Why we label sulci

The cortical surface is characterized by alternating ridges and furrows. The sulci (singular sulcus), as the Latin suggests, are the fissures or grooves; they serve as counterpoint to
the raised gyri (see the figure; some of the primary sulci are highlighted in color). The sulci, in a sense, exist because they do not exist. Their utility derives from this fact as demonstrated by the following:

1) In neurosurgery they function as channels which give a surgeon access to parts of the brain even deep within the subcortex. As M.G. Yaşargil, a noted neurosurgeon, writes in the foreword to the Ono atlas [1]: "any point within the cranium can be reached by following the corridors of the sulci." Tissue damage from incisions is thus minimized.

2) They also serve as orienting landmarks in neurosurgery. The major sulci, in addition, partition important functional areas of the brain. This information reinforces their usefulness as landmarks. The central sulcus, for instance, demarcates the sensory-motor cortex. The sylvian fissure, one of the most identifiable cortical features, is the locus of language cortex. Both these sulci are important reference points in a variety of contexts and applications.

3) The sulcal grooves are filled with cerebrospinal fluid (CSF) which make them easy to identify in T1-weighted images (where CSF is dark in contrast to the brighter gray/white matter.)

To be useful in the neurosurgical applications described, we first need to identify and label the sulci. There are other applications that would also benefit from a reliable labeling scheme. Internal changes in the brain, either due to aging or pathology, for instance, alter the cortical surface. Labeling is the first step in a systematic study that allows us to quantify these changes for the differential diagnosis of disease.


Reference
1) Ono, M., Kubic, S. & Abernathy, C. (1990) "Atlas of the Cerebral Sulci", (Thieme, New York).


Posts on Sulcal Labeling
1) Why we label sulci
2) Why is sulcal labeling difficult ?
3) The use of a spatial distribution model in labeling sulci

Soutenance

My thesis defense was yesterday. So now I have a degree certificate in French in addition to certificates in English and Latin.
At Takshashila, which Panini and Chanakya both attended, the certification would have been a Sanskrit title--acharya perhaps. At Nalanda, which resembeled a public university far more, it may have been in Pali or some other Prakrit. These were the oldest universities in the world by, as Amartya Sen puts it, "a long margin".

A round of acknowldegement:
I'd have to credit the disembodied collective of knowledge that is the internet as coauthor.

The open curve representation and brain image analysis

There are about a hundred sulci in the human brain and over a hundred billion white matter fibers. These two anatomical structures differ in many respects but they share a common geometric description: they are both open continuous curves.
To study how the physical attributes of open curves can be used to advantage in the many varied quantitative applications of white matter fibers and sulci, we begin by considering the open curve. The brain imaging landscape from this window has its own particular challenges, open problems and possibilities. The view offers the following:

Related to the representation
The choice of representation restricts our analysis to sulci, white matter fibers and the veins and arteries of the vascular system in the brain. These are the three anatomical structures that can be modeled by open curves in 3D.

An open curve is one of the simplest ways to mathematically represent an individual sulcus or white mater fiber tract. An (open) surface, which may also be used, is more difficult to model and implement. The closed curve is a third option, but it is also a mathematically more complex representation. Anatomical structures are not easy abstractions of geometrical primitives. Were this the case, a closed curve representation of a square or circle would be the right choice.

The details of the mathematical modeling aside, for the purposes of analysis (of sulci or white matter fibers), an open curve is the more useful representation. In the case of sulci, the fact that they are highly variable in shape and structure also means that there is no additional information in a surface representation. The bottom curve (the section of the sulcus closest to the inner cortex) is the most stable part of a sulcus and this is what we use in our analysis. For white matter fiber analysis too, a curve model enables us to analyze individual tracts and there are benefits to this analysis over a volume-based one.

An open curve may be represented in a Euclidean or Riemannian space. In a Euclidean space, a spline (a parameterized curve defined piecewise by polynomials) can easily approximate shapes of complex curves. The L2 distance we compute between two such curves can be used for clustering or other classification tasks. The main advantage with this approach is the ease of implementation and the fact that existing methods from machine learning and statistics can be readily incorporated into the analysis.

In a Riemannian analysis, the open curve can be represented in different ways. In the model I use, the open curve is represented as a point in the space of continuous functions.This space of functions is infinite dimensional and nonlinear and advanced methods such as those from functional analysis, differential geometry and group theory are needed to represent and analyze curves. The advantage of a Riemannian representation is that we can use the intrinsic (Riemannian) geometry of the space to compute distances between two points (curves). Open curves are essentially nonlinear structures. The space of open curves is also nonlinear. Linear
measurements in such spaces are not well-defined and the results may not be consistent.

