On the estimation and correction of bias in local atrophy estimations using example atrophy simulations

Abstract

Brain atrophy is considered an important marker of disease progression in many chronic neuro-degenerative diseases such as multiple sclerosis (MS). A great deal of attention is being paid toward developing tools that manipulate magnetic resonance (MR) images for obtaining an accurate estimate of atrophy. Nevertheless, artifacts in MR images, inaccuracies of intermediate steps and inadequacies of the mathematical model representing the physical brain volume change, make it rather difficult to obtain a precise and unbiased estimate. This work revolves around the nature and magnitude of bias in atrophy estimations as well as a potential way of correcting them. First, we demonstrate that for different atrophy estimation methods, bias estimates exhibit varying relations to the expected atrophy and these bias estimates are of the order of the expected atrophies for standard algorithms, stressing the need for bias correction procedures. Next, a framework for estimating uncertainty in longitudinal brain atrophy by means of constructing confidence intervals is developed. Errors arising from MRI artifacts and bias in estimations are learned from example atrophy simulations and anatomies. Results are discussed for three popular non-rigid registration approaches with the help of simulated localized brain atrophy in real MR images.

Introduction

Brain atrophy is a conspicuous marker for monitoring disease progression and is also known to be related to cognitive impairment in many chronic neuro-degenerative diseases including multiple sclerosis (MS) [1] and Alzheimer's disease [2]. Consequently, a lot of attention has been paid to the development of approaches that enable the estimation of longitudinal change in brain volume for an individual. Typically, one is interested in brain atrophy analysis on a global as well as on a local level. Global measures [“Boundary Shift Integral” (BSI) [3], “Structural Image Evaluation, using Normalization of Atrophy” (SIENA) [4] or segmentation based approaches (such as SIENAX) [4]] are used in population studies for separating patient groups from normals or in cases when one just wants to know whether the brain has undergone atrophy globally. Local measurements (Jacobian integration, i.e. integration of the determinant of the Jacobian matrix of a deformation field over a region of interest [5]) make it possible to obtain a regional (or voxel by voxel) estimate of brain atrophy. The accuracy of estimates of volume change is questionable when they are obtained from images corrupted with artifacts. Bias field inhomogeneities and noise in MR images have been identified as major determinants of errors in atrophy estimations [5], [6]. In addition, inaccuracies of registration and segmentation procedures employed during an atrophy estimation algorithm may also result in biased estimates of atrophy, even in the absence of any image artifacts. Validation studies [5], [6], [7], [9] have confirmed the existence of bias in atrophy estimations. Another critical consideration that is generally overlooked is model error, i.e. errors in the mathematical model of brain atrophy. It has also been shown that Jacobian maps obtained from non-rigid registration (NRR) algorithms may be biased [10]. Due to these reasons, it may become difficult to separate real anatomical changes from spurious ones in local as well as global estimations of atrophy. It is thus important to make such maps of change more reliable by providing estimates of uncertainties in atrophy estimations. This will aid end users in decision making.

In recent years, several articles have addressed the problem of uncertainty estimation in non-rigid registration algorithms often used in atrophy measurements. Existing approaches of uncertainty estimation can be classified depending on whether they employ a ground truth [9], [11], manipulate the similarity criterion chosen during the registration step [12] or the deformation field [13], [14] or are based on Bayesian formulations [15], [16], [17]. All these existing methods, though developed and tested for other applications, can be employed or extended for brain atrophy estimations. The mean registration errors are reported to be around 2.5 mm [16], [18] in image guided surgical applications. It should be mentioned here that these errors are large from the point of view of atrophy, where often deformations of this order are seen and must be quantified for increasing the reliability of volume change measurements. Except for the approach of Jalobeanu and Fitzenz [14] targeting satellite imagery, uncertainty estimation approaches have not attempted to examine the bias in registration parameters’ estimates. Bias in atrophy estimations from deformable registration algorithms arise from the asymmetric estimation as well as asymmetric application of global and local transforms. Ensuring symmetry of estimated transformations is a way of minimizing unwanted bias in atrophy measurements [19]. Rectification of biased measurements has been performed on the same lines, by introducing symmetric regularization terms [20], [21] and estimation of symmetric transforms [22], [23]. Asymmetry in the application of the global transform results in a bias of 2–3% when Avants et al.'s [24] NRR method is used [22]. Symmetrizing the similarity metric and regularization term has been shown to reduce the bias (additive offset) from 1.4% to 0.28% [23].

In this article, we show that depending on the NRR algorithm used, the bias in atrophy estimations is of the order of the expected atrophy, emphasizing the need for correction of bias in estimations. Plus, bias may or may not be proportional to the true atrophy value. This is an important finding as, for such cases, bias cannot be learned from scan-rescan images as is the case addressed by Yushkevich et al. [22]. To address this point, a generic framework for estimating uncertainties in longitudinal brain atrophy by means of constructing confidence intervals for any atrophy estimation method is presented. A simulated ground truth acts as an independent source of learning errors in atrophy measurements. Example simulations on multiple subjects allow us to estimate uncertainties that are stored for future measurements in a “learning database”. Wang et al. [25] have presented an analogous framework in the context of performance improvement of segmentation algorithms. Their method is based on training of the segmentation data with manual segmentations in order to adjust for bias in segmentation results.

The interest of such a framework lies in its ability to be applied to any atrophy estimation approach as the uncertainty calculations are performed separately and do not need to be incorporated in the atrophy estimation method. Plus, errors originating from MRI artifacts and method specific biases are considered in the constructed confidence intervals. Three non-rigid registration (NRR) approaches are chosen [26], [24], [27] to demonstrate their behavior in the presence of noise and confidence intervals are constructed for atrophies estimated by them. Hippocampal atrophies are simulated in real MR brain images while estimated confidence intervals are quantitatively evaluated on the basis of coverage probability and length. Results show that Avants et al.'s [24] method provides meaningful confidence intervals as compared to the NRR methods of Noblet et al. [26] and Vemuri et al. [27]. We will refer to the NRR approaches of Noblet et al. [26] as NobletNRR, Avants et al. as ANTS (Advanced Normalization Tools) and Vemuri et al. [27] as VemuriNRR in the forthcoming discussion.