Construction of a neuroanatomical shape complex atlas from 3D MRI brain structures Ting Chen a , Anand Rangarajan a, ⁎, Stephan J. Eisenschenk b , Baba C. Vemuri a, ⁎ a Department of CISE, University of Florida, Gainesville, FL 32611-6120, USA b Department of Neurology, University of Florida, Gainesville, FL 32611, USA abstract article info Article history: Received 23 June 2011 Revised 14 January 2012 Accepted 18 January 2012 Available online 28 January 2012 Keywords: Brain MRI Shape complex atlas Epilepsy Lobectomy Distance transform Schrödinger equation Karcher mean Level set Square-root density Brain atlas construction has attracted significant attention lately in the neuroimaging community due to its application to the characterization of neuroanatomical shape abnormalities associated with various neurodegen- erative diseases or neuropsychiatric disorders. Existing shape atlas construction techniques usually focus on the analysis of a single anatomical structure in which the important inter-structural information is lost. This paper proposes a novel technique for constructing a neuroanatomical shape complex atlas based on an information geometry framework. A shape complex is a collection of neighboring shapes – for example, the thalamus, amygdala and the hippocampus circuit – which may exhibit changes in shape across multiple structures during the progression of a disease. In this paper, we represent the boundaries of the entire shape complex using the zero level set of a distance transform function S(x). We then re-derive the relationship between the stationary state wave function ψ(x) of the Schrödinger equation -ℏ 2 ∇ 2 ψ +ψ =0 and the eikonal equation ‖ ∇S‖ =1 satisfied by any distance function. This leads to a one-to-one map (up to scale) between ψ(x) and S(x) via an explicit relationship. We further exploit this relationship by mapping ψ(x) to a unit hypersphere whose Riemannian structure is fully known, thus effectively turn ψ(x) into the square-root of a probability density function. This allows us to make comparisons – using elegant, closed-form analytic expressions – between shape complexes represented as square-root densities. A shape complex atlas is constructed by computing the Karcher mean  ψ x ðÞ in the space of square-root densities and then inversely mapping it back to the space of distance transforms in order to realize the atlas shape. We demonstrate the shape complex atlas computation technique via a set of experiments on a population of brain MRI scans including controls and epilepsy patients with either right anterior medial temporal or left anterior medial temporal lobectomies. © 2012 Elsevier Inc. All rights reserved. Introduction Human brain MRI analysis is an important problem due to its application in the diagnosis and treatment of neurological diseases. In this context, the construction of neuroanatomical atlases of the human brain is of particular interest and its importance has been emphasized in a number of recent studies (Aljabar et al., 2009; Sabuncu et al., 2009; Shattuck et al., 2008; Yeo et al. 2008). In brief, an atlas provides a reference for a population of shapes/images which is useful in numerous applications: (i) statistical analysis of volumetric changes in control and patient populations, (ii) atlas-guided segmenta- tion of structures of interest which is needed in further diagnostic procedures, and (iii) automated detection of disease regions based on shape variations between the atlas and individual subjects. Most existing shape atlases are based on isolated, single anatomical shapes (Fletcher et al., 2004; Liu et al., 2008; Wang et al., 2006) which do not contain any inter-structural information. For example, the spatial relationships among different neighboring structures may change due to the effect of non-uniform volume shrinkage or expansion of neigh- borhood structures. Furthermore, many neurological disorders are diagnosed by the structural abnormalities (e.g. volume change) ascribed to several brain structures rather than a single structure. Alzheimer's disease is an example of such a neurological disorder—a morphological marker for which is the enlargement of ventricles and the shrinkage of the entorhinal cortex, amygdala and hippocampi (Brice, 2009). Mania, which is most often associated with bipolar disor- der serves as another example. In Strakowski et al. (1999), all the brain structures associated with the neural pathways were examined and the authors claimed that patients with mania have a significant overall volume difference in the regions including the thalamus, hippocampi and the amygdala. In Seidman et al. (1999), the authors concluded that the structural abnormalities in the thalamus and the amygdala– hippocampus regions represent remarkable anatomical vulnerabilities in schizophrenia subjects. Therefore, a neuroanatomical shape complex atlas which captures anatomical connectivity as well as inter-structural relationships is of primary clinical importance. NeuroImage 60 (2012) 1778–1787 ⁎ Corresponding authors. E-mail addresses: tichen@cise.ufl.edu (T. Chen), anand@cise.ufl.edu (A. Rangarajan), stephan.eisenschenk@neurology.ufl.edu (S.J. Eisenschenk), vemuri@cise.ufl.edu (B.C. Vemuri). 1053-8119/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2012.01.095 Contents lists available at SciVerse ScienceDirect NeuroImage journal homepage: www.elsevier.com/locate/ynimg