Related to the geometrical analysis

The open curve is a geometric structure and we use it in analysis in this context. Geometrical analysis in medical imaging is most commonly a shape study although the term itself covers other possibilities. We use a mathematical model for open curves that allows us to incorporate shape, orientation, scale and position, the physical features associated with the curve. We expect to be able to apply this mathematical model to new kinds of problems or to reinterpret old ones. (I have done some work on a new study design which I will be publishing shortly. It is essentially a study of morphological changes, but my tools enable me to study the problem in a new way.)

White matter DTI fibers bundles have a well-defined shape and structure which is determined by the regions they connect and the constraints of the surrounding anatomy. Sulci, on the other hand, are not easily interpreted in a geometrical study. This is due to the variability which we can also interpret as a problem of selecting a feature that is unique to a class of sulci. The possibility of shape analysis has been explored before. It is my view that sulcal variability, as it relates to identication and labeling, is better addressed by a graph pattern matching paradigm.

Offshoring work for medical diagnosis

A Wall Street Journal blog article headline states: India is benign for radiologists. It turns out that:

... reading such images relies heavily on what the two economists call “tacit knowledge.” Pattern-recognition software, which could make the work routine, doesn’t work very well in identifying malignancies and other problems,

The paper the blog refers to, was of course, talking about clinical diagnosis but one step removed from this is medical image analysis. Studies usually require databases (some very large) of subjects/cohorts who fit a common description. We might be quite far from full automatic diagnosis, but tasks can be broken down and piecemeal solutions can cetainly be outsourced. (I've been tempted more than once to go this route with the data I use or would use if only I could process them all.)

Suppose we want to build a database of twin brains or multiple sclerosis brains or whatever. Some of the tasks that are routine and could be outsourced are: registering the brains to a common template, segmenting specific sections of anatomy, representing the data in a certain way (in a representation space). Now with close supervision someone with basic training could easily do this. So the problem might be that there are not enough people who can supervise such work. And the people who can have not thought of setting up shop in India. Big pharma is interested in such studies so it could be a lucrative outsourcing venture if someone could put it all together.

Tract-based white matter fiber analysis: what to study

The difficult problem in white matter fiber analysis (obtained from DTI or HARDI sources) is to efficiently handle the large volume of fibers. This may be achieved by clustering the fibers, by clustering using approximations such as the Nyström method, by sampling from the data in a fiber bundle or computing representative means as suggested here.

Volume-based methods side-step this issue by treating a white matter structure as a single anatomical unit. With tract-based analysis, however, processing individual fibers is usually a requirement. Tractography and clustering algorithms have made this possible. Improvements are needed but it is my view that making the pre-processing pipeline more efficient is an engineering effort.

While tedious processing is a disadvantage, tract-based methods offer the potential to study local parameters along a tract. This is important in the study of white matter disease
and, at this juncture, it is here that the maximum contributions to the medical community can be made.

References
C. Fowlkes, S. Belongie, F. Chung, J. Malik. "Spectral Grouping Using the Nyström Method", TPAMI. 26 (2) p.214-225.

White matter fiber analysis: why we need statistical summaries (means)

A key concept in medical image analysis is the idea of a mean template, i.e., a statistical average around which deviations can be assessed. In the context of white matter fiber analysis, we seek to represent an anatomically defined fiber bundle with a mean and variance that describes its essential characteristics. This mean is of interest for practical reasons that go beyond atlas construction. From my hard-won experience working with DTI fibers, these are 3 reasons we need to compute means:

1.There are a large number of fibers involved in white matter fiber analysis. The corpus callosum has over 300 million fibers alone, the whole brain, 100 billion! fibers--source: Mori's atlas. The tractography output which is some fraction of this can still be several thousand fibers. Due to this large volume, a practical first step in any study, and one that is advocated by me, is to compute a representative mean of a fiber bundle.

2. The tractography output is subject to error. Noise, imperfections in the image and the presence of regions of low anisotropy due to fiber crossings all contribute to this.
In order to make the streamline output more robust we can average over the fiber bundle. This strategy is also useful when a representative bundle is sought and there are discontinuities and other fiber damage due to disease.

3. To facilitate statistical analysis for population studies where the underlying problem is one of assigning membership to a group.

Means may be computed for the following situations:

i)for a group of fibers within a fiber bundle
ii) for an intra- or inter-subject collection of fibers from many bundles
iii)for an intra- or inter-subject collection of means of fiber bundles

The computation of statistical summaries is usually part of any framework for white matter
fiber analysis. Some of these frameworks are listed in my previous post on Mathematical frameworks.

