Estimating Uncertainty in Brain Region Delineations

This paper presents a method for estimating uncertainty in MRI-based brain region delineations provided by fully-automated segmentation methods. In large data sets, the uncertainty estimates could be used to detect fully-automated method failures, identify low-quality imaging data, or endow downstream statistical analyses with per-subject uncertainty in derived morphometric measures. Region segmentation is formulated in a statistical inference framework; the probability that a given region-delineating surface accounts for observed image data is quantified by a distribution that takes into account a prior model of plausible region shape and a model of how the region appears in images. Region segmentation consists of finding the maximum a posteriori (MAP) parameters of the delineating surface under this distribution, and segmentation uncertainty is quantified in terms of how sharply peaked the distribution is in the vicinity of the maximum. Uncertainty measures are estimated through Markov Chain Monte Carlo (MCMC) sampling of the distribution in the vicinity of the MAP estimate. Experiments on real and synthetic data show that the uncertainty measures automatically detect when the delineating surface of the entire brain is unclear due to poor image quality or artifact; the experiments cover multiple appearance models to demonstrate the generality of the method. The approach is also general enough to accommodate a wide range of shape models and brain regions. 1 Uncertainty in Brain Region Delineations 1.1 Importance of Uncertainty Estimation in Brain Region Delineations Structural magnetic resonance imaging (MRI) is a technology for measuring biological properties of the brain. A widespread methodology for large-scale epidemiological studies is to collect MRI scans of a cohort of subjects, delineate brain regions on those scans using manual or automated methods, and relate morphometric measures derived from those region delineations to clinical variables of interest. Studies of this sort have played an important role in clarifying the biological course of a range of neurological disorders, including multiple sclerosis and dementia [9] [7]. This paper provides a method for quantifying uncertainty in brain region delineations. Once the uncertainty in a brain region delineation is known, we can use this information to identify and possibly discard images whose region delineations have high uncertainty and therefore may have been segmented imprecisely. Our formulation of uncertainty is chiefly concerned with the precision, rather than the accuracy, of the delineating boundaries of brain regions. As with every measurement, both the accuracy and precision of the delineating surface determine the validity of inference made from their derived measures: accuracy is the degree to which the measurement is close to the true value of the quantity being measured, while precision is the degree to which repeated measurements provide similar values [8]. If a measurement is imprecise, we cannot be certain whether the measurement value arises from underlying biological phenomena or random fluctuations in the measurement process. Our approach is to use a statistical sampling procedure to simulate the repeated measurement of the same brain region boundary by an automated segmentation method, and assess whether the segmentation method tells us that a diverse, widely-scattered set of boundary surfaces delineate the region equally well. The uncertainty in a brain region delineations obtained via automated methods can be affected by several factors. Poor image quality or low contrast between two brain regions can both lead to images that are hard to segment precisely. Further, measurement uncertainties are exacerbated when the models of brain appearance and shape used by an automated method are oversimplified or invalid. In these cases, a measurement can be imprecise; to our knowledge, there are currently no automated tools for quantifying measurement precision so these errors may be left undetected in the absence of time-consuming, tedious manual checking of segmentation results. For example, imagine that the area of the ellipses in figure 1 are to be used to estimate the volume of the brain in the image. When making this derived measurement for the blurred image in figure 1, we could not be sure what size the ellipse should be in order to best approximate the brain contour since the image is blurry and the boundary between the skull and the brain is uncertain. This is an important concern because the size of the ellipse will determine the computed volume, which in turn will be used in a statistical analysis to test a hypothesis about relationships between brain volume and other clinical measures. Thus, uncertainty in derived measures give rise to errors in the data which in turn result in errors in the statistical analyses. To our knowledge, current fully automated segmentation methods cannot detect corrupted images or grossly incorrect delineations. Our hypothesis is that this is partly due to the fact that modern segmentation methods suffer from the inability to incorporate a quantitative estimate for the precision of reported measurements. Though the uncertainty in this example was synthetically produced, there are many real sources of uncertainty in image delineation. For example, blurry or ”‘ghosted”’ images can lead to measurement uncertainty as well as images of brains with large pathologies which can sometimes deviate from the model assumptions of the segmentation method being used for the study. Fig. 1. The image on the right is the result of applying heavy white noise to the same image shown on the left. The ellipse on the left image is a region-delineation that partitions the image into brain region(inside the ellipse) and the non-brain region(outside of the ellipse). The ellipse on the right is the same as that on the left but due to the noise, it is no longer clear that the ellipse on the right truly does partitions the image correctly. Given the noisy image, different ellipses with different areas seem to fit the brain equally well. 1.2 Overview of our approach A convenient way of formulating region-based delineation that encompasses many popular methods involves a Bayesian framework. The Bayesian approach seeks an answer to the question, ’what is the most probable delineating contour or surface of a desired region given the current image?’ If we let I represent the image to be segmented and θ be a vector of parameters used to represent the shape of the delineating surface, then we can mathematically phrase this question by asking for a solution to the following conditional probability equation: θ̂ = argmax θ P (θ|I) (1) P (θ|I) is the posterior probability of the parameters θ given the image, I. By Bayes Theorem equation 1 can be written as θ̂ = argmax θ L(I|θ)q(θ). (2) L(I|θ) is the image model: it is the likelihood of obtaining the image I given that the shape parameters are θ. The q(θ) is the shape model; it is the prior probability that the region will take on the shape described by θ, regardless of how I looks. We estimate uncertainty in region delineation by using Markov Chain Monte Carlo (MCMC) to sample a series of θ from P (θ|I) and using the samples to approximate the solution to expectation integrals that give measures of uncertainty in the position of θ̂. Because morphometric measures such as volume and surface area are often derived from θ̂ and related to clinical variables in statistical analyses, the uncertainty measures can provide measures of uncertainty in the derived measures. We will demonstrate that the method is modular in the sense that it can be used in combination with a broad range of currently existing region delineation methods to estimate an optimal region-delineating surface and uncertainty in the surface.