFI, etc.). For example, in one study considered in this dissertation, multivariate measurements (Retinal Nerve Fiber Layer (RNFL) thickness and Ganglion Cell Complex (GCC) thickness) were repeatedly measured on each eye, within each patient (cluster). When we detect influential observations for longitudinal ophthamology data, our trajectory model must take into account three kinds of correlations: (1) correlation among different characteristics measured at the same time point within the same eye; (2) correlation among different time points; (3) correlation between characteristics in the two eyes. In the first part of my dissertation, we propose a multivariate conditional version of Cook's distance for multivariate mixed effect models. Some research has shown that, in mixed effect models, influential observations having a large effect on subject-specific parameters cannot always be detected by the original Cook's distance due to large between-subject variation. Hence, in the multivariate longitudinal setting, the influential observation problem is better approached by conditioning on subjects and characteristics. Repeated simulations within this dissertation show that multivariate conditional Cook's distance successfully detected most 92.5% influential observations, but unconditional Cook's distance only detected 7.5%. In the second part of the dissertation, we extend the multivariate conditional Cook's distance to multilevel multivariate mixed effect model. In this model, there are two levels of random effects to handle the subject level and cluster level correlations among different time points, and the residual covariance matrix to handle correlations among different responses. Also, the two-level multivariate conditional Cook's distance can be decomposed into six parts, indicating the influences of fixed effects, 1st and 2nd level of random effects, and the co-variation between them, respectively. Examples are given to illustrate how the influential observation in one characteristic changes the effects of both characteristics. This research has public health implications because the influence of outliers can bias the results of any longitudinal study in public health. Hence, recognizing observations which have undue influence on study results ensures that reliable conclusions can be obtained in medical and public health research settings

Medical image analysis via Fréchet means of diffeomorphisms

The construction of average models of anatomy, as well as regression analysis of anatomical structures, are key issues in medical research, e.g., in the study of brain development and disease progression. When the underlying anatomical process can be modeled by parameters in a Euclidean space, classical statistical techniques are applicable. However, recent work suggests that attempts to describe anatomical differences using flat Euclidean spaces undermine our ability to represent natural biological variability. In response, this dissertation contributes to the development of a particular nonlinear shape analysis methodology. This dissertation uses a nonlinear deformable model to measure anatomical change and define geometry-based averaging and regression for anatomical structures represented within medical images. Geometric differences are modeled by coordinate transformations, i.e., deformations, of underlying image coordinates. In order to represent local geometric changes and accommodate large deformations, these transformations are taken to be the group of diffeomorphisms with an associated metric. A mean anatomical image is defined using this deformation-based metric via the Fréchet mean—the minimizer of the sum of squared distances. Similarly, a new method called manifold kernel regression is presented for estimating systematic changes—as a function of a predictor variable, such as age—from data in nonlinear spaces. It is defined by recasting kernel regression in terms of a kernel-weighted Fréchet mean. This method is applied to determine systematic geometric changes in the brain from a random design dataset of medical images. Finally, diffeomorphic image mapping is extended to accommodate extraneous structures—objects that are present in one image and absent in another and thus change image topology—by deflating them prior to the estimation of geometric change. The method is applied to quantify the motion of the prostate in the presence of transient bowel gas