Tract-based quantitative white matter fiber analysis: Mathematical frameworks

This is a standalone post. It is also part of the Quantitative white matter fiber analysis
series posted here.


There are three steps in tract-based data analysis: pre-processing, modeling and statistical analysis. The pre-processing may involve registration, tractography segmentation or clustering and the information contained in these results is often incorporated into the data analysis model. Fractional anisotropy (FA) is one such artifact and the analysis of FA profiles along tracts is usually included in most studies. The quantitative frameworks use both simple ideas such as the alignment of fibers for statistical analysis and more advanced mathematical concepts such as currents or techniques in Riemannian geometry.
These methods for tract-based analysis are listed below:

1) Fiber-tract oriented statistics
This model by Corouge et al.[1] is a prototype for computing statistics along a fiber bundle. The core idea was to give a compact description of the geometry and diffusion properties of a fiber bundle along its length profile. Corresponding points for a set of fibers are aligned and averages of DTI indices along the cross-section are computed. This can be done with linear or nonlinear Riemannian spaces.
This basic idea has also been extended to a population atlas where information contained in tractography results[2] or registration results[3] are evaluated along a tract-based coordinate system.

2) Statistical modeling and EM clustering of white matter fiber bundles
Individual fibers are parameterized and aligned with the goal of performing tract-oriented analysis over a population. A statistical model of pre-segmented fiber bundles is then calculated which serves as a prior for mixture model clustering. The Expectation-Maximization algorithm is used to infer membership probabilities and cluster parameters. Atlas guided clustering gives anatomically meaningful bundles. This is work by Maddah et al. [4].

3) Statistical model of white matter fiber bundles based on currents
This is a flexible framework where fiber bundles (a collection of curves) are modeled as currents. The space of currents is a vector space equipped with an inner product and norm which defines the distance between two bundles. This is a global distance that does not use point-wise correspondences [5]. Durrleman et al. give a statistical model for a fiber bundle atlas and it variability in a population.

4) Comprehensive Riemannian Framework
In this framework, Mani et al.[6] use various combinations of shape, scale, orientation and position, the physical features associated with white matter fibers, to define Riemannian feature spaces. This is useful since applications have different objectives and often need different sets of tools and metrics for optimal results. For each joint manifold, a (geodesic) distance metric that quantifies differences between fibers as well as tools for computing statistical summaries of samples are defined. Correspondences between fibers are implicitly established during these pairwise comparisons. The framework may be extended to accomodate features such as scalar diffusion indices.

5) Statistical mapping of medial models of white matter tracts
Yushkevich et al.[7] use a different geometric model by giving major fiber tracts a medial representation. While this is strictly not a tract-based model, it is included here because local statistics of tensor-based features can be computed.

Bibliography:

1) Isabelle Corouge, P.Thomas Fletcher, Sarang Joshi, Sylvain Gouttard, Guido Gerig, "Fiber Tract-Oriented Statistics for Quantitative Diffusion Tensor MRI Analysis," Medical Image Analysis 10 (2006), pp. 786-798.

2) Lauren J. O'Donnell, Carl-Fredrik Westin and Alexandra J. Golby. "Tract-Based Morphometry for White Matter Group Analysis," NeuroImage. Volume 45, Issue 3, (2009), pp. 832-844.

3) Casey B. Goodlett, P. Thomas Fletcher, John H. Gilmore, and Guido Gerig. "Group Analysis of DTI Fiber Tract Statistics with Application to Neurodevelopment," NeuroImage. (2009), 45(1 Suppl): S133–S142.

4) Maddah M., Grimson W.E.L., Warfield S.K. "Statistical Modeling and EM Clustering of White Matter Fiber Tracts", Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging. (2006);1, pp. 53-56.

5) M. Vaillant and J. Glaunès, "Surface matching via Currents", Proceedings of Information Processing in Medical Imaging, Lecture Notes in Computer Science vol. 3565, Springer (2005), pp. 381–392.

6)Meena Mani, Sebastian Kurtek, Christian Barillot, Anuj Srivastava. "A Comprehensive Riemannian Framework for the Analysis of White Matter Fiber Tracts," In ISBI'2010, pp. 1101-1104.

7) Yushkevich P.A., Zhang H., Simon T.J., Gee J.C., "Structure-specific statistical mapping of white matter tracts," (2008) NeuroImage, 41 (2), pp. 448-461.