Why is sulcal labeling difficult?

This is a follow-up to an earlier post Why we label sulci. There will be two or three more posts; taken altogether, they will describe the sulcal labeling problem.

The labeling of sulci is a challenging problem. This is because, cortical sulci are highly variable. Sulci vary not just across individuals but even between the hemispheres of a single brain [1]. It might be useful when looking for ways to address this variability to classify this variation as follows:

Variation in physical features
Sulci vary in shape, in scale and in their placement (i.e. position and orientation) The figure below illustrates how the variability can make feature selection difficult.

The boxplot shows the length distribution for 18 subjects. The 10 types or classes of sulci shown cannot be identified solely on a length measurement. This poses a problem for feature selection and classification.
Figure credit: Meena Mani




Variation in branching
19th century illustrations such as those from Horsley [2], trace the wide variations along a sulcal fold. A whole nomenclature has developed since then to account for the branch variations possible along a single sulcus. (An example from the Ono atlas is illustrative--see figure below). For this reason, there is no gold standard in sulcal labeling; one neuroanatomist may disagree with another.

The figure to the left shows the pattern variations for a single sulcus (the posterior end of the superior frontal sulcus). Types A, B, C, D, are possible variations for this sulcus (for the 25 postmortem brains examined, 4 variations were found). The pattern in the two hemispheres of a single subject may differ; the left may be Type B and the right may be Type C. The lengths of the small segments and the connections they make to other sulci may also vary. Reproduced from Ono et al. [1].

Variation in number
Sulci may be continuous (present as one uninterrupted segment) in some individuals, fragmented (exist as multiple segments) in others and altogether absent in yet others. The larger primary sulci which start forming early in fetal development are the most consistent; the secondary and tertiary sulci are not always expressed.

References
1) Ono, M., Kubic, S. & Abernathy, C. (1892) "Atlas of the Cerebral Sulci", (Thieme, New York).
2) Horsley, V. (1892) "On the topographical relations of the cranium and surface of the cerebrum", In "Contribution to the surface anatomy of the cerebral hemispheres", pp.306-355, (Royal Irish Academy).


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

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

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.