Health-e-Child - IST-2004-027749 - Deliverable D.11.3

Brain Tumours

Computational Anatomy of the Brain

1. Computational Anatomy: Aims and Methods

1.1 Evolution and Revolutions in Anatomy

According to the dictionary, anatomy is the science that studies the structure and the relationship in space of different organs and tissues in living systems. During the antiquity and middle-ages, anatomical descriptions were mainly based on animal models and the physiology was more philosophical than scientific. Modern anatomy really began at the renaissance with the authorisation of the dissection of human cadavers, giving birth to the "De humani corporis fabrica" published by in 1543 by Vesale (1514-1564), and was strongly pushed by the progresses in surgery, as exemplified by the "Universal anatomy of the human body" (1561-62) of the great surgeon Ambroise Paré(1509-1590).

Figure 1. Left: first cerebral atlas, Vesale, 1543. Middle: human anatomy (skeleton) by Paré, 1585. Right: Atlas of the brain by Talairach and Tournoux.

During the following centuries, many progresses were done in anatomy thanks to the introduction of new observation tools like microscopy and histology, going down to the level of cells in the 19th and 20th centuries. However, in-vivo and in-situ imaging is radically renewing the field since the 1980ies. An ever growing number of imaging modalities allows to observe both the anatomy and the function at many spatial scales (from cells to the whole body) and at multiple time scales: milliseconds (e.g. beating heart), years (growth or ageing), or even ages (evolution of species). Moreover, the non-invasive aspect allows repeating the observations on multiple subjects, giving rise to large databases. This has a strong impact on the goals of the anatomy which are changing from the description of a representative individual to the description of the structure and organisation of organs at the population level. The huge amount of information generated (typically 50 to 150 Mb for a clinical MRI exam, much more for a CT one) also raises the need for computerised methods to extract and structure information. This led in the last 10 to 20 years to the gradual evolution of descriptive atlases into interactive and generative models, allowing the simulation of new observations. Typical examples are given for the brain by the MNI 305 ([Evans et al, 1993]) and ICBM 152 ([Mazziotta et al, 2001] ) templates that are the basis of the Brain Web MRI simulation engine [Collins et al, 1998]. In the orthopaedic domain, one may cite the "bone morphing" method [Fleute and Lavallée, 1998,Rajamani et al, 2004] that allows to simulate the shape of bones.

1.2 Aims of Computational Anatomy

The combination of these new observation means and of the computerised methods is at the heart of computational anatomy, an emerging discipline at the interface of geometry, statistics and image analysis whose goal is to develop algorithms to model and analyse the biological shape of tissues and organs. The goal is not only to estimate representative organ anatomies across species, populations, diseases, ageing, ages but also to model the organ development across time (growth) and to establish their variability. Another goal is to correlate this variability information with other functional, genetic or structural information (e.g. fibre bundles extracted from diffusion tensor images). From an applicative point of view, a first objective is to understand and to model how life is functioning at the population level, for instance by classifying pathologies from structural deviations (taxonomy) and by integrating individual measures at the population level to relate anatomy and function. For instance, the goal of spatial normalisation of subjects in neuroscience is to map all the anatomies into a common reference system. A second application objective is to provide better quantitative and objective measures to detect, understand and correct dysfunctions at the individual level in order to help therapy planning (before), control (during) and follow-up (after). This generally implies mapping some generic (atlas-based) knowledge to patients-specific data through atlas-to-patient registration. Computational anatomy is currently a very active research field, as exemplified by the IPAM's Graduate Summer School on Mathematics in Brain Imaging organised by P. Thompson, M. Miller in 2004 or the workshop on the Mathematical Foundations of Computational Anatomy (MFCA'06), organised in September 2006 in conjunction with MICCAI'06, the 9th International Conference on Medical Image Computing and Computer Assisted Intervention.

1.3 Methods of Computational Anatomy

In the case of observations of the same subject, many geometrical and physically based registration methods were proposed to faithfully model and recover the deformations. However, in the case of different subjects, the absence of physical models relating the anatomies leads to a reliance on statistics to learn the geometrical relationship from many observations. The general method is to identify anatomically representative geometric features (points, tensors, curves, surfaces, volume transformations), and to model their statistical distribution across the population. This can be done, for instance, via a mean shape and covariance structure after a group-wise matching. In the case of the brain, one can rely on a hierarchy of structural models ranging from anatomical or functional landmarks like the AC and PC points ([Talairach and Tournoux, 1988; Bookstein, 1978]), curves (e.g. crest lines, sulcal lines) ([Mangin et al 2004; Le Goualher et al. 1999, Fillard et al. 2005]), surfaces like sulcal ribbons ([Thompson et al., 1997; Andrade et al., 2001; Vaillant et al., 2007]), images seen as 3D functions, which lead to voxel-based morphometry (VBM [Ashburner and Friston, 2000]), diffusion imaging or rigid, multi-affine or diffeomorphic transformations ([Trouvé 1998; Miller et al., 2002; Arsigny et al., 2006]), leading to Tensor-based morphometry (TBM).

Figure 2. Top: Surface of the cortex extracted from 4 different subjects with the Anatomist/BrainVISA package. Bottom: example extraction and labelling of the sulci with the same tools. One can see that finding the invariants of the cortex shape is a difficult problem. (Courtesy of J.-Fr. Mangin, CEA, Laboratoire de neuroimagerie (LNAO), Neurospin, Saclay, France)

1.4 Computing on Manifolds

One crucial point is that these features usually belong to curved manifolds rather than to Euclidean spaces, which precludes the use of classical linear statistics. For instance, the average of points on a sphere is located inside the sphere and not on its surface. To address this problem, one has to rely on statistical tools that work directly on Riemannian manifolds in an intrinsic way. A consistent set of such tools including the mean value, covariance matrix, Normal law, Mahalanobis distance were for instance proposed in [Pennec, 2006] along with efficient algorithms and tractable approximations for small variances. These tools were later extended to a complete algorithmic framework to compute on manifolds with the example of so-called "tensors" (symmetric positive definite matrices) [Pennec et al, 2006]. In particular, it was showed that one can perform interpolation, filtering, isotropic and anisotropic regularisation and restoration of missing data (extrapolation or in-painting) on manifold valued images by using generalised weighted means and partial derivative equations (PDEs).

1.5 References