Saturday, June 4, 2011

Shape analysis: on selecting a set of points or landmarks

One approach to shape analysis (after Kendall) uses a fixed number of points to define a shape. The points may describe an object boundary or an interior morphology such as the veins on a leaf. They may be selected randomly; alternatively they may be landmarks--i.e. points of significance. A set of points, so selected, constitutes a shape summary and the original shape data, that had been extracted from the raw image, is discarded.

One problem with representing a shape in this way is that it introduces a source of variability. Different shapes can be reconstructed with the same set of points. This is especially true if the number of points selected for that particular shape are few.

There are, of course, other approaches to shape analysis that do not involve selecting points (at least not at this stage of shape representation). Deformable templates is one such methodology and its use is quite common in medical image analysis. The shape could also be represented by a continuous function (see for example Younes et al. [1]). But these methods are also computationally expensive.

1) Younes, Laurent; Michor, Peter W.; Shah, Jayant; Mumford, David. A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), no. 1, 25--57.

Thursday, April 7, 2011

Anatomy of a good talk

I listened to a very interesting talk recently (via GoogleTech Talks--if the link is broken, search for Breakthroughs in Imaging Neurovascular diseases such as Multiple Sclerosis ...). Mark Haacke, the speaker, had some years ago developed Susceptibility Weighted Imaging (SWI)--an MR modality that uses the phase information in a signal to image iron deposits in the brain. His talk is interesting because there is a story, a narrative, built around this endeavour and his efforts to find important clinical applications. Indeed, he has linked his research to Paolo Zamboni's work with multiple sclerosis (MS). (There is a Facebook group advocating a Nobel prize for Zamboni so a significant impact is projected.)

For those of us in research it is always rewarding to witness--even from afar--the full lifecycle of an idea or concept. Haacke shows how he develops the method (it is really just another case of using parts of the signal that were being filtered out as noise); verifies (with Xray fluorescence--XRF) that what they are looking at is iron; looks for and finds iron deposits in the veins and brain tissue of patients with MS and other neurological diseases (which suggests that excess iron is a biomarker for these kinds of conditions); makes a case for the quantitative analysis of the venous system (the other half of the better explored arterial system); identifies new imaging applications such as the previously unseen microbleeds in patients with traumatic brain injury (TBI); presents temporal data to show the build-up of iron and finally links this aggregation with Zamboni's hypothesis that the narrowing or stenosis of veins (such as the internal jugular vein--IJV) that drain out of the brain, creates a reflux which subsequently results in the accumulation of iron.
It remains to be seen if multiple sclerosis and other neurodegenerative diseases have a vascular origin but at the very least, the evidence--presented in various ways in this talk--shows a strong cause.

Haacke ends by listing specific ways in which "technical people" can get involved. He suggests ways to quantify blood flow, develop biomarkers, track patients, develop databases and develop new sponsership models to fund all this work. It's a very complete talk in this sense and the right way to invite people with different kinds of expertise in.

The talk is pitched at a general audience but there is enough detail so that someone like me, who works in medical imaging and with MS datasets, can also benefit. I'm a street kid; there've been no mentors. And talks like this give a perspective I haven't been able to get anywhere else. But more pampered academics can also benefit. I've been reading a paper where the author dabbles in a whole lot of esoteric math but is unable to construct a biomarker that appeals to common sense. Well, having perspective is one way to compensate for a lack of common sense.

Friday, February 11, 2011

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. Ya┼čargil, 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.

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

Tuesday, February 1, 2011


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.

Wednesday, January 5, 2011

